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Horizon Risk and Interest Rate Risk

Horizon Risk and Interest Rate Risk. Chapter 21. 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

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Horizon Risk and Interest Rate Risk

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  1. Horizon Risk and Interest Rate Risk Chapter 21 Chapter 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 Chapter 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 Chapter 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: Chapter 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 Chapter 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 Chapter 21: Interest Rate Risk and Horizon Risk

  7. Convexity in the Price-Yield Relationship 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. Illustrates the price-yield relationships on previous slide. Chapter 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 Chapter 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 Chapter 21: Interest Rate Risk and Horizon Risk

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

  18. Example: Calculation of MAC and MOD • MAC can be calculated using the previous present value calculations MOD = MAC  (1+YTM) = 2.84899  1.10 = 2.5899 Chapter 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. Chapter 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 Notice the maximum value is the same for all the bonds. Chapter 21: Interest Rate Risk and Horizon Risk

  21. Contrasting Time Until Maturity and Duration Chapter 21: Interest Rate Risk and Horizon Risk

  22. Contrasting Time Until Maturity and Duration Chapter 21: Interest Rate Risk and Horizon Risk

  23. Contrasting Time Until Maturity and Duration As horizon increases, bond’s MAC, MOD and convexity increase. Chapter 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 Chapter 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 Chapter 21: Interest Rate Risk and Horizon Risk

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

  28. Results in an EL of –0.01713  0.00909 = -1.90 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% Chapter 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 Chapter 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 Chapter 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 Chapter 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 As time passes, the interest on interest component has a greater impact on total return. Chapter 21: Interest Rate Risk and Horizon Risk

  33. Example: Immunizing the Palmer Corporation’s $1,000 Liability Note that the total yield is 9% regardless of the reinvestment rate. Chapter 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 Chapter 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 Chapter 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 Chapter 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 • DurationAssets = DurationLiabilities • PVAssets = PVLiabilities • DispersionAssets = DispersionLiabilities Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 21: Interest Rate Risk and Horizon Risk

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