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Introduction to Mathematics of Finance
Dr. Tsang
Chapter 6
Bonds & fixed income
securities - I
1
Introduction
• In this Chapter, we extend the concepts and
techniques covered to date and applies them to
common financial securities such as bonds and
preferred stocks.
• An investment that provides a return in the form of
fixed periodic payments and the eventual return of
principal at maturity is called Fixed-Income
Security.
• Taxes and investment expenses are to be ignored
unless stated.
2
What are Bonds ?
• A debt investment in which an investor loans
money to an entity (corporate or governmental)
that borrows the funds for a defined period of
time at a fixed interest rate.
• Bonds are used by companies, municipalities,
states and U.S. and foreign governments to
finance a variety of projects and activities.
• Bonds and stocks are both securities, but the major difference
between the two is that stockholders have an equity stake in the
company (i.e., they are owners), whereas bondholders have a
creditor stake in the company (i.e., they are lenders).
3
Why study the bond market?
• NYSE, the world's largest stock exchange by market
capitalization of its listed companies at US$28.5 trillion as
of May 2008
• Amounts outstanding on the global bond market
increased 6% in 2008 to $83 trillion
– Average daily trading volume in the U.S. bond market is
$822 billion ($100 billion for NYSE)
• Bond market is the largest of all capital markets (for
the raising of capital).
– China has about $2.45 trillion (March 2010) in foreign reserves.
– Most of them are invested in the US government-issued
debt securities for the safety and liquidity.
4
Classification of Fixed-Income
Securities
• 1. Treasury Securities
– U.S. Treasury securities (bills, notes, bonds)
– Bunds, JGBs, U.K. Gilts . . .
• 2. Federal Agency Securities
– Securities issued by federal agencies (FHLB, FNMA . . .)
• 3. Corporate Securities
– Commercial paper
– Medium-term notes (MTNs)
– Corporate bonds . . .
• 4. Municipal Securities
• 5. Mortgage-Backed Securities
5
More detail
• The US was the largest market for bonds
in 2008 accounting for 43% of amounts
outstanding followed by Japan with 16%.
• A quarter of amounts outstanding in the
US were in mortgage backed bonds, a fifth
in corporate debt and 18% in Treasury
bonds with most of the remainder in
Federal Agency securities and municipal
bonds.
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roughly $19 trillion
According to the European Capital Markets Institute: There is
€5 trillion ($6.5tn)
of euro-zone government debt outstanding
7
8
INVESTORS in bond market
1. Governments
2. Pension funds
3. Insurance companies
4. Commercial banks
5. Mutual funds
6. Foreign institutions
7. Individual investors
9
Interesting books on the bond
market by Michael Lewis
• The Big Short: Inside the
Doomsday Machine, how a 32year-old investor spotted the huge bubble in the
subprime-mortgage bond market in 2004
• Liar's Poker, a semi-autobiographical
book by Michael Lewis describing the author's
experiences as a bond trader on Wall Street
during the late 1980s, one of the books that
define Wall Street during that periods.
10
Terminologies
• A bond is a promise by the issuing company (issuer)
to repay (redemption) an original indebtedness of a
fixed amount (the face value or par value of the bond)
at a given time in the future (the maturity date). The
time span is called the term of the bond.
• The bond pays interest periodically at a prescribed
rate, known as the coupon rate.
• Bonds are commonly referred to as fixed-income
securities and are one of the three main asset classes,
along with stocks and cash equivalents.
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Types of bonds
• Accumulation (zero-coupon) bond – redemption
price includes the principal plus all interest.
• Coupon bond – coupons are periodic interest
payments (usually every six months, at least in
US) made by the issuer prior to the redemption.
• Callable bond – may be redeemed early at the
discretion of the issuer to provide a degree of
protection against interest rate risk.
• Putable bond – lender has an option to redeem prior to
maturity date. Much less commonly used.
12
Preferred stocks
• It is a type security provides a fixed rate of return
(dividend) similar to bonds.
• It is supposedly an ownership security of a
corporation, not a debt security.
• In reality, it is a second-class ownership security
of a corporation, because preferred stockholders
have very limited voting-right.
• Some of them can be converted to
regular/common stock and is called convertible
preferred stocks.
13
Common stock
• It is a type of ownership security that does
not earn a fixed dividend, set by the
corporation’s board of directors.
• Stockholders have voting rights.
• They are the true owners of the
corporation.
14
Example6.1: zero-coupon bonds
A zero-coupon bond is a corporate, Treasury, or municipal
debt instrument that pays no periodic interest. Typically,
the bond is redeemed at maturity for its full face value. It
will be a security issued at a discount from its face value,
or it may be a coupon bond stripped of its coupons and
repackaged as a zero-coupon bond.
A zero-coupon bond will pay $1000 at the end of 10 years
and is selling for $400 now. Find the yield rate (yield to
maturity) convertible semiannually that would be earned by
a buyer.
400 (1 + j)^20 = 1000 ,
j=0.04688
The yield rate convertible semiannually is 2j.
15
Example6.2
• A 13-week Treasury bill matures for
$10000 and is bought at a discount to
yield an annual rate of 7.5%. What is the
price to purchase it?
Using actual/360, the price is
p * (1+ 13*7*0.075/360) = 10000
p = 10000/ (1+ 13*7*0.075/360) = 9813.94
If the discount rate is 7.5%, the purchase price is
10000* (1- 13*7*0.075/360) = 9810.42
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Zero coupon bonds
• Zero coupon bonds are the simplest fixed-income
securities.
• Prices of zero coupon bonds provide information about spot interest
rates and vise versa.
• On 2001.08.01, STRIPS (Separate Trading of Registered
Interest and Principal of Securities, which are the securities obtained
when trading the coupons and principal of bonds separately) are
traded at the following prices:
For the 5-year STRIPS, we have
17
Coupon bonds
18
Pricing the Bond coupons
19
Pricing the Cash Flows of a Bond
20
Example6.3: How to value a
coupon bond?
1
Suppose a bond has a face value of $1000, will
pay semi-annual interest at the (coupon) rate of
5%, and will be due 10 years after issue.
This means that the owner of one such bond will
receive $25 (=1000*5%/2) every six months for a
total of 20 payments, together with a payment of
$1000 10 years after the date of issue.
21
Example6.3: How to value a
coupon bond?
2
If the purchaser wishes to receive a 6% return, compounded
semiannually, then the present value at issue is
A purchaser content with a return of 4% compounded semi-annually
would value the bond at
Price of a bond is equal to the present value of future coupons plus that of the
redemption value.
22
Coupon Bond valuation equation
Price of a bond is equal to the sum of present values of future coupons
plus that of the redemption value.
For a typical bond that promises to pay a fixed
coupon payment of C/2 every six months and
repay the par amount FV (face value) at maturity,
current price P is:
where r /2 is the appropriate semiannual yield rate and n is the remaining
life of the bond measured in the unit of the coupon payment period (six months).
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The five key bond variables
• Price, par value, coupon rate, remaining life, and
the rate (r)used in the discount factor—are tied
together.
• Any four of them determine the fifth.
• “r” is also called yield to maturity (YTM) or the
internal rate of return (IRR).
• There is no easy way to calculate the IRR or
YTM. Usually a computer will solve the equation
numerically. [Excel has an IRR function which
solves the equation numerically.]
24
Example6.3: How to value a
coupon bond?
3
Three years after issue, the bonds are priced to
yield 7% (compounded semiannually). The price per $1000
bond is
25
MATLAB Financial Toolbox
bndprice -- Price a fixed income security from yield to maturity (SIA compliant)
Syntax
[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity)
[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity,
Period, Basis, EndMonthRule, IssueDate, FirstCouponDate,
LastCouponDate, StartDate, Face)
Parameters with
default values
Period--Coupons per year of the bond. Default: 2
Face--Face or par value. Default: 100
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Using the “bndprice” function - example
27
Coupon rate=5% maturity=5yr period=2
Price
Yield
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What are the factors that
determine the price of a bond?1
• The price of a bond is a function of prevailing
interest rates. Interest rate risk
• As rates go up, the price of the bond goes down,
because that particular bond becomes less
attractive (i.e., pays less interest) when
compared to current offerings.
• As rates go down, the price of the bond goes up,
because that particular bond becomes more
attractive (i.e., pays more interest) when
compared to current offerings.
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What are the factors that
determine the price of a bond?2
• When you buy a bond, you may pay a premium. In other
words, you may pay more than the face (par) value. For
example, a bond with a face value of $1,000 might sell
for $1050, meaning at a $50 premium. Or, depending on
the markets and such, you might buy a bond for less
than face value, which means you bought it at a
discount.
• Bond price also fluctuates in response to the risk
perceived for the debt of the particular organization.
Credit risk
• For example, if a company is in bankruptcy, the price of
that company's bonds will be low because there may be
considerable doubt that the company will ever be able to
redeem the bonds.
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Credit rating agency - credit risk
• The credit worthiness (i.e., the ability to pay back
a loan) of a bond issuer (from companies to
national governments) is rated by credit rating
agency, a company that assigns credit ratings to
provide independent, easy-to-use
measurements of relative credit risk to increases
the efficiency of the bond market.
• Moody’s Investors Service, Standard & Poor’s,
Fitch Ratings (US), Japan Credit Rating Agency,
Dominion Bond Rating Service (Canada)
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32
33
Britain and the PIGS
By Ambrose Evans-Pritchard Economics Last updated: February 15th, 2010
As of today, the British government must pay a higher interest rate to borrow money for ten years
than either the Italian or the Spanish governments, despite the extraordinary ructions going on
within the eurozone.
The yields on 10-year British Gilts have risen to 4.06pc, compared to 4.05pc and
4.01pc for Spain. So if international bond markets are turning wary of Club Med
sovereign bonds, they seem even more distrustful of British bonds.
…
…
…
While Britain went in to this crisis with a much lower public debt than Greece or Italy (though
higher total debt than either), it now has the highest budget deficit in the OECD (Organisation for
Economic Co-operation and Development) rich club — and perhaps the world — at 13pc of
GDP.
http://blogs.telegraph.co.uk/finance/ambroseevans-pritchard/100003763/britain-and-the-pigs/
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Greece’s fortunes were dealt yet another blow as Standard &
Poor’s slashed its credit rating to junk status - BB+ - the first time
that has happened to a euro member since the single currency
was created, pushing yields on 10-year Greek bonds up to a
record 9.73pc.
The credit-rating agency also cut Portugal’s sovereign debt
ratings by two notches to A-, as the swirling storm hit the country
with full-force. … Yields on 10-year Portuguese bonds spiked
48 basis points to 5.67pc, replicating the pattern seen as the
Greek crisis started.
Portugal’s public debt will be just 84pc of GDP by the end of this
year, far lower than that of Greece, at 124pc. However, its private
debt is much higher and data from the IMF shows that its external
debt position is worse.
27 Apr 2010
http://www.telegraph.co.uk/finance/economics/7640783/ECB-may-have-to-turn-tonuclear-option-to-prevent-Southern-European-debt-collapse.html
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When the bond market effectively forced former US
president Bill Clinton to balance the US budget deficit in
the 1990s with a massive sell-off of US Treasury bonds,
his adviser James Carville said: "I used to think that
if there was reincarnation, I wanted to come
back as the president or the pope or as a 400
baseball hitter. But now I would like to come
back as the bond market. You can intimidate
everybody".
“Bond traders who could decide Portugal's fate tell of a 'crazy, fun' week”
http://www.guardian.co.uk/business/2010/apr/30/debt-crisis-bond-traders-profile
36
37
Athens in turmoil as
riot police and
protesters clash in
Greece
Protestors clash with riot police
during a protest outside the
Parliament in Athens
38
Book value of a bond
• In accounting, book value is the value of an asset
shown on a firm's balance sheet.
• When a bond is purchased at a discount/premium, each
subsequent coupon payment can be broken down to
interest and principal, in much the same way that a loan
is amortized.
• The book value of a bond is simply the present value
of the remaining coupon payments and redemption
value computed using the yield rate at which the bond
was purchased.
• The interest part of the current coupon is simply the
book value at the previous time point multiplied by the
yield rate;
• The mark up or mark down is the remainder of the
coupon.
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Example: Book value of a bond
1
The bond of the previous example(6.3-3) has an initial book value equal
to the purchase price of 890.79. The semi-annual coupon is 25, so the interest
part of the first coupon after purchase is 890.79(.035) = 31.18 and the mark up is
31.18-25 = 6.18. The book value after the payment of the coupon is 890.79+6.18 =
896.97.
The computation above shows that the interest payment is larger than the coupon.
The appreciation in the book value of the bond is attributable to the unpaid interest
to the bond holder.
Writing the purchase price as 890.79 = 1000 + (25 - 35)a(14, .035) shows the
discount at purchase can be viewed as a loan which is amortized by payments of 25
- 35 = -10. The principal part of each loan payment is then the amount of markup.
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Example: Book value of a bond
2
If the same bond (example6.3-3) had been purchased to
yield 3% , the purchase price would have been
25 a(14, 0.015) + 1000(1.015)^(-14) = 1125.43
In this case, the interest part of the first coupon after purchase is
1125.43(.015) = 16.88 and the mark down is 25 - 16.88 = 8.12. The book
value after the payment of the coupon is 1125.43 - 8.12 = 1117.31.
In this case, the excess of the coupon over the interest is a return of the
buyer’s capital.
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Example6.4: known yield rate
A bond with redemption value of 1000 and coupon rate of
6% payable semi-annually for 20 years is sold 5 years after
issue at a price to yield 4%. What price was paid for the
bond?
42
Example6.5: unknown yield rate
• Suppose a $10,000 par bond has 6%
semi-annual coupons and matures in 10
years. What is the yield rate if the price is
$9,000?
43
Yield rates are not necessarily unique
• Suppose payments of 100 now and
109.20 two years from now are to be
made in return for receiving 209 one year
from now.
• The yield rate i then satisfies
100(1 + i)^2 + 109.20 = 209(1 + i)
from which i = .04 or i = .05.
44
Price between coupon
payments dates 1
• Preceding valuation assumed the price (book value) is
calculated just after a coupon has been paid.
• What price that a purchaser would pay for the bond a
fractional time t through a coupon period?
• Assume that the purchaser will
– obtain a yield rate equal to that of the current
bond holder.
– receive all of the next coupon.
• The current holder would expect to receive part of this
coupon as interest for the period.
45
How to allocate value between accrued
interest and price for the bond?
The purchase price including accrued interest is
known as the "full" or "dirty price". The price
excluding accrued interest is known as the "flat" or
"clean price“, which is the book value and is usually
quoted in the financial press. It changes smoothly
through time.
The full price fluctuates due to coupon accrual.
46
Graph of the purchase price of a bond
which is equal to the flat price + accrued interest (assumed
that the flat price remains constant over the 2 years,) The
accrued interest must be calculated according to formula.
Note that the purchase price steadily increases each day
until reaching a peak the day before an interest payment,
then drops to minimum on the day of the payment.
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Accrued interest
Between payment dates, the price of the bond will be the
flat price + the accrued interest. Accrued interest is the
interest that has been earned, but not paid, and is most
simply calculated by the following formula:
Accrued Interest = Interest Payment x
Number of Days
Since Last Payment
─────────────
Number of days
between payments
48
Determining the full price of
a bond: Three methods
The theoretical method argues that the flat price should be the book
value B after the preceding coupon payment, accumulated by (1+i)^t,
where i is the yield rate, giving the flat price of B(1+i)^t. The accrued
coupon is the coupon amount c accumulated [by S(t,i)] c((1+i)^t -1)/i.
The book value is then the difference B(1+i)^t -c((1+i)^t -1)/i.
The practical method argues that the flat price is the book value B after
the preceding coupon accumulated at simple interest, giving B(1 + it).
The accumulated coupon is tc, and the book value is B(1 + it) - tc.
The semi-theoretical method is the most widely used method, and has
been accepted as the standard method of calculation by the securities
industry. The flat price is determined as in the theoretical method, and
the accrued coupon is determined as in the practical method.
49
[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity)
[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity,
Period, Basis, EndMonthRule, IssueDate, FirstCouponDate,
LastCouponDate, StartDate, Face)
Price - NUMBONDS-by-1 vector for the clean price of the bond. The dirty price
of the bond is the clean price plus the accrued interest. It equals the present
value of the bond cash flows of the yield to maturity with semiannual
compounding.
AccruedInt - NUMBONDS-by-1 vector for the accrued interest payable at
settlement.
50
51
Assignment 6.1
Calculate & graph of the purchase price ("full" or
"dirty price“) of the following bond as function of
time from 1-Jun-2010 to 1-Jun-2012, compounded
semi-annually:
coupon rate=0.04, yield rate=0.06, face value=100,
maturity date=1-Jun-2016,
52
Callable bond
• Some corporate bonds have a call provision that gives the
issuer the right to call or redeem the bonds after a few
years.
• For example, the issuer may have the right to call an
issue of bonds at any time after five years at par value.
Alternately, there may be a declining schedule of
redemption price such that the issuer pays 105% of par
value to call during the sixth year, 104% to call in the
seventh year, and so forth.
• If interest rates go down sufficiently, issuer can call and
replace the bonds with new ones or other forms of debt at
a lower rate.
• Callable bond is a regular bond with an enbedded call
option for the lender. So the pricing of callable bond is
more complicated.
53
Example - callable bond 1
• A 100 par value 4% bond with semiannual
coupons callable at $109 starting 5 years after
issue for the next 5 years, at $104.50 starting 10
years after issue for the next 5 years, and
maturing at $100 at the end of 15 years. Find the
highest price which an investor can pay and still
be certain of a yield of 5% convertible
semiannually.
• Since the bond will not be called at a yield of 5%
(higher yield causes the bond price to move down), the
price is
100v30 + 2*a(30, 0.025) = 89.53
where v=1/1.025
54
Example - callable bond 2
Same problem. Find the highest price which an investor can
pay and still be certain of a yield of 3% convertible
semiannually.
Since lower yield causes the bond price to move up, the bond will be called at
some point. So we calculate the lowest possible PV of all call schemes.
v=1/1.015, v30=0.63976
109vn + 2*a(n, 0.015) for 9<n<20
n=10
112.37
104.5vn + 2*a(n, 0.015) for 19<n<30
n=20
111.93
100vn + 2*a(n, 0.015)=112.01
for n=30
The answer is the lowest of these, $111.93, for redemption 10 years after issue.
55
Assignment 6.2
A 100 par value bond with 5% semi-annual coupons is
callable at $110 starting 5 years after issue for the next 5
years, then at $105 starting 10 years after issue for the
next 5 years, and maturing at $100 at the end of 15 years.
Find the highest price which an investor can pay and still
be certain of a yield of 3% convertible semi-annually.
Verify the procedure used in previous example is equally
applicable for this problem by evaluating (using MatLab)
the lowest possible PV of all call schemes.
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