Download Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA Actuarial Program

Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA
Actuarial Program Director and Professor of Mathematics
Illinois State University, Normal, IL 61790-4520, U.S.A.
Tel. 1-309-438-7226
Fax: 1-309-438-5866
http://www.math.ilstu.edu/krzysio
E-mail: [email protected]
Valuation and Pricing of
Bank and Insurance Products
Copyright © 2006-2007 by Krzysztof M. Ostaszewski.
All rights reserved.
Problems from the past actuarial examinations are copyrighted by the Society of
Actuaries and/or Casualty Actuarial Society and are used here with permission.
1. Financial Assets
A real asset is defined as an asset that produces income, or is utilized in production of
income. A factory is a real asset, machinery in a factory is a real asset, computer is a real
asset. Precious metals are real assets. It is a common misconception that real assets must
be tangible. They need not be. For example, a patent is a real asset. Human capital
(ability to earn income) is a real asset.
A financial asset (also called a security, or a capital asset) is defined as a claim on the
income produced by real assets. In other words, financial assets direct the distribution of
income produced in the economy. For example, you may use your human capital to get a
paycheck, but then pay a portion of it to your mortgage, that mortgage is a financial asset
of its owner, giving that owner the right to a portion of the income produced by your
human capital.
There are two main types of financial assets:
- Bonds (or loans): assets that specify in advance the amount of income that will be
forwarded to the financial asset holder.
- Stocks (or shares): assets that allow the asset holder to share in both good and bad
fortunes of income producer, but do not specify in advance what the amount of
income forwarded to the financial asset holder will be.
Technically, government’s rights to incomes of its citizens, i.e., taxes, are also financial
assets, but governments rarely present balance sheets, instead producing only income
statements (and those on cash basis, as opposed to accrual basis used by businesses).
While gold, silver, or other precious metals, are treated as real assets, because they are
used in the process of producing income, money is treated as a financial asset, as it
represents a claim to consumption of goods and services, and income really means ability
to obtain goods and services.
You might want to ask what money is. Economists usually do not pose this question
directly, as it is probably too philosophical for most of them, and their philosophical
approaches to money differ. This author believes (and believes that his belief is derived
directly from Adam Smith’s An Inquiry Into The Origin of the Wealth of Nations) that:
money is an objective measure of value of human labor derived from continuously
negotiated subjective evaluations of such value by all interested human parties. Most
excellent classical economist, David Ricardo, is known as the creator of the labor theory
of value, which proclaimed that value of a good or a service is the accumulated amount of
labor used in its production. The labor theory of value was later adopted by Marx and by
Marxists. Subsequently, this labor theory of value was sharply criticized by the Austrian
School of Economics, notably Ludwig von Mises, who pointed out that value exists only
subjectively: as no matter how hard you work on something, if there is no willing buyer
for your product or service, its value is zero. The definition given here attempts to point
out that, as in all things, Mises was right and Marx was wrong, but David Ricardo was
not all that wrong, because while the evaluation process is subjective, it is not one-sided:
it is a process of negotiation between the buyer and the seller, and the value established in
the negotiation is in fact the value of the labor input in a good or a service traded.
However, in this work, we are not interested in money, but in financial assets.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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The two main types of financial assets: bonds and stocks, are not completely mutually
exclusive. If you give a loan to a business, you are promised specific payments of interest
and a repayment of principal. If however, the borrower business is unable to make the
promised payments and becomes bankrupt, you, the lender, are likely to end up with a
share of the business, instead of the promised payments of interest and principal, thus
becoming a shareholder of the business, albeit inadvertently. Thus your bond (loan) is
partially a stock, and the riskier the bond is, the more it becomes like a stock in the
borrowing company.
There is also a third group of financial assets: derivative securities. A derivative security
has its cash flows derived from cash flows of other securities. It does not relate directly to
income produced in economic activity, but rather, it relates to it indirectly, through other
securities. Assets issued by financial intermediaries: banks, insurance companies,
investment companies, etc. are derivatives. A bank deposit is a derivative security. But
the most important derivatives are those that are simple, building blocks for more
complicated securities, and they include: forwards, options, futures, swaps. Those
“building blocks” derivatives are commonly embedded in derivatives issued by financial
intermediaries, and even securities issued by businesses (e.g., bonds are commonly issued
with a right to pay off the loan early, which is a form of a call option).
Derivatives are used for a variety of purposes: risk management, speculation, as an
alternative to investment in primary securities (especially if investment in derivatives
reduces costs), or to address/manage regulatory requirements.
Financial engineering is the science of design and pricing of financial assets out of other
financial assets. Financial assets can be purchased in individual transactions or in
financial markets (or capital markets), i.e., markets created specifically for financial
assets. Examples of financial markets are: New York Stock Exchange (located at Wall
Street in New York City), or Giełda Papierów Wartościowych in Warsaw, Poland. The
risk of investing in financial assets is divided into diversifiable risk and non-diversifiable
risk (also known as market risk). Diversifiable risk is called so because it can be reduced
or managed by combining securities in a larger portfolio (this process is called
diversification). The main purpose of existence of financial markets is to create a place
for sharing of non-diversifiable risk within the society. Such risk can be either avoided
(thus reducing economic activity, and reducing society’s overall welfare), accepted by the
government (resulting in government ownership of means of production and subsequent
risks of political influences on the economic process, with likely reduction in economic
output and threat to domestic tranquility), or traded in the markets (resulting in unequal
distribution of risks, and likely accumulation of wealth to those bearing higher risks, i.e.,
unequal distribution of risk causing unequal distribution of wealth). Financial assets can
also be purchased from and sold to dealers, who hold inventory of them, instead of
trading in an exchange. This is called the over-the-counter market.
The price at which a financial asset is offered for sale in the market (or by a dealer, so by
any market-maker in general) is called the ask price, while the purchase price offered (by
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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a market-maker) for it is called the bid price. The difference between the two is called the
bid-ask spread.
Exercise 1.1
Mr. Romuald Carcosheek has proposed Ms. Berlin Marriott an unusual real estate
transaction. Mr. Carcosheek believes that the prices of real estate in New York City will
decline substantially over the next year, and he has asked Ms. Marriott to let him borrow
a hotel in New York City she owns, sell it short, and repurchase it in the future, returning
it to Ms. Marriott. Ms. Marriott requires a 10% haircut (deposit) on the transaction, and
she will hold it, together with the proceeds of the short sale, in trust, earning 5% on the
funds, and reinvesting interest. They agree to proceed with the transaction. Mr.
Carcosheek sells the hotel for $25 million today, and has to pay a 3% commission on the
transaction out of his own pocket, because the proceeds of the sale will be held in trust. In
three years, Mr. Carcosheek buys the hotel back for $20 million, with the commission
paid by the seller. Calculate the difference between the annual rate of return earned by
Mr. Carcosheek over those three years, and the annual rate of return earned by Ms.
Marriott (including her unrealized gain/loss on the market value of the hotel).
Solution.
Mr. Carcosheek has to put down $2.5 million for the haircut. He sells the hotel for $25
million and has to pay 0.03 ! 25, 000, 000 = 750, 000 in commission. Thus his total cash
outlay is $3.25 million. In three years, he buys the hotel back realizing a profit on the
transaction of $5 million. Moreover, he only gets $2.5 million, the haircut, back, but the
sales commission is not returned. Thus his annual rate of return is
1
! 7.5 $ 3
#"
& ' 1 ( 32.1476%.
3.25 %
Ms. Marriott owns a $25 million building. After the short sale, she receives $25 million
plus $2.5 million haircut, for a total of $27.5 million. In three years, this will accumulate
to $27.5 !1.05 3 " $31.83 million. She will return $27.5 million of that, and keep $4.33
million. But her $25 million building is now a $20 million building, so she has an
unrealized loss of $5 million, for a net loss of $0.67 million on a $25 million capital over
three years and the annual rate of return of
1
" 25 ! 0.67 % 3
$#
' ! 1 ( !0.9014%.
25 &
The difference is 32.1476% ! ( !0.9014 ) % = 33.0490%.
Valuation of Stocks
We generally assume that the time horizon for stocks (also called shares, or equity) is
infinite. Stocks represent fractional ownership in limited liability corporations. Cash
flows generated from holding stocks are uncertain: dividends may or may not be paid out,
and dividends amounts may vary because they depend on a company’s growth,
profitability and investment opportunities, as well as behavior of management, taxes, and
regulatory environment. If you are an investor in stocks, you must decide whether you
want to receive income from owning the stock or whether you want to capitalize on
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company’s growth. In the first case, you prefer companies that pay their profits out in
dividends, while in the second case you prefer companies that reinvest profits in their
operations. Companies that pay stable dividends usually try to keep their dividend level
or increasing, within a certain percentage range of their earnings per share (EPS). For
example, a company has a policy of paying 40% of its earnings in dividends, then
dividends per share are 40% of EPS. In the case of such policy, dividends tend to grow at
the same rate as earnings. Some corporations use up their retained earnings before paying
dividends. This is called a residual dividend policy. Companies that follow a residual
dividend policy only pay dividends if they have some leftover income that they do not
need to invest in projects that will result in growth of their business.
If a dividend paid at time t is denoted by Dt and the amount of dividend is known with
certainty, the price P0 of stock at time 0 is
D
D2
D3
P0 = 1 +
+ …,
2 +
1 + k (1 + k ) (1 + k )3
where k is the applicable discount rate. If dividend grows at a constant rate g, then
D1
P0 =
.
k!g
The model giving this last formula is called the Dividend Discount Model, and the
formula is also called the dividend discount formula.
Exercise 1.2
The cash flows of the stock of Universe Inc. are discounted by the rate k = 10%, and pays
a dividend that grows at a rate of g = 6%. If the dividend paid at the time t = 0 is $1,
what is the price of the stock based on the Dividend Discount Model?
Solution.
Since the dividend is $1 now, it will be $1.06 at the end of the year. Therefore,
D1
$1.06
$1.06
P0 =
=
=
= $26.50.
k ! g 0.10 ! 0.06 0.04
Exercise 1.3
Assume current market price is $23, the stock will pay a dividend of $1.242 one year
from now, and it will grow at a constant rate of 8% in the future. What is the rate of
return k expected on this stock?
Solution.
We solve the dividend discount formula for k and obtain
D
1.242
k= 1 +g=
+ 8% = 5.4% + 8% = 13.4%.
P0
23
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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D1
is called the dividend yield of the stock. If a stock’s dividend does not
P0
increase, or decrease, then a stock is valued as a perpetuity. This can be handled also by
simply assuming that g is 0% in the Dividend Discount Model.
The ratio
Price-Earnings Ratio
The price-earnings ratio of a company is defined as the ratio of the price per share to the
earnings per share (the earnings are generally profits expected at the end of the year). The
P/E ratio is generally used for relative valuation of firms, i.e. to compare financial
performance among firms. High P/E ratio means higher value for the same amount of
profits. Some research shows that investors can be interested in firms that have a lower
P/E ratio based on the belief that these companies might be under-priced by the market.
Generally, firms that focus more on growth and re-invest more of their earnings tend to
have a higher P/E ratio than firms that don’t reinvest as much. Firms that have a higher
Return on Equity (ROE) tend to have a higher P/E ratio. Finally, firms perceived as being
“riskier” (the ones for which investors require a higher rate of return k) have a smaller
P/E ratio.
Fixed-income securities
In the United States bonds and other types of fixed-income securities (such as mortgagebacked securities or asset-backed securities) are issued by corporations, federal, state and
local governments, and federal agencies that wish to borrow money from investors. There
is no centralized place or exchange on which bonds are traded. They are traded generally
over-the-counter via a dealer system, electronically. It is not possible to issue bearer
bonds in the United States, as all bonds must have registered owners. Coupons are
typically paid twice a year.
In Poland bonds are traded on the Warsaw Stock Exchange. There are two types of debt
instruments: state Treasury bonds and corporate bonds. State Treasury bonds have 2-year,
3-year, 5-year and 10-year maturities, and they all have a nominal value of 1000.
Coupons can be paid quarterly or once a year.
Bonds with variable coupon payments are called floating-rate bonds. Bonds that pay a
variable coupon have coupon rates periodically reset according to a specified market rate,
e.g., Treasury Bill rate or LIBOR (London Inter-Bank Offered Rate, the most widely
short-term interest rate index in U.S. dollars). In the US, Treasury Bonds and Notes are
issued with a fixed coupon rate. Corporate bonds can be issued with fixed or variable
rates. In Poland, state Treasury Bonds can pay a fixed coupon rate (paid once a year) or a
floating coupon rate (paid quarterly). The floating rate is based on the average yield
(interest rate) of 13-week Treasury Bills if the coupon rate is paid quarterly, or on the
average yield of 52-week Treasury Bills if the coupon rate is paid once a year.
A call provision in a bond allows a corporation to “recall” (or re-buy) all or part of the
debt issue prior to maturity. In the U.S., Treasury-issued securities are not callable, but
corporate or municipal issues typically are. Generally, the issuer will pay a premium over
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the nominal or face value of the bond when the bond is called. A bond with a call
provision is called a callable bond.
A put provision gives the bondholder the right to sell a bond back to the corporation at
the nominal or face value, generally after a relatively short period (about 3 to 5 years
after the bond was first issued). A bond with a put provision is called a putable bond.
A convertible bond gives bondholders the option to exchange the bond for a pre-specified
number of shares of the company.
Exercise 1.4
Assume a $10,000 face value bond is convertible into 40 shares of stock. At which stock
price does it become beneficial for the investor to convert the bonds into stock?
Solution.
Actually, this question is a bit complicated, because the answer depends on the current
level of interest rate in relation to the coupon rate of the bond. If this bond is trading at
par, the answer would be
10, 000
= 250.
40
Bond provisions for risk of default
• Sinking fund: allows the issuer to periodically repurchase some proportion of the
outstanding bonds prior to maturity.
• Subordination clause: refers to the situation when a firm re-issues additional bonds or
notes (“junior issues), these stipulate that previous “senior bondholders” will be paid first
in the event of bankruptcy and that “junior” bondholders will be paid only after that.
• Collateral: bonds can be issued with a specific asset pledged against possible default of
the company. Collaterals are generally assets of the firm that investors in the bond will
receive if the firm defaults in its payment: property, equipment, and so on.
Risks associated with investing in bonds
• Inflation risk.
• Liquidity risk: refers to the risk associated to re-trading a bond in the secondary
market (no one wants it)
• Maturity risk: refers to the fact that longer-term bonds prices react more to interest
rate changes that shorter-term bonds.
• Default risk: risk that the company will default on the coupon payment or on the face
value of the bond
Bond Ratings
Since the early 1900s, bonds have been assigned quality ratings that reflect their
probability of going into default. In the U.S, the three major rating agencies are:
Moody’s, Standard & Poor’s (S&P) and Fitch Investors Services. These agencies also
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analyze the credit quality of non-U.S. corporations. Bond rating criteria are both
qualitative and quantitative. They can be based on:
o The corporation’s various financial ratios (debt, liquidity or profitability ratios).
o Subordination provisions. Is the bond subordinated to another debt instrument?
o Maturity of the bond.
o Stability: are the bond issuer’s sales and revenues stable?
o Overseas: political climate in other countries?
o Environmental factors.
o Product liability: are the firm’s products safe?
o Labor unrest.
Bond ratings are important to both the firm and the investors in the firm, because bond
ratings assess a firm’s risk of default. As a result, the lower the ratings, the higher the
risk, the higher the return required by the investor, the more costly it will be for the firm
to issue debt.
2. Forwards and Options
Forwards
A forward contract is a sale transaction, which is consummated in the future, but with all
details of the transaction specified in the present. The time at which the contract settles is
called the expiration date. The asset (often a commodity) on which the forward is based
is called the underlying. A forward contract generally requires no upfront payment. It
should be noted that both sides of a forward are exposed to a significant risk of nonperformance of the other party (i.e., credit risk), and for that reason, forwards are rarely
used in practice, and even if they are used, they are private transactions, not traded on an
exchange. Forwards are negotiated and customized for specific parties. They are created
on commodities, financial assets, indices, or currency exchange rates. They are arranged
by brokers or dealers, who make money on the spread.
Note that the price paid in the market for a given asset is called the spot price (as opposed
to the price named in the forward contract, the forward price). The person buying the
underlying in a forward contract is said to be long forward, and the person selling the
underlying is said to be short forward. We have the following
Payoff to long forward = Spot price at expiration – Forward price
Payoff to short forward = Forward price – Spot price at expiration
By analyzing the payoffs of a forward we can also conclude that
Long forward = Underlying’s Spot Price – PV(Forward Price)
Forward price (price at which the contract is agreed upon) depends on the spot price of
the underlying, and converges to it as the contract nears maturity (otherwise arbitrage
opportunities would be present). If forward is created so that no cash changes hands at
inception, then
Forward price = Accumulated value (Spot Price),
F = S ! e" t .
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Consider a forward on an n-year zero-coupon bond, for the time k in the future. Let us
write F ( 0, k ) for the forward price. Let P ( n + k ) be the price of a $1 risk-free zero-
coupon bond maturing at time n + k, and let P ( k ) be the price of a $1 risk-free zero
coupon bond maturing at time k. This case does not seem to allow for a straightforward
application of the formula above. However, we have the following:
PV(Forward Price) = PV ( F ( 0, k )) = F ( 0, k ) ! P ( k ) .
The underlying in this case is a zero-coupon bond maturing at time n + k, issued at time
n, which effectively is a unit monetary amount paid at time n + k, so that its present value
is P ( n + k ) . Because we assume that a long forward, with
Long forward = Underlying – PV(Forward Price),
is costless to enter, we must have
0 = P ( n + k ) ! F ( 0, k ) " P ( k ) ,
so that
P (n + k )
F ( 0, k ) =
.
P(k)
You also can show this by using a no-arbitrage argument (assume all maturity values are
units):
- Buy a k-year zero-coupon bond and then buy a forward contract to purchase an n-year
bond k years from now; you are thus entitled to a unit payment at time n + k.
- Purchase an (n + k)-year zero coupon bond now with the same unit terminal value.
- Since these strategies would produce identical payoffs, they must cost the same:
P (n + k )
P ( k ) ! F ( 0, k ) = P ( n + k ) , or F ( 0, k ) =
.
P(k)
Note that the key formula
P (n + k )
F ( 0, k ) =
P(k)
effectively says that the forward price is the accumulated price of the underlying,
accumulated at a risk free interest force of interest r (credit risk is ignored in this model).
This is equivalent to the standard relationship of interest rates:
1
F ( 0, k ) =
or
1
(1 + f
n,n + k
P ( n + k ) (1 + sn + k )
=
=
1
P(k)
(1 + sk )k
n+ k
)
k
(1 + sn + k )n + k = (1 + sk )k ! (1 + fn,n + k )
k
,
.
General relationship between spot and forward prices
Let S be the current spot price at time. Then the relationship between spot and forward
prices is:
F ( t ) = S ! ert " D ( t ) ,
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where F ( t ) is the futures price to be paid at time t agreed upon at time 0, D ( t ) is the
cash flows produced by the asset from time 0 to time t, accumulated to time t, and r is the
risk-free force of interest. This relationship is known as the forward-spot parity (it also
applies to futures). If the cash flow is a dividend, and the dividend is payable at a
continuous fashion, with the force of interest ! for the rate of dividend payment, then
F ( t ) = S ! e"# t ! ert .
(
)
For commodities, the underlying does not produce income, but it requires the cost of
carry (the total cost of “carrying”, i.e. storing, transporting etc.), and if that cost is
expressed as a continuously compounded annualized rate c, we obtain this relationship
F ( t ) = S ! ect ! ert .
This relationship is known as the cost of carry relation between forward (also applies to
futures, see below) and spot prices.
(
)
A futures contract is basically like a forward, but with a lot of things added in order to
make it tradable on the exchange, and eliminate the counterparty credit risk. Futures are
exchange-traded and are standardized with regard to maturity, size, and the underlying
asset. Futures are also marked-to-market each day (with cash flows required from all
parties): this means that each side of the trade must deposit a certain required margin and
the margin must be sufficient for the position held not just at the beginning, but each day
the position is held. Also, the exchange may impose a price limit on a daily movement in
the futures price (once that limit is reached, trading is halted for a specified period of
time).
While there may be some slight differences between prices of forwards and futures in
practice, due to margin requirement and intermediate cash flows in future contracts, we
will generally assume that pricing formulas for futures are the same as pricing formulas
for forwards that we developed above. We will discuss possible divergence between
forward price and futures price later.
Forwards versus futures overview:
Security feature
Forward
Type of market
Dealer or broker
Liquidity
Low (almost zero)
Contract form
Custom
Performance guarantee
Creditworthiness
Transaction costs
Bid-ask spread
Futures
Exchange
Very high
Standard
Mark-to-market
Fees or commission
Because futures are usually traded for hedging or speculation, without actual intent of
buying or selling the underlying, many futures contracts are settled in cash, without the
delivery of the underlying. Stock index futures, for example for the S&P 500 index, are
always settled in cash.
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Hedging spot prices with futures
Assume a company knows now that it will want to issue bonds at a later date and it is
afraid that rates will rise between now and that date, forcing it into higher cost of debt.
The company can hedge its position by holding a short position in Treasury-Bond
futures:
- If rates go up, it will be forced to issue debt at a higher rate but this loss will be
offset by a gain from the short futures position.
- If rates go down, it can issue debt at a lower rate, but this gain will be offset by a
loss from the short futures position.
- This kind of a hedge is called a cross hedge, because the underlying of the hedge
(futures) is different than the position hedged. As a result the company may still
be assuming basis risk: the risk of divergence between the hedge underlying and
the security hedged; even if Treasury rates remain level, the company may be
forced into higher rates if corporate bonds spreads widen.
Stock index futures
No futures are written on the DJIA (Dow Jones Industrial Average), but there are futures
contracts on the S&P 500, the NYSE composite, the MMI (Major Market Index, closely
correlated with DJIA), and the Nikkei (Tokyo market index). MMI futures contracts sell
for 250 times the index value. Suppose you want to hedge a portfolio of X dollars of a
diversified portfolio of stocks of large companies (commonly called blue chips). You
can’t use DJIA futures, but MMI is available. You can hedge the portfolio by shorting n
futures contracts, where n is obtained from n ! 250 ! S = X where S is the spot index
level. This creates a hedge: if the index rises, the loss on the futures contracts is offset by
a gain in the portfolio, and if the index falls, the loss on the portfolio is offset by a gain in
the futures contracts. In general, for index futures, F ( t ) = S ! e( r " # )t , where ! is the
dividend yield of the index. This shows us that when the dividend yield is greater than the
risk-free rate, the futures price will be less than the current index price.
Exercise 2.1
Assume it is now June 30, 2002, and a Treasury Bill maturing September 30, 2002 with
$10,000 face value is selling for $9,955.20. Current spot price for gold is $315 per ounce.
What is the price of the futures contract for gold with September 30, 2002 delivery?
Assume that there is neither any cost of storage for gold, nor any convenience yield to
owning it.
Solution.
The information about the Treasury Bill gives us information about the risk-free interest
rate, i.e,, force of interest r applicable must satisfy:
r!
3
9, 955.20 ! e 12 = 10, 000.
Since gold does not pay any dividends, all we have is this relationship between spot and
futures price (spot-futures parity):
3
r!
10, 000.00
F = S ! e 12 = $315 !
= $316.42.
9955.20
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Options
In general, an option is a contract in which one side acquires the right to buy (or sell, but
only one of these two) the underlying at a predetermined price (might be a function of
something, but the conditions are stated in advance) at a predetermined time or by
predetermined time. In hedging with forwards and futures, upside potential must be
sacrificed in order to receive downside protection. When hedging with options, this is not
the case. Options have asymmetrical payoff patterns such that they only pay off when the
index/security/commodity price moves in a specific direction. In an option contract, the
long side had the right, but not the obligation, to purchase or sell (depending on the
option) a security at a specified price. The other (short) side must be the counterparty and
provide the market for the right of the long side. This sounds good for the long side, bad
for the short side, but … the long side must pay an up-front premium to the short side.
Option terminology
- American options are exercisable at any time prior to expiration.
- European options can be exercised only at maturity.
- Bermuda options can only be exercised during prescribed periods before maturity,
but not the entire period from now till maturity.
- A call option gives the long side the right to buy the underlying at a fixed price,
called the strike price, or exercise price.
- A put option gives the long side the right to sell the underlying at a fixed price,
called the strike price, or exercise price.
Remember that every right (to buy or to sell) of the long side originates from the
obligation of the short side to accommodate the right granted to the long side. The
process of creation an option by the short side is called option writing.
Let us now assume that the underlying is a stock. If St is the stock price at the time of
expiration of a European option, and K is the strike price, then the following table
summarizes the long side payoffs of call and put options:
Call Option Payoff
Put Option Payoff
St < K
0
K ! St
St = K
0
0
St > K
St ! K
0
Equivalently (for the long side),
Call option payoff = max ( St ! K, 0 ) = ( St ! K ) .
+
Put option payoff = max ( K ! St , 0 ) = ( K ! St ) .
The above are true for American options, but the stock price need not be the price at
expiration of the option; it can be the price at any time until the expiration.
+
The value that an option has if it were exercised instantly is known as its intrinsic value.
Before its expiration, the option may have a price higher than intrinsic value, and the
difference between the two is called the time value of the option.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 11 -
An option whose intrinsic value is positive is said to be in-the-money. If the underlying is
trading at exactly the exercise price, then the option is said to be at-the-money. An option
that is at-the-money has intrinsic value of zero, but the opposite is not necessarily true. If
an option has intrinsic value of zero, but it is not at-the-money, we say that the option is
out-of-the-money. Options generally have positive time value before expiration. This is,
in fact, a must for an American option. The price of American options can never be less
than their intrinsic value, since arbitrage opportunities would exist otherwise (someone
could buy the American option and immediately exercise it, profiting a gain).
What happens if an option is issued on a stock that pays a dividend? In a sense, nothing.
The dividend has no effectively no influence on the situation of the parties involved in
the option trade. The only influence of the dividend is that affected on the stock price
itself. When a dividend is paid, stock price is reduced by the amount of the dividend.
Effectively, before the time of the payment, the stock trades with a dividend, and after
that, without a dividend. But the price that applies to the option contract is the price of the
stock, either the one with the dividend, or without it.
Exercise 2.2
Mr. Romuald Carcosheek writes a six-month put option on the MIDWIG (an index of the
Warsaw Stock Exchange in Warsaw, Poland) index with the exercise price of 2750. In
order to be able to write options, he must put down a margin deposit of 1000. He does not
receive interest on his margin deposit. The put option he writes on MIDWIG sells for a
premium of 75. In six months, MIDWIG index stands at 2700. Assuming that Mr.
Carcosheek earns interest on the option premium at a nominal annual rate of 4%
compounded semiannually, and assuming that at option expiration he buys the index at
2750 from the long side of the option contract and sells it immediately in the market,
calculate the effective annual rate of return Mr. Carcosheek will have earned over the six
month period.
Solution.
Mr. Carcosheek puts down an investment in the amount of 1000. He receives the option
premium of 75, and that premium, with interest, after six months is worth
2!
1
0.04 % 2
"
75 ! $ 1 +
' = 75 !1.02 = 76.50.
#
2 &
When he buys MIDWIG for 2750 and sells it for 2700, he suffers a loss of 50. Thus his
net cash flow at the end of the six-month period is 26.50, and his effective annual rate of
return earned is
2
26.50 $
!
#" 1 +
& ' 1 ( 5.37%.
1000 %
Exercise 2.3
Mr. Romuald Carcosheek owns a home that costs 20,000,000 PLN. The house is so well
built and so well protected that it can only suffer damage from a fire or an earthquake.
Mr. Carcosheek purchases an insurance policy that will pay for the damage to his house
due to fire or earthquake any time during the next year, with a 1,000,000 PLN deductible,
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 12 -
for a premium of 100,000 PLN. Assuming that the value of the house does not change for
any other reason than a fire or an earthquake, describe an option contract that Mr.
Carcosheek can use to obtain the same protection as that given by the insurance policy.
Solution.
Let us write S for the value of the house. The payoff (not counting the insurance
premium) of the insurance policy is
if S ! 19, 000, 000, &
# 0,
$
' = max (19000000 " S, 0 ) .
%19, 000, 000 " S, if S < 19, 000, 000, (
This is the same as the payoff of an American put option on the house, with exercise
price of K = 19,000,000 and expiration date equal to the last day the insurance policy is
valid.
Exercise 2.4
Countrybank is entering the Polish market and has decided to offer a new attractive
Certificate of Deposit, tied to the WIG index (an index of the Warsaw Stock Exchange in
Warsaw, Poland). The certificate is issued for two years, and it promises to pay the full
amount deposited plus 70% of the performance of the WIG index over that period,
assuming that performance is positive, and zero otherwise. Express the payoff of the
certificate in terms of an appropriate option contract.
Solution.
Assume for simplicity that the amount of the initial deposit K in the certificate is the
current value of WIG. Let S stand for the value of the index in two years. In two years,
the certificate will be worth
$ K + 0.7 ! ( S " K ) , if S > K, '
%
( = K + 0.7 ! max ( S " K, 0 ) .
if S # K, )
& K,
Therefore, Countrybank certificate simply pays 70% of the payoff of a European call
option on WIG with exercise price of K, which expires when the certificate expires, as
the return on the deposit.
3. Insurance Created With Options, Collars, Floors, Caps, Spreads, Straddles
As we had already pointed out in an exercise in Section 2, some options positions result
in the same payoffs as certain insurance policies. For example, a put option with the
exercise price of $50 on a stock selling now for $60 provides insurance against the fall of
the price below $50. This insurance provided by a put is sometimes called a floor, as it
puts a “floor” under losses that the long position can suffer.
On the other hand, there are situations when an opposite form of insurance is needed.
Suppose that an insurance company promises to pay its customer the rate of return on a
stock market index, and the customer holds an account valued at $1,000,000, and the
current value of the index is 1000. Thus the customer holds 1000 units of the stock
market index. In order to protect itself (i.e., insure itself) against a sharp increase in the
stock market index (which would result in a corresponding sharp increase in the liability
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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to the customer), the insurance company can buy 1000 calls on a unit of the stock market
with the exercise price of 1000. Then any increase in the stock market index above the
current value of customer’s account would be fully covered by the increase in the value
of these calls, and the company’s liability would be fully insured. This form of insurance
is sometimes called a cap, as it “caps” the value of the liability of the company.
Caps and Floors
Caps and floors, however, are most commonly used in the context of insuring against
interest rate risk. They provide one-sided protection against movements in a floating
interest rate. They are commonly embedded in adjustable rate mortgages in the United
States. The cost can either be paid for up-front or embedded in the interest rate paid. An
interest rate cap consists of a series of caplets, and an interest rate floor consists of a
series of floorlets.
An interest rate caplet is analogous to a call option on the level of interest rates: at
expiration, if the interest rate is above the strike rate, the caplet pays ( i ! k ) times the
notional, and zero otherwise ( i is the index rate, k is the strike rate).
An interest rate floorlet is analogous to a put option on the level of interest rates; at
expiration, if the interest rate is below the strike rate, the caplet pays ( k ! i ) times the
notional, and zero otherwise ( i is the index rate, k is the strike rate).
An interest rate collar is a long position in a cap plus a short position in a floor; it
effectively makes a payment to its holder whenever the index rate is outside the
boundaries set by the strike rates (the floor rate is below the cap rate, and the range
between them leaves the floating rate alone).
Covered and naked option writing
If you own the underlying and write an option, this is called covered writing. If you do
not own the underlying, and write an option, this is called naked writing.
Exercise 3.1
Mr. Romuald Carcosheek owns 100,000 shares of Megabank, a Polish bank whose shares
are traded on the Warsaw Stock Exchange. Mr. Carcosheek is concerned that the current
price of PLN 50 of the Megabank stock is too high, and it is likely to decline to PLN 45
range. He decides to protect himself, at least partially, by writing at-the-money calls on
his shares, which currently sell for 1 PLN per share. He is able to invest the premium
received from writing these calls in a risk-free one-year Polish government bond earning
6%. In one year, the shares of Megabank are selling for PLN 47. Calculate Mr.
Carcosheek’s total rate of return over that one-year period, including his unrealized loss
on the shares, premiums received for options written, and interest earned on the
premiums invested in a risk-free one-year Polish government bond.
Solution.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 14 -
Mr. Carcosheek begins the year with 100,000 shares of Megabank worth PLN 50 each,
for a total initial investment of PLN 5,000,000. By writing covered calls, he earns PLN
100,000, which accumulate to PLN 106,000 by the end of the year by being invested in a
government bond. But at the end of the year, his 100,000 shares are worth only PLN
4,700,000, so that his total amount at the end of the year is PLN 4,806,000. His rate of
return is
4, 806, 000
! 1 = !3.88%.
5, 000, 000
While this may look bad, note that the shares declined by 6%, so was able to cushion his
loss by utilizing the strategy of writing covered calls.
An important observation to make concerning European calls and puts is that if a
European call and a European put have the same underlying, same maturity date, and the
same exercise price, then
Call ! Put = Forward.
In other words, a investor who buys a European call with exercise price K and writes a
European put on the same underlying, same maturity date, and the same exercise price,
will experience exactly the same payoff as an investor who enters into a long forward
position with the same maturity date and the same exercise price K. Short forward
position is replicated by a portfolio of a long put and short call (both European).
Put-Call Parity
Let us make the following assumptions:
- No dividends payable on the underlying.
- All investors may borrow and lend at the risk-free rate.
- There are no transaction fees or taxes.
- Short selling and borrowing are allowed, fractional shares may be traded.
- There are no arbitrage opportunities.
Under these assumptions, consider the following two portfolios:
Portfolio 1: Long European call option with maturity date t years from now, plus
Ke! rt invested in the risk-free asset.
Portfolio 2: Long European put and one share of the underlying.
We assume that the call and put have the same exercise price and maturity.
Then the value of these two portfolios is as follows:
ST < K
Long call
0
! rt
K
Ke in risk-free asset
Total Portfolio 1
K
Long put
K ! ST
One share of stock
ST
Total Portfolio 2
K
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
ST > K
ST ! K
K
ST
0
ST
ST
- 15 -
We see that these two portfolios produce identical payoffs regardless of the price of the
underlying. Therefore, these two portfolios must sell for the same price, i.e.,
c + Ke! rt = S + p
where c is the call price, and p is the put price. This relationship is called the put-call
parity.
Exercise 3.2
The current price, as of June 30, 2002, of a $325 call on September 30, 2002, gold is $12,
with the spot price being $315 per ounce, and a three-month Treasury-Bill maturing on
September 30, 2002 with $10,000 maturity value selling now for $9,955.20. Find the
price of a September 30, 2002, $325 (exercise price) gold put as of June 30, 2002.
Assume that all conditions for put-call parity to hold are satisfied.
Solution.
We use the put-call parity formula
c ! p = S ! PV (K ).
In this case, c = 12, S = 315, and the present value of a cash flow paid on September 30,
9955.20
2002, as of June 30, 2002, is established by multiplying that cash flow by
. We
10000
get
9955.20
12 ! p = 315 !
" 325 # !8.54.
10000
This gives us p ! 20.54.
Exercise 3.3
You are given the following information:
- An option market satisfies the condition for put-call parity.
- The current underlying security price is 100.
- A call option with a strike price of 105 and maturity one year from now has a current
price of 4.
- A put option with a strike price of 105 and maturity one year from now has a current
price of 6.
Determine the one-year risk-free interest rate.
Solution.
We use the put-call parity relationship
c ! p = S ! PV ( K ) .
Substituting the data given in the problem we get
105
4 ! 6 = 100 !
,
1+ i
105
3
or 102 =
. The solution is i =
! 2.94%.
1+ i
102
Spreads, Collars, Straddles
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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An option spread is a position of only calls or only puts, in which some options are
purchased and some are written. Here are some of these strategies:
• A bull spread consists of a long call with a lower exercise price and short call with
higher exercise price, both calls expiring at the same time.
• A bear spread consists of a long call with a higher exercise price and a short call with a
lower exercise price, both calls expiring at the same time.
• A box spread consists of a long call and short put with the same exercise price, and
another position of a long put and a short call with the same exercise price, but different
than the previous one. This amounts to being long one forward and short another forward.
The strategy is purely a means of borrowing or lending money.
• A ratio spread is constructed by buying by buying m calls at one exercise price and
selling n calls at a different exercise price, with same maturity and same underlying.
• A collar is created by purchasing a put option and writing a call option. A reverse
position is a short collar. A collar width is the difference between the call and put strike
prices. A zero-cost collar is created by adjusting strike prices so that there is no cost or
income to the transaction.
• A straddle is a position consisting of a long call and a long put with the same exercise
price. This position benefits from high volatility. Short straddle benefits from low
volatility.
• A strangle consist of a long out-of-the-money call and a long out-of-the-money put.
This is a strategy similar to a straddle, but at a lower cost.
• A butterfly spread consists of a short straddle combined with a long out-of-the-money
put and a long out-of-the-money call. The position could be symmetric or asymmetric.
4. Managing Risk With Derivatives
• Hedging with forwards or futures
A long position in stocks or commodities can be hedged by holding a short position in a
forward or in futures.
• Portfolio insurance
You can insurance a minimum value of a portfolio by buying a put option.
• Insurance by selling a call
Used to hedge the risk of a long portfolio of stocks or commodities. This does not
provide complete protection, but reduces risk with premium income. But recall put-call
parity:
c ! p = S ! PV ( K ) ,
so that
S ! c = PV ( K ) ! p.
Thus the long position insured with a short call can be replicated by buying a bond and a
selling a put option.
Why do firms manage risk?
Financial risk management should be viewed from the perspective of the ModiglianiMiller Theorem: financing mode affects value only if it
• increases firm’s output (without increasing costs, or at least not increasing them above
the benefit created),
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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• reduces taxes,
• reduces bankruptcy costs, or
• reduces agency costs.
But there may be good reasons not to hedge:
• Derivatives involve high transaction costs.
• Hedging requires costly expertise.
• Hedging increases agency costs (e.g., rogue traders).
• Hedging increases taxes or other government costs (e.g., regulatory).
Cross-Hedging
There are situations when hedging is achieved not by a position in a security identical or
directly related to the originally position held, but a more remotely related security. This
means practically that the movements of prices of the hedged security will not be
mimicked by the hedge, but will only be related somehow. Suppose that an investor holds
Q units of a security whose price is S and hedges it with a short position in H units of a
security whose price is F, with ! S being the standard deviation of the changes in price of
a unit of the hedged instrument, ! F being the standard deviation in the changes in price
of the hedging instrument, and ! being the correlation of the price changes of the two
instruments. Then the investor’s total position is
! = QS " HF.
The variance of this position is
Var ( ! ) = Var (QS " HF ) = Q 2# S2 + H 2# F2 " 2QH $# S# F .
We want to derive the value of H, which minimizes this variance. Note that the value of
Q is given. We take the derivative of Var ( ! ) with respect to H and set it equal to 0
!Var ( " )
= 2H # F2 $ 2Q%# S# F = 0.
!H
Note that
! 2Var ( " )
= 2# F2 > 0.
2
!H
The value of H at which the derivative of Var ( ! ) with respect to H is equal to 0, i.e.,
!"
H =Q S,
"F
produces the lowest variance portfolio. Also,
H !" S
h=
=
Q
"F
is called the optimal hedge ratio or minimum variance hedge ratio.
Consider a company that is hired to repair roads in a small town every year. The
materials used in the process of road repair are not traded in financial markets, but they
are produced from crude oil, an asset for which futures contracts are readily available.
Suppose that this company is paid Pr per kilometer of a road repair, and that N c barrels
of crude oil are needed to repair N r kilometers of roads. Let Pc be the crude oil price per
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 18 -
barrel. Then this company’s profit (we ignore other possible expenses and sources of
profits for now) is
! = N r " Pr # N c " Pc .
Now suppose that this company enters into H futures contracts on crude oil, assuming for
simplicity that each contract covers one barrel. If F is the futures price, the profit on such
a hedged position is
! H = N r " Pr # N c " Pc + H ( Pc # F ) .
Now suppose that the objective is to minimize the variance of the hedged position. The
variance of the hedged position is (we make a simplifying assumption that F does not
vary, as we are doing this calculation for this moment in time, based on historical
estimates of volatility and correlations)
2
! "2 H = N r2 # ! P2r + ( H $ N c ) # ! P2c + 2 ( H $ N c ) # N r # Cov ( Pr , Pc ) .
The variance-minimizing hedge ratio is
Cov ( Pr , Pc )
H * = Nc ! Nr "
.
# c2
In this hedge ratio, we can interpret the first term as hedging costs, and the second term
as hedging revenue. The coefficient that N r is multiplied by is the result of regression of
the price of a kilometer of a road on the price of a barrel of oil. The resulting minimum
variance is
(
)
! "2 H * = N r2 # ! P2r # 1 $ %P2r , Pc .
This positive variance indicates that there is risk left in this position, due to possible
divergence of the revenue earned from the road and the price of oil. This kind of risk due
to divergence of the value of the portfolio hedged and the value of the hedge used is
called the basis risk.
5. More on Forwards and Futures
Alternative ways to buy a stock
• Buy the stock directly, for cash.
• Fully leveraged purchase: borrow all money used to buy the stock.
• Prepaid forward contract: pay for the stock today, receive it at time t in the future.
• Forward contract.
Prepaid forward contract
In the absence of dividends, the price on the prepaid forward contract is today’s stock
price S. The reason is simple: you will own the stock anyway, so you should pay its
market price. We will denote the prepaid forward price by F0,tP . If we use a subscript in
the stock price notation to denote the time of the price, we have F0,tP = S0 . If the prepaid
forward is on a stock that pays a dividend, then the forward contract holder does not
receive that dividend, and the prepaid forward price is the price of the stock without any
of the dividends paid through time t.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 19 -
Forward contract
Let us denote by F0,t the forward price for transaction to occur at time t. As we had
already noted before, F0,t = S0 ! ert , where r is the risk-free force of interest between time
0 and time t. If the stock pays a dividend, the price of the stock should be used without
any of the dividends payable between time 0 and time t. The forward premium is defined
F
1 "F %
as the ratio 0,t . The annualized forward premium is ! ln $ 0,t ' , and, in the absence of
t
S0
# S0 &
arbitrage opportunities, it equals the risk-free force of interest (or the difference between
the risk-free force and the force of dividend).
Note that the payoff of a long forward position is St ! F0,t . This payoff can be replicated
by buying 1 share of stock with borrowed funds of e! rt " F0,t . This shows again that
F0,t = S0 ! ert . It also illustrates the fact that
Forward = Stock – Risk-Free Zero-Coupon Bond in the amount e! rt " F0,t ,
and
Stock = Forward + Risk-Free Zero-Coupon Bond in the amount e! rt " F0,t ,
and
Risk-Free Zero-Coupon Bond in the amount e! rt " F0,t = Stock – Forward.
Note that given the market prices of a stock and of a forward on that stock, we get the
implied risk-free interest rate from the last identity. That implied interest rate is called the
implied repo rate.
A transaction in which you buy the underlying and short the offsetting forward is called a
cash-and-carry. As we see from the above, this is equivalent to a purchase of a risk-free
zero-coupon bond. Market makers in forwards often offset their position by buying the
underlying, and creating the cash-and-carry position. A reverse cash-and-carry is created
by being short underlying and long forward, and is equivalent to borrowing at the riskfree rate. We should note that the rate on cash-and-carry is the risk-free rate only if the
forward is prices by the familiar equation F0,t = S0 ! ert . Otherwise, arbitrage can be
utilized to earn riskless profit. Recall that:
Arbitrage: Creation of a portfolio requiring no capital outlay, but never losing
money, and producing positive payoff with positive probability.
In practice, however, an attempt to arbitrage the difference between F0,t and S0 ! ert
involve transaction costs. Suppose that the spot bid and ask prices are S b < S a and that
the forward bid and ask prices are F b < F a . Assume also that there is a transaction cost k
for a transaction in the stock or its forward, and that the force of interest for borrowing
and lending differs, with r b > r l . We assume all transaction costs to occur at time 0, and
none at time t. Consider an arbitrageur who sells the forward and buys the stock (assume
no dividends for simplicity, or consider the stock without the dividend). The arbitrageur
will have an upfront cash cost of k plus S a + k . We assume he/she borrows to finance
that position. At time t the payoff is
(
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
)
- 20 -
(
)
(
)
! S a + 2k " er t + ( F0,t ! St ) + St = F0,t ! S a + 2k " er t .
!#"#
$
!##"##$
!#
#"##
$
b
Repayment of borrowing
Value of short forward
b
Denoted by F +
The quantity F + is the upper bound for current forward price so that the arbitrage is not
profitable. In the same fashion, the lower bound F ! below which arbitrage is profitable
is given by the formula F ! = S b ! 2k " erl "t . It should be noted that this analysis may
actually underestimate all costs involved.
(
)
Quasi-arbitrage is a substitution of a lower-yielding position by a higher-yielding
position. If a company can borrow at 8.5% and lend at 7.5%, clearly there is no arbitrage
possible. But if a company is already lending at 7.5%, and it is possible to arrange for a
cash-and-carry with implied repo rate of 8%, an arbitrage becomes possible.
Does the forward price predict the future price?
When you invest in stock, you expect to earn the risk-free rate plus the risk premium. But
when buy enter into a forward, you put down no money, so you should not get the riskfree rate, just the risk premium (as you are still exposed to the risk of the underlying).
Consider a one-year forward. Let r be the risk-free force of interest for that year. Let !
be the expected force of return of the stock. Then we have
F0,1 = S0 ! er
and
E ( S1 ) = S0 ! e" .
This tells us that
E ( S1 ) = F0,1 ! e" # r .
The expression ! " r is the risk-premium for the underlying. This tells us that the price
of a forward is a downward (assuming positive risk premium) biased predictor of the
future price of the underlying, and the degree of the bias is determined by the riskpremium of the asset. Recall that a forward can be replicating by a long position in the
underlying (earning the risk-free rate plus the risk-premium) and a short position in a
zero-coupon risk-free bond (earning the risk-free rate). As the risk-free rate is paid, only
the risk-premium accrues to the forward.
More on Futures
Forward and futures prices may differ. The reason is that with the futures contracts,
interest is earned on any mark-to-market proceeds, and lost on any required margin
deposits. The required margin during the life of a contract (called the maintenance
margin) is generally lower than the initial margin, but is nevertheless required. We will
illustrate this in the exercise now.
Exercise 5.1
Consider a futures contract on the stock market index in a hypothetical country called
Cuba Libre, such index being called Viva Cuba Libre, or VCL for short. Suppose that
you enter into a contract with a notional value of 1 million libretas (currency of Cuba
Libre). The margin required is 7%, or 70,000 libretas. Each contract correspondes to
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 21 -
1000 units of VCL index, and that 1000 is used as the contract multiplier. Futures are
initially trading at 1000 libretas. Calculate the dollar-weighted rate of return over a week
on this contract, assuming the following:
• The maintenance margin is 5%.
• Risk-free interest rate is 0.02% per day.
• When additional margin deposit is required, you pay it in.
• When funds become available to withdraw, you take them out immediately.
• Prices at the close of the days of the week for the futures on the VCL index are:
Monday: 990 libretas;
Tuesday: 1025 libretas;
Wednesday: 1075 libretas;
Thursday: 920 libretas;
Friday: 1000 libretas.
• Position is costless to enter into and to close (other than the margin deposit), and it is
closed at the end of the week.
Compare the net cash flow from the futures contract to that for a costless forward entered
at the beginning of the week, for the purchase at the end of the week.
Solution.
We have the following history of this position
Week beginning
Futures
price
Price
change
Margin
balance
1000
---
70000
70000 !1.0002 " 1000 !10 = 60014
Monday close
990
– 10
As of Monday close, maintenance margin required is 5% of 990,000, i.e., 49,500, so that
10,514 libretas are withdrawn and 49,500 libretas remain in the margin account
49500 !1.0002 + 1000 ! 35 = 84509.9
Tuesday close
1025
35
As of Tuesday close, maintenance margin required is 5% of 1025,000, i.e., 51,250, so
that 33,259.90 libretas are withdrawn and 51,250 libretas remain in the margin account
51250 !1.0002 + 1000 ! 50 = 101260.25
Wednesday close 1075
50
As of Wednesday close, maintenance margin required is 5% of 1075,000, i.e., 53,750, so
that 47,510.25 libretas are withdrawn and 53,750 libretas remain in the margin account
53750 !1.0002 " 1000 !155 = "101239.25
Thursday close
920
– 155
As of Thursday close, maintenance margin required is 5% of 920,000, i.e., 46,000, so that
147,239.25 libretas are deposited and 46,000 libretas are in the margin account at the end
of the day Thursday
46000 !1.0002 + 1000 ! 80 = 126009.20
Friday close
1000
80
As of Friday close, the position is liquidated and the margin balance of 126009.20
libretas is collected
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 22 -
Therefore, the dollar-weigthed rate of return over the trading week (five days) was:
!70000 + 10514 + 33259.90 + 47510.25 ! 147239.25 + 126009.20
=
5
4
3
2
1
70000 " ! 10514 " ! 33259.90 " ! 47510.25 " + 147239.25 "
5
5
5
5
5
54.10
=
# 0.1039%.
52076.61
Let us now compare this to a forward contract. We enter a forward at the beginning of the
week. Assuming that transaction is costless, the price at which we set the purchase is the
same as the initial futures price, i.e., 1000, if we take the futures-spot parity F = Se! t and
the forward-spot parity formula F = Se! t to hold identically for both contracts. Then at
the end of the week the price is 1000, and the transaction results in a zero return. This is
quite interesting: we have a net positive cash flow of 54.10 from the futures contract, but
a zero return from the forward. The futures part is more interesting: you had several cash
flows for the week. Those cash flows end up netting to a positive number, 54.10, and this
is what created your return. The structure of the futures contract and changes in the price
in this problem forces you to adopt the strategy of buying when the price is down, and
selling when the price is up, creating a positive return, in a sense, out of nothing. Good
for you!
One important practical implication of the fact that in general futures tend to benefit from
volatility of the underlying, as in the long run most volatility in on the upside, when we
hedge with futures, fewer futures contracts than equivalent forwards are commonly used.
Nevetheless, most of the time we will assume the same pricing formula for forwards and
futures. This is illustrated by the following exercises.
Exercise 5.2
Consider a May futures contract that calls for delivery of 1000 ounces of gold in July
(July 1) of the same year. Suppose that the current quoted spot price is $680 per ounce,
and the current annual risk-free continuously compounded interest rate is 4.75%. Assume
that gold carrying cost is zero. What is the current futures price assuming that the
fundamental relation between cash and future prices holds?
Solution.
The fundamental relation is F = Sert . We are given that r = 0.0475, t =
2
, S = $680,000
12
(1000 oz.) so that
2
F = Sert = 680, 000 ! e12
!0.0475
" 685, 404.70.
Exercise 5.3
Suppose that the value of the S&P 500 stock index is currently $1,300. If one year
Treasury Bill interest rate is 6%, and the expected dividend yield on the S&P 500 Index
is 2%, both of these interest rates expressed as annual effective rates of return, what
should the one-year maturity futures price be?
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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Solution.
The spot-futures parity relationship with adjustment for dividends, assuming annual
effective risk-free interest rate, and annual effective dividend yield, is (approximately):
F0 = S0 (1 + rF ! d ) ,
where d is the dividend yield of the underlying. The exact formula would be
S
F0 = 0 ! (1 + rF ) ,
1+ d
Substituting problem data we get:
F0 = $1, 300 (1 + 0.06 ! 0.02 ) = $1, 300 "1.04 = $1, 352.
You might wonder whether we can use the expected dividend yield in this exercise, if the
actual dividend yield is uncertain. If you do so wonder, you are most certainly raising a
valid point. This formula assumes that the dividend yield is known with certainty, and in
real life applications you would have to be more careful about using such a formula.
One more question about the above is whether the Treasury-Bill rate is appropriate for
the risk-free rate used in the calculations above. The problem with the Treasury-Bill rate
is that it tends to be relatively lower than other short-term interest rates, and some believe
it to be unnaturally so. One possible explanation is that the margin in transactions such as
buying on margin, short-selling, option writing, or purchases of futures, can be posted in
cash, in which case it generally does not earn interest, or in Treasury-Bills, which earn
interest automatically, as they are discounted zero-coupon bonds of short maturity (up to
one year). Thus traders may buy up Treasury-Bills, bidding up their prices, and bidding
down their yields. For that reason, some prefer using the short-term LIBOR (London
Inter-Bank Offer Rate) index for pricing forwards and futures, as those instruments are
also nearly completely free of credit risk.
Quanto index futures
The Nikkei 225 futures contract traded on the Chicago Mercantile Exchange is quite
peculiar. Its values are derived from the Nikkei 225 index, but the currency in which it is
expressed is the U.D. dollar, not the Japanese yen. The size of that futures contract is $5
times the numerical reading of the Nikkei 225 index. It is cash-settled based on the
opening Osaka quotation of the Nikkei 225 index on the second Friday of the expiration
month. A contract of that nature is called a quanto contract: referring to an index in one
currency, but traded in another currency.
What are the uses of stock index futures?
• Asset allocation. This can mean switching from stocks to Treasury-Bills or the other
way around. But using longer term Treasury-Bond futures together with stock index
futures we can also construct allocations between stock index and a bond portfolio. If we
own a diversified stock portfolio and short S&P 500 futures ot reduce stock allocation,
while going long Treasury-Bond futures, the first transaction converts stocks into T-Bills,
and the second one converts T-Bills into T-Bonds. This combined use of futures is called
futures overlay. You can also use futures to convert a bond-portfolio manager into a stock
portfolio manager. Suppose that you have found a bond portfolio manager who can beat
the Treasury-Bond index consistently, but you want to invest in stocks. You can hire that
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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manager, while simultaneously shorting Treasury Bond futures (thus converting the
manager to T-Bill plus this manager’s outperformance of the index) and going long S&P
500 futures (thus converting T-Bill return into stock return).
• Cross-hedging. This refers to using a futures contract on a different security than the
one we are hedging. It is quite common to hedge diversified stock portfolios with S&P
500 futures, after adjusting for the size of the portfolio in relation to the size of the
contract, and for the beta of the portfolio.
Currency futures and forwards
These instruments are used to hedge against changes in currency exchange rates. The
simplest instrument is a currency prepaid forward. Suppose that in one year you want to
have 1 liberta. Suppose that the risk-free force of interest in libertas is rFor and the riskfree force of interest in U.S. dollars is rDom . To obtain 1 liberta in one year, we must have
e! rFor today. Suppose that the exchange rate at time t is Et dollars needed to purchase
one liberta. We conclude that in order to assure a purchase of one liberta a year from now
P
we need F0,1
= E0 ! e" rFor dollars. This is the price of a prepaid forward.
In general, the currency forward price (assuming a costless contract to be entered an time
0 and realized at time t) is
F0,t = E0 ! e" rFor !t ! erDom !t = E0 ! e( rDom " rFor )t .
Covered interest arbitrage is a transaction consisting of borrowing in domestic currency,
lending in a foreign currency, and entering into a forward transaction to purchase the
foreign currency. The principle behind this is that a position in foreign risk-free bonds,
with the currency risk hedged, pays the same return as domestic risk-free bond.
Eurodollar futures
The Eurodollar contract is based on a $1 million 3-month deposit earning LIBOR. There
is also a 1-month contract, handled similarly. Suppose that the current LIBOR is 1.25%
over 3 months. By convention, this is annualized by multiplying by 4, to the quoted
annualized LIBOR rate of 5.0%. The Eurodollar futures price at expiration of the contract
is 100 – Annualized 3-month LIBOR. The settlement is based on then current LIBOR
(i.e., for the future three months counting from the date of settlement). This futures
contract is used for hedging interest rate risk.
6. Swaps
Swaps are generally derivatives that trade cash flows between counterparties based on
two pieces of underlying. Such a contract is written for a specific term, called the swap
term, or swap tenor. An unusual, and simplest type of a swap is an exchange of a single
payment for multiple payments (or, possibly, multiple deliveries of some form of
underlying) in the future. This is called a prepaid swap.
Swaps are commonly settled financially, i.e., currency payments are exchanged. But it is
also possible to create a physical delivery swap. For example, one could arrange for a
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 25 -
delivery of an ounce of gold every year for the next twenty years in exchange for a fixed
payment of $600 every year. Note that this transaction can be actually decomposed into a
series of forwards (and that is not a coincidence). Such physical delivery swap can also
be settled financially, of course, by the side delivering the gold paying the other party the
difference between $600 and the price of an ounce of gold at the time of delivery.
It is rare that the two counterparties of a swap are both market participants. Swaps are
arranged as private transactions and not traded on exchanges. They are typically arranged
by a dealer, for a client of that dealer. The dealer ends up holding one side of the swap
deal, but usually seeks to make another swap arrangement that would at least partially
offset that position. That second transaction is commonly called back-to-back
transaction, or matched book transaction. The dealer can also seek to hedge the exposure
using traded derivative instruments. It is generally difficult, however, to find an exact
hedge in the market, and the dealer may have to look at the dealer’s entire portfolio
exposure and hedge pieces of it, combining exposures from various transactions.
The market value of a swap
When a swap is entered into, it is standard that no payments change hands and future
payments committed to by each party have the same market value. As time passes, the
value of each counterparty’s position changes. Because what one party pays, the other
one receives, the total of the values remains zero. This may create credit risk for the
party, which has a positive value of the swap.
Interest rate swaps
A “plain vanilla” (i.e., the simplest kind, as in the “plain vanilla” ice cream) interest rate
swap is a contract between two counterparties, requiring them to make them interest
payments to each other over the term of the contract, based on different types of
underlying bonds or interest rate indices. The long party (fixed-rate payer) pays interest
to the second at a fixed rate, while the short party (floating-rate payer) pays interest to the
first at a rate that changes (“floats”) according to a specified index. This kind of a swap is
also called a pay-fixed swap, because the long party pays fixed. The actual cash payments
are determined by multiplying the relevant rate of interest by a face amount, or principal,
which is called the notional principal, or just the notional. For example, consider a
semiannual swap with a notional of $10 million:
Long: Pay 4%, nominal compounded semiannually, and receive 6-month LIBOR,
Short: Pay 6-month LIBOR and receive 4%, compounded seminannually.
If the 6-month LIBOR on the first payment date were 3% (nominal compounded
semiannually), then:
0.04
Fixed-rate payer pays floating-rate payer $10M !
= $200, 000,
2
0.03
Floating-rate payer pays fixed-rate payer $10M !
= $150, 000.
2
This would be actually settled by the fixed-rate payer paying the floating-rate payer
$50,000 rather than the two parties paying offsetting amounts.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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Swaps are commonly used to manage interest rate risk exposure. An insurance company
can use swaps to produce an income stream that better matches its liability structure.
Suppose that you are managing a company that issued fixed-rate liabilities that credit
5.50% (e.g., GIC or SPDA), and backs them with a bond that pays 3-month LIBOR +
1%. There is a mismatch creating interest rate risk exposure. Now suppose that this
company enters into a swap where it pays 3-month LIBOR and receives 5.50%. This
would result in a net 1% profit to the company, without any interest rate risk. This, of
course, is an idealized example, but you should understand how it works. The company
used a swap to convert its floating-rate asset to a fixed-rate asset that better matched its
liabilities; in doing so, it locked in a spread of 100 bps. In practice, most commonly, life
insurance companies seek to convert their fixed coupon bonds income into floating rate
income. A swap in which a party receives a floating rate in exchange for fixed payments
on bonds that this party already holds is called an asset swap.
Swap rate
A short (pay-variable) swap position is equivalent to buying a fixed coupon bond with
funds borrowed at the swap’s floating rate. If an interest rate swap is arranged in such a
way that no payment is made upfront, and in exchange for fixed rate payment, the long
side receives the market floating rate, the resulting fixed interest rate is called the swap
rate. The swap rate turns out to be simply the coupon rate on a par coupon bond, with
that bond’s maturity equal to the swap term.
The swap curve
Thanks to Eurodollar futures on 3-month LIBOR, 3-month LIBOR forward rates can be
found for up to 10 years. The swap rates established against the floating rates equal to
those LIBOR forward rates constitute the swap curve. The swap spread is the difference
between a given swap rate and a Treasury Bond yield for the corresponding maturity.
The swap’s implicit loan balance
At inception of an interest rate swap, the swap has zero value to both sides. As time
passes, and interest rates change, the value to each counterparty changes. Even in the
absence of interest rate changes, the value will change if the yield curve is not flat, as one
side makes a fixed rate payment, and the other one pays, implicitly, based on forward
rates. The present value of future payments is an asset to one party, and a liability to the
other party. We can call it the implicit loan balance.
Deferred swap
A deferred swap is a swap that begins at some date in the future, but for which the swap
rate is agreed upon today.
Why do firms use interest rates swaps?
Short-term borrowing is generally only available to firms with very good credit rating.
Even such high quality firms may have a hard time issuing large amounts of short-term
debt. Longer-term (five, possibly to ten years) debt is generally easier to issue for firms
with lower credit rating, and can be issued in larger amounts. Longer-term fixed interest
rate debt is easier to handle for a borrower, as payments are more predictable. Floating
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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rates can change significantly, and having a large amount of debt issued at floating rates
can be quite risky, even for a firm with good operating results. Given this situation, some
firms issue large amounts of debt in five to ten year fixed coupon bonds and then swap
them to floating, hoping that they will save in interest expenses in the long run, as the
yield curve is typically upward sloping.
Amortizing and accreting swaps
A swap in which the notional amount declines over time is called an amortizing swap. A
swap is which the notional amount increases over time is called an accreting swap.
Currency swaps
In a currency swap, the notional amounts are specified in different currencies. This is
handled by exchanging the notional amounts at the beginning of the swap and returning
them at the end. Currency swaps are used to manage foreign exchange exposures.
Consider, for example, a British company that wants to issue debt in Japan, but finds the
process of registration of securities and issuing them in Japan too much to handle. Instead
the company can issue floating debt in Great Britain, and enter into a swap to pay fixed
long-term Japanese Yen rate, and receive British Pound floating.
The relationship used for pricing currency swaps is the Covered Interest Rate Arbitrage.
Let:
• E0 be the current exchange rate in dollars per unit of foreign currency,
• Et is the foreign currency exchange rate
• rDom be the U.S. dollar-denominated (domestic) risk-free force of interest,
• rFor be the risk-free force of interest in foreign currency,
• F0,t be the forward exchange rate in dollars per one unit of foreign currency at time t.
Take one unit of foreign currency now. You can then do one of these two things:
• Convert it immediately into U.S. dollars and invest in the U.S. risk-free asset; this
would give you E0 ! erDom !t foreign currency units at time t.
• Invest in the foreign risk-free asset and enter a forward contract to convert to dollars at
time t. This would give you erFor !t ! F0,t at time t.
In the absence of arbitrage, your final payoff is the same, and the prices of the two
instruments should be the same. The resulting relation is:
E0 ! erDom !t = erFor !t ! F0,t .
This is, of course, the same relationship we discussed previously.
A differential swap is a swap in which payments are made based on the difference in
floating interest rates in two different currencies, with the notional in a single currency.
Commodity swaps
A swap in which the payments (or at least one side payments) are expressed in terms of a
commodity is called a commodity swap. Those swaps may be arranged with varying
quantities and prices, depending on seasonal factors.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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Swaptions
An option to enter into a swap is called a swaption. A payer swaption is a right, but not
an obligation, to enter into a swap to pay fixed and receive floating. A receiver swaption
is a right to enter swap to pay floating and receive fixed.
A total return swap is a swap in which one party pays the realized total return (dividends
plus capital appreciation) on a reference asset or index, and the other side pays some
floating rate, e.g., LIBOR, possibly plus a spread. For example, total return on S&P 500
can be swapped into LIBOR plus a spread. Total return swaps can be used to invest in a
stock index in a country requiring withholding taxes on dividends.
A default swap is an instrument in which one side pays regular payments, similar to
insurance premiums, and the other party makes a payment (or payments) when a
specified credit event (or specified credit events) happens (happen).
7. No-Arbitrage Pricing Models
Arbitrage: Investment requiring no capital outlay, but never losing money, and producing
positive payoff with positive probability.
In a one-period market, assuming a finite number of states of the world ! 1 , ! 2 ,..., ! M , the
condition of the market being arbitrage-free means that any security, which makes only
zero or positive payments in each state of the world, with positive payments made with
positive probability, cannot have a zero or negative price. The simplest type of such a
!
security is an Arrow-Debreu security ei making the following payments:
…
…
!1 ! 0
!2 ! 0
!i ! 1
!M ! 0
We cannot be certain that all Arrow-Debreu securities are available for trading in the
market, but we do know this: linear combinations of securities (portfolios) generally
(assuming short selling is allowed) are also securities. This implies that some ArrowDebreu securities can be replicated with securities traded in the market. How many – that
depends on the dimension of the vector space spanned by the securities traded in the
market. Can this space be not of maximum (M) dimension? Yes, we do not know, in
general, that we can place bets on all possible states of the world.
The way we constructed this one-period model, all securities are elements of the Mdimensional Euclidean vector space. And their prices? It is important to note that if we
write P for the price of a security, then for a linear combination of securities S1 and S2 ,
we have:
P (! S1 + " S2 ) = ! P ( S1 ) + " P ( S2 ) .
Therefore, price is a linear functional on the space of securities. We can work on infinitedimensional generalizations of this, but practical applications have only finitedimensional Euclidean spaces.
If the dimension of the space spanned by traded securities is equal to the number of the
states of the world, then we can find prices of all Arrow-Debreu securities – in this case
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 29 -
the market is called complete (assuming it is arbitrage-free). If all of their prices are
positive, the market is arbitrage-free, and the vector of those prices is called the stateprice vector. If the dimension of the space spanned by the traded securities is less than
the number of the states of the world, then we can always extend the linear functional of
existing prices to the whole space. Since the space is finite dimensional, so we can just do
this step-by-step, adding a price value in each missing dimension, by pricing an
appropriate Arrow-Debreu security, until we reach a functional defined on the whole
space. This also shows that if the initial model was arbitrage-free, we can make the final
model also arbitrage-free, by making sure that the prices of Arrow-Debreu securities we
are adding are positive. Once all Arrow-Debreu securities are priced by the market and
have positive prices, the market must be arbitrage-free and complete.
In such a complete and arbitrage-free market a security, which is a sum of all ArrowDebreu securities becomes available for trade. This security pays the same amount at the
end of the period regardless of the state of the world. Such a security is called a risk-free
security. We will denote its price by v and the value of a unit amount 1 invested in that
security S1 at the end of the period by 1 + i. Given that the Arrow-Debreu securities
form a basis for the vector space of all securities, the price of a security S j paying
S j (1, ! k ) = S j (! k ) in the state ! j is:
M
!
!
!
!
!" P ( ek ) #$ % !" S j (& k ) #$ = S j (& 1 ) % P ( e1 ) + S j (& 2 ) % P ( e2 ) + … + S j (& M ) % P ( eM ) =
k =1
= S j (& 1 ) % ' 1 + S j (& 2 ) % ' 2 + … + S j (& M ) % ' M .
In particular,
!
M
1 = P ( S1 ) = !" P ( ek ) #$ % [1 + i ]k =1 = (1 + i ) % & 1 + (1 + i ) % & 2 + … + (1 + i ) % & M .
If we define
!
P ( ek ) ! (1 + i ) = " k ! (1 + i ) = qk ,
then the qk ’s can be considered probabilities, the so-called risk-neutral probabilities.
Fundamental Theorem of Asset Pricing. The single period securities market is arbitrage
free if, and only if, there exists a state price vector, and this is equivalent to existence of a
risk-neutral measure.
Assume frictionless market for securities in which trades occur only at the times
t = 0,1, 2... Let it be the one-period risk-free interest rate, short rate. Let Vj,t be the j-th
primitive security value (ex-dividend) at time t, and C j,t be the dividend (cash-flow
payment) at time t for it. Assumption of no arbitrage in the market is equivalent to the
existence of risk-neutral probability measure under which:
" Vj,t +1 + C j,t +1
%
Vj,t = Et $
Vk,t ,it ,1 ! k ! N ' .
1 + it
#
&
Going back by induction:
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 30 -
$
'
&
)
C
Vj ( 0 ) = E & # t t +1 )
& t "0
)
&% ! 1 + i j )(
k=0
The model should be calibrated to existing default-free multi-period bonds. You can
rewrite the above formula as:
'
*
)
,
Ct +1
Vj ( 0 ) = $ Pr (! ) ) $ t
,
! "#
) t &0
,
)( % 1 + i j (! ) ,+
k=0
Each ! "# represents a possible interest rate scenario (in the risk-neutral world). So …
what’s the point? The point is: you must run interest rates scenarios, and scenarios of the
future in general, modeling your financial intermediary via simulation, in order to
properly establish valuation of derivatives issued by you.
(
)
(
)
Exercise 7.1
The market model consists of 2 securities and a bank account. The bank account pays
interest of 10% per year and is risk-free. The price of each security today is 100. There
are three possible scenarios for the prices of the securities in 1 year:
Security X
Security Y
Scenario 1
220
0
Scenario 2
55
0
Scenario 3
0
250
Calculate the state price vector for this securities market model, if one exists. If there is
no state price vector, explain why.
Solution.
We have the following system of equations implied by the existing price structure:
# 110! 1 + 110! 2 + 110! 3 = 100,
%
$ 220! 1 + 55! 2 + 0 " ! 3 = 100,
%0 " ! + 0 " ! + 250! = 100.
1
2
3
&
24
4
2
This solves to: ! 1 =
, !2 =
, ! 3 = . The state price vector is:
55
55
5
24
4
2
[! 1 ! 2 ! 3 ] = "$ 55 55 5 %' .
#
&
Note that
24
4
2
[ q1 q2 q3 ] = "$ 55 !1.10 55 !1.10 5 !1.10 %' =
#
&
" 24 4 22 %
=$
.
# 50 50 50 '&
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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Exercise 7.2
You are given a securities market described by:
!1 1.20 $
S ( 0 ) = [1 1],
S (1) = #
&.
"1 0.90 %
Calculate the arbitrage-free price of a European put on the second asset with the strike
(exercise) price of 1.10.
Solution.
We find the state price vector from the system of equations:
#1! " 1 + 1! " 2 = 1,
$
%1.2 ! " 1 + 0.9 ! " 2 = 1.
This gives
1 1
#
%
%! = 1 0.9 = "0.1 = 1 ,
1
1
% 1
"0.3 3
%
1.2 0.9
%
$
1 1
%
%
1.2 1
"0.2 2
=
= .
%! 2 =
1
1
"0.3 3
%
%
1.2 0.9
&
The solution is
[!
1
"1
!2]= $
#3
2%
.
3 '&
! 0 $
The put described in this problem has cash flows #
& at time 1, and its price equals
" 0.20 %
1
2
0.4 2
! 0 + ! 0.2 =
= .
3
3
3 15
Exercise 7.3
You are given a securities market described by:
!1 1.50 $
S ( 0 ) = [1 1],
S (1) = #
&.
"1 0.90 %
Find a state price vector.
Solution.
Let [! 1 ! 2 ] be the state price vector. Then we must have:
"! 1 + ! 2 = 1,
#
$1.5! 1 + 0.90! 2 = 1.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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Hence 0.60! 2 = 0.50, so that ! 2 =
5
1
and ! 1 = . The state price vector is:
6
6
"1 5%
!2]= $
.
# 6 6 '&
We also have
#
& #1
1 + i ) ! 2 " (1 + i ) ( = %
[ q1 q2 ] = %! 1 " (!
!
%$
(' $ 6
=1
=1
[!
1
5&
.
6 ('
Exercise 7.4
You are given securities market described by:
!1.10 1.20 $
S ( 0 ) = [1 1],
S (1) = #
&.
"1.10 0.90 %
Find the risk-neutral probabilities.
Solution.
We can see that the risk-free rate is 10%. To find state-price vector, we write
" 1.1! 1 + 1.1! 2 = 1,
#
$1.2! 1 + 0.9! 2 = 1.
The solution
1 1.1
#
%
%! = 1 0.9 = "0.2 = 20 ,
% 1 1.1 1.1 "0.33 33
%
1.2 0.9
%
$
1.1 1
%
%
1.2 1
"0.1 10
=
= .
%! 2 =
1.1 1.1 "0.33 33
%
%
1.2 0.9
&
Using the notation: u = value of a unit invested in the risky instrument in the “up” state, d
= value of a unit invested in the risky instrument in the “down” state, risk-neutral
probabilities are:
(1 + i ) # d = 0.20 = 2 ,
q1 = (1 + i ) ! " 1 =
u#d
0.30 3
u # (1 + i ) 0.10 1
q2 = ! 2 " (1 + i ) =
=
= .
u#d
0.30 3
Exercise 7.5
You are given a securities market described by:
! 2 3$
S ( 0 ) = [1 2 ],
S (1) = #
&.
"2 1%
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 33 -
Is this market arbitrage-free? Is it complete?
Solution.
The risk-free security doubles, while the risky security appreciates by 50% or depreciates.
Thus the risk-free security always beats the risky security, the market is not arbitragefree. Because it is not arbitrage-free, it is not complete. You can also solve this by solving
the system of equations for the state price vector, and you will see that the solution
contains negative entries:
"2! 1 + 2! 2 = 1,
#
$ 3! 1 + ! 2 = 2,
so that
1 2
#
%
% ! = 2 1 = 3,
% 1 2 2 4
%
3 1
%
$
2 1
%
%
3 2
1
=" .
%! 2 =
2 2
4
%
%
3 1
&
Exercise 7.6
You are given a securities market described by:
!1 0 $
S ( 0 ) = [1 1],
S (1) = ##1 2 && .
#"1 0 &%
This market is not complete. Find an Arrow-Debreu security, and its price, such that
adding it to this market makes the market complete, while keeping it arbitrage-free.
Solution.
!0 $
Note that adding this vector ## 0 && as a third column makes the matrix S(1) invertible.
#" 1 &%
Therefore it would be best to price this e3 Arrow-Debreu security. State price vector
must satisfy the equations ! 1 + ! 2 + ! 3 = 1 and 2! 2 = 1. This implies that ! 1 + ! 3 =
and ! 2 =
1
2
1
1
1
1
. Therefore we must have 0 < ! 1 < , ! 2 = , and 0 < ! 3 = " ! 1 . The
2
2
2
2
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 34 -
!0 $
price of ## 0 && must equal ! 3 . In fact, we can pick ! 3 to be any number between 0 and
#" 1 &%
1
1
1
, and then let ! 1 = " ! 3 . We can, for example pick ! 3 = , and the market
2
2
4
becomes
!1 0 0 $
1$
!
S ( 0 ) = #1 1
,
S (1) = ##1 2 0 && ,
&
4%
"
"# 1 0 1 %&
clearly complete and arbitrage-free.
8. Option pricing
Assume the underlying is a stock, although the reasoning works for any underlying. Let t
be the time till expiration of an option, and assume that the stock price’s probability
distribution over the period is a discrete distribution, defined as follows:
with probability q, and
St = Seut
with probability 1 ! q.
St = Sedt
We assume that d ! r ! u , where r is the risk-free force of interest, and that if the stock
price goes up, the option is worth fu , and if the stock price goes down, the option value is
worth fd . We would like to construct a portfolio consisting of stock and risk-free asset
(Treasury Bills), whose payoffs will equal the payoffs of the option in both the up and the
down scenario. Since there are no arbitrage opportunities, the price of the option would
have to equal the price of such portfolio. Let n be the number of shares of stock bought
and B be the dollar amount of the risk-free asset, assumed to be short position. Then the
price of the portfolio is nS ! B. In the up scenario, the stock part of the portfolio is worth
nSeut , and in the down scenario, the stock part of the portfolio is worth nSedt . This gives
us the following system of equations:
nSeut ! Bert = fu ,
nSedt ! Bert = fd .
By solving it we get the portfolio value (and thus the option value) as:
nS ! B = e! r" # ( qfu + (1 ! q ) fd )
ert ! edt
. This is the basic binomial option pricing formula. It can be
eut ! edt
generalized to the trinomial case (with three possible prices) following the same
reasoning.
where q =
If the interest rate is stated as an annual rate, the notation used is somewhat different: u
denotes then the value of a dollar invested in the underlying in the “up” state, and d
denotes the value of a dollar invested in the underlying in the “down” state, with i being
the risk-free interest rate. Then the risk-neutral probability of the “up” state is
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 35 -
q=
1+ i ! d
.
u!d
Black-Scholes formula
Assumptions: All put-call parity assumptions, plus:
• The continuously compounded rate of return on the underlying asset over the period of
time of length t has normal distribution with mean µt and variance of ! 2t.
• Returns over disjoint time intervals are independent (this is also called the random walk
model).
Then the price of a European call with exercise price K and maturity time t
c = Se!" t N ( d1 ) ! Ke! rt N ( d2 )
and the price of a European put with the same exercise price and maturity time is
p = Ke! rt N ( !d2 ) ! Se!"t N ( !d1 )
where
)2 $
! S$ !
ln # & + # ( r ' ( ) +
t
"K% "
2 &%
d1 =
,
d2 = d1 ! " t ,
) t
and ! is the dividend force of interest on stock, r is the risk-free force of interest, N is
the CDF of the standard normal distribution, and S is the current stock price. For a stock
without a dividend you can also write the formulas for d’s in this nice form:
# S & 1 2
# S & 1
+ ) t
ln % ! r" ( + ) 2t ln %
$ Ke ' 2
$ PV ( K ) (' 2
d1 =
=
,
) t
) t
# S & 1 2
# S & 1
! ) t
ln % ! r" ( ! ) 2t ln %
$ Ke ' 2
$ PV ( K ) (' 2
d2 =
=
.
) t
) t
Exercise 8.1
A multi-period securities market model follows the binomial stock price model. You are
given that u = 1.2, d = 0.92, i = 0.04, and the initial stock price is $20. Compute the
arbitrage-free price of a European call option on the stock that expires in two period and
has a strike price of $21.
Solution.
One period risk-neutral probabilities are:
1 + i ! d 0.12 3
qu =
=
= ,
u!d
0.28 7
u ! (1 + i ) 0.16 4
qd =
=
= .
u!d
0.28 7
Over two periods, the stock can go:
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 36 -
3 3 9
! =
.
7 7 49
• Up-down or down-up to 1.2 ! 0.92 ! $20 = $22.08, with call payoff of $1.08, and
3 4 24
probability 2 ! ! =
.
7 7 49
16
• Down-down to 0.92 2 ! $20 = $16.93, with call payoff or $0.00, and probability
.
49
The expected present value, which is the arbitrage-free price of the call, equals
1 " 9
24
16
%
! $7.80 +
! $1.08 +
! $0.00 ' ( $1.81.
2 $
&
1.04 # 49
49
49
• Up-up to 1.2 2 ! $20 = $28.80, with call payoff of $7.80, and probability
Exercise 8.2
You are given the following binomial model for the value of the short-term interest rate,
risk-free over a one-year period:
• One year from now this short-term interest rate is either r1u = r0 (1 + ! ) with probability
r
0.50, or r1d = 0 with probability 0.50, where r0 in the initial short-term rate, and ! is
1+ !
a parameter. The probabilities given are risk-neutral probabilities.
• Annualized volatility of this short-term interest rate is ! = 25%. You are also given that
in this binomial model u = e! t (the value of a dollar invested in the underlying in the
“up” state) and d = e! " t (the value of a dollar invested in the underlying in the “down”
state).
• The current value of the short-term rate is 4%.
A 2-year European (meaning that it pays only if the short-rate breaches the floor at the
end of two years) interest rate floor with a 3.5% strike level and a notional amount of 100
is issued. This derivative security will pay the difference between 3.5% and then current
short-term interest rate as calculated for the 100 notional amount, if such difference is
positive.
(a) Calculate the value of this interest rate floor.
(b) Now assume that the floor only pays the difference between 3.5% and the current
short-term rate at the end of the first year, if such difference is positive. What is the value
of such a one-year floor?
Solution.
(a) When t = 1, we have u = e! and d = e! " . In terms of the notation of this problem,
r1u
2
= e2 ! = (1 + " ) .
d
r1
We are given ! = 25%. This means that
1 + ! = e" = 1.28402542.
Given this, the short-term rate will be in two years:
2
with probability 0.25,
4% ! (1.28402542 ) = 6.59488508%
4%
with probability 0.50,
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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and
with probability 0.25.
4% ! (1.28402542 ) = 2.42612264%
Only the third outcome produces positive cash flows from the floor, which is then worth
100 ! ( 3.50% " 2.42612264%) = 100 !1.07387736% = 1.07387736.
This cash flow is paid with probability 0.25, and its expected present value is the price of
the floor. We use the risk-free rate over the next year as the rate for discounting for that
year. Therefore the value of the floor is:
1.07387736
! 0.25 = 0.25034485.
0.04
"
%
1.04 ! $ 1 +
# 1.28402542 '&
(b) At the end of the first year, the short rate is:
4% ! (1.28402542 ) = 5.13610167% with probability 0.50,
and
"1
4% ! (1.28402542 ) = 3.11520313% with probability 0.50.
Only the second outcome gives rise to a payment by the floor, such payment being
0.38479687 on the 100 notional amount. Its expected present value equals:
0.38479687
! 0.50 = 0.1849985.
1.04
If the floor were a two-year floor, inclusive of the first and second year, its total value
would be 0.25034485 + 0.18653868 = 0.43688353.
"2
Exercise 8.3
You are given the following information about a call option on a certain stock: current
stock price: S0 = 100, exercise price: X = 95, interest rate: r = 10%, continuously
compounded (i.e., force of interest), time to expiration: t = 0.25, standard deviation of
the rate of return of the underlying stock: ! = 0.50. Find the price of this call option
using the Black-Scholes formula.
Solution.
We have
d1 =
100 &
0.50 2
#
ln %
+
0.25
"
$ 95e!0.25"0.10 ('
2
) 0.43,
0.50 0.25
100 &
0.50 2
#
ln %
! 0.25 "
$ 95e!0.25"0.10 ('
2
d2 =
) 0.18,
0.50 0.25
so that N ( d1 ) = 0.6664, and N ( d2 ) = 0.5714. Thus the value of the call option is:
c = 100 ! 0.6664 " 95e"0.25!0.10 ! 0.5714 = $13.70.
Exercise 8.4
On January 1, 2013, $10,000 is invested in a Cuba Libre stock selling at 50 libretas per
share. The exchange rate is 2 libretas per dollar. At the same time, a contract is entered
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 38 -
into to deliver 20,000 libretas for dollars on December 31, 2013, at the forward rate of
1.90 libretas per dollar. On December 31, 2013, the stock is selling at 60 libretas, and the
exchange rate is 1.80 libretas per dollar. Ignoring transaction costs, calculate the dollardenominated rate of return, and the libretas-denominated rate of return.
Solution.
One important thing to remember is that forward contracts do not require a cash outlay,
except for a commission, if any. To calculate the rate of return, we need to concentrate on
cash flows. Initial $10,000 investment bought 20,000 libretas, and this purchased 400
shares. At the end of the year, these 400 shares were worth 24,000 libretas. The investor
will deliver 20,000 libretas of that amount as fulfillment of the forward contract, and
receive for it $10,526.32 (at the exchange rate of 1.90). The remaining 4,000 libretas will
have to be exchanged at them prevailing rate of 1.80, giving $2,222.22, and the total
dollar amount received will be $12,748.54. On a $10,000 investment, this represents a
total rate of return of 27.49%.
For the second part, January 1, 2013, libretas cash outlay was 20,000. The investor
bought 400 shares of the stock, as it was selling for 50 libretas. On December 31, 2013,
there is a libretas cash outlay of 20,000 as required by the forward contract. The investor
also receives the payment in dollars for that delivery, at the exchange rate of 1.90, which
means that $10,526.32 is paid to the investor, and at the current rate of 1.80, this is worth
18,947.37 libretas. There is also a cash flow of 24,000 libretas from the proceeds of the
sale of the stock, so the net cash flow in libretas is: 18,947.37 + 24,000 – 20,000 =
22,947.37. The rate of return is 22,947.37/20,000 – 1 = 14.73684%. This is very different
than the rate of return in U.S. dollars above, and that is understandable, because libreta
has appreciated sharply during the period under consideration.
Exercise 8.5
The following two securities coexist in the market with $1000 Treasury Bill maturing one
year from now with 6% annual yield, and have the same price as that T-Bill:
• A European call option on 10,000 shares of stock in Company ABC at a strike price of
10 with expiration date exactly one year from now. The risk-neutral probability of stock
price being 10.50 at expiration date is p1 . Otherwise, the stock price will be 10 or less.
• A one-year forward on 2500 bushels of wheat that will enable purchase at 30 per
bushel at that date. Analysts expect that the price of wheat will be at 37 with probability
p2 or at 31 with probability p1 . Otherwise, the price of wheat will be at 28.
Calculate the value of p1 .
Solution.
$1 invested in Treasury Bill will be worth $1.06 a year from now no matter what, so
that 1 + i = 1.06 . If on the exercise date, the price of the stock is $10.50, the call on 10,000
shares will be worth 10,000($0.50) = $5,000. If the price of the stock is $10 or less, call
will expire worthless. Thus a $1 invested in the call will mature with the value of $5 in
the up-state, and $0 in the down state.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 39 -
p1 =
1 + i ! d 1.06 ! 0
=
= 0.212.
u!d
5!0
Exercise 8.6
The following two securities have the same current price of 1000 as a Treasury Bill
maturing one year from now with 6% annual coupons and a face amount of 1000.
• A European call option on 10,000 shares of stock in Company ABC at a strike price of
10 with expiration date exactly one year from now. The risk-neutral probability of stock
price being 10.50 at expiration date is p1 . Otherwise, the stock price will be 10 or less.
• A one-year forward on 2500 bushels of wheat that will enable purchase at 30 per
bushel at that date. Analysts expect that the price of wheat will be at 37 with probability
p2 or at 31 with probability p1 . Otherwise, the price of wheat will be at 28.
Calculate the value of p2 .
Solution.
We know from the previous problem that p1 =
53
= 0.212. The forward contract will be
250
worth
• $17,500 with probability p2 ,
• $2,500 with probability p1 ,
• – $5,000 with probability 1 – p1 – p2 .
We have
17, 500
2, 500
5000
1000 = p2
+ p1
! (1 ! p2 ! p1 )
.
1.06
1.06
1.06
Therefore, substituting the value of p1
6060 = 22, 500 p2 + 7, 500 p1 = 22, 500 p2 + 7, 500
53
= 22, 500 p2 + 1590,
250
resulting in
p2 =
4470
= 0.1987.
22, 500
9. Insurance Pricing and Valuation
All insurance contracts, in one form or another, transfer the risk of the financial
consequences of a future uncertain event from one party (the insured) to another party
(the insurer), in exchange for some form of financial consideration. Examples include:
- Life insurance, where a fixed payment is made upon death of the insured.
- Life annuity, where a prescribed series of payments is made until the death of the
insured.
- Property insurance, e.g., automobile insurance or home insurance, where
reimbursement for losses is made upon the occurrence of accidental events.
- Liability insurance, e.g. auto bodily injury liability or medical malpractice
insurance, where the insurer covers the financial consequences of at-fault events.
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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-
Reinsurance, where the insured party (the ceding company) itself is an insurance
firm, and the insurer (reinsurer) is a firm in business of providing insurance
coverage to other insurers.
The business of insurance is viable only if the risk transfer resulting from the existence of
the insurance contract brings about some reduction in the severity of the financial
consequences of future events. But as the events themselves are unchanged by the
existence of insurance contracts (unless the contract changes the participants’ incentives
to prevent losses, which in fact is a serious practical issue in the business of insurance),
the reduction occurs in the variance of the financial position of the insured. In fact, the
reduction is achieved by changing the relationship of the consequences of risk to the
participants’ capacity to bear it. The insurer removes the unknown consequences, or at
least a part of them, from the insured party’s future for a known price in the present. The
insurer combines the risk exposures of numerous parties insured, and collects their
payments (premiums), thus making the risk more predictable (because of combination of
many, mostly uncorrelated, risks) and more bearable (because of the combined financial
resources from collected premiums). There are also more consequences in the longer
run. Because of less exposure to unexpected financial consequences of risk, the insured
parties can now assume a more risky posture in their business and personal activities (this
is commonly referred to as moral hazard when the altered activity affects the probability
of loss, sometimes deliberately: in the case of fraud), thus undertaking more projects and
projects of larger scale. This, in turn, can benefit the entire society if it expands the set of
opportunities in terms of production output, or other desirable activities, but it may also
create more risk, because of less restraint on the part of the insured participants.
The science of mathematical models applicable to insurance, and financial risk modeling
in general, is known as actuarial science. Actuaries are financial professionals who are
typically employed by insurance companies, and other entities involved in financial
consequences of risk (e.g., consulting companies, governments, etc.). Actuaries perform
the following key duties:
- Setting the premiums for insurance products. While the process of pricing
insurance is a dynamic one, involving many parties, including the market for
insurance products, insurance company management and marketing divisions,
insurance regulators, and others, actuaries create the core portion of this process
by balancing the premium income of insurance firms and the payments made by
the insurance firms for benefits, claims, expenses, and distribution of profits to the
firm owners.
- Setting insurance reserves. Reserves are defined as expected future payments on
policies already underwritten and currently in force. They are the liabilities of
insurance companies that represent claim and expense payments, or benefits
promised to the insured parties, when the events named in the insurance policies,
occur.
- Assuring solvency of the insurance firm, in both short-term context and in the
long-run. This solvency requirement does not just mean the standard ability to
make scheduled payments, which applies to all firms, but also an additional
requirement of appropriate level of surplus (or capital), defined as the excess of
the firm’s assets over the liabilities, and needed to assure the payment of all
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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insured obligations with high probability. Insurance firm’s liabilities consist
nearly entirely of reserves. When an insurance premium is collected by an
insurance company, a portion of it is used to pay expenses, such as marketing, or
administrative, or salaries of actuaries (this is a sizable expense, as actuaries are
consistently among the highest paid professionals) and other personnel, while the
rest is placed in reserve to pay future claims or benefits. In addition to making
certain that those reserves are large enough to cover actual promised payments,
actuaries have to assure that assets and liabilities of the firm are managed
properly, so that the difference between them (the surplus) remains at a level
required by insurance regulators, or higher. This form of management of an
insurance firm (or any other entity involved in risk management) is termed assetliability management, and is a part of a larger field of enterprise risk
management, a field of study of proper management of risky activities in which
any business entity can and should be engaged.
Insurance Pricing Fundamentals
The key principle of private insurance contracts is a form of a “law of conservation”: the
totality of funds that are paid out to the insured parties must be originally obtained from
the same insured parties in the form of premium payments. While the owners of the
insurance firm do provide capital to start the company, and possibly additional capital for
its continuing functioning, they do expect the opportunity to earn a return on their
investment comparable with that available from other sources in the markets for capital
(capital markets) of similar risk. Thus insurance consumers as a whole cannot expect to
be subsidized by the insurance firm, but rather should expect to pay for their own claims
and expenses and, additionally, for the cost of the capital supplied to the insurance firm
by its owners.
Actuaries begin the process of pricing insurance by forecasting the future costs of claims
in property, casualty and liability insurance, or benefits, in the case of life, disability, or
health insurance, and life annuities. For example, in the case of life insurance, this begins
with the study of the future lifespan of a given insured. If we denote by T the future
length of life (a real number) of an insured aged x, then the cost of a life insurance policy
in a specified amount is the present value of that amount discounted from the moment of
payment (upon death or just immediately following it), T years in the future (or a number
close to T) to the present moment. This cost is likely to be increased by all policy
expenses, such as marketing, administrative, and settlement expenses. Furthermore,
starting an insurance enterprise, or issuance of new policies, requires an outlay of capital
by the enterprise owner or owners, and that capital must be paid for. This cost of capital
becomes an additional expense added to the cost charged to the insured party. If the
insurance premium is set as the expected value of the random variable describing future
payment of benefits/claims and expenses, this method of pricing is called the Equivalence
Principle. This most basic principle of pricing insurance calls for the premium to be set
at the level equal to the expected value (or mean) of future payouts, modeled as random
variables. The process of diversification of many risks combined from various insured
parties makes the average payout approximately equal to the theoretical expected value,
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 42 -
as a consequence of limit theorems from the probability theory, such as the Central Limit
Theorem, and the Law of Large Numbers. But this diversification may not always be
enough, and some provision for what actuaries call the adverse deviation (deviation of
the actual amount of total claims or benefits from the expected amount, in a manner
detrimental to the insurance company) must be made by appropriately increasing the
insurance premium to cover this risk to the insurer’s capital. Thus the premium is
typically set as the expected value of claims or benefits, adjusted for discounted value of
money, plus the expected value of all expenses and taxes, plus a provision for risk of
adverse deviation, commonly called the risk loading, covered by the proper estimate of
the cost of capital commensurate with the total risk of the insurer.
The key part of the calculation of the insurance premium is the estimate of the future
losses or benefits. In the case of life insurance, since the amount to be paid is set in
advance, the uncertainty is only twofold: unknown time of death and unknown rate of
return that the firm will earn on the premium or premiums collected. While the simplest
method of calculation of life insurance would use a one amount paid upfront (single
premium), real life policies are nearly always paid for with a series of periodic (annual,
quarterly, or monthly) premiums. Those periodic premiums are paid for a set period of
time (e.g., five or ten years) but only if the insured is alive, or for the entire remaining life
of the insured person. This means that in the calculation of premium not only the time of
payment of the death benefit is uncertain (modeled as a random variable T, remaining
time until death), but so is the length of time over which the premium will be paid.
Luckily, the underlying random variable, T, is the same for both phenomena, although its
practical implications on the present value of the death benefit, and the present value of
the remaining premium payments, are different. Ideally, estimates involving the random
variable T would be based on data concerning exact length of life of all people in the
population. But historically, accurate data about the exact length of life has not always
been easy to collect, and instead annual data (expressing the length of life in full years)
has been common. A table starting with a given population of newborn persons in a
given year, and then showing the population alive at any future age is called a mortality
table. Population alive at age x is denoted by lx and number dying between ages x and
d
x + 1 is denoted by d x . The ratio qx = x is a natural estimate of the probability of dying
lx
between ages x and x + 1, and px = 1 ! qx is the natural estimate of the probability of
surviving that year. The first mortality table is generally attributed to Sir Edmund Halley,
who in 1683 created it for the city of Breslau (now Wroclaw, in Poland). In the United
States, commonly used tables are generally created by the Society of Actuaries, usually
utilizing data collected in the National Census. Insurance companies commonly
undertake their own mortality studies, in order to better understand the risks of the
populations they insure, and subsequently modify the published tables.
The other uncertain element in pricing of life insurance (or other products related to
human mortality) is the interest rate that will be earned on the insurance company
investments. Modeling of that rate of return is complicated enough if the period over
which the return is considered is fixed, e.g., one year. One period models of rates of
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 43 -
return are usually derived based on an equilibrium approach, or arbitrage-free approach.
The equilibrium models take into consideration consumption and risk preferences of the
participants in the economy, and derive expected rates of return corresponding to varying
levels of risk. Its crowning achievement is the Capital Asset Pricing Model (CAPM),
widely utilized, but derived under severely restrictive assumptions, and thus limited in its
applicability. CAPM is basically a one-period model, and thus of very limited
applicability to business of life insurance, life annuities and pensions, which are, by
nature, long-term and multi-period. CAPM states that the expected rate of return on a
stock is given by the formula
E ( R ) = rF + ! " E ( RM ) # rF ,
where R is the random rate of return of the stock under consideration, RM is the random
rate of return of the overall market of all risky assets, and the coefficient beta is
Cov ( RM , R )
!=
.
Var ( RM )
(
)
The second approach to modeling rates of return, the no-arbitrage approach, is rooted in
the idea that capital markets would not allow arbitrage to exist, at least not in any
persistent fashion. Arbitrage is defined as creation of an investment portfolio, which does
not require any outlay of funds, yet allows positive returns with positive probability, and
never loses any money. In other words, arbitrage is a “free lunch.” Given the noarbitrage condition, two investment portfolios, which generate the same cash flows in the
future, must have identical prices today. The no-arbitrage approach starts with observed
prices and rates of return of assets traded in capital markets, and attempts to derive
appropriate rates of return for other assets of comparable risk not directly priced by the
market. These two methodologies are of significance to insurance, because insurance
products are not continuously traded in the markets, but they are priced by insurance
firms, and their prices must relate to risk tolerance and other preferences of market
participants, as well as prices of similarly risky capital assets traded in the markets. In
other words, an actuary deriving an insurance premium must be aware of the value placed
on the insurance product by the firm’s customers, and prices of capital assets that can be
possibly used to replicate some, or even all, of the features of the insurance product under
consideration. The actuary must make certain that the estimate of the interest rate used in
the calculations of present values of future cash flows of the policy under consideration
correspond realistically to the interest rate that will be earned on the company’s
investments, after consideration for possible additional investment expenses and taxes.
Methodologies of life insurance and life annuities are quite naturally extended to the area
of pricing and planning of retirement. Retirement is usually expected to be funded by a
combination of government social insurance pension, employer-sponsored pension or
savings plan, and private savings. The problem of providing appropriate amounts of
savings for the purpose of obtaining desirable level of income upon retirement, is a
natural actuarial model problem, but compounded not just by the uncertainty of the length
of life, but also the length of period of employment, date of retirement, desired level of
income replacement (in relation to pre-retirement income) upon retirement, as well as
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
- 44 -
additional complications of possible disability, and provision for the spouse and the
survivors of the individual under consideration.
Life insurance models do not, however, extend naturally to insurance coverages of
property, protection against liability, group insurance including group health insurance
policies, or workers compensation insurance. Life insurance policies are typically issued
for long-time periods, possibly the entire life of the insured party. In contrast with that
approach, automobile insurance or homeowner’s insurance, is usually issued for a
relatively short period of time to individuals (personal) and to corporations (commercial),
half a year to a year, and require a different modeling approach. Actuarial models for
these forms of insurance covering accidental and at-fault events forecast the random
variable counting the number of claims per insured received in the period of insurance,
termed frequency, and separately the probability distribution of the size of those claims,
termed severity. The two random variable so modeled are then combined in a collective
risk model of the firm
S = X1 + X2 + … + X N ,
where S is the aggregate claims random variable, while N is the claim frequency random
variable, and each Xi represents individual claim severity. Examples of commonly used
probability distributions describing N are: the Poisson distribution, the binomial
distribution, and the negative binomial distribution. If the random variables Xi are
assumed to be identically distributed and independent, the resulting distribution of S is
termed a compound distribution, derived from combining the distribution of N and the
distributions of Xi .
Estimation of the frequency and severity distributions is an integral part of the work of an
actuary in the areas of property, casualty, liability insurance, and other similar forms of
insurance. These estimates are continuously updated based on the claim data, as well as
other data collected by insurance enterprises. The challenge is additionally complicated
by the fact that not all losses are covered by insurance contracts, and even those covered
are typically not covered in full (with the use of deductibles, i.e., amounts paid by the
insured party before insurance coverage starts, or co-insurance, requiring the insured
party to share in the payment for the losses), thus the actuary does not always have full
access to the data describing the losses. Furthermore, the cost of items or events insured
changes continuously. This is due to inflation, but also due to changes in relative prices
in items or events insured. In the United States, for example, health insurance industry
struggles with increases in costs of health care well in excess of overall inflation, as well
as with nearly continuous introduction of new medical technologies and new prescription
drugs, which may have not been considered in historical models used for pricing of
health insurance. Liability insurance, especially policies covering general (pain and
suffering) and punitive damages, are not tied closely to general inflation but, rather, to
current and future laws and their interpretations by the court. Thus the actuary must
consider not just the estimates for frequency and severity based on historical data, but
also adjustments to those estimates for the trend, or changes, of the cost of coverages
provided. This requires development of forecasting methodologies for projecting future
costs of claims. Standard forecasting methodologies are typically based on either
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regression or time series, both created within probability theory. Basic linear regression
models assume that a predicted random variable Y is related to a predictor variable X via
a linear model of the form
Y = a + bX + ! ,
where a and b are parameters derived in the model estimation, and ! is a residual
random variable, typically assumed to be normally distributed with mean zero and
relatively small standard deviation. If empirical values of the predictor variable X are
x1 , x2 ,…, xn and the corresponding value of the predicted variable Y are y1 , y2 ,…, yn then
the standard methodology for estimation of parameters a and b is to minimize the
Euclidean distance or mean square error
n
" ( y ! ( ax
i =1
i
i
+ b )) .
2
This approach dates back to Gauss. The relationship between the predictor and the
predicted variable can be generalized to allow either X or Y to be replaced by functions of
them. For example, if we know that Y is expected to grow exponentially with X then it
would be natural to consider a model of the form
lnY = a + bX + ! .
One can also have a general multivariate model of the form
Y = a + b1 X1 + b2 X2 + … + bm X m + ! .
Note that in this general model, given values X1 = x1 , X2 = x2 , …, X m = xm , the
predicted value of Y given those predictor variables values is
y = a + b1 x1 + b2 x2 + … + bm xm ,
as the expected value of the residual is zero. Thus this model gives the predicted value as
the mean of the probability distribution of Y given that X1 = x1 , X2 = x2 , …, and
X m = xm . One can, in fact, generalize this approach to regression analysis to nonparametric regression models, under which the distribution of the residuals is allowed to
be completely arbitrary, instead of the normal distribution, and if that arbitrary
distribution can be somehow estimated or theoretically established, the predicted value of
Y is the mean of its conditional distribution.
The second set of methodologies deals with the situation when the variables modeled are
time-dependent, so that their historical observations do not constitute independent
observations of the same random variable. Time series analysis takes into consideration
the time structure of data, and it accounts for phenomena such as autocorrelation, trend
or seasonal variation, all of which are common in real life insurance data. For example,
an autoregressive (AR) model, which assumes a regression-type relationship of the value
of a variable Xt at time t to the preceding values:
p
$
'
Xt = & 1 ! # " i ) µ + "1 Xt !1 + … + " p Xt ! p + * t .
%
(
i =1
One additional set of prediction methodologies has been created with the arrival of the
fuzzy set theory. A fuzzy set E! is defined by its membership function µ E in the universe
of consideration U, so that for every element u of U, there is a value µ E ( u ) , where
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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0 ! µ E ( u ) ! 1. Prediction methodologies typically utilize the concept of a fuzzy number,
for which U is the set of all real numbers, and use special generalizations of arithmetic
operations developed for fuzzy numbers.
A vitally important area of actuarial science for both life and non-life insurance is risk
classification. If two insured parties have significantly different risk profiles in relation to
expected claims, yet are charged the same premium for insurance, one of them (the low
risk party) effectively subsidizes the other one (the high risk party), and if the insurance
contracts are voluntary, the low risk party will avoid obtaining insurance or minimize its
amount, while the high risk party will seek to maximize the coverage. If the insurance
coverage is desirable for public policy reason, this adverse selection is commonly
resolved by making the purchase of the insurance contract compulsory, and often
administered by a government entity. But in private voluntary markets, when
competition of insurance providers is present, the insured parties must be classified in
reasonably homogeneous risk classes, within each of which the twisted incentives of
inequitable premiums are no longer present. This requires that actuaries collect data
concerning potential risk classes, and classify insured parties accordingly. Various
approaches have been developed to address this problem. Bayesian methodologies adjust
the premium based on observed experience. For contracts that bundle insured events
(perils), such as auto liability, damage to vehicles, and theft, the diversification benefit
(low cross correlations) becomes an important risk class pricing variable. Credibility
theory treats the premium rate for a given insured party (usually a group) as a weighted
average of a premium derived based on that party’s experience, and of a premium rate
derived for a general population. The credibility weight assigned to the party’s
experience is reflective of the accuracy of the empirical sample mean as a predictor of the
true mean. Classification methods can also be derived from methodologies used in other
areas of mathematics. General and fuzzy clustering, principal component, and kernel
smoothing algorithms have been proposed and used for risk classification in insurance.
Reserves
Once an insurance contract is in places, and premium is collected, some portion of that
premium must be placed in reserve for the purpose of payment of future benefits, claims
and expenses. Life insurance, life annuities, and pensions, as well as long-term health
insurance contracts (such as disability insurance, and non-statutory private health
insurance in Germany, which is a type of contract not in existence in North America) all
have a long-term nature, with risks generally increasing with age, but with premium set in
advance, and rarely changed over time, and even when changed, generally not changed as
rapidly as the increase in risk occurs. Such contracts must effectively have a level of
premium, which is too large in relation to risk in the early part of the policy, and too
small in the later part. As a result, a reserve must account for this divergence between the
premiums and the payments made by the insurance company. In actuarial terminology,
the expected value (i.e., the probability mean) of the present value of future cash flows
(i.e., accounting for the time value of money and the risk of adverse development) is
termed the actuarial present value. The most standard formula for the reserve in all
forms of long-term insurance contracts is the difference between the actuarial present
value of the future benefits, claims, and expenses to be paid, and the actuarial present
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value of future premiums to be collected. Of course, if the future premiums were
sufficient to pay future benefits, claims and expenses (including the cost of capital) at all
times, reserves would not be needed. The only long-term contract, which does not use
such approach to reserving, is the deferred annuity contract in the accumulation phase in
the United States, which requires the reserve to be the highest possible present value (not
adjusted for any probability of occurrence) of guaranteed future account balances under
the contract. The Commissioners Annuity Reserve Valuation Method required for
deferred annuities by the insurance regulators in the United States is a unique exception
in the actuarial methodology of reserving: it actually never uses any probability concepts.
The process of calculation of reserves for long-term contracts is commonly called
valuation. The interest rate used in the process of calculation of present values in that
process is called the valuation interest rate and the mortality table used is the valuation
mortality table. In combination, the interest rate and mortality table form the valuation
basis. It would seem natural that the judgment concerning the mortality table and the
valuation rate belongs with the actuarial professional. This is generally the approach
adopted in Great Britain, Australia and Canada. However, in many countries, including
the United States (until 1980, and, to lesser degree, still so), this decision is taken away
from the actuary, or even from the insurance firm management, and instead, the mortality
and interest rate parameters are prescribed by law. The process of valuation based on the
methodology prescribed by law is called the statutory valuation and is required in the
United States of all insurance companies for the purpose of submission of their financial
statements to the insurance regulators. Insurance is regulated in the United States
separately in each state, and statutory valuation reports must be submitted to each state in
which an insurance firm is engaged in business of insurance. Interestingly enough, the
valuation methodology required of insurance companies, which issue their shares for
trading in public stock exchanges in the United States (e.g., the New York Stock
Exchange) is different, and prescribed by the Generally Accepted Accounting Principles
(GAAP). To make things even more complicated, and possibly to create more
employment opportunities for actuaries, the accounting rules for the calculation of the
income tax due to the federal Government of the United States are different than the
statutory, or GAAP rules, and prescribed separately in the tax laws and their
interpretations by the tax agency, Internal Revenue Service. These peculiar regulatory
and accounting complexities in the insurance industry in the United States are perceived
by some as barriers to entry for foreign insurers.
Private pensions in the United States are, however, regulated by the federal Government
(Department of Labor) and generally use a valuation basis chosen by the pension plan
actuary, on the basis of that actuary’s professional judgment. The GAAP rules for
pension plans are, however, different than the valuation for regulatory purposes.
The last quarter century has witnessed increased interest in making the reserving
methodologies less command-based (with formulas and valuation bases prescribed by
law) and more principle-based. This has been especially important in view of dramatic
changes in the level of interest rates experienced in the 1970s and 1980s, as well as
improvements in longevity of general population, making the older mortality tables, still
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used for older policies (as the valuation basis for a long-term policy in the United States
is assigned to a policy based on the date of its issue, and remains unchanged unless a
change would result in a increase in statutory reserves), quite obsolete in many cases.
The principle-based approach to reserving usually requires a complex long-term model of
the insurance company. Since 1991 in the United States, such long-term models are
effectively required for most long-term contracts companies, i.e., life insurance, life
annuities, etc., and the process of creating them is called cash flow testing. In the longterm model, the set of cash flows generated by an insurance firm is generally treated as a
stochastic process (i.e., a series of time-dependent random variables), with financial
outcomes, such as payments of benefits and claims, payments of expenses, profits
generated, and the level of surplus held (assets minus liabilities) modeled in each
realization of the stochastic process generated.
The stochastic process under
consideration is influenced the most by the future scenarios of interest rates, but it is also
affected by random outcomes of mortality or other basis for benefit payments (e.g.,
payments of any amounts, known as nonforfeiture amounts) upon policy termination. All
of these phenomena must be modeled by the valuation actuary. The resulting set of
generated scenarios of the future creates an empirical distribution of financial outcomes
describing company’s solvency under all of the scenarios. Actuaries say that the
company passes a scenario if it remains solvent during its entire duration. The minimum
period modeled is ten years, although the frequency of cash flows considered during
those years need not be very high (quarterly cash flows can be acceptable, and hourly, or
even daily or weekly, cash flows are generally not required). Long-term insurance firms
are required to pass seven scenarios of the future prescribed by the regulators (those
scenarios are termed the New York 7, because they originate from seven scenarios
considered in the Regulation 126 in the State of New York), and an overwhelming
majority (e.g., 95%) of random scenarios generated by the insurance company internal
model. In effect, the regulators want the insurance company to remain solvent with 95%
probability, based on the large random sample of the probability distribution describing
future financial situation of the company. This regulatory approach has created
significantly increased demand for applications of stochastic processes to the modeling of
insurance firms.
For the short-term insurance policies, such as automobile insurance or homeowner’s
insurance, or workers’ compensation, the emphasis in reserving is not on the long-term
discrepancy between payouts to be made and premiums collected, but rather on the claim
payments that must be made within the remaining short-term of the policy. Because for
short-term policies the premium is typically paid upfront (e.g., for a six-month
automobile policy the premium payment occurs at the beginning of the six-month
period), payments to be made cannot be offset by any future premiums. The emphasis is
therefore on forecasting the payments. Those payments come in two major categories:
- Payment yet to be made on claims that have already occurred, and have been
reported to the insurance company, and
- Payments yet to be made for claims that have already occurred, and have not yet
been reported (commonly called Incurred But Not Reported, or IBNR).
If a claim has already been reported to the insurance company, and reasonably well
evaluated, it is generally not necessary to use any probability-based methods to estimate
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their values. But for a claim that is entirely unknown to the company, some form of
estimation must be made. The traditional approach to this problem has been completely
deterministic (i.e., devoid of any probability applications). An actuary establishing IBNR
considers the period of time since a theoretical claim has occurred and, based on the
company’s own data (from historical experience) estimates how long it will take for that
claim to be reported and fully paid, and what portion of it will be paid at what moment in
time. Then the actuary applies the knowledge so obtained to all data about claims already
known and claims not yet known, assuming that the estimates can be applied to the
current situation. By applying these estimates, the actuary projects what the ultimate full
amount will be paid, and compares it to the amount that has already been paid. The
difference of the two represents the amount that will be paid in the future on the claims
already in existence. That difference is the IBNR reserve, the largest liability item in a
typical property/casualty insurance company financial statement. The process of paying
the claim from the date when the claim is incurred to the date when it is fully settled is
called development (in property/casualty insurance) or completion (in health insurance).
The last quarter century has witnessed a gradual increase in interest in applying
probability-based methodologies to estimation of the IBNR reserves and loss
development in general. In such probability-based approaches, the final amount to be
paid is typically modeled as a random variable dependent on, at the very minimum, time,
and then typically in addition to time, other variables describing the process of
development, or completion, the entity insured, the nature of the claim, etc. Regressionbased models are most common. In property-casualty insurance reserving, models often
use hundred of variables, including interactions of those variables, and all such variables
must be always very carefully examined for multi-colinearity, i.e., dependence of the
variables on each other, which reduces or even eliminated model’s predictive power.
Asset-Liability Management
The cash flow testing models required in life insurance and life annuities in the United
States for the purpose of establishing statutory reserves are an example of the expansion
of sophisticated asset-liability management models that gradually have entered insurance
practice in the last quarter-century. The practice of asset-liability management began in
response to increased volatility of assets held in insurance companies’ portfolios in the
1970s and 1980s. Varying interest rates were initially the greatest concern. Insurance
companies have traditionally provided long-term guarantees of interest rates paid on
policies used for accumulation of wealth for retirement, but those guarantees were at
relatively low levels. When interest rates rose, those policies became unattractive, and
were abandoned by their owners in the process of disintermediation, i.e., flight from lowinterest rate insurance and bank products to higher-return investment products, or from
low credit quality intermediaries to those perceived as higher credit quality. Insurance
companies responded by offering higher rates of return, and pursuing higher returns
themselves by investing in riskier bonds and mortgages. High rates of return in the stock
markets have been countered by offering new variable annuity products tied to the
performance of the stock market. But some of those strategies of the insurance firms
have resulted in a significant increase of their risk exposure, and this was a new type of
exposure: not to diversifiable risk of insuring individuals or firms against death, or perils,
but to nondiversifiable risk of the bond market (both in the form of the risk of changing
Copyright © 2006-2007 by Krzysztof M. Ostaszewski
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interest rates and the risk of default of risky bonds) and the stock market. In 1991, the
United States life insurance industry experienced two insolvencies of large and
established insurance firms: Executive Life and Mutual Benefit. Both of these companies
had sizable portfolios of risky assets that resulted in a panic of withdrawals from their
retirement-type policies when risky investments declined in value and rumors of
insolvency spread.
Those developments illustrated the dangers of any divergence between the value of
insurance firm’s assets and liabilities. The difference between assets and liabilities, the
surplus (or capital) is carefully watched by insurance regulators. Since the mid-1990s,
the level of surplus required is a function of risks undertaken by the company, in its asset
portfolio, in the structure of the insurance products it offers, and in the interaction of its
assets and liabilities. While early models used for asset-liability management called for
elimination of risks of divergence of assets and liabilities, by matching assets and
liabilities cash flows, in a process called immunization, or matching the values of assets
and liabilities under changes of interest rates, recent developments in this area are more
significantly based in probability models.
One particularly important area of significance for understanding of asset-liability
management is the study of options embedded in the insurance contracts and the
relationship between the insured party and the insurer. Life insurance policies, as well as
life annuities, traditionally contain minimum interest rate guarantees. When interest rates
fall, such guarantees are equivalent to the option to purchase a bond with a coupon at the
level of minimum interest rate guarantee, regardless of the current level of interest rates.
Life insurance policy or disability insurance policy can be viewed as an option to receive
a certain monetary value when the human capital (the ability to generate income through
work) of the insured person disappears due to death or disability.
But most importantly, the insurance company creates an option by the very process of
issuance of insurance contracts. The policyholder is promised certain monetary values
upon occurrence of insured events. This promise will be kept only if the insurance
company is still in business. If the company is not in business, the insured can only make
a claim on company assets. In effect, the insurance company holds an insolvency put (an
option to sell at a predetermined price) on its own assets. If the value of the assets
exceeds the obligation to the policyholder, the insurance company has an incentive to
make good on that obligation. But as soon as the value of the assets falls below the value
of the obligation, under limited liability the insurance company can just walk away from
the obligation and let the policyholders take and divide the assets instead. This is, of
course, the very reason for the existence of insurance regulation. Government regulates
insurance firms, and prescribes their (minimum) level of surplus, because as soon as that
surplus becomes negative, the insurance firm has little financial incentive to serve the
best interests of its policyholders.
Practical evaluations of various options embedded in insurance contracts generally follow
some variation of the Black-Scholes methodology, or its simplified version, the binomial
model, using a stochastic process derived from the binomial probability distribution. This
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is also the underlying theoretical justification of the cash-flow testing methodologies used
by long-term insurance companies, and dynamic solvency testing models used for longterm models of companies issuing short-term insurance contracts.
Property/Casualty Insurance Pricing Models
Insurance pricing begins with the fact that both the cost and the expected return (profit)
are not known with certainty. Rather, the ultimate cost derives from a stochastic process
that commences when the insurance is purchased and is resolved when final claim,
expense and tax payments are made. The ultimate profit too results from an interrelated
stochastic process, which depends on the ultimate costs, the premium charged at the time
of sale, and the results of invested premiums and capital. Thus, the determination of an
appropriate premium by the actuary or available in the market, prior to sale, is key to the
opportunity to earn a profit consistent with the risk of adverse development. Of course,
in complete or workably competitive markets, where equilibrium supply and demand
prices are equal, those prices may or may not agree with the actuary’s calculation of
expected profits. We turn next to two comprehensive pricing models developed for
insurance in the past thirty to forty years and illustrate their use in the property-casualty
context.
Policyholder Demand Side Model
A policyholder purchases insurance to trade risky and uncertain future adverse financial
events for the almost (because of insolvency potential) certainty of a premium payment.
In decision theory, the premium would be the certainty equivalent of the uncertain future
liabilities. Insureds should be willing to pay a present value of premiums equal to the
present value of expected future claims, expenses and taxes, adjusted in the discount rate
for the risk of adverse deviations from the expectations. The key to the demand model is
then the identification of all expected payments to (claims), or on behalf of (expenses and
taxes), the policyholder. Proper policyholder demand pricing models should not directly
include consideration of the insurer’s invested assets and expected investment income
above the risk-free rate. Policyholders are generally unwilling, except in explicit
investment-insurance linked products such as variable annuities, to trade uncertain
adverse accidental events for certain premiums plus uncertain insurer investment returns.
Policyholder demand models assume the presence of a supplier at the policyholder
demand price.
The paradigm demand model, developed at MIT by Stewart C. Myers and Richard A.
Cohn, posits that the appropriate premium P at the beginning of the policy equals the net
present value (NPV) of losses, L, expenses E, and taxes T (because of double taxation in
the U.S., i.e., taxation of profits of corporation, and then taxation of distributions of those
profits when received by the company’s shareholders)
PV ( L + E + T )
where PV incorporates a negative adjustment to the risk-free rate to provide the necessary
profit incentive for the insurer assuming the risk. In practice, those risk-adjustments have
been difficult to model and calculate from empirical data.
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Shareholder Supply Side Model
Insurers offer policies precisely to capitalize on the diversification benefits of pooling
uncertain adverse future financial event consequences from personal and commercial
risks. Insurers count on low levels of adverse event correlations among risks to provide a
substantial risk spreading benefit across the insured population. A certain premium is
traded by the insurer for the risk of collective adverse events in excess of the premiums
and investment income from premiums over the life of the policy. If the certain
premium, combined with the after-tax investment income from the insurer’s asset
portfolio, is expected to provide returns to shareholders (investors of capital in the
insurer) commensurate with the combined underwriting and investment risk, then the
insurer offers the insurance contract at the price that produces expected returns equal to
the cost of capital for those risks.
The paradigm supply model assumes that the flow of invested capital and the return of
that capital with realized profit, if any, should be expected to have a net present value of
zero when discounted at the cost of capital of the insurance enterprise. Nominal
policyholder premium, loss, and expense flows and company invested asset flows with
after-tax returns are used to estimate
• The size and timing of the shareholder investment to back the outstanding liabilities,
• The size and timing of the return of that invested capital to shareholders as liabilities are
resolved and paid, and
• The size and timing of any assets to be returned as income to shareholders.
There are two principal methods for estimating cost of capital for the entire firm, CAPM
and the Gordon Growth (presented in the notes previously as Dividend Discount Model)
model. At this point in time, the use of empirical data and the simple CAPM discussed
above suffers from an omitted or confounding variable problem. The extended three
factor CAPM model of Fama and French developed in the 1990s included an important
variable omitted in the simple formulation, the size of the insurer. This is well known in
finance as the size effect on stock market returns: smaller capitalization stocks need to
earn higher percentage rates of return than large cap stocks. Recent research shows that
once the omitted CAPM variables are introduced, the market beta for property-liability
companies is about one; i.e., P&C companies are about average risk.
The Gordon Growth Model (GGM), also known as Dividend Discount Model, is built on
the common assumption that the current per share price P is equal to the present value of
all future dividend payments D, discounted at the same cost of capital rate. In a simple
GGM formulation, the growth rate of dividends and the cost of capital are assumed
constant in perpetuity leading to a simple estimation equation:
D
k= +g
P
where D/P is the current dividend rate, k is the cost of capital, and g is the dividend
growth rate. The dividend growth rate g is, of course, key and many ways of estimation
have been used. Most often, the growth rate is estimated as an average of some historical
rate and some forecasted rate.
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