Bond Portfolio Management Strategies: Basics II

02/25/09

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Bond Portfolio Management Strategies

• What are theoretical spot rates and forward rates and how do we compute them?

• When the bond’s yield changes, what characteristics of a bond cause differential price changes for individual bonds?

• What is modified duration and what is the relationship between a bond’s modified duration and its volatility?

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Bond Portfolio Management Strategies

• What is the convexity for a bond, how do you compute it, and what factors affect it?

• Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?

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Theoretical spot rates

• We have seen that using STRIPS we can determine the spot rate for a particular maturity.

• However, the theoretical spot rates may be slightly different from those observed in STRIPS because the stripped securities are not as liquid as the current Treasury issues.

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Theoretical spot rates

• We can compute a set of theoretical spot rates through a process referred to as boot-strapping.

• With this process, we assume that the value of a Treasury coupon security should equal the value of a package of zero coupon securities that duplicates the coupon bond’s cash flows.

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Forward rates

• Forward rates represent the market’s expectation of future short-term rates.

• For example, the yield on a 6-month Treasury bill six months from now would be a forward rate.

• Given the current rate for the 6-month and 1-year T-bills, we can extrapolate this forward rate.

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Interest Rate Sensitivity

• Interest rate sensitivity is the amount of bond price change for a given change in yield.

• This sensitivity is a function of:• Coupon rate• Maturity• Direction and level of yield change.

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Trading strategies based on interest rate sensitivity

• If you expect a decline (increase) in interest rates, you want a portfolio of bonds with maximum (minimum) interest rate sensitivity.

• Duration measures provide composite measures of interest rate sensitivity based on coupon and maturity.

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Macaulay Duration Measure

• The Macaulay Duration can be calculated as:

• Where t =time period in which the coupon or principal payment

occursCt = interest or principal payment that occurs in period t

price

)(1

n

ttCPVt

D

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Characteristics of Macaulay Duration

• Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments• A zero-coupon bond’s duration equals its maturity

• There is an inverse relationship between duration and coupon

• There is a positive relationship between term to maturity and duration, but duration increases at a decreasing rate with maturity

• There is an inverse relationship between YTM and duration

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Determining interest rate sensitivity

• An adjustment of Macaulay duration called modified duration can be used to approximate the bond price change to changes in yield.

• Where:m = number of payments a yeari = yield to maturity (YTM)

mi

1

durationMacaulay duration modified

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Modified Duration and Bond Price Volatility• Bond price movements will vary proportionally with

modified duration for small changes in yields.

• We can estimate the change in bond prices as:

PiD *)(price bondin change mod

Where:

P = beginning price for the bond

Dmod = the modified duration of the bond

i = yield change

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Trading Strategies Using Modified Duration

• Longest-duration security provides the maximum price variation

• If you expect a decline in interest rates, increase the average modified duration of your bond portfolio to experience maximum price volatility

• If you expect an increase in interest rates, reduce the average modified duration to minimize your price decline

• Note that the modified duration of your portfolio is the market-value-weighted average of the modified durations of the individual bonds in the portfolio

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Bond Convexity

• Modified duration is a linear approximation of bond price change for small changes in market yields

• However, price changes are not linear, but a curvilinear (convex) function.

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Determinants of Convexity

The convexity is the measure of the curvature and can be calculated as:

n

t

ti

tti

CF

1

2

)1(1 )(

)1(Convexity 2

The change in price due to convexity is then:

2i)(*convexity*price*2/1

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Determinants of Convexity

• There exists a(n):• Inverse relationship between coupon and convexity

• Direct relationship between maturity and convexity

• Inverse relationship between yield and convexity

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Modified Duration-Convexity Effects• Changes in a bond’s price resulting from a change in yield are

due to:• Bond’s modified duration• Bond’s convexity

• Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change

• Convexity is desirable

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Limitations of Macaulay and Modified Duration

• Percentage change estimates using modified duration only are good for small-yield changes.

• It is difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift.

• Initial assumption that cash flows from the bond are not affected by yield changes. This may not be true for bonds with options attached.

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Effective Duration

• Effective duration is also a measure of the interest rate sensitivity of an asset but adjusts for limitations of modified duration.

• It uses a pricing model to estimate the market prices surrounding a change in interest rates.

• Many practitioners use this direct measure to estimate interest rate sensitivity of bonds.

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Readings

• RB 18 (pgs. 704 – 711, 716-730, 733-734)