Yields & Prices:
Continued
Chapter 11
Learning Objectives

Understand interest rate risk and the key
bond pricing relation

Compute and understand the valuation
implications of:
 Duration
 Modified
Duration, and
 Convexity of a bond portfolio

Construct immunized bond portfolios
2
Prices and Yields
Remember: Yield changes have a larger
impact on longer maturity bonds
 All else equal price changes are larger the
lower the coupon rate
 SO: The longer the maturity and the lower the
coupon rate the greater the price fluctuation
when interest rates change

3
Bond Prices as a Function of Change in YtM
4
Rate Changes and Bond Prices
Known as interest rate risk
 Consider three bond

 A:
8% Coupon Annual, 4 Years till maturity
 B: 8% Coupon Annual, 10 Years till maturity
 A: 4% Coupon Annual, 4 Years till maturity

Calculate the change in the price of each bond
if:
 Interest
rates fall from 8% to 6%
 Interest rates rise from 8% to 10%
5
Measuring Interest Rate Risk
We can measure a bond’s interest rate risk
with DURATION
 Duration: Measures a bond’s effective
maturity

 Can
tells us the effective average maturity of a
portfolio of bonds
 The weighted average of the time until each
payment is received
 Weights are proportional to the payment’s PV
 Duration is shorter than maturity for coupon bonds
 Duration is equal to maturity for zeros.
6
Duration Calculation
wt 
 CFt =
y =
1  y 
CFt
t
Price
Cash flow at time t
YTM
T
D   t wt
t 1
7
Duration Example

What is the duration of a 2 year 12% annual
bond? The YTM is 10%.
Price?
T
(Years)
1
CF
P.V.
Wt
t*Wt
2
Duration?
8
Duration as a Risk Measure

When yields change the resulting price change
is proportional to Duration
 1  y  
P
 D

P
 1 y 

Practitioners generally Modify Duration
 Modified
Duration = D* = D/(1+y)
P
  D * y
P
9
Duration Example 2

Two bonds have a duration of 1.8852 years
8% 2-year bond with YTM=10%
2. Zero coupon bond maturing in 1.8852 years
1.


Semiannual compounding
Duration in semi annual periods
1.8852 yrs x 2 = 3.7704 semiannual periods
Modified D = 3.7704/(1+0.05) = 3.591 periods
 What happens if interest rates increase by 0.01%?

16-10
Duration Determinants
1.
2.
3.
4.
5.
The duration of a zero-coupon bond equals its time
to maturity
Holding maturity constant, a bond’s duration is
higher when the coupon rate is lower
Holding the coupon rate constant, a bond’s duration
generally increases with its time to maturity
Holding other factors constant, the duration of a
coupon bond is higher when the bond’s yield to
maturity is lower
The duration of a level perpetuity is equal to:
(1 + y) / y
11
Duration & Maturity
12
Portfolio Duration Example

You are managing a $1 million portfolio. Your
target duration is 10 years. You can choose
from two bonds: a zero-coupon bond with a
maturity of 5 years and a perpetuity, each
currently yielding 5%.
 How
much of each bond will you hold in your
portfolio? (Hint: Start with the perpetuity’s duration)
 How do these fractions change next year if target
duration is now 9 years?
13
Immunization

A strategy to shield the net worth of a bond

Control interest rate risk
 Widely
used by pension funds, insurance companies,
and banks
 Basics:
Match the duration of the assets
and liabilities
As
a result, value of assets will track the
value of liabilities whether rates rise or fall
14
Immunization Example


We need $14,693.28 in five years (Liability)
 Received $10,000 and guaranteed an 8% return
We can invest $10,000 in a 6yr 8% (an) bond (Asset)
 Duration of the obligation and asset is 5 years
Cashflows
Yr 5 Value @ 8%
Yr 5 Value @ 7%
Yr 5 Value @ 9%
1
800
1,088.39
1,048.64
1,129.27
2
800
1,007.77
980.03
1,036.02
3
800
933.12
915.92
950.48
4
800
864.00
856.00
872.00
5
800
800.00
800.00
800.00
6
10,800
10,000.00
10,093.46
9,908.26
14,693.28
14,694.05
14,696.03
15
Tuition

You have tuition expense of $18,000 per
semester (assume semi-annual) for the next
two years. Bonds currently yield 8%.
 What
is the duration of your obligation?
 What is the duration of a zero that would
immunize you, and its future redemption value?
 What happens to your net position if yields
increase to 9%?

Difference between obligation and asset
16
Breaking Down Interest Effects

When interest rates change it affects the bond
investor in two ways
 Affects

the price of the bond (Price Risk)
Negative relation
 Affects
the investment opportunities available for
coupon payments (Reinvestment Risk)


Positive relation
When a portfolio is immunized the Price risk
and reinvestment rate risk exactly cancel out
17
Immunization Example 2
Suppose you are managing a pension’s obligation to
make perpetual $2M payments. The YTM on all
bonds is 16%.
 5 yr 12% (annual) bonds have a 4yr duration
 20 yr 6% (annual) bonds have an 11yr duration
 What are the weights of your immunized
portfolio?
 What is the par value of your holdings in the 20year bond?
18
Immunization Example 3
Your pension plan will pay you $10,000 per year
for 10 years. The first payment will be in 5 years.
The pension fund wants to immunize its position.
The current interest rate is 10%
 What
is the duration of its obligations to you?
 If the plan uses 5-year and 20-year zero coupon
bonds to construct the immunized position, how
much money ought to be placed in each position?
19
Rebalancing

An bond’s duration will change as yields
changes, rebalancing is the practice of altering
our weights in the portfolio to keep the
durations matched
20
Immunization Alternative

Cash Flow Matching
 Match
the cash flows from the fixed income assets
with obligation
 Automatic immunization
 Dedication is cash flow matching over multiple
periods

Not widely used because of constraints
associated with bond choices
21
Actual vs Duration Approx Price Change
30 yr, 8% Coupon, 8% YTM
22
The Real Price Yield Relation



Bond prices are not linearly related to yields
Duration is a good approximation only for small
yields changes
Convexity is the measure of the curvature in the
price-yield relation
 Bonds with
greater convexity have more curvature in
the price-yield relationship.
Convexity Correction
P
1
2
 ( D*)y  [Convexity * (y ) ]
P
2
16-23
Convexity of Two Bonds
Which bond is more Convex?
24
Why do Investors Like Convexity?

Bonds with greater curvature gain more in
price when yields fall than they lose when
yields rise.
 This
asymmetry becomes more attractive as
interest rates become more volatile

Bonds with greater convexity tend to have
higher prices and/or lower yields, all else
equal.
16-25
Convexity Example
A 12% (Annual) 30-year bond has a duration of
11.54 years and convexity of 192.4. The bond
currently sells at a yield to maturity of 8%.
 Find
the bond price changes if YTM falls to 7% or
rises to 9%.
 What is the price change according to the duration
rule, and the duration-with-convexity rule
26
Convexity Example 2
A 12.75-year zero-coupon bond has a YTM=8% (effective
annual) has convexity of 150.3 and modified duration of
11.81 years.
A 30-year, 6% coupon (annual) bond also has YTM=8% has
nearly identical duration = 11.79, but higher convexity=231.2.
a)
b)
c)
YTM of both bonds increases to 9%. What is the percentage
loss on each bond? What percentage loss is predicted by
duration with convexity rule?
What if YTM decreases to 7%?
Given the above results, what is the attraction of convexity?
27
Active Bond Management

There are two ways to make money
 Interest

rate forecasting
Anticipating changes in the whole market
 Identifying

relative mispricings
However, you must be right and first
 If
everyone already knows it, then its already
priced
28
Active Bond Strategies

Substitution Swap
 Switch

one bond for a nearly identical (mispriced)
Intermarket Spread Swap
 Switching
two bonds from different market
segments (mispriced)

Rate Anticipation Swap
 Changing

between bond duration (Rate Forecasting)
Pure Yield Swap
 Moving
rate
into longer duration bonds for the higher
29