Yields & Prices: Continued
Chapter 11
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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
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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
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Bond Prices as a Function of Change in YtM
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Rate Changes and Bond Prices Known as interest rate risk Consider three bond
A: 8% Coupon Annual, 4 Years till maturityB: 8% Coupon Annual, 10 Years till maturityA: 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%
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Measuring Interest Rate Risk We can measure a bond’s interest rate risk with
DURATION Duration: Measures a bond’s effective
maturityCan tells us the effective average maturity of a
portfolio of bondsThe weighted average of the time until each
payment is receivedWeights are proportional to the payment’s PVDuration is shorter than maturity for coupon bondsDuration is equal to maturity for zeros.
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Duration Calculation
CFt = Cash flow at time t
y = YTM
1
Price
t
tt
CF yw
twtDT
t
1
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Duration Example What is the duration of a 2 year 12% annual
bond? The YTM is 10%.
Price?
Duration?
T (Years)
CF P.V. Wt t*Wt
1
2
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Duration as a Risk Measure When yields change the resulting price change
is proportional to Duration
Practitioners generally Modify DurationModified Duration = D* = D/(1+y)
y
yD
P
P
1
1
yDP
P
*
16-10
Duration Example 2 Two bonds have a duration of 1.8852 years
1. 8% 2-year bond with YTM=10%
2. Zero coupon bond maturing in 1.8852 years Semiannual compounding
Duration in semi annual periods1.8852 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%?
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Duration Determinants1. The duration of a zero-coupon bond equals its time
to maturity
2. Holding maturity constant, a bond’s duration is higher when the coupon rate is lower
3. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity
4. Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
5. The duration of a level perpetuity is equal to:
(1 + y) / y
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Duration & Maturity
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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?
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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 liabilitiesAs a result, value of assets will track the
value of liabilities whether rates rise or fall
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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
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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
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Breaking Down Interest Effects When interest rates change it affects the bond
investor in two waysAffects the price of the bond (Price Risk)
Negative relationAffects 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
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Immunization Example 2Suppose 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 20-
year bond?
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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?
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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
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Immunization Alternative
Cash Flow MatchingMatch the cash flows from the fixed income assets
with obligationAutomatic immunizationDedication is cash flow matching over multiple
periods Not widely used because of constraints
associated with bond choices
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Actual vs Duration Approx Price Change30 yr, 8% Coupon, 8% YTM
16-23
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
])(*[2
1*)( 2yConvexityyD
P
P
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Convexity of Two Bonds
Which bond is more Convex?
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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.
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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
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Convexity Example 2A 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) 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?
b) What if YTM decreases to 7%?
c) Given the above results, what is the attraction of convexity?
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Active Bond Management
There are two ways to make moneyInterest rate forecasting
Anticipating changes in the whole marketIdentifying relative mispricings
However, you must be right and firstIf everyone already knows it, then its already
priced
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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 SwapChanging between bond duration (Rate Forecasting)
Pure Yield SwapMoving into longer duration bonds for the higher
rate
