1. Slides by: Pamela L. Hall, Western Washington University Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk1 Horizon Risk and Interest Rate Risk Chapter 21

2. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 2 Background  This chapter analyzes default-free bonds –Evaluates prices relative to changing interest rates and maturity  Horizon risk increases with the time remaining until a bond matures  Interest rate risk increases with the size of a bond’s price fluctuation when its YTM changes

3. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 3 Present Value of a Bond  The present value of a bond is determined by the following equation:  Even though a bond’s par, maturity and coupon rate may be fixed  The bond’s price varies over time

4. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 4 Present Value of a Bond  Example –Given information A U.S. Treasury bond pays an annual coupon rate of 5%, has a life of 12 years and a $1,000 par –At a discount rate of 10% the bond’s present value is:

5. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 5 Present Value of a Bond  Bond prices vary due to fluctuating market interest rates –As a bond’s YTM increases its price decreases The size of the fluctuations depends on the bond’s time horizon and coupon rate

6. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 6 Par Vs. Price  The following characterizes a bond’s relationship between coupon and YTM –If a bond is default-free then this relationship is the only thing that determines whether it sells above or below par Coupon and Price Relationship Price CategoryYield RelationshipPrice Relationship Premium bondYTM < coupon ratePrice > par Par bondYTM = coupon ratePrice = par Discount bondYTM > coupon ratePrice < par Zero coupon bondZero couponsPrice < par

7. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 7 Convexity in the Price-Yield Relationship Illustrates the price-yield relationships on previous slide. At low discount rates the prices of the 4 bonds are far apart— but the spread narrows toward zero as discount rate rises. However, price curves will never intersect.

8. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 8 Convexity in the Price-Yield Relationship  The shape of a bond’s price-yield relationship offers information about bond’s interest rate risk –Interest rate risk—variability in a bond’s price due to fluctuating interest rates  Price-yield relationship is more convex for –Longer maturities –Lower coupons

9. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 9 The Coupon Effect  A bond’s coupon rate impacts its YTM  YTM depends on: –Term structure of interest rates –Size and timing of coupons –Bond’s time horizon  Bonds with low coupons receive more of their value from its principal payments –Involve more interest rate risk Thus have more convexity

10. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 10 The Horizon Effect Bonds with longer horizons are more risky than short- term bonds. Bonds intersect at 5% because they have identical coupon rates of 5%--so their YTMs are equal at 5%.

11. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 11 Hedging Fixed Income Instruments  Even someone who invests in default- free fixed-income securities face risk –Reinvestment risk –Price fluctuation risk

12. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 12 Reinvestment Risk  Variability of return resulting from reinvestment of a bond’s coupon at fluctuating interest rates –Can be avoided by investing in zero coupon bonds

13. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 13 Hedging Bond Price Fluctuation Risk  Rational bond investors may wish to hedge price fluctuation risk –Hedge—a combination of investments designed to reduce or avoid risk Hedged portfolios usually earn lower rates of return than unhedged portfolios –Perfect hedges result when returns from long and short positions of equal value exactly offset each other

14. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 14 Derivation of Formula For Macauley’s Duration  The slope of a bond’s price-yield relationship measures the bond’s sensitivity to YTM –Thus, the first derivative of a bond’s present value formula with respect to YTM is  Multiplying both sides by (1/P) results in

15. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 15 Derivation of Formula For Macauley’s Duration  Rearranging the previous equation gives us:  Multiplying by (1+YTM) results in:  Which measures the percentage change in a bond’s price resulting from a small percentage change in its YTM

16. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 16 Derivation of Formula For Macauley’s Duration  MAC and MOD are similar measures of a bond’s time structure –MAC: average number of years the investor’s money is invested in the bond –MOD: average number of modified years the investor’s money is invested in the bond

17. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 17 Example: Calculation of MAC and MOD  Given information –A $1,000 par bond with a YTM of 10% has three years to maturity and a 5% coupon rate –Currently sells for $875.657

18. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 18 Example: Calculation of MAC and MOD  MAC can be calculated using the previous present value calculations TPV of CF Each CFs PV as fraction of Price T weighted by CF 145.4550.051911 × 0.05191 = 0.05191 241.3220.047182 × 0.04718 = 0.09438 3788.8810.90093 × 0.9009 = 2.70270 1.00000MAC = 2.84899 MOD = MAC  (1+YTM) = 2.84899  1.10 = 2.5899

19. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 19 Macaulay Duration  Macaulay (1938) suggested studying a bond’s time structure by examining its average term to maturity –Macauley’s duration (MAC) represents the weighted average time until the investor’s cash flows occur For zeros the weights are zero, making this term = 0. Thus, for zeros, MAC = t.

20. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 20 Contrasting Time Until Maturity and Duration  MAC  T –For zeros MAC = T –For non-zeros MAC < T Earlier and/or larger CFs result in shorter MAC Macauley durations for a bond with a 6% YTM at various times to maturity Coupon Rate T2%4%6%8% 10.9950.9900.9850.981 24.7564.5584.3934.254 108.8918.1697.6627.286 2014.98112.98011.90411.232 5019.45217.12916.27315.829 10017.56717.23217.12017.064  17.667 Notice the maximum value is the same for all the bonds.

21. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 21 Contrasting Time Until Maturity and Duration

22. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 22 Contrasting Time Until Maturity and Duration

23. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 23 Contrasting Time Until Maturity and Duration Duration and convexity for a bond with a 5% coupon at YTM of 10% 2-year bond12-year bond22-year bond PV$913.22$659.32$561.42 MAC1.958.5811 MOD1.777.810 Convexity0.020.210.58 As horizon increases, bond’s MAC, MOD and convexity increase.

24. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 24 Contrasting Time Until Maturity and Duration  A bond’s duration is inversely related to its coupon rate Duration and convexity for a bond with a 12 year maturity at YTM of 10% Zero coupon 5% coupon 10% coupon 15% coupon PV$318.63$659.32$1,000$1,340.68 MAC128.587.506.96 MOD10.917.86.816.33 Convexity0.100.210.200.19

25. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 25 MACLIM Defines a Boundary for MAC  A bond’s MAC will never exceed this limit: –MACLIM = (1+YTM)  YTM  MAC for a a perpetual bond will be equal to MACLIM –Regardless of coupon rate  For a coupon bond selling at or above par, MAC increases with the term to maturity  For a coupon bond selling below par, MAR hits a maximum and then decreases to MACLIM

26. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 26 Duration is a Linear Approximation of the Curvilinear Price-Yield Relationship Bonds A and B have positive convexity The straight line approximation becomes less accurate the further from the tangency point we go. At this point, the three bonds have the same duration.

27. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 27 Interest Rate Risk  Interest rate elasticity measures a bond’s price sensitivity to changes in interest rates Always negative because a bond’s price moves inversely to interest rates.

28. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 28 Example: Evaluating a Bond’s EL  Given information: –A bond has a 10% coupon rate and a par of $1,000. Its current price is $1,000 as the YTM is 10% –If interest rates were to rise from 10% to 11%, what would the new price be?  The price drops by $17.13 or 1.713%  -$17.13  $1,000 = -0.01713 or –1.713%  The increase in YTM from 10% to 11% is a percentage change of  (0.11 – 0.10)  1.1 = 0.0090909 or 0.9% Results in an EL of –0.01713  0.00909 = -1.90

29. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 29 Interest Rate Risk  MAC can also be used to calculate a bond’s elasticity –MAC = [(t=1)($90.909  $1,000] + [(t=2)($909.091)  $1,000] = 1.90 years  Interest rate elasticity and MAC are equally good measures of interest rate risk  Also good measures of total risk –Because all bonds are impacted by systematic fluctuations in interest rates

30. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 30 Immunizing Interest Rate Risk  Immunization—procedure designed to reduce or eliminate interest rate risk –Purchase an offsetting asset or liability with the same duration and present value Creates a portfolio that will earn the same rate of return expected prior to immunization, regardless of interest rate fluctuations

31. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 31 Example: Immunizing the Palmer Corporation’s $1,000 Liability  Given information –Palmer Corporation has a $1,000 liability due in 6.79 years  If Palmer purchased a default-free bond with a 9% coupon rate, par of $1,000 and maturity of 10 years for $1,000 [has a duration of 6.79 years] to repay the liability due in 6.79 years –Would have to deal with reinvestment risk—if interest rates drop below the original YTM of 9% this is a problem

32. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 32 Example: Immunizing the Palmer Corporation’s $1,000 Liability  The total return from this bond held under different reinvestment assumptions Holding Period (years) Return SourcesRate1356.79910 Coupon income5%$90$270$450$611$810$900 Capital gain$287$234$175$100$39$0 Interest on Interest$1.25$17$54$105$191$241 Total Return$378$521$679$816$1040$1141 Total yield37.0%15.0%11.0%9.00%8.5%8.2% Coupon income7%$90$270$450$611$810$900 Capital gain$132$109$83$56$19$0 Interest on Interest$2$25$78$149$279$355 Total Return$92$302$554$816$1197$1395 Total yield22.0%12.0%10.0%9.0%8.6%8.5% As time passes, the interest on interest component has a greater impact on total return.

33. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 33 Example: Immunizing the Palmer Corporation’s $1,000 Liability Holding Period (years) Return SourcesRate1356.79910 Coupon income9%$90$270$450$611$810$900 Capital gain$0 Interest on Interest$2$32$103$205$387$495 Total Return$92$302$554$816$1197$1395 Total yield9.0% Coupon income11%$90$270$450$611$810$900 Capital gain-$112-$95-$75-$56-$18$0 Interest on Interest$2$40$129$261$502$647 Total Return$20$215$504$816$1294$1547 Total yield2.0%6.7%8.5%9.0%9.7%9.8% Note that the total yield is 9% regardless of the reinvestment rate.

34. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 34 Example: Immunizing the Palmer Corporation’s $1,000 Liability  A bond’s total return is impacted by –Its interest income and interest-on-interest –Its price fluctuations  These two forces work in the opposite direction –Is there some point where they exactly offset each other? Yes, when the bond has been held for the length of the bond’s duration

35. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 35 Maturity Matching  Palmer Corporation could purchase a bond with a maturity exactly equal to the maturity of its liability, 6.79 years –However, ignores the coupon and interest on invested coupons  What if Palmer bought a zero-coupon bond? –There would be no need to worry about coupons and reinvestment  These methods are impractical –Extremely difficult to find zeros with needed maturity date –Extremely difficult (impossible) to find fixed-income securities with needed maturity date Also difficult to match a single bond’s duration with the liability’s duration  Due to these problems the more practical duration-matching strategy was developed

36. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 36 Duration Matching  Can immunize against interest rate risk by matching the weighted average MAC of a portfolio’s assets and liabilities –The MAC of a portfolio is equal to a weighted average of the individual MACs

37. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 37 Duration Matching  Financial institutions routinely perform duration matching strategies –Called asset-liability management (ALM)  Duration matching is necessary but not sufficient to achieve immunization –If CFs are spread over a wide range of times must meet all of these conditions to effectively immunize Duration Assets = Duration Liabilities PV Assets = PV Liabilities Dispersion Assets = Dispersion Liabilities

38. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 38 Duration Wandering and Portfolio Rebalancing  A bond’s duration does not decrease on a one-to-one basis with time  Market interest rates impact durations –For these reasons portfolios must be rebalanced to maintain a duration that will eliminate interest rate risk Annual or semi-annual rebalancing may be sufficient for certain assets/liability characteristics

39. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 39 Duration Wandering and Portfolio Rebalancing  For example: –Palmer Corporation originally wanted to match a liability with a life of 6.79 years So perhaps it bought a bond with a duration of 6.79 –After 1 year the maturity of its liability has decreased to 5.79 years »However the duration of the matched bond has declined by a smaller amount »Portfolio needs to be rebalanced to maintain the duration-matching strategy

40. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 40 Problems with Duration  Changes in term structure of interest rates cause stochastic process risk –Alternative duration measures have been developed to deal with this Macaulay Duration (MAC)—simplest and most popular measure of duration –Implicit assumptions »Yield curve is horizontal at the level of the bond’s YTM »Yield curve only experiences horizontal shifts

41. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 41 Problems with Duration Fisher-Weil Duration (FWD) –Produces similar value as MAC but is superior because »Considers each time period’s forward interest rate Modified Duration (MOD) –Different from MAC because MAC measures the percentage change in a bond’s price resulting from a percentage change in the market interest rate »MOD’s denominator is d(YTM)  (1+YTM) Cox, Ingersoll & Ross Duration (CIR) –More difficult to calculate than MAC and never been as popular  Results of tests indicate that MAC works about as well as the other measures –Is also cost effective, because of its simplicity

42. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 42 Problems with Duration  MAC, FWD & CIR are one-factor models –Based on fluctuations in a single interest rate  Other researchers are developing two- factor interest rate risk models –Use a short-term and a long-term interest rate None of these models are popular

43. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 43 Horizon Analysis  A bond buyer’s investment horizon is often different from a bond’s maturity horizon –Investor should perform a horizon analysis for every potential bond investment Horizon return—a bond’s total return including CFs and price changes over relevant investment horizon

44. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 44 Horizon Analysis  Some investors rely only upon a bond’s YTM –Don’t calculate horizon return because it requires estimates about future interest rates Horizon analysis is important—need to analyze different interest rate scenarios  Contingent immunization –Combines active management and immunization Portfolio manager may actively manage a portfolio so long as it earns a minimum safety net return –If safety net return is not earned manager is terminated and remaining assets are immunized

45. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 45 The Bottom Line  Behavior of bond prices –Bond prices move inversely to YTM –A bond’s interest rate risk usually increases with the time to maturity (horizon risk) However, risk increases at a decreasing rate –Price changes resulting from an equal-size change in a bond’s YTM are asymmetrical A decrease in YTM increases prices by more than an equal increase in YTM decreases prices –Coupon-paying bonds are influenced by the size of their coupon rates

46. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 46 The Bottom Line  Duration Axioms –Duration measures the average length of time funds are tied up in an investment –MAC is less than maturity for a coupon-paying bond and equals maturity for a zero MOD is less than MAC –Duration always varies directly with a bond’s maturity for zeros and bonds selling above or at par, and usually for bonds selling at a discount –All other factors equal, duration varies inversely with YTM for a non-zero –MAC equals a bond’s interest rate elasticity –Duration is a linear forecast of a bond’s price movement relative to YTM changes Only accurate for small changes in YTM –MAC has a limiting value

47. Francis & IbbotsonChapter 21: Interest Rate Risk and Horizon Risk 47 The Bottom Line  Interest rate risk axioms –Interest rate risk usually increases directly with MAC, MOD, elasticity and term to maturity –Immunization is used to reduce or eliminate interest rate risk –Asset-liability management may also be used to manage interest rate risk as well as market and/or credit risk –Positive convexity exists for option-free bonds but some embedded bonds may have negative convexity –If a bond will not be held to its maturity a horizon analysis should be performed