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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Other duration “rules”Other duration “rules”

• A bond’s duration (and interest sensitivity) are A bond’s duration (and interest sensitivity) are higher the lower is the coupon rate (all else the higher the lower is the coupon rate (all else the same)same)

• Duration and interest rate sensitivity Duration and interest rate sensitivity usuallyusually increase with maturity (all else the same) increase with maturity (all else the same)

• Duration and interest rate sensitivity are higher Duration and interest rate sensitivity are higher when yields are lower (all else the same)when yields are lower (all else the same)

• Duration for perpetuity = 1/(1+y)Duration for perpetuity = 1/(1+y)

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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 2yConvexityyD

P

P

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for convexity:

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ConvexityConvexity

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

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

1 $100.00 $87.72 87.7193 2 $175.442 $100.00 $76.95 153.8935 6 $461.683 $100.00 $67.50 202.4915 12 $809.974 $100.00 $59.21 236.8321 20 $1,184.165 $1,100.00 $571.31 2856.528 30 $17,139.17

Total $862.68 3537.464 $19,770.41D = 4.10 C = 17.6343

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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

• Consider the two components of Consider the two components of interest rate riskinterest rate risk– price riskprice risk– reinvestment rate riskreinvestment rate risk

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

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

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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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Target or Horizon Date ImmunizationTarget or Horizon Date Immunization

• Set DSet Dpp = Horizon Date or Target date = Horizon Date or Target date

– then price risk (sale price of the bond) then price risk (sale price of the bond) and reinvestment risk (accumulated and reinvestment risk (accumulated value of the coupon payments) offset value of the coupon payments) offset one anotherone another

• Rebalancing (must monitor & update)Rebalancing (must monitor & update)– 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 matching (horizon matching (not analysisnot 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– D = HD => immunized D = HD => immunized – D > HD => net price riskD > HD => net price risk– D < HD => net reinvestment rate riskD < HD => net reinvestment rate risk

• Pure yield pickupPure yield pickup

• Tax swapTax swap

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Rate anticipationRate anticipation

• Consider a bond with 10 years to Consider a bond with 10 years to maturity, 8% coupon (paid annually), maturity, 8% coupon (paid annually), priced at $877.11. Current interest rates priced at $877.11. Current interest rates = 10%= 10%– Duration = 7.04Duration = 7.04– Suppose your HD = 4 yearsSuppose your HD = 4 years– You expect interest rates will decline You expect interest rates will decline – Since D > HD => net price riskSince D > HD => net price risk

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Rate anticipationRate anticipation

Future Value(CF)Time Cash Flow @10% @8% @12%

1 80 106.48 100.78 112.392 80 96.80 93.31 100.353 80 88.00 86.40 89.604 80 80.00 80.00 80.00

Price 912.89 1000.00 835.54

Interest Earned 51.28 40.49 62.35Change: Price 87.11 -77.35

Interest -10.79 11.07Net Price Effect 76.32 -66.28

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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