Chapter 11Chapter 11

Managing Fixed-Income Managing Fixed-Income InvestmentsInvestments

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Managing Fixed Income Managing Fixed Income Securities: Basic StrategiesSecurities: Basic Strategies

• Active strategyActive strategy

– Trade on interest rate Trade on interest rate predictionspredictions

– Trade on market inefficienciesTrade on market inefficiencies

• Passive strategyPassive strategy

– Control riskControl risk

– Balance risk and returnBalance risk and return

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Bond Pricing RelationshipsBond Pricing Relationships

• Inverse relationship between price and Inverse relationship between price and yieldyield

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

• Long-term bonds tend to be more price Long-term bonds tend to be more price sensitive than short-term bondssensitive than short-term bonds

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Bond Pricing Relationships Bond Pricing Relationships (cont.)(cont.)

• As maturity increases, price As maturity increases, price sensitivity increases at a sensitivity increases at a decreasing ratedecreasing rate

• Price sensitivity is inversely Price sensitivity is inversely related to a bond’s coupon raterelated to a bond’s coupon rate

• Price sensitivity is inversely Price sensitivity is inversely related to the yield to maturity at related to the yield to maturity at which the bond is sellingwhich the bond is selling

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DurationDuration

A measure of the effective maturity of a bondA measure of the effective maturity of a bond

The weighted average of the times until each payment is received, The weighted average of the times until each payment is received, with the weights proportional to the present value of the paymentwith the weights proportional to the present value of the payment

Duration is shorter than maturity for all bonds except zero coupon Duration is shorter than maturity for all bonds except zero coupon bondsbonds

Duration is equal to maturity for zero coupon bondsDuration is equal to maturity for zero coupon bonds

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Duration: CalculationDuration: Calculation

t tt

w CF y ice ( )1 Pr

D t wt

T

t

1

CF CashFlow for period tt

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Duration CalculationDuration Calculation

8%Bond

Timeyears

Payment PV of CF(10%)

Weight C1 XC4

1 80 72.727 .0765 .0765

2 80 66.116 .0690 .1392

Sum

3 1080 811.420

950.263

.8539

1.0000

2.5617

2.7774

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Consider a 5-year, 10% coupon Consider a 5-year, 10% coupon bond. Yield = 14%.bond. Yield = 14%.

DurationTime CF PV(CF) Weight W*T

1 $100.00 $87.72 0.101683 0.1016832 $100.00 $76.95 0.089195 0.1783913 $100.00 $67.50 0.078242 0.2347254 $100.00 $59.21 0.068633 0.2745325 $1,100.00 $571.31 0.662248 3.311238

Total $862.68 4.100567

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Duration/Price RelationshipDuration/Price Relationship

Price change is proportional to Price change is proportional to duration and not to maturityduration and not to maturity

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

DD* * = = modified durationmodified duration

DD* * = D / (1+y)= D / (1+y)

P/P = - DP/P = - D* * x x yy

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Approximating price changesApproximating price changes

• Consider our 10%, 5-year bond. Consider our 10%, 5-year bond. Yields are initially at 14% and the Yields are initially at 14% and the duration of the bond is 4.1.duration of the bond is 4.1.

• Suppose rates fall by 200 basis Suppose rates fall by 200 basis points. Estimate the percentage points. Estimate the percentage change in the bond’s price. change in the bond’s price. Estimate the price (& compare to Estimate the price (& compare to actual price).actual price).

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Estimating price sensitivityEstimating price sensitivity

Price

Duration

Pricing Error from convexity

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Using duration and convexity to Using duration and convexity to estimate price changes.estimate price changes.

])([21 2yConveixityyD

P

P

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for convexity:

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ConvexityConvexity

ConvexityTime CF PV(CF) T*(T+1) T*(T+1)*PV(CF)

1 $100.00 $87.72 2 $175.442 $100.00 $76.95 6 $461.683 $100.00 $67.50 12 $809.974 $100.00 $59.21 20 $1,184.165 $1,100.00 $571.31 30 $17,139.17

Total $862.68C = 15.28733847

Estimate of percentage price change = 0.074997Estimate of price = $927.3751

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Uses of DurationUses of Duration

• Summary measure of length or Summary measure of length or effective maturity for a portfolioeffective maturity for a portfolio

• Immunization of interest rate risk Immunization of interest rate risk (passive management)(passive management)

– Net worth immunizationNet worth immunization

– Target date immunizationTarget date immunization

• Measure of price sensitivity for Measure of price sensitivity for changes in interest ratechanges in interest rate

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Target date immunizationTarget date immunization

• Suppose rates are at 14% and you Suppose rates are at 14% and you have a 4.1 year horizon.have a 4.1 year horizon.

• Consider the bond with a 10% Consider the bond with a 10% annual coupon, 5 years, and annual coupon, 5 years, and duration of 4.1 yearsduration of 4.1 years

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Target date immunizationTarget date immunization

Future Value(CF)Time Cash Flow @14% @12% @16%

1 100 150.11 142.09 158.422 100 131.67 126.87 136.573 100 115.50 113.28 117.734 100 101.32 101.14 101.505 1100 977.64 993.34 962.46

HD Value 1476.24 1476.72 1476.68

Realized Return 14.00% 14.01% 14.01%

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Immunization and rebalancingImmunization and rebalancing

• Changing interest ratesChanging interest rates

• The passage of timeThe passage of time

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Other approachesOther approaches

• Cash flow matchingCash flow matching

• dedication strategydedication strategy

• horizon analysishorizon analysis

• contingent immunizationcontingent immunization

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Active Bond Management: Active Bond Management: Swapping StrategiesSwapping Strategies

• Substitution swapSubstitution swap

• Intermarket swapIntermarket swap

• Rate anticipation swapRate anticipation swap

• Pure yield pickupPure yield pickup

• Tax swapTax swap

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Interest rate swapInterest rate swap

• Contract between two parties to trade Contract between two parties to trade the cash flows corresponding to the cash flows corresponding to different securities without actually different securities without actually exchanging the securities directly.exchanging the securities directly.

• Plain vanilla: convert interest payments Plain vanilla: convert interest payments based on a floating rate into payments based on a floating rate into payments based on a fixed rate (or vice versa)based on a fixed rate (or vice versa)– NotionalNotional