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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 2

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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 3

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Bond Prices as a Function of Change in YtM 4

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5 Rate Changes and Bond Prices Known as interest rate risk Consider three bond A: 8% Coupon Annual, 4 Years till maturity B: 8% Coupon Annual, 10 Years till maturity A: 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 bonds interest rate risk with DURATION Duration: Measures a bonds effective maturity Can tells us the effective average maturity of a portfolio of bonds The weighted average of the time until each payment is received Weights are proportional to the payments PV Duration is shorter than maturity for coupon bonds Duration is equal to maturity for zeros. 6

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Duration Calculation CF t = Cash flow at time t y = YTM 7

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Duration Example What is the duration of a 2 year 12% annual bond? The YTM is 10%. Price? Duration? 8 T (Years) CFP.V.WtWt t*W t 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 Duration Modified Duration = D * = D/(1+y) 9

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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 periods 1.8852 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 Determinants 1. The duration of a zero-coupon bond equals its time to maturity 2. Holding maturity constant, a bonds duration is higher when the coupon rate is lower 3. Holding the coupon rate constant, a bonds duration generally increases with its time to maturity 4. Holding other factors constant, the duration of a coupon bond is higher when the bonds yield to maturity is lower 5. The duration of a level perpetuity is equal to: (1 + y) / y 11

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Duration & Maturity 12

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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 perpetuitys duration) How do these fractions change next year if target duration is now 9 years? 13

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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 liabilities As a result, value of assets will track the value of liabilities whether rates rise or fall 14

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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 15 CashflowsYr 5 Value @ 8%Yr 5 Value @ 7%Yr 5 Value @ 9% 18001,088.391,048.641,129.27 28001,007.77980.031,036.02 3800933.12915.92950.48 4800864.00856.00872.00 5800800.00 610,80010,000.0010,093.469,908.26 14,693.2814,694.0514,696.03

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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 16

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Breaking Down Interest Effects When interest rates change it affects the bond investor in two ways Affects the price of the bond (Price Risk) Negative relation Affects 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 17

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Immunization Example 2 Suppose you are managing a pensions 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? 18

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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? 19

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Rebalancing An bonds duration will change as yields changes, rebalancing is the practice of altering our weights in the portfolio to keep the durations matched 20

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Immunization Alternative Cash Flow Matching Match the cash flows from the fixed income assets with obligation Automatic immunization Dedication is cash flow matching over multiple periods Not widely used because of constraints associated with bond choices 21

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Actual vs Duration Approx Price Change 30 yr, 8% Coupon, 8% YTM 22

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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

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Convexity of Two Bonds 24 Which bond is more Convex?

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16-25 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 26

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Convexity Example 2 A 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? 27

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Active Bond Management There are two ways to make money Interest rate forecasting Anticipating changes in the whole market Identifying relative mispricings However, you must be right and first If everyone already knows it, then its already priced 28

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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 Swap Changing between bond duration (Rate Forecasting) Pure Yield Swap Moving into longer duration bonds for the higher rate 29