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Bond Risk

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Bond Risk

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    1. Chapter 4 Bond Risk

    2. Bond Risk Investment risk is the uncertainty that the actual rate of return realized from an investment will differ from the expected rate. There are three types of risk associated with bonds and fixed income securities: Default Risk Call Risk Market Risk

    3. Default Risk Default Risk: Uncertainty that the realized return will deviate from the expected return because the issuer will fail to meet the contractual obligations specified in the indenture. A failure to meet any of the interest payments, the principal obligation, or other terms specified in the indenture (e.g., sinking fund arrangements, collateral requirements, or other protective covenants) places the borrower/issuer in default. When issuers default they can file for bankruptcy, their bondholders/creditors can sue for bankruptcy, or both parties can work out an agreement.

    4. Default Risk Many large institutional investors have their own credit analysis departments to evaluate bond issues in order to determine the abilities of companies to meet their contractual obligations. However, individual bond investors, as well as some institutional investors, usually do not make an independent evaluation of a bond's chance of default. Instead, they rely on bond rating companies.

    5. Default Risk Currently, the major rating companies in the United States are Moody's Investment Services Standard and Poor's Fitch Investors Service

    6. Default Risk Moody's Investment Services, Standard and Poor's, and Fitch Investors Service evaluate bonds by giving them a quality rating in the form of a letter grade: The grades start at ‘A’ with three groups Grade A bonds are followed by B-rated bonds Finally, there are C-grade and lower-grade bonds

    8. Default Risk Bonds with relatively low chance of default are referred to as investment-grade bonds, with quality ratings of Baa (or BBB) or higher. Bonds with relatively high chance of default are referred to as speculative-grade or junk bonds and have quality ratings below Baa.

    9. Default Risk The term junk bond became part of the financial vernacular in the 1980s to refer to low graded bonds sold by companies with financial problems, often referred to as Fallen Angels.

    10. Historical Default Rates Since World War II, the percentage of the dollar value of bonds defaulting has been quite low, averaging .12% per year. During the 1980s, the default rate became relatively high for junk bonds (3.27%). During that decade, there was a rapid growth in low-rated debt issues. Slow economic growth in 1990 and the recession in 1991 resulted in a high incidence of defaults by many companies that had issued junk bonds in the 1980s. In 1990, the default rate for junk bonds was 8.74%, and in 1991, it was 9%.

    11. Default Risk Premium Because there is a default risk on corporate, municipal, and other non-U.S. Treasury bonds, they trade with a default risk premium (also called a quality or credit spread). This premium is often measured as the spread between the rates on a non-Treasury security and a U.S. Treasury security that are the same in all respects except for their default risk.

    12. Studies of Default Risk Salomon Brothers and Hutch Studies Usually adverse economic conditions result in a greater default risk premium. A number of empirical studies have looked at the relation between the default risk premium (RP) and the state of the economy. Studies by Salomon Brothers and Hutch have found that the RP widens in recession and narrows in periods of economic expansion.

    13. Studies of Default Risk Johnston Study The Johnston study looked at the yield curves for different quality bonds. In the Johnston study, the spread between moderate-grade and high-grade bonds was found to increase as maturity increased, while the spread between low and moderate or high-grade bonds was found decreasing as maturity increased.

    14. Studies of Default Risk Johnston Study

    15. Studies of Default Risk McEnnally-Boardman Study In a 1979 study, McEnnally and Boardman examined the relationship between portfolio risk and size for bonds grouped in terms of their quality ratings. Using the same methodology employed by Evans and Archer in their portfolio risk and size study on stocks, McEnnally and Boardman collected monthly rates of return for over 500 corporate and municipal bonds with quality ratings of Baa or greater. For each quality group (Aaa, Aa, A, and Baa), they randomly selected portfolios with n-bonds (n = 2, 3, 4, ... 40) and calculated the n-bond portfolio's average standard deviation.

    16. Studies of Default Risk McEnnally-Boardman Study McEnnally and Boardman found that as the size of the bond portfolio increased, the portfolio risk decreased asymptotically, with the maximum risk reduction being realized with a portfolio size of 20. More interestingly, though, McEnnally and Boardman also found that the portfolios consisting of the lowest-quality bonds had the lowest portfolio risk when sufficiently diversified.

    17. Studies of Default Risk McEnnally-Boardman Study McEnnally and Boardman seemingly counterintuitive result can be explained in terms of the correlation between bonds in the same quality groups.

    18. Studies of Default Risk McEnnally-Boardman Study

    19. Call Risk Call Risk is the uncertainty that the realized return will deviate from the expected return because the issuer calls the bond, forcing the investor to reinvest in a market with lower rates.

    20. Call Risk Note: When a bond is called the holder receives the call price (CP). Since the CP usually exceeds the principal, the return the investor receives over the call period is often greater than the initial YTM. The investor, though, usually has to reinvest in a market with lower rates that often causes his return for the investment period to be less than the initial YTM.

    21. Call Risk Compare the ARR for the call period with the ARR for the investment period for a bond that is called. Example: Buy: 10-year, 10% annual coupon bond at par ($1000) callable at 110: CP = $1100 Assume: HD = 10 years Flat YC at 10% YC stays at 10% until the end of year 3 Year 3, the YC shifts down to a flat 8% and the bond is called Investor reinvests at 8% for the next 7 years.

    22. Call Risk

    23. Call Risk Premium Because of this call risk, there is usually a lower market demand and price for callable bonds than noncallable bonds, resulting in a higher rate of return or interest premium on callable over noncallable bonds. That is, the call risk premium (RP) is positive. The call RP is also referred to as the option adjusted spread (OAS)

    24. Call Risk Premium The size of the call risk premium, in turn, depends on investors' and borrowers' expectations concerning interest rates: When interest rates are high and expected to fall, bonds are more likely to be called; thus, in a period of high interest rates, a relatively low demand and higher rate on callable over noncallable bonds would occur. In contrast, when interest rates are low and expected to rise, we expect the effect of call provisions on interest rates to be small.

    25. Call Risk Premium Call Risk Premium:

    26. Call Risk: Price Compression Call features put limitations on the price-yield curve. At the rate where the bond could be called, the price-yield curve flattens, with the price equal to the CP. This limitation on the price is referred to as price compression. This limitation is illustrated in the figure.

    27. Call Risk: Price Compression

    28. Call Risk: Price Compression In the figure, the price-yield curve AA is shown for a noncallable bond. This curve is negatively sloped and convex from below. The curve AA/ represents the price-yield curve for a comparable callable bond. As shown, this curve flattens out and becomes concave (negative convexity) at rate y*, where y* represents a threshold rate that corresponds to a bond price equal, or approximately equal, to the call price. Since the callable bond would likely be called if rates are at y* or less, we would not expect investors to pay a price for such bond greater than the call price. Thus, the price-yield curve for the callable bond would tend to flatten out at y*.

    29. Valuation of Callable Bonds When valuing a callable bond, one needs to take into account the possibility that interest rates could decrease, leading to the bond being called. If called, the bond's cash flow patterns would be different than if rates increased and the bond was not called. Given the uncertainty of the bond's cash flows, valuing callable bonds and other bonds with embedded option features is more difficult than valuing option-free bonds.

    30. Valuation of Callable Bonds One approach to valuing callable bonds is to incorporate interest rate volatility by using a binomial interest rate tree. Another valuation approach is to determine the value of the call feature. Conceptually, when an investor buys a callable bond, she implicitly sells a call option to the bond issuer, giving the issuer the right to buy the bond from the bondholder at a specified price before maturity.

    31. Valuation of Callable Bonds Theoretically, the price of a callable bond, PC, should therefore be equal to the price of an identical, but noncallable bond, PNC, minus the value of the call feature or call premium, VC. The value of the call feature can be estimated using the option pricing model developed by Black and Scholes.

    32. Market Risk Market Risk is the uncertainty that the realized return will deviate from the expected return because of interest rate changes.

    33. Market Risk Recall, the return on a bond comes from: Coupons Interest earned from reinvesting coupons: interest on interest Capital gains or losses A change in rates affects interest on interest and capital gains or losses.

    34. Market Risk A change in interest rates has two effects on a bond's return: price effect and interest-on-interest effect.

    35. Market Risk

    36. Market Risk One obvious way an investor can eliminate market risk is to purchase a pure discount bond with a maturity that is equal to the investor's horizon date. If such a bond does not exist (or does, but does not yield an adequate rate), a bondholder will be subject (in most cases) to market risk.

    37. Market Risk: Example Consider the following case: Investor with a horizon date of 3.5 years Investor buys a 10-year, 10% annual coupon bond at its par value of $1,000 to yield 10%.

    38. Market Risk: Example If the yield curve were initially flat at 10% and if there were no changes in the yield curve in the ensuing years, then the investor would realize a rate of return (as measured by her ARR) of 10%.

    39. Market Risk

    40. Market Risk: Example Suppose that shortly after the investor purchased the bond, the flat 10% yield curve shifted up to 12% and remained there for the 3.5 years. At her HD the investor would be able to sell the bond for only $961.70, resulting in a capital loss of $38.30. This loss would be partly offset, though, by the gains realized from reinvesting the coupons at 12%. Combined, the investor's HD value would be $1,318.81 and her ARR would be only 8.23%.

    41. Market Risk

    42. Market Risk: Example Suppose that shortly after the investor purchased the bond, the flat 10% yield curve shifted down to 8% and remained there for the 3.5 years. At her HD the investor would be able to sell the bond for $1147.44. This gain would be partly offset, though, by the loss realized from reinvesting the coupons at 8%. Combined, the investor's HD value would be $1,484.82 and her ARR would be 11.96%.

    43. Market Risk

    44. Market Risk: Example Summary

    45. Market Risk The interest rate change has two opposite effects: Inverse price effect Direct interest on interest effect Whether the ARR varies directly or inversely to rate changes depends on which effect dominates. In this case (10-year, 10% coupon bond with HD = 3.5 yrs), the inverse price effect dominates. This causes the ARR to vary inversely with rate changes.

    46. Market Risk Bond with interest-on-interest effect that dominates price effect: Suppose the investor purchased a four-year, 20% annual coupon bond when the yield curve was flat at 10% (price of $1,317). If the yield curve shifted up to 12% shortly after the purchase and remained there, then the investor would realize an ARR of 10.16%. If the yield curve shifted down to 8%, the investor would realize a lower ARR of 9.845%. With an HD of 3.5 years, the four-year, 20% bond has an interest-on-interest effect that dominates the price effect, resulting in the direct relationship between the ARR and interest rate changes.

    47. Market Risk

    48. Market Risk

    49. Market Risk It is possible to select a bond in which the interest on interest and price effects exactly offset each other. When this occurs, the ARR will not change when there is a yield curve shift just after the purchase. Example: Suppose the investor purchased a four-year, 9% annual coupon for $968.30 to yield 10%. If the flat yield curve immediately shifted to 12%, 8%, or any other rate, the ARR would remain at 10%.

    50. Market Risk

    51. Market Risk

    52. Market Risk A 4-year, 9% coupon bond purchased when the yield curve is flat at 10% (P0 = 968.30) would yield an HD value at year 3.5 of 1351 and an ARR of 10% regardless of the shift in the yield curve occurring shortly after the purchase

    53. Market Risk What is distinctive about a 4-year, 9% coupon bond when the YTM is 10% is that it has a duration (D) equal to 3.5 years That is: Duration can be defined as the weighted average of the time periods. Duration also can be defined for a portfolio of bonds:

    54. Duration of 4-year, 9% coupon Bond

    55. Market Risk: Bond Immunization Definition: Bond immunization is a bond strategy of minimizing market risk. A bond immunization strategy is buying a bond or a portfolio of bonds with a duration equal to the HD.

    56. Duration as a Price Sensitivity Measure Duration is also an important parameter in describing a bond or bond portfolio's volatility in terms of its price sensitivity to interest rate changes.

    57. Duration as a Price Sensitivity Measure Like e (defined in Chapter 2 PPT), duration is a measure of a bond’s price sensitivity to interest rate changes: This measure of duration is known as the modified duration. The measure of duration as a weighted average of the time periods is known as Macauley’s duration.

    58. Duration as a Price Sensitivity Measure

    59. Modified Duration A modified duration measure for a bond that pays coupons each period and its principal at maturity (note: let y = YTM):

    60. Annualized Duration Duration is defined in terms of the length of the period between payments: If the CFs are distributed annually, then duration is in years. If CFs are semi-annual, then duration is measured in half years.

    61. Annualized Duration The convention is to expressed duration as an annual measure. The annualized duration is obtained by dividing duration by the number of payments per year (n):

    62. Modified Duration Example: The duration in half-years for a 10-year, 9% coupon bond selling at par (F = 100) and paying coupons semiannually is -13 and its annualized duration is -6.5:

    63. Duration Uses Descriptive Parameter: Measure of a bond’s price sensitivity to interest rate changes -- a measure of a bond’s volatility. Note, duration is consistent with the bond price relations discussed in Chapter 2:

    64. Duration Uses Define Strategies: Use duration to define active (speculative) and passive bond strategies. Examples: Rate-Anticipation Swap: Rates expected to decrease across all maturities, go long in high duration bonds. Rates expected to increase across all maturities, change bond portfolio composition so that it has lower duration bonds. Bond Immunization Strategy: Select bond or portfolio with duration that matches HD to minimized market risk.

    65. Duration Uses Estimate the percentage change in a bond’s price for a given change in rates:

    66. Duration Uses For 10-year, 9% bond, an increase in the annualized yield by 10BP (.09 to .0910) would lead to a .65% decrease in price (the actual is .6476%): A 200BP increase (9% to 11%) would lead to an estimated price decrease of 13%: Note: The actual decrease is only 12%.

    67. Convexity Duration is a measure of the slope of the price-yield curve at a given point -- first-order derivative. Convexity is a measure of the change in the slope of the price-yield curve -- second-order derivative. Convexity measures how bowed-shaped the price-yield curve is.

    68. Convexity Property: The greater a bond’s convexity, the greater its capital gains and the smaller its capital losses for given absolute changes in yields.

    69. Convexity Measures:

    70. Convexity (4) The convexity measure for a bond that pays coupons each period and its principal at maturity:

    71. Convexity Example: The convexity in half-years for a 10-year, 9% coupon bond selling at par (F = 100) and paying coupons semiannually is 225.43 and its annualized convexity is 56.36:

    72. Convexity Uses Descriptive Parameter: Convexity measures the asymmetrical gain and loss relation of a bond: Greater k-gains and smaller k-losses the greater a bond’s convexity. Estimation of price sensitivity to rate changes: Using Taylor Expansion, a better estimate of percentage price changes to discrete yield changes than the duration measure can be obtained by combining duration and convexity measures.

    73. Convexity Uses Taylor Expansion: For 10-year, 9% bond, an increase in the annualized yield by 200 BP (9% to 11%) would lead to an estimated 11.87% decrease in price using Taylor Expansion (the actual is 12%):

    74. Convexity Uses Note: Using Taylor Expansion the percentage increases in price are not symmetrical with the percentage decreases for given absolute changes in yields.

    75. Alternative Duration and Convexity Formulas Duration and convexity can also be estimated by determining the price of the bond when the yield increases by a small number of basis points (e.g., 2-10 basis points), P+, and when the yield decreases by the same number of basis points, P-. These measures are referred to as approximate duration and convexity and can be estimated using the following formulas:

    76. Caveats: Two Problems with Duration and Convexity Measures We've assumed a constant YTM. For yield curves that are not flat, one can use appropriate spot rates in determining the present values of the bond's cash flows instead of the same YTM. Our measures apply only to option-free bonds. Call, put, and sinking fund arrangements alter a bond's cash flow patterns and can dramatically change a bond's duration and convexity. Moreover, since many bonds have option features, adjusting their cash flow patterns to account for such features is important in measuring a bond's duration and convexity.

    77. Derivation of Duration and Convexity Duration: Take derivative with respect to y:

    78. Derivation of Duration and Convexity Factor out

    79. Derivation of Duration and Convexity Divide through by P:

    80. Derivation of Duration and Convexity Convexity: Take the derivative of Doing that yields: Divide through by PB and express as a summation:

    81. Web Sites For information on bond ratings: www.moodys.com ww.standardandpoors.com www.fitichratings.com

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