Chapter 11

Managing Bond

Portfolios

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

11.1 Interest Rate Risk

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Interest Rate Sensitivity1. Inverse relationship between bond price and

interest rates (or yields)

2. Long-term bonds are more price sensitive than short-term bonds

3. Sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases

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Interest Rate Sensitivity (cont)

4. A bond’s price sensitivity is inversely related to the bond’s coupon

5. Sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling

6. An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield

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Summary of Interest Rate Sensitivity

The concept: • Any security that gives an investor more money back

sooner (as a % of your investment) will have lower price volatility when interest rates change.

• Maturity is a major determinant of bond price sensitivity to interest rate changes, but

• It is not the only factor; in particular the coupon rate and the current ytm are also major determinants.

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Change in Bond Price as a Function of YTM

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Duration

1 2 3 4 5

$100 $100 $100 $100 $1100

Consider the following 5 year 10% coupon annual payment corporate bond:

• Because the bond pays cash prior to maturity it has an “effective” maturity less than 5 years.

• We can think of this bond as a portfolio of 5 zero coupon bonds with the given maturities.

• The average maturity of the five zeros would be the coupon bond’s effective maturity.

• We need a way to calculate the effective maturity.

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Duration• Duration is the term for the effective maturity of a bond

• Time value of money tells us we must calculate the present value of each of the five zero coupon bonds to construct an average.

• We then need to take the present value of each zero and divide it by the price of the coupon bond. This tells us what percentage of our money we get back each year.

• We can now construct the weighted average of the times until each payment is received.

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Duration Formula

icePr)ytm1(

CF

W

N

1tt

t

t

N

1tt tWDur

Wt = Weight of time t, present value of the cash flow earned in time t as a percent of the amount invested

CFt = Cash Flow in Time t, coupon in all periods except terminal period when it is the sum of the coupon and the principal

ytm = yield to maturity; Price = bond’s price

Dur = Duration

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Year (T) Cash Flow PV @8%

CFT / (1+ytm)T % of Value PV/Price

Weighted % of Value (PV/Price)*T

1 $ 90 2 90 3 90 4 $1090

Totals

Calculating the duration of a 9% coupon, 8% ytm, 4 year annual payment bond priced at $1033.12,

$1,033.12 100.00% 3.5396 yrs

$ 83.33

77.16

71.45

$801.18

8.06%

7.47%

6.92%

77.55%

0.0806

0.1494

0.2076

3.1020

Duration = 3.5396 years

icePr)ytm1(

CF

W

N

1tt

t

t

N

1tt tWDur

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Using Excel to Calculate Duration

Excel can be used to calculate a bond’s duration.

Usage notes:

•The dates should be entered using the formulas given

•If you don’t know the actual settlement date and maturity date, set the 6th term in the duration formulae to 0 as shown and pick a maturity date with the same month and day as the settlement date and the correct number of years after the settlement date.

•The par is not needed

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More on Duration1. Duration increases with maturity

2. A higher coupon results in a lower duration

3. Duration is shorter than maturity for all bonds except zero coupon bonds

4. Duration is equal to maturity for zero coupon bonds

5. All else equal, duration is shorter at higher interest rates

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More on Duration5. The duration of a level payment perpetuity

is ytmy ;y

y1Dperpetuity

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Figure 11.2 Duration as a Function of Maturity

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Duration/Price Relationship• Price change is proportional to duration

and not to maturity

P/P = -D x [y / (1+y)]

D* = modified duration

D* = D / (1+y)

P/P = - D* x y

D = Duration

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11.2 Passive Bond Management

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Interest Rate RiskInterest rate risk is the possibility that an investor does not earn the promised ytm because of interest rate changes.

A bond investor faces two types of interest rate risk:

1.Price risk: The risk that an investor cannot sell the bond for as much as anticipated. An increase in interest rates reduces the sale price.

2.Reinvestment risk: The risk that the investor will not be able to reinvest the coupons at the promised yield rate. A decrease in interest rates reduces the future value of the reinvested coupons.

The two types of risk are potentially offsetting.

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Immunization• Immunization: An investment strategy

designed to ensure the investor earns the promised ytm.

• A form of passive management, two versions1. Target date immunization

• Attempt to earn the promised yield on the bond over the investment horizon.

• Accomplished by matching duration of the bond to the investment horizon

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Terminal Value of an Immunized Portfolio over a 5 year Horizon

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Figure 11.3 Growth of Invested Funds

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Immunization

2. Net worth immunization

• The equity of an institution can be immunized by matching the duration of the assets to the duration of the liabilities.

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Figure 11.4 Immunization

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Cash Flow Matching and Dedication

• Cash flow from the bond and the obligation exactly offset each other

– Automatically immunizes a portfolio from interest rate movements

• Not widely pursued, too limiting in terms of choice of bonds

• May not be feasible due to lack of availability of investments needed

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Problems with Immunization

1. May be a suboptimal strategy

2. Does not work as well for complex portfolios with option components, nor for large interest rate changes

3. Requires rebalancing of the portfolio periodically, which then incurs transaction costs– Rebalancing is required when interest rates move– Rebalancing is required over time

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11.3 Convexity

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The Need for Convexity

• Duration is only an approximation

• Duration asserts that the percentage price change is linearly related to the change in the bond’s yield– Underestimates the increase in bond prices

when yield falls– Overestimates the decline in price when the

yield rises

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Pricing Error Due to Convexity

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Convexity: Definition and Usage

2yConvexity2/1)y1(

yD

P

P

n

1t

2t

t2

)tt()y1(

CF

)y1(P

1Convexity

Where: CFt is the cash flow (interest and/or principal) at time t and y = ytm

The prediction model including convexity is:

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Bond Price Convexity

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Convexity of Two Bonds

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Prediction Improvement With Convexity

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11.4 Active Bond Management

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Swapping Strategies1. Substitution swap

– Exchanging one bond for another with very similar characteristics but more attractively priced

2. Intermarket spread swap– Exploiting deviations in spreads between two market segments

3. Rate anticipation swap– Choosing a duration different than your investment horizon to

exploit a rate change.• Rate increase: Choose D > Investment horizon• Rate decrease: Choose D < Investment horizon

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Swapping Strategies4. Pure yield pickup

– Switching to a higher yielding bond, may be longer maturity if the term structure is upward sloping or may be lower default rating.

5. Tax swap– Swapping bonds for tax purposes, for example

selling a bond that has dropped in price to realize a capital loss that may be used to offset a capital gain in another security

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Horizon Analysis

• Analyst selects a particular investment period and predicts bond yields at the end of that period in order to forecast the bond’s HPY

11-35