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1
Chapter 5
Bond Prices and Interest Rate Risk
This Chapter …
2
Explain how interest rate movements affect the
prices of assets and liabilities of investors and
financial institutions.
Time Value of Money
3


A dollar today is worth more than a dollar
received at some future date
Positive time preference for consumption
Time Value of Money
4
Future Value: the value of a given amount of money
invested today at a given point in the future at a
given rate of interest
The future value (FV) of a sum (PV) is:

FV = PV (1+i)n
where
i is the periodic interest rate and
n is the number of compounding periods.
Time Value of Money
5
Example: $100 in savings for 5 years, and the bank
pays 4% interest. Future value?
Now assume interest is paid quarterly?
The difference is because of compounding periods.
Explain
Time Value of Money
6



Present Value: the value today of a given sum of
money to be received at given point in the future
Risk-free interest rate
Finding the present value is called discounting
Time Value of Money
7

Present Value: Cont’d
1
PV = FV
n
(1 + i)

Opportunity cost: Interest rate on the next best
alternative investment
Time Value of Money
8


Risk-free interest rate
With risk present, a premium may be added to
the risk-free rate. The higher the discount rate,
the lower the present value.
Bond Pricing
9


Focus on how investors and financial institutions
price bonds
An application of present value formula
Bond Pricing
10



It is a contractual obligation of a borrower to
make periodic cash payments to a lender over
a given number of years
Borrower  issuer “ of bond”
Lender  holder “of bond”
11

1.
2.
3.
Terms of the bond contract:
Principle
Coupon rate
Term to maturity
12
Bond Pricing
13





Two types of cash flows:
At maturity, holder receives principal (or face
value or par value).
Periodically before maturity, holder receives
interest (coupon) payments determined by
coupon rate, original interest rate promised as
percentage of par on face of bond.
c = C/F
Term to maturity: timing of the cash flows
Bond Pricing
14
Par value
Coupon Rate
Issued
Matures
$1,000
5%
Today
30 years from today
Scheduled
Payments: $50/year interest for 30 years
$1,000 par at end of year 30
Bond Pricing
15
Notes:
 It is assumed for most bonds that payment of both
coupon and principle are made at maturity
 It can semiannually, quarterly
 Coupon rate and face value are fixed
 a similar bond: close substitute, same maturity and
risk
 Coupon rate and interest rate may differ
Bond Pricing
16


Bondholder thus owns right to a stream of cash
flows:

Ordinary annuity of interest payments and

Future lump sum in return of par value,
Discountable to a present value at any time while
bond is outstanding.
Bond Pricing
17

The value (price) of a bond is the present value of
the future cash flows promised, discounted at the
market rate of interest

(the required rate of return on this risk class in today’s
market)
Bond Pricing
18

Bond Price Formula:
C1
C2
CN + F N
PB =
+
+
...
1
2
N
(1 + i) (1 + i)
(1 + i)
Where PB = price of bond or present value of promised payments;
Ct = coupon payment in period t, where t = 1, 2, 3,…, n;
Fn = par value (principal amount) due at maturity;
i = market interest rate (discount rate or market yield); and
n = number of periods to maturity.
Bond Pricing
19

Cash flows are assumed to flow at end of the
period and to be reinvested at i. Bonds
typically pay interest semiannually.

Increasing i decreases price (PB); decreasing i
increases price; thus bond prices and interest
rates move inversely.
Bond Pricing
20



If market rate equals coupon rate, bond trades at
par.
If coupon rate exceeds market rate, the bond
trades above par—at a premium.
If market rate exceeds coupon rate, bond trades
below par—at a discount.
Bond Pricing
21
Par value
Coupon Rate
Issued
Matures
$1,000
5%
Today
30 years from today
Scheduled
Payments: $50/year interest for 30 years
$1,000 par at end of year 30
Bond Pricing
22
Par value
Coupon Rate
Interest Rate
Matures
$1,000
8%
10%
3 years from today
Bond Pricing
23
In your calculator:
Enter the number of years over which the bond contract extends,
n, the interest rate, i, the par value received at maturity, F (usually
the FV key on your financial calculator), and the annual coupon
payment amount Ct (usually the PMT key on your financial
calculator)
Bond Pricing
24
Price < face value  Discount bond
Price > face value  Premium bond
Price = face value  Par bond
Bond Pricing
25
Par bond:
Par value
Coupon Rate
Interest Rate
Matures


$1,000
5%
5%
3 years from today
Bond Pricing
26

a.
b.
What is the price if the market interest rate was:
8%?
2% ?
Bond Pricing
27

Semiannual Compounding
C1 m
C2 m
C N m+ F Nm
PB =
+
+
...
1
2
Nm
(1 + i) (1 + i)
(1 + i)
m
m
m: is the number of times coupon payments are made each year.
m
Bond Pricing
28
Par value
Coupon Rate
Interest Rate
Matures
$1,000
5% paid semi-annually
6%
3 years from today
Bond Pricing
29
Zero Coupon Bonds:
“have no coupon payments but promise a single
payment at maturity”
Examples: treasury bills, U.S savings bonds
Issued at discount from par.
Return on security is the difference between the
purchase price and the face value.

Bond Pricing
30

Zero Coupon Bond:
F
PB =
mn
i
(1 + )
m
Bond Pricing
31
Par value
Interest Rate
Matures
$1,000
10%
10 years from today
Recap..
32
Already covered:
1.
Time Value of Money
2.
Bond Pricing
We will cover:
3. Bond Yields
4. Important Bond Pricing Relationships
5. Interest rate and Duration
Bond Yields
33
Yield: is the return on any investment
Bonds Yield: Return on bond
The yield should take into consideration the three
sources of cash flow of a bond:
1.
Coupon payment
2.
Interest income from reinvesting coupon payment
3.
Any capital gain or loss
Bond Yields
34


1.
2.
3.
Coupon rate : annual cash flow promised by the
borrower to the lender
Actual rate of return (for the lender), depends on
risks:
Credit or default risk
Reinvestment risk
Price risk
Bond Yields
35
Important note:
Reinvestment risk: change in market interest rate that
might cause the lender to have to reinvest coupon
payments at interest rates different from the interest
rate at the time the bond was purchased.
Price risk: change in interest rate that cause market
value of the bond to change resulting in capital gains
or losses (invers relationship)

Bond Yields
36

1.
2.
3.
4.
We will discuss four yield measures:
Yield to Maturity
Expected Yield
Realized Yield
Total Return
Bond Yields
37

1.
2.
3.
4.
We will discuss four yield measures:
Yield to Maturity
Expected Yield
Realized Yield
Total Return
Bond Yields
38
Yield to Maturity:
“Investor's expected yield if bond is held to maturity
and all payments are reinvested at same yield”
1.
25
25
1,025
951.90 =
+
+ ...
1
2
6
(1 + (i / 2) ) (1 + (i / 2) )
(1 + (i / 2) )

Normally determined by trail and error
Bond Yields
39
1.

Yield to Maturity (Cont’d):
It is cumbersome and difficult method, but it can be
solved using financial calculator. Calculation need
the following input:
1.
2.
3.
4.
Current bond price or PV
number of periods over which the bond contract extends
Coupon payments
Bond face value
Bond Yields
40
1.

1.
2.
3.
Yield to Maturity (Cont’d):
Important assumptions to calculate Yield to
Maturity:
Borrower makes all cash payments
Interest rates does not change over the maturity
Investors holds the bond to maturity
Bond Yields
41
1.

Yield to Maturity (Cont’d):
the coupon payments are reinvested at a lower
rate, the bondholder’s actual yield is less than the
promised yield.
Bond Yields
42
1.
Yield to Maturity (Cont’d):
Investor buys 5% percent coupon (semiannual
payments) bond for $951.90; bond matures in 3
years. Solve the bond pricing equation for the
interest rate (i) such that price paid for the bond
equals PV of remaining payments due under the
bond.
Bond Yields
43

1.
2.
3.
4.
We will discuss four yield measures:
Yield to Maturity
Expected Yield
Realized Yield
Total Return
Bond Yields
44
2. Expected Yield
“Predicted yield for a given holding period (same
procedure as YTM, but for some holding period
shorter than maturity)”
 Sell before maturity  to know potential impact of
interest rate change on returns on bonds investments
Bond Yields
45
2. Expected Yield (Cont’d):
Must forecast—
Expected interest rate(s)
Bond price at end of holding period
Plug forecast results into bond pricing
formula
Bond Yields
46
2. Expected Yield (Cont’d):


Given the future price, the investor can calculate the
expected yield; that reflects the expected sale price
Example: Book page 114
47
2. Expected Yield (Cont’d):
Steps for solving for expected yield:
- Find the expected future interest rate
- Find the future price using the computed interest
rate
Bond Yields
48

1.
2.
3.
4.
We will discuss four yield measures:
Yield to Maturity
Expected Yield
Realized Yield
Total Return
Bond Yields
49
3. Realized Yield
“The return earned on a bond given the cash flows actually received
by the investor and assuming that the coupon payments are
reinvested at the realized yield”

It might differ from yield to maturity due to:
a. change in the amount or timing of promised
payments (e.g. default).
b.
change in market interest rates affecting premium or
discount.
Bond Yields
50
3. Realized Yield (Cont’d):
Terminal price of a bond: the market price of the bond
on the date you sell it. ( in other words, it is the price of
remaining coupon payments and the final principle
repayment)
 Realized yield is useful because it allows investors to
evaluate the return on a bond ex-post (after the
end of the holding period or investment horizon)
Bond Yields
51
3. Realized Yield (Cont’d):
Book example page 115:
Investor pays $1,000 for 10-year 8% coupon bond;
sells bond 3 years later for $902.63.
Solve for i such that $1,000 (the original
investment) equals PV of 2 annual payments of
$80 followed by a 3rd annual payment of $902.63
(the actual cash flows this investor received).
Bond Yields
52
3. Realized Yield (Cont’d):
80
80
902.63
1000 =
+
+
...
1
2
3
(1 + i) (1 + i)
(1 + i)
Solving either by trial and error or with a financial calculator
results in a realized yield of 4.91%.
Bond Yields
53

1.
2.
3.
4.
We will discuss four yield measures:
Yield to Maturity
Expected Yield
Realized Yield
Total Return
Bond Yields
54
4. Total Return:
 Both expected and realized yield calculation
assumes that we will be able to reinvest coupon
payments at the calculated yield.
 That’s not always true
 However, if we know or can explicitly predict the
reinvestment rate, we will be able to calculate the
Total Return on a bond.
Bond Yields
55
4. Total Return (Cont’d):
“It is the return we receive on a bond that considers
capital gain or losses and changes in the reinvestment
rate”
To calculate the total return we need to determine two
things:
1.
2.
The terminal value of the bond
The accumulated future value of all the coupon
payments based on a known reinvestment rate
Bond Yields
56
4. Total Return (Cont’d):
Calculated using this formula:
Initial Purchase
Price
=
Terminal
Value
+
Accumulated
Future Value
(1+i)
Bond Yields
57
4. Total Return (Cont’d):


We have learned how to calculate the terminal value
when calculating expected and realized yield
To calculate the accumulated future values using this
formula:
FVc = C[(1+i) -1] / i
n
Important Bond Pricing Relationships
58
3.
There are three important relationship between
bond prices and the change in the level of
interest rates:
Bond Prices and Yield
Bond Price Volatility and Maturity
Bond Price Volatility and Coupon Rate

They apply to all fixed-income securities

1.
2.
Important Bond Pricing Relationships
59
Bond Prices and Yield:
They vary inversely (Negative Relationship):
As the market rate of interest (yield) rises, a bond’s
market price declines and vice-versa
The negative relationship exists because the coupon
rate ( interest on bond) is fixed when the bond is
issued.
Decreasing or increasing the bond’s price is the only
way to adjust for changes in market interest rates
1.
Important Bond Pricing Relationships
60
2. Bond Price Volatility and Maturity
 Bond Price Volatility: “the percentage change in
bond prices for a given change in interest rates
(yield) ”
 It is also a measure of how sensitive a bond’s price
is to changes in yields.
 Long term bonds have greater price volatility than
short term bonds, holding other bond characteristics
constant.
Important Bond Pricing Relationships
61
Important Bond Pricing Relationships
62
2. Bond Price Volatility and Maturity (Cont’d):
Calculating Bond Price Volatility:
Where
Pt - Pt-1
%DPB =
´100
P
t-1 change in price
%∆PB = percentage
Pt = new price in period t
P t – 1 = bond’s price one period earlier
Important Bond Pricing Relationships
63
3. Bond Price Volatility and Coupon Rate
“ the lower a bond’s coupon rate, the greater the
percentage price change (price volatility) for a given
change in yield”
Important Bond Pricing Relationships
64
Interest Rate Risk and Duration
65

1.
2.
3.
This section will include the following topics:
Interest Rate Risk
Duration and bond properties
Use of Durations:
a.
b.
c.
4.
To calculate bond price volatility
As a measure of interest rate risk
To measure and manage interest rate risk
Duration of bond portfolios
Interest Rate Risk and Duration
66
Interest Rate Risk
“ Risk related to changes in the interest rates that causes
a bond’s total return to differ from the promised yield
or yield-to-maturity”
- It includes two different but related risks:
a.
Price Risk
b.
Reinvestment Risk

Interest Rate Risk and Duration
67
Interest Rate Risk: (Cont’d)

Price risk: change in interest rate that cause market
value of the bond to change resulting in capital gains
or losses (inverse relationship)
Interest Rate Risk and Duration
68
Interest Rate Risk: (Cont’d)
Reinvestment risk: change in market interest rate that
might cause the lender to have to reinvest coupon
payments at interest rates different from the interest
rate at the time the bond was purchased.
 The change in a bond’s total return caused by
changing coupon reinvestment rate is what constitute
reinvestment risk.

Interest Rate Risk and Duration
69
Interest Rate Risk: (Cont’d)
It is important to note that price risk and reinvestment
risk partially offset each other.
Understand this carefully:
- When interest rate decline  bond’s price increase
 result in capital gain  but lower coupon
reinvestment income

Interest Rate Risk and Duration

-
-
-
Duration and Bond Properties
It is important to evaluate the effect of interest rate
risk on bond investments.
Problem: bond price volatility varies directly with
maturity and inversely with coupon rate.
A good measure should take both into
consideration.
Interest Rate Risk and Duration
Duration and Bond Properties: (Cont’d)
Duration can be defined as :
“ A measure of interest rate risk (bond price volatility)
that considers both coupon rate and maturity”
“ It is a weighted average of the number of years until
each of the bond’s cash flow is received”

Interest Rate Risk and Duration
Duration and Bond Properties: (Cont’d)
It can be computed using:

n
CFt * t

t
t 1 (1  i )
D n
CFt

t
(
1

i
)
t 1
where:
D = duration of the bond
CFt = interest or principal payment at time t
t = time period in which payment is made
n = number of periods to maturity
i = the yield to maturity (interest rate)
Interest Rate Risk and Duration


Duration and Bond Properties: (Cont’d)
Suppose we have a bond with a 3-year term to
maturity, an 8% coupon paid annually, and a
market yield of 10%. Duration is:
Interest Rate Risk and Duration
Duration and Bond Properties: (Cont’d)
If the yield increases to 15%:

Interest Rate Risk and Duration
Interest Rate Risk and Duration

1.
Duration and Bond Properties: (Cont’d)
Important properties of Duration:
High coupon rates  shorter duration
-
2.
When bonds have the same maturity
because bond holder will receive more of the total cash flow
earlier
Long maturity  High bond duration
- maturity can be called term-to-maturity
Interest Rate Risk and Duration
Duration and Bond Properties: (Cont’d)
3. Bonds with a single payment (with or without
coupon)  duration = term to maturity

-
example: zero coupon bond
Bonds with interim payments always have durations less
than their final maturity
Interest Rate Risk and Duration
78


Duration and Bond Properties: (Cont’d)
4. All other factors held constant, the higher the
market interest rate  shorter duration. Because in
this case coupon reinvestment income accumulates
faster.
Interest Rate Risk and Duration
Duration and Bond Properties: (Cont’d)
5. Direct relationship between duration and bond
price volatility. Greater bond duration  greater the
percentage change in the bond’s price for a given
change in interest rate.

- Important for managers specifically
Interest Rate Risk and Duration
80
Use Duration
To calculate
bond price
volatility
As a measure
of interest rate
risk
To measure and
manage
interest rate risk
Interest Rate Risk and Duration
81

-
Use Duration to Calculate Bond Price Volatility
“how to use duration to estimate the
percentage change in the bond’s price”
 i 
%PB   D 

100

 (1  i ) 
Interest Rate Risk and Duration
82

-
-
Use Duration to Calculate Bond Price Volatility:
(Cont’d)
It is important to note that this formula, work best
for small changes in interest.
Graph in the book page 125
To solve this issue, investors will add an adjusting
factor to the formula called “convexity factor”
Interest Rate Risk and Duration
83

Use Duration to Calculate Bond Price Volatility:
(Cont’d)
Interest Rate Risk and Duration
84
Use Duration to Calculate Bond Price Volatility:
(Cont’d)
To illustrate the preceding, work out this example:
A $1,000, 3-year bond, has a coupon payment of
4%. It has a duration of 2.88 and market interest
rates increased from 10% to 10.25%. Compute the
bond price volatility.

Interest Rate Risk and Duration
85
Use Duration to Calculate Bond Price Volatility:
(Cont’d)
Assume, that the rate actually increased from 10% to
12%, what would be the percentage change in the
bond price?
Now, find prices of each bond and find the
percentage change, do you get the same answers?

Interest Rate Risk and Duration
86
Using Duration as measure of interest rate risk
We already studied that:
1.
Long term bonds have higher interest rate risk
2.
Low coupon bonds have more interest rate risk
However, these properties cannot allow me to rank
bonds of basis of riskiness.

Interest Rate Risk and Duration
87
Using Duration as measure of interest rate risk:
(Cont’d)
For example: consider these bonds:
a.
10 year, c=7% bond and,
b.
8 year, c= 5% bond
Which one is more risky in-terms of interest rate risk ?

Interest Rate Risk and Duration
88

-
-
-
Using Duration as measure of interest rate risk:
(Cont’d)
It is not possible to tell using the given information.
Duration solves the problem.
Low-duration bonds face low interest rate risk and
vice-versa.
Duration is relate to the price-yield profile
steeper the price-yield profile  greater duration
 greater exposure to interest rate risk
Interest Rate Risk and Duration
89

Use Duration to Calculate Bond Price Volatility:
(Cont’d)
Back to the two bonds described earlier:
Market interest rate is 5%
Bond
Coupon Rate
Maturity
A
B
7%
5%
10 years
8 years
Now weDuration
can conclude that the
to lower interest
7.71bond
yearsthat is exposed
6.79 years
rate risk is the one with the lower duration.
Interest Rate Risk and Duration
90

-
-
Use Duration to measure and manage interest rate
risk
Duration is used as a tool for reducing or
eliminating interest rate risk over a given holding
period.
There are three possible approaches to deal (or
manage) interest rate risk:
1.
2.
3.
A Zero Coupon Approach
The Maturity-Matching Approach
The Duration-Matching Approach
Interest Rate Risk and Duration
91

Use Duration to measure and manage interest rate
risk: (Cont’d)
1. A Zero Coupon Approach:
-
Price risk: Eliminated by holding the bond for the whole
maturity
-
Reinvestment risk: Eliminate because no coupon payment
-
Return: the difference between purchase price and maturity
value.
Interest Rate Risk and Duration
92

Use Duration to measure and manage interest rate risk:
(Cont’d)
2. The Maturity-Matching Approach:
-
It is a naïve alternative
-
Price risk: Eliminated by holding the bond for the whole maturity
-
Reinvestment risk: Still there
-
A Dramatic change in interest rates would have a grate effect
Interest Rate Risk and Duration
93

Use Duration to measure and manage interest rate
risk: (Cont’d)
3. The Duration-Matching Approach:
-
A surefire way to eliminate both risks
-
Duration of the bond equals the required holding period
-
By doing that, capital gains and losses caused by changes in
interest rates are exactly offset by changes in reinvestment
income
Interest Rate Risk and Duration
94

-
Duration of Bond Portfolio
Investors rarely hold only one asset
Bond portfolio, or bond mutual fund?
Duration can help rank bond portfolios as it help us
rank individual bonds in terms of exposure to
interest rate risk
95

Duration of Bond Portfolio: (Cont’d)
Federal
bonds
(15%)
Shell
bonds
(50%)
Bond
Portfolio
Google
bonds
(35%)
Interest Rate Risk and Duration
96


Duration of Bond Portfolio: (Cont’d)
Duration of a bond portfolio is a weighted average of the
individual bond durations. The weight is according to the
proportion of the portfolio accounted for each bond.
Interest Rate Risk and Duration
97
Duration of Bond Portfolio: (Cont’d)
Formula to calculate bond portfolio duration:
i 1
Portfolio Duration   wi Di
n
where: wi = proportion of bond i in portfolio and
Di = duration of bond i.
Interest Rate Risk and Duration
98
Duration of Bond Portfolio: (Cont’d)
Example:
Suppose a bond portfolio contains four bonds, A,B,C, and
D. Bond A has duration of 15.7 years and makes up 20
percent of the portfolio. Bond B has duration of 22.3
years and makes up 40 percent of the protfolio. Bond
C has duration of 10.2 years and makes up 15 percent
of the portfolio. Finally, Bond D has duration of 7.6
years and makes up 25 percent of the portfolio.
Calculate the duration of the portfolio.

Interest Rate Risk and Duration
99
Duration of Bond Portfolio: (Cont’d)
- Managers can change the duration of their bond
portfolio by changing the proportions of bonds
in the portfolio.

100
End of Chapter 5