Fixed Income Analysis (2019) - PDF

CHAPTER 5 UNDERSTANDING FIXED INCOME RISK AND RETURN James F. Adams, PhD, CFA Donald J. Smit...

CHAPTER 5 UNDERSTANDING FIXED INCOME RISK AND RETURN James F. Adams, PhD, CFA Donald J. Smith, PhD LEARNING OUTCOMES After completing this chapter, you will be able to do the following: calculate and interpret the sources of return from investing in a fixed-rate bond; define, calculate, and interpret Macaulay, modified, and effective durations; explain why effective duration is the most appropriate measure of interest rate risk for bonds with embedded options; define key rate duration and describe the use of key rate durations in measuring the sensitiv- ity of bonds to changes in the shape of the benchmark yield curve; explain how a bond’s maturity, coupon, and yield level affect its interest rate risk; calculate the duration of a portfolio and explain the limitations of portfolio duration; calculate and interpret the money duration of a bond and price value of a basis point (PVBP); calculate and interpret approximate convexity and distinguish between approximate and effective convexity; estimate the percentage price change of a bond for a specified change in yield, given the bond’s approximate duration and convexity; describe how the term structure of yield volatility affects the interest rate risk of a bond; describe the relationships among a bond’s holding period return, its duration, and the investment horizon; explain how changes in credit spread and liquidity affect yield-to-maturity of a bond and how duration and convexity can be used to estimate the price effect of the changes. © 2019 CFA Institute. All rights reserved. 203 204 Fixed Income Analysis 1. INTRODUCTION It is important for analysts to have a well-developed understanding of the risk and return characteristics of fixed-income investments. Beyond the vast worldwide market for publicly and privately issued fixed-rate bonds, many financial assets and liabilities with known future cash flows may be evaluated using the same principles. The starting point for this analysis is the yield-to-maturity, or internal rate of return on future cash flows, which was introduced in Chapter 3. The return on a fixed-rate bond is affected by many factors, the most important of which is the receipt of the interest and principal payments in the full amount and on the sched- uled dates. Assuming no default, the return is also affected by changes in interest rates that affect coupon reinvestment and the price of the bond if it is sold before it matures. Measures of the price change can be derived from the mathematical relationship used to calculate the price of the bond. The first of these measures (duration) estimates the change in the price for a given change in interest rates. The second measure (convexity) improves on the duration estimate by taking into account the fact that the relationship between price and yield-to-maturity of a fixed-rate bond is not linear. Section 2 uses numerical examples to demonstrate the sources of return on an investment in a fixed-rate bond, which includes the receipt and reinvestment of coupon interest payments and the redemption of principal if the bond is held to maturity. The other source of return is capital gains (and losses) on the sale of the bond prior to maturity. Section 2 also shows that fixed-income investors holding the same bond can have different exposures to interest rate risk if their investment horizons differ. Discussion of credit risk, although critical to investors, is postponed to Section 5 so that attention can be focused on interest rate risk. Section 3 provides a thorough review of bond duration and convexity, and shows how the statistics are calculated and used as measures of interest rate risk. Although procedures and for- mulas exist to calculate duration and convexity, these statistics can be approximated using basic bond-pricing techniques and a financial calculator. Commonly used versions of the statistics are covered, including Macaulay, modified, effective, and key rate durations. The distinction is made between risk measures that are based on changes in the bond’s yield-to-maturity (i.e., yieldd duration and convexity) and on benchmark yield curve changes (i.e., curvee duration and convexity). Section 4 returns to the issue of the investment horizon. When an investor has a short- term horizon, duration (and convexity) are used to estimate the change in the bond price. In this case, yield volatility matters. In particular, bonds with varying times-to-maturity have different degrees of yield volatility. When an investor has a long-term horizon, the interaction between coupon reinvestment risk and market price risk matters. The relationship among in- terest rate risk, bond duration, and the investment horizon is explored. Section 5 discusses how the tools of duration and convexity can be extended to credit and liquidity risks and highlights how these different factors can affect a bond’s return and risk. A summary of key points and practice problems in the CFA Institute multiple-choice format conclude the chapter. 2. SOURCES OF RETURN An investor in a fixed-rate bond has three sources of return: (1) receipt of the promised coupon and principal payments on the scheduled dates, (2) reinvestment of coupon payments, and (3) potential capital gains or losses on the sale of the bond prior to maturity. In this section, it is Chapter 5 Understanding Fixed-Income Risk and Return 205 assumed that the issuer makes the coupon and principal payments as scheduled. This chapter focuses primarily on interest rate risk (the risk that interest rates will change), which affects the reinvestment of coupon payments and the market price if the bond is sold prior to maturity. Credit risk is considered later in this chapter and is the primary subject of Chapter 6, “Funda- mentals of Credit Analysis.” When a bond is purchased at a premium or a discount, it adds another aspect to the rate of return. Recall from Chapter 3 that a discount bond offers the investor a “deficient” coupon rate, or one below the market discount rate. The amortization of the discount in each period brings the return in line with the market discount rate as the bond’s carrying value is “pulled to par.” For a premium bond, the coupon rate exceeds the market discount rate and the amortization of the premium adjusts the return to match the market discount rate. Through amortization, the bond’s carrying value reaches par value at maturity. A series of examples will demonstrate the effect of a change in interest rates on two in- vestors’ realized rates of return. Interest rates are the rates at which coupon payments are reinvested and the market discount rates at the time of purchase and at the time of sale if the bond is not held to maturity. In Examples 1 and 2, interest rates are unchanged. The two in- vestors, however, have different time horizons for holding the bond. Examples 3 and 4 show the impact of an increase in interest rates on the two investors’ total return. Examples 5 and 6 show the impact of a decrease in interest rates. In each of the six examples, an investor initially buys a 10-year, 8% annual coupon payment bond at a price of 85.503075 per 100 of par value. The bond’s yield-to-maturity is 10.40%. 8 8 8 8 8 85.503075 = + + + + + (1 + r )1 (1 + r )2 (1 + r )3 (1 + r )4 (1 + r )5 8 8 8 8 108 6 + 7 + 8 + 9 + , r = 0.1040 (1 + r ) (1 + r ) (1 + r ) (1 + r ) (1 + r )10 EXAMPLE 1 A “buy-and-hold” investor purchases a 10-year, 8% annual coupon payment bond at 85.503075 per 100 of par value and holds it until maturity. The investor receives the series of 10 coupon payments of 8 (per 100 of par value) for a total of 80, plus the re- demption of principal (100) at maturity. In addition to collecting the coupon interest and the principal, the investor has the opportunity to reinvest the cash flows. If the cou- pon payments are reinvested at 10.40%, the future value of the coupons on the bond’s maturity date is 129.970678 per 100 of par value. 8 × (1.1040 )9  + 8 × (1.1040 )8  + 8 × (1.1040 )7  + 8 × (1.1040 )6  +         8 × (1.1040 )5  + 8 × (1.1040 )4  + 8 × (1.1040 )3  + 8 × (1.1040 )2  +         8 × (1.1040 )1  + 8 = 129.970678   The first coupon payment of 8 is reinvested at 10.40% for nine years until maturity, the second is reinvested for eight years, and so forth. The future value of the annuity 206 Fixed Income Analysis is obtained easily on a financial calculator, using 8 for the payment that is received at the end of each of the 10 periods. The amount in excess of the coupons, 49.970678 (= 129.970678 − 80), is the “interest-on-interest” gain from compounding. The investor’s total return is 229.970678, the sum of the reinvested coupons (129.970678) and the redemption of principal at maturity (100). The realized rate of return is 10.40%. 229.970678 85.503075 = , r = 0.1040 (1 + r )10 Example 1 demonstrates that the yield-to-maturity at the time of purchase measures the investor’s rate of return under three assumptions: (1) The investor holds the bond to maturity, (2) there is no default by the issuer, and (3) the coupon interest payments are reinvested at that same rate of interest. Example 2 considers another investor who buys the 10-year, 8% annual coupon payment bond and pays the same price. This investor, however, has a four-year investment horizon. Therefore, coupon interest is only reinvested for four years, and the bond is sold immediately after receiving the fourth coupon payment. EXAMPLE 2 A second investor buys the 10-year, 8% annual coupon payment bond and sells the bond after four years. Assuming that the coupon payments are reinvested at 10.40% for four years, the future value of the reinvested coupons is 37.347111 per 100 of par value. 8 × (1.1040 )3  + 8 × (1.1040 )2  + 8 × (1.1040 )1  + 8 = 37.347111       The interest-on-interest gain from compounding is 5.347111 (= 37.347111 − 32). After four years, when the bond is sold, it has six years remaining until maturity. If the yield-to-maturity remains 10.40%, the sale price of the bond is 89.668770. 8 8 8 8 1+ 2 + 3 + + (1.1040 ) (1.1040 ) (1.1040 ) (1.1040 )4 8 108 5 + = 89.668770 (1.1040 ) (1.1040 )6 The total return is 127.015881 (= 37.347111 + 89.668770) and the realized rate of return is 10.40%. 127.015881 85.503075 = , r = 0.1040 (1 + r )4 Chapter 5 Understanding Fixed-Income Risk and Return 207 In Example 2, the investor’s horizon yield d is 10.40%. A horizon yield is the internal rate of return between the total return (the sum of reinvested coupon payments and the sale price or redemption amount) and the purchase price of the bond. The horizon yield on a bond in- vestment is the annualized holding-period rate of return. Example 2 demonstrates that the realized horizon yield matches the original yield-to- maturity if: (1) coupon payments are reinvested at the same interest rate as the original yield-to-maturity, and (2) the bond is sold at a price on the constant-yield price trajectory, which implies that the investor does not have any capital gains or losses when the bond is sold. Capital gains arise if a bond is sold at a price above its constant-yield price trajectory and capital losses occur if a bond is sold at a price below its constant-yield price trajectory. This tra- jectory is based on the yield-to-maturity when the bond is purchased. The trajectory is shown in Exhibit 1 for a 10-year, 8% annual payment bond purchased at a price of 85.503075 per 100 of par value. EXHIBIT 1 Constant-Yield Price Trajectory for a 10-Year, 8% Annual Payment Bond Price 102 100 98 96 94 Capital Gain If the Bond Is Sold 92 at a Price Above the Trajectory 90 Capital Loss If 88 the Bond Is Sold at a Price Below the Trajectory 86 84 0 2 4 6 8 10 Year Note: Price is price per 100 of par value. A point on the trajectory represents the carrying value of the bond at that time. The carrying value is the purchase price plus the amortized amount of the discount if the bond is purchased at a price below par value. If the bond is purchased at a price above par value, the carrying value is the purchase price minus the amortized amount of the premium. The amortized amount for each year is the change in the price between two points on the trajectory. The initial price of the bond is 85.503075 per 100 of par value. Its price (the carrying value) after one year is 86.393394, calculated using the original yield-to-maturity of 10.40%. Therefore, the amortized amount for the first year is 0.890319 (= 86.393394 − 85.503075). The bond price in Example 2 increases from 85.503075 to 89.668770, and that increase over the four years is movement alongg the constant-yield price trajectory. At the time the bond is sold, its carrying value is also 89.668770, so there is no capital gain or loss. 208 Fixed Income Analysis Examples 3 and 4 demonstrate the impact on investors’ realized horizon yields if interest rates go up by 100 basis points (bps). The market discount rate on the bond increases from 10.40% to 11.40%. Coupon reinvestment rates go up by 100 bps as well. EXAMPLE 3 The buy-and-hold investor purchases the 10-year, 8% annual payment bond at 85.503075. After the bond is purchased and before the first coupon is received, interest rates go up to 11.40%. The future value of the reinvested coupons at 11.40% for 10 years is 136.380195 per 100 of par value. 8 × (1.1140 )9  + 8 × (1.1140 )8  + 8 × (1.1140 )7  + 8 × (1.1140 )6  +         8 × (1.1140 )5  + 8 × (1.1140 )4  + 8 × (1.1140 )3  + 8 × (1.1140 )2  +         8 × (1.1140 )1  + 8 = 136.380195   The total return is 236.380195 (= 136.380195 + 100). The investor’s realized rate of return is 10.70%. 236.380195 85.503075 = , r = 0.1070 (1 + r )10 In Example 3, the buy-and-hold investor benefits from the higher coupon reinvestment rate. The realized horizon yield is 10.70%, 30 bps higher than the outcome in Example 1, when interest rates are unchanged. There is no capital gain or loss because the bond is held until ma- turity. The carrying value at the maturity date is par value, the same as the redemption amount. EXAMPLE 4 The second investor buys the 10-year, 8% annual payment bond at 85.503075 and sells it in four years. After the bond is purchased, interest rates go up to 11.40%. The future value of the reinvested coupons at 11.40% after four years is 37.899724 per 100 of par value. 8 × (1.1140 )3  + 8 × (1.1140 )2  + 8 × (1.1140 )1  + 8 = 37.899724       The sale price of the bond after four years is 85.780408. 8 8 8 8 + + + + (1.1140 )1 (1.1140 )2 (1.1140 )3 (1.1140 )4 8 108 5 + = 85.780408 (1.1140 ) (1.1140 )6 Chapter 5 Understanding Fixed-Income Risk and Return 209 The total return is 123.680132 (= 37.899724 + 85.780408), resulting in a realized four-year horizon yield of 9.67%. 123.680132 85.503075 = , r = 0.0967 (1 + r )4 In Example 4, the second investor has a lower realized rate of return compared with the in- vestor in Example 2, in which interest rates are unchanged. The future value of reinvested cou- pon payments goes up by 0.552613 (= 37.899724 − 37.347111) per 100 of par value because of the higher interest rates. But there is a capital losss of 3.888362 (= 89.668770 − 85.780408) per 100 of par value. Notice that the capital loss is measured from the bond’s carrying value, the point on the constant-yield price trajectory, and not from the original purchase price. The bond is now sold at a price below the constant-yield price trajectory. The reduction in the realized four-year horizon yield from 10.40% to 9.67% is a result of the capital loss being greater than the gain from reinvesting coupons at a higher rate, which reduces the investor’s total return. Examples 5 and 6 complete the series of rate-of-return calculations for the two investors. Interest rates decline by 100 bps. The required yield on the bond falls from 10.40% to 9.40% after the purchase of the bond. The interest rates at which the coupon payments are reinvested fall as well. EXAMPLE 5 The buy-and-hold investor purchases the 10-year bond at 85.503075 and holds the security until it matures. After the bond is purchased and before the first coupon is received, interest rates go down to 9.40%. The future value of reinvesting the coupon payments at 9.40% for 10 years is 123.888356 per 100 of par value. 8 × (1.0940 )9  + 8 × (1.0940 )8  + 8 × (1.0940 )7  + 8 × (1.0940 )6  +         8 × (1.0940 )5  + 8 × (1.0940 )4  + 8 × (1.0940 )3  + 8 × (1.0940 )2  +         8 × (1.0940 )1  + 8 = 123.888356   The total return is 223.888356, the sum of the future value of reinvested coupons and the redemption of par value. The investor’s realized rate of return is 10.10%. 223.888356 85.503075 = , r = 0.1010 (1 + r )10 In Example 5, the buy-and-hold investor suffers from the lower coupon reinvestment rates. The realized horizon yield is 10.10%, 30 bps lower than the result in Example 1, when interest rates are unchanged. There is no capital gain or loss because the bond is held until 210 Fixed Income Analysis maturity. Examples 1, 3, and 5 indicate that the interest rate risk for a buy-and-hold investor arises entirely from changes in coupon reinvestment rates. EXAMPLE 6 The second investor buys the 10-year bond at 85.503075 and sells it in four years. After the bond is purchased, interest rates go down to 9.40%. The future value of the reinvested coupons at 9.40% is 36.801397 per 100 of par value. 8 × (1.0940 )3  + 8 × (1.0940 )2  + 8 × (1.0940 )1  + 8 = 36.801397       This reduction in future value is offset by the higher sale price of the bond, which is 93.793912 per 100 of par value. 8 8 8 8 + + + + (1.0940 )1 (1.0940 )2 (1.0940 )3 (1.0940 )4 8 108 5 + = 93.793912 (1.0940 ) (1.0940 )6 The total return is 130.595309 (= 36.801397 + 93.793912), and the realized yield is 11.17%. 130.595309 85.503075 = , r = 0.1117 (1 + r )4 The investor in Example 6 has a capital gain of 4.125142 (= 93.793912 − 89.668770). The capital gain is measured from the carrying value, the point on the constant-yield price tra- jectory. That gain offsets the reduction in the future value of reinvested coupons of 0.545714 (= 37.347111 − 36.801397). The total return is higher than that in Example 2, in which the interest rate remains at 10.40%. In these examples, interest income for the investor is the return associated with the passage of time. Therefore, interest income includes the receipt of coupon interest, the reinvestment of those cash flows, and the amortization of the discount from purchase at a price below par value (or the premium from purchase at a price above par value) to bring the return back in line with the market discount rate. A capital gain or loss is the return to the investor associated with the change in the valuee of the security. On the fixed-rate bond, a change in value arises from a change in the yield-to-maturity, which is the implied market discount rate. In practice, the way interest income and capital gains and losses are calculated and reported on financial statements depends on financial and tax accounting rules. This series of examples illustrates an important point about fixed-rate bonds: The in- vestment horizon is at the heart of understanding interest rate risk and return. There are two offsetting types of interest rate risk that affect the bond investor: coupon reinvestment risk and market price risk. The future value of reinvested coupon payments (and, in a portfolio, the principal on bonds that mature before the horizon date) increasess when interest rates go Chapter 5 Understanding Fixed-Income Risk and Return 211 up and decreasess when rates go down. The sale price on a bond that matures after the horizon date (and thus needs to be sold) decreasess when interest rates go up and increasess when rates go down. Coupon reinvestment risk matters more when the investor has a long-term horizon relative to the time-to-maturity of the bond. For instance, a buy-and-hold investor only has coupon reinvestment risk. Market price risk matters more when the investor has a short-term horizon relative to the time-to-maturity. For example, an investor who sells the bond before the first coupon is received has only market price risk. Therefore, two investors holding the same bond (or bond portfolio) can have different exposures to interest rate risk if they have different investment horizons. EXAMPLE 7 An investor buys a four-year, 10% annual coupon payment bond priced to yield 5.00%. The investor plans to sell the bond in two years once the second coupon payment is received. Calculate the purchase price for the bond and the horizon yield assuming that the coupon reinvestment rate after the bond purchase and the yield-to-maturity at the time of sale are (1) 3.00%, (2) 5.00%, and (3) 7.00%. Solution: The purchase price is 117.729753. 10 10 10 110 + + + = 117.729753 (1.0500 )1 (1.0500 )2 (1.0500 )3 (1.0500 )4 1. 3.00%: The future value of reinvested coupons is 20.300. (10 × 1.0300) + 10 = 20.300 The sale price of the bond is 113.394288. 10 110 + = 113.394288 (1.0300 )1 (1.0300 )2 Total return: 20.300 + 113.394288 = 133.694288. If interest rates go down from 5.00% to 3.00%, the realized rate of return over the two-year investment horizon is 6.5647%, higher than the original yield-to-maturity of 5.00%. 133.694288 117.729753 = , r = 0.065647 (1 + r )2 2. 5.00%: The future value of reinvested coupons is 20.500. (10 × 1.0500) + 10 = 20.500 The sale price of the bond is 109.297052. 10 110 1+ = 109.297052 (1.0500 ) (1.0500 )2 212 Fixed Income Analysis Total return: 20.500 + 109.297052 = 129.797052. If interest rates remain 5.00% for reinvested coupons and for the required yield on the bond, the realized rate of return over the two-year investment horizon is equal to the yield-to-maturity of 5.00%. 129.797052 117.729753 = , r = 0.050000 (1 + r )2 3. 7.00%: The future value of reinvested coupons is 20.700. (10 × 1.0700) + 10 = 20.700 The bond is sold at 105.424055. 10 110 + = 105.424055 (1.0700 )1 (1.0700 )2 Total return: 20.700 + 105.424055 = 126.124055. 126.124055 117.729753 = , r = 0.035037 (1 + r )2 If interest rates go up from 5.00% to 7.00%, the realized rate of return over the two- year investment horizon is 3.5037%, lower than the yield-to-maturity of 5.00%. 3. INTEREST RATE RISK ON FIXEDRATE BONDS This section covers two commonly used measures of interest rate risk: duration and convexity. It distinguishes between risk measures based on changes in a bond’s own yield to maturity (yield duration and convexity) and those that affect the bond based on changes in a benchmark yield curve (curve duration and convexity). 3.1. Macaulay, Modified, and Approximate Duration The duration of a bond measures the sensitivity of the bond’s full price (including accrued interest) to changes in the bond’s yield-to-maturity or, more generally, to changes in bench- mark interest rates. Duration estimates changes in the bond price assuming that variables other than the yield-to-maturity or benchmark rates are held constant. Most importantly, the time-to-maturity is unchanged. Therefore, duration measures the instantaneouss (or, at least, same-day) change in the bond price. The accrued interest is the same, so it is the flat price that goes up or down when the full price changes. Duration is a useful measure because it represents the approximate amount of time a bond would have to be held for the market discount rate at purchase to be realized if there is a single change in interest rate. If the bond is held for the duration period, an increase from reinvesting coupons is offset by a decrease in price if interest Chapter 5 Understanding Fixed-Income Risk and Return 213 rates increase and a decrease from reinvesting coupons is offset by an increase in price if interest rates decrease. There are several types of bond duration. In general, these can be divided into yield du- ration and curve duration. Yield duration is the sensitivity of the bond price with respect to the bond’s own yield-to-maturity. Curve duration is the sensitivity of the bond price (or more generally, the market value of a financial asset or liability) with respect to a benchmark yield curve. The benchmark yield curve could be the government yield curve on coupon bonds, the spot curve, or the forward curve, but in practice, the government par curve is often used. Yield duration statistics used in fixed income analysis include Macaulay duration, modified dura- tion, money duration, and the price value of a basis point (PVBP). A curve duration statistic often used is effective duration. Effective duration is covered in Section 3.2. Macaulay duration is named after Frederick Macaulay, the Canadian economist who first wrote about the statistic in a book published in 1938.1 Equation 1 is a general formula to calculate the Macaulay duration (MacDur) of a traditional fixed-rate bond. MacDur =  (1 − t T ) × PMT ( 2 − t T ) × PMT ( N − t T ) × ( PMT + FV )   1− t T + 2 −t T +⋯ +  (1 + r ) (1 + r ) (1 + r )N −t T   PMT PMT PMT + FV  (1)  1− t T + 2 −t T +⋯ + N −t T   (1 + r ) ( 1 + r ) (1 + r )  where t = the number of days from the last coupon payment to the settlement date T = the number of days in the coupon period t/T = the fraction of the coupon period that has gone by since the last payment PMT = the coupon payment per period FV = the future value paid at maturity, or the par value of the bond r = the yield-to-maturity, or the market discount rate, per period N = the number of evenly spaced periods to maturity as of the beginning of the current period The denominator in Equation 1 is the full price (PV VFulll) of the bond including accrued inter- est. It is the present value of the coupon interest and principal payments, with each cash flow discounted by the same market discount rate, r. PMT PMT PMT + FV PV Full = 1− t T + 2 −t T +⋯ + (1 + r ) (1 + r ) (1 + r )N −t T Equation 3 combines Equations 1 and 2 to reveal an important aspect of the Macaulay duration: Macaulay duration is a weighted average of the time to receipt of the bond’s prom- ised payments, where the weights are the shares of the full price that correspond to each of the bond’s promised future payments. 1 Frederick R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 18566 (New York: National Bureau of Economic Research, 1938). 214 Fixed Income Analysis   PMT   PMT     (1 + r )1−t T   (1 + r )2 −t T   (1 − t T )  Full  + ( 2 − t T )  Full  +⋯ +   PV   PV         MacDur =   (3)   PMT + FV     (1 + r )N −t T   ( N − t T )  Full     PV       The times to receipt of cash flow measured in terms of time periods are 1 − t/TT, 2 − t/T T, …, N − t/T. T The weights are the present values of the cash flows divided by the full price. Therefore, Macaulay duration is measured in terms of time periods. A couple of examples will clarify this calculation. Consider first the 10-year, 8% annual coupon payment bond used in Examples 1–6. The bond’s yield-to-maturity is 10.40%, and its price is 85.503075 per 100 of par value. This bond has 10 evenly spaced periods to maturity. Settlement is on a coupon payment date so that t/T = 0. Exhibit 2 illustrates the calculation of the bond’s Macaulay duration. EXHIBIT 2 Macaulay Duration of a 10-Year, 8% Annual Payment Bond Period Cash Flow Present Value Weight Period × Weight 1 8 7.246377 0.08475 0.0847 2 8 6.563747 0.07677 0.1535 3 8 5.945423 0.06953 0.2086 4 8 5.385347 0.06298 0.2519 5 8 4.878032 0.05705 0.2853 6 8 4.418507 0.05168 0.3101 7 8 4.002271 0.04681 0.3277 8 8 3.625245 0.04240 0.3392 9 8 3.283737 0.03840 0.3456 10 108 40.154389 0.46963 4.6963 85.503075 1.00000 7.0029 The first two columns of Exhibit 2 show the number of periods to the receipt of the cash flow and the amount of the payment per 100 of par value. The third column is the present value of the cash flow. For example, the final payment is 108 (the last coupon payment plus the redemption of principal) and its present value is 40.154389. 108 = 40.154389 (1.1040 )10 The sum of the present values is the full price of the bond. The fourth column is the weight, the share of total market value corresponding to each cash flow. The final payment of 108 per 100 of par value is 46.963% of the bond’s market value. Chapter 5 Understanding Fixed-Income Risk and Return 215 40.154389 = 0.46963 85.503075 The sum of the weights is 1.00000. The fifth column is the number of periods to the receipt of the cash flow (the first column) multiplied by the weight (the fourth column). The sum of that column is 7.0029, which is the Macaulay duration of this 10-year, 8% annual coupon payment bond. This statistic is sometimes reported as 7.0029 years, although the time frame is not needed in most applications. Now consider an example between coupon payment dates. A 6% semiannual payment corporate bond that matures on 14 February 2027 is purchased for settlement on 11 April 2019. The coupon payments are 3 per 100 of par value, paid on 14 February and 14 August of each year. The yield-to-maturity is 6.00% quoted on a street-convention semiannual bond basis. The full price of this bond comprises the flat price plus accrued interest. The flat price for the bond is 99.990423 per 100 of par value. The accrued interest is calculated using the 30/360 method to count days. This settlement date is 57 days into the 180-day semiannual period, so t/T t = 57/180. The accrued interest is 0.950000 (= 57/180 × 3) per 100 of par value. The full price for the bond is 100.940423 (= 99.990423 + 0.950000). Exhibit 3 shows the calculation of the bond’s Macaulay duration. EXHIBIT 3 Macaulay Duration of an Eight-Year, 6% Semiannual Payment Bond Priced to Yield 6.00% Period Time to Receipt Cash Flow Present Value Weight Time × Weight 1 0.6833 3 2.940012 0.02913 0.019903 2 1.6833 3 2.854381 0.02828 0.047601 3 2.6833 3 2.771244 0.02745 0.073669 4 3.6833 3 2.690528 0.02665 0.098178 5 4.6833 3 2.612163 0.02588 0.121197 6 5.6833 3 2.536080 0.02512 0.142791 7 6.6833 3 2.462214 0.02439 0.163025 8 7.6833 3 2.390499 0.02368 0.181959 9 8.6833 3 2.320873 0.02299 0.199652 10 9.6833 3 2.253275 0.02232 0.216159 11 10.6833 3 2.187645 0.02167 0.231536 12 11.6833 3 2.123927 0.02104 0.245834 13 12.6833 3 2.062065 0.02043 0.259102 14 13.6833 3 2.002005 0.01983 0.271389 15 14.6833 3 1.943694 0.01926 0.282740 16 15.6833 103 64.789817 0.64186 10.066535 100.940423 1.00000 12.621268 There are 16 semiannual periods to maturity between the last coupon payment date of 14 February 2019 and maturity on 14 February 2027. The time to receipt of cash flow in semiannual periods is in the second column: 0.6833 = 1 − 57/180, 1.6833 = 2 − 57/180, etc. 216 Fixed Income Analysis The cash flow for each period is in the third column. The annual yield-to-maturity is 6.00%, so the yield per semiannual period is 3.00%. When that yield is used to get the present value of each cash flow, the full price of the bond is 100.940423, the sum of the fourth column. The weights, which are the shares of the full price corresponding to each cash flow, are in the fifth column. The Macaulay duration is the sum of the items in the sixth column, which is the weight multiplied by the time to receipt of each cash flow. The result, 12.621268, is the Macaulay duration on an eight-year, 6% semiannual payment bond for settlement on 11 April 2019 measured in semiannual periods. Similar to coupon rates and yields-to-maturity, duration statistics invariably are annualized in practice. Therefore, the Macaulay duration typically is reported as 6.310634 yearss (= 12.621268/2).2 (Such precision for the duration statistic is not needed in practice. Typically, “6.31 years” is enough. The full precision is shown here to illus- trate calculations.) Another approach to calculating the Macaulay duration is to use a closed-form equation derived using calculus and algebra. Equation 4 is a general closed-form formula for deter- mining the Macaulay duration of a fixed-rate bond, where c is the coupon rate per period (PMT/FV V).3 1 + r 1 + r + N × ( c − r )   MacDur =  − [ ]  − (t T )  (4)  r c × (1 + r ) − 1 + r   N   The Macaulay duration of the 10-year, 8% annual payment bond is calculated by entering r = 0.1040, c = 0.0800, N = 10, and t/T = 0 into Equation 4. 1 + 0.1040 1 + 0.1040 + [10 × ( 0.0800 − 0.1040 )] MacDur = − = 7.0029 0.1040 0.0800 × (1 + 0.1040 )10 − 1 + 0.1040 Therefore, the weighted average time to receipt of the interest and principal payments that will result in realization of the initial market discount rate on this 10-year bond is 7.00 years. The Macaulay duration of the 6% semiannual payment bond maturing on 14 February 2027 is obtained by entering r = 0.0300, c = 0.0300, N = 16, and t/T = 57/180 into Equation 4.  1 + 0.0300 1 + 0.0300 + [16 × ( 0.0300 − 0.0300 )]  MacDur =  −  − ( 57 180 )  0.0300 0.0300 × (1 + 0.0300 )16 − 1 + 0.0300    = 12.621268 Equation 4 uses the yield-to-maturity per periodd, the coupon rate per periodd, the number of periodss to maturity, and the fraction of the current periodd that has gone by. Its output is the Macaulay duration in terms of periodss. It is converted to annual duration by dividing by the number of periods in the year. 2 Microsoft Excel users can obtain the Macaulay duration using the DURATION financial function: DURATION(DATE(2019,4,11),DATE(2027,2,14),0.06,0.06,2,0). The inputs are the settlement date, maturity date, annual coupon rate as a decimal, annual yield-to-maturity as a decimal, periodicity, and the code for the day count (0 for 30/360, 1 for actual/actual). 3 The step-by-step derivation of this formula is in Donald J. Smith, Bond Math: The Theory behind the Formulas, 2nd edition (Hoboken, NJ: John Wiley & Sons, 2014). Chapter 5 Understanding Fixed-Income Risk and Return 217 The calculation of the modified duration (ModDur) statistic of a bond requires a simple adjustment to the Macaulay duration. It is the Macaulay duration statistic divided by one plus the yield per period. MacDur ModDur = (5) 1+ r For example, the modified duration of the 10-year, 8% annual payment bond is 6.3432. 7.0029 ModDur = = 6.3432 1.1040 The modified duration of the 6% semiannual payment bond maturing on 14 February 2027 is 12.253658 semiannual periods. 12.621268 ModDur = = 12.253658 1.0300 The annualized modified duration of the bond is 6.126829 (= 12.253658/2).4 Although modified duration might seem to be just a Macaulay duration with minor ad- justments, it has an important application in risk measurement: Modified duration provides an estimate of the percentage price change for a bond given a change in its yield-to-maturity. VFull ≈ −AnnModDur %∆PV − × ∆Yield (6) The percentage price change refers to the full price, including accrued interest. The AnnMod- Dur term in Equation 6 is the annuall modified duration, and the ∆Yield term is the change in the annuall yield-to-maturity. The ≈ sign indicates that this calculation is an estimation. The minus sign indicates that bond prices and yields-to-maturity move inversely. If the annual yield on the 6% semiannual payment bond that matures on 14 February 2027 jumps by 100 bps, from 6.00% to 7.00%, the estimated loss in value for the bond is 6.1268%. VFull ≈ −6.126829 × 0.0100 = −0.061268 %∆PV If the yield-to-maturity were to drop by 100 bps to 5.00%, the estimated gain in value is also 6.1268%. VFull ≈ −6.126829 × −0.0100 = 0.061268 %∆PV Modified duration provides a linearr estimate of the percentage price change. In terms of absolute value, the change is the same for either an increase or decrease in the yield-to-maturity. Recall from “Introduction to Fixed-Income Valuation” that for a given coupon rate and time-to-maturity, the percentage price change is greater (in absolute value) when the market 4 Microsoft Excel users can obtain the modified duration using the MDURATION financial function: MDURATION(DATE(2019,4,11),DATE(2027,2,14),0.06,0.06,2,0). The inputs are the same as for the Macaulay duration in Footnote 2. 218 Fixed Income Analysis discount rate goes down than when it goes up. Later in this chapter, a “convexity adjustment” to duration is introduced. It improves the accuracy of this estimate, especially when a large change in yield-to-maturity (such as 100 bps) is considered. The modified duration statistic for a fixed-rate bond is easily obtained if the Macaulay du- ration is already known. An alternative approach is to approximatee modified duration directly. Equation 7 is the approximation formula for annual modified duration. ApproxModDur = ( PV− ) − ( PV+ ) (7) 2 × ( ∆Yield ) × ( PV0 ) The objective of the approximation is to estimate the slope of the line tangent to the price– yield curve. The slope of the tangent and the approximated slope are shown in Exhibit 4. EXHIBIT 4 Approximate Modified Duration Price Price–Yield Curve PV– Approximation for the Line Tangent to the Price–Yield Curve PV0 Line Tangent to the PV+ Price–Yield Curve Yield-to-Maturity ∆ Yield ∆ Yield To estimate the slope, the yield-to-maturity is changed up and down by the same amount—the ∆Yield. Then the bond prices given the new yields-to-maturity are calculated. The price when the yield is increased is denoted PV V+. The price when the yield-to-maturity is reduced is denoted PV V−. The original price is PVV0. These prices are the full prices, including accrued interest. The slope of the line based on PV V+ and PV− is the approximation for the slope of the line tangent to the price–yield curve. The following example illustrates the remarkable accuracy of this approximation. In fact, as ∆Yield approaches zero, the approximation ap- proaches AnnModDur. Consider the 6% semiannual coupon payment corporate bond maturing on 14 February 2027. For settlement on 11 April 2019, the full price (PV V0) is 100.940423 given that the yield-to-maturity is 6.00%.  3 3 103  16  × (1.03) 57 180 PV0 =  1+ 2 +⋯+ = 100.940423  (1.03 ) (1.03 ) (1.03 )  Raise the annual yield-to-maturity by five bps, from 6.00% to 6.05%. This increase corre- sponds to an increase in the yield-to-maturity per semiannual period of 2.5 bps, from 3.00% to 3.025% per period. The new full price (PVV+) is 100.631781. Chapter 5 Understanding Fixed-Income Risk and Return 219  3 3 103  16  × (1.03025) 57 180 PV+ =  1+ 2 +⋯+ = 100.631781  (1.03025) (1.03025) (1.03025)  Lower the annual yield-to-maturity by five bps, from 6.00% to 5.95%. This decrease corre- sponds to a decrease in the yield-to-maturity per semiannual period of 2.5 bps, from 3.00% to 2.975% per period. The new full price (PV−) is 101.250227.  3 3 103  16  × (1.02975) 57 180 PV− =  1+ 2 +⋯+ = 101.250227  (1.02975) (1.02975) (1.02975)  Enter these results into Equation 7 for the 5 bp change in the annual yield-to-maturity, or ∆Yield = 0.0005: 101.250227 − 100.631781 ApproxModDur = = 6.126842 2 × 0.0005 × 100.940423 The “exact” annual modified duration for this bond is 6.126829 and the “approximation” is 6.126842—virtually identical results. Therefore, although duration can be calculated using the approach in Exhibits 2 and 3—basing the calculation on the weighted average time to receipt of each cash flow—or using the closed-form formula as in Equation 4, it can also be estimated quite accurately using the basic bond-pricing equation and a financial calculator. The Ma- caulay duration can be approximated as well—the approximate modified duration multiplied by one plus the yield per period. ApproxMacDur = ApproxModDur × (1 + r) (8) The approximation formulas produce results for annualizedd modified and Macaulay durations. The frequency of coupon payments and the periodicity of the yield-to-maturity are included in the bond price calculations. EXAMPLE 8 Assume that the 3.75% US Treasury bond that matures on 15 August 2041 is priced to yield 5.14% for settlement on 15 October 2020. Coupons are paid semiannually on 15 Febru- ary and 15 August. The yield-to-maturity is stated on a street-convention semiannual bond basis. This settlement date is 61 days into a 184-day coupon period, using the actual/actual day-count convention. Compute the approximate modified duration and the approximate Macaulay duration for this Treasury bond assuming a 5 bp change in the yield-to-maturity. Solution: The yield-to-maturity per semiannual period is 0.0257 (= 0.0514/2). The cou- pon payment per period is 1.875 (= 3.75/2). At the beginning of the period, there are 21 years (42 semiannual periods) to maturity. The fraction of the period that has passed is 61/184. The full price at that yield-to-maturity is 82.967530 per 100 of par value.  1.875 1.875 1.875  PV0 =  + +⋯ + 42  × (1.0257 )61 184 = 82.96753 ( ) ( ) ( )  1 2 1.0257 1.0257 1.0257 220 Fixed Income Analysis Raise the yield-to-maturity from 5.14% to 5.19%—therefore, from 2.57% to 2.595% per semiannual period—and the price becomes 82.411395 per 100 of par value.  1.875 1.875 1.875  PV+ =  + +⋯ + 42  × (1.02595 )61 184  (1.02595 ) (1.02595 ) (1.02595 )  1 2 = 82.411395 Lower the yield-to-maturity from 5.14% to 5.09%—therefore, from 2.57% to 2.545% per semiannual period—and the price becomes 83.528661 per 100 of par value.  1.875 1.875 1.875  42  ( × 1.02545 ) 61 184 PV− =  1 + 2 +⋯ +  (1.02545 ) ( 1. 02545 ) (1.02545 )  = 83.528661 The approximate annualized modified duration for the Treasury bond is 13.466. 83.528661 − 82.411395 ApproxModDur = = 13.466 2 × 0.0005 × 82.967530 The approximate annualized Macaulay duration is 13.812. ApproxMacDur = 13.466 × 1.0257 = 13.812 Therefore, from these statistics, the investor knows that the weighted average time to receipt of interest and principal payments is 13.812 years (the Macaulay duration) and that the estimated loss in the bond’s market value is 13.466% (the modified duration) if the market discount rate were to suddenly go up by 1% from 5.14% to 6.14%. 3.2. Effective Duration Another approach to assess the interest rate risk of a bond is to estimate the percentage change in price given a change in a benchmark yield curve—for example, the government par curve. This estimate, which is very similar to the formula for approximate modified duration, is called the effective duration. The effective duration of a bond is the sensitivity of the bond’s price to a change in a benchmark yield curve. The formula to calculate effective duration (EffDur) is Equation 9. EffDur = ( PV− ) − ( PV+ ) (9) 2 × ( ∆Curve ) × ( PV0 ) The difference between approximate modified duration and effective duration is in the denominator. Modified duration is a yield duration statistic in that it measures interest rate risk in terms of a change in the bond’s own yield-to-maturity (∆Yield). Effective duration is a curve duration statistic in that it measures interest rate risk in terms of a parallel shift in the benchmark yield curve (∆Curve). Effective duration is essential to the measurement of the interest rate risk of a complex bond, such as a bond that contains an embedded call option. The duration of a callable bond Chapter 5 Understanding Fixed-Income Risk and Return 221 is nott the sensitivity of the bond price to a change in the yield-to-worst (i.e., the lowest of the yield-to-maturity, yield-to-first-call, yield-to-second-call, and so forth). The problem is that future cash flows are uncertain because they are contingent on future interest rates. The issuer’s decision to call the bond depends on the ability to refinance the debt at a lower cost of funds. In brief, a callable bond does not have a well-defined internal rate of return (yield-to-maturity). Therefore, yield duration statistics, such as modified and Macaulay durations, do not apply; effective duration is the appropriate duration measure. The specific option-pricing models that are used to produce the inputs to effective dura- tion for a callable bond are covered in later chapters. However, as an example, suppose that the full price of a callable bond is 101.060489 per 100 of par value. The option-pricing model inputs include (1) the length of the call protection period, (2) the schedule of call prices and call dates, (3) an assumption about credit spreads over benchmark yields (which includes any liquidity spread as well), (4) an assumption about future interest rate volatility, and (5) the level of market interest rates (e.g., the government par curve). The analyst then holds the first four inputs constant and raises and lowers the fifth input. Suppose that when the government par curve is raised and lowered by 25 bps, the new full prices for the callable bond from the model are 99.050120 and 102.890738, respectively. Therefore, PV V0 = 101.060489, PV V+ = 99.050120, PV V− = 102.890738, and ∆Curve = 0.0025. The effective duration for the callable bond is 7.6006. 102.890738 − 99.050120 EffDur = = 7.6006 2 × 0.0025 × 101.060489 This curve duration measure indicates the bond’s sensitivity to the benchmark yield curve—in particular, the government par curve—assuming no change in the credit spread. In practice, a callable bond issuer might be able to exercise the call option and obtain a lower cost of funds if (1) benchmark yields fall and the credit spread over the benchmark is unchanged or (2) benchmark yields are unchanged and the credit spread is reduced (e.g., because of an upgrade in the issuer’s rating). A pricing model can be used to determine a “credit duration” statistic—that is, the sensitivity of the bond price to a change in the credit spread. On a tradi- tional fixed-rate bond, modified duration estimates the percentage price change for a change in the benchmark yield and/or the credit spread. For bonds that do not have a well-defined internal rate of return because the future cash flows are not fixed—for instance, callable bonds and floating-rate notes—pricing models are used to produce different statistics for changes in benchmark interest rates and for changes in credit risk. Another fixed-income security for which yield duration statistics, such as modified and Macaulay durations, are not relevant is a mortgage-backed bond. These securities arise from a residential (or commercial) loan portfolio securitization. The key point for measuring interest rate risk on a mortgage-backed bond is that the cash flows are contingent on homeowners’ ability to refinance their debt at a lower rate. In effect, the homeowners have call options on their mortgage loans. A practical consideration in using effective duration is in setting the change in the benchmark yield curve. With approximate modified duration, accuracy is improved by choosing a smaller yield-to-maturity change. But the pricing models for more complex secu- rities, such as callable and mortgage-backed bonds, include assumptions about the behavior of the corporate issuers, businesses, or homeowners. Rates typically need to change by a minimum amount to affect the decision to call a bond or refinance a mortgage loan because issuing new debt involves transaction costs. Therefore, estimates of interest rate risk using 222 Fixed Income Analysis effective duration are not necessarily improved by choosing a smaller change in benchmark rates. Effective duration has become an important tool in the financial analysis of not only traditional bonds but also financial liabilities. Example 9 demonstrates such an application of effective duration. EXAMPLE 9 Defined-benefit pension schemes typically pay retirees a monthly amount based on their wage level at the time of retirement. The amount could be fixed in nominal terms or indexed to inflation. These programs are referred to as “defined-benefit pension plans” when US GAAP or IFRS accounting standards are used. In Australia, they are called “superannuation funds.” A British defined-benefit pension scheme seeks to measure the sensitivity of its re- tirement obligations to market interest rate changes. The pension scheme manager hires an actuarial consultancy to model the present value of its liabilities under three interest rate scenarios: (1) a base rate of 5%, (2) a 100 bp increase in rates, up to 6%, and (3) a 100 bp drop in rates, down to 4%. The actuarial consultancy uses a complex valuation model that includes assump- tions about employee retention, early retirement, wage growth, mortality, and longevity. The following chart shows the results of the analysis. Interest Rate Assumption Present Value of Liabilities 4% GBP973.5 million 5% GBP926.1 million 6% GBP871.8 million Compute the effective duration of the pension scheme’s liabilities. Solution: PV V0 = 926.1, PVV+ = 871.8, PV− = 973.5, and ∆Curve = 0.0100. The effective duration of the pension scheme’s liabilities is 5.49. 973.5 − 871.8 EffDur = = 5.49 2 × 0.0100 × 926.1 This effective duration statistic for the pension scheme’s liabilities might be used in asset allocation decisions to decide the mix of equity, fixed-income, and alternative assets. Although effective duration is the most appropriate interest rate risk measure for bonds with embedded options, it also is useful with traditional bonds to supplement the information provided by the Macaulay and modified yield durations. Exhibit 5 displays the Bloomberg Yield and Spread (YAS) Analysis page for the 2.875% US Treasury note that matures on 15 May 2028. Chapter 5 Understanding Fixed-Income Risk and Return 223 EXHIBIT 5 Bloomberg YAS Page for the 2.875% US Treasury Note Used with permission of Bloomberg Businessweek. Copyright © 2018. All rights reserved. In Exhibit 5, the quoted (flat) asked price for the bond is 100–07, which is equal to 100 and 7 32nds per 100 of par value for settlement on 13 July 2018. Most bond prices are stated in decimals, but US Treasuries are usually quoted in fractions. As a decimal, the flat price is 100.21875. The accrued interest uses the actual/actual day-count method. That settlement date is 59 days into a 184-day semiannual coupon payment period. The accrued interest is 0.4609375 per 100 of par value (=59/184 × 0.02875/2 × 100). The full price of the bond is 100.679688. The yield-to-maturity of the bond is 2.849091%, stated on a street-convention semiannual bond basis. The modified duration for the bond is shown in Exhibit 5 to be 8.482, which is the conventional yieldd duration statistic. Its curvee duration, however, is 8.510, which is the price sensitivity with respect to changes in the US Treasury par curve. On Bloomberg, the effective duration is called the “OAS duration” because it is based on the option-pricing model that is also used to calculate the option-adjusted spread. The small difference arises because the gov- ernment yield curve is not flat. When the par curve is shifted in the model, the government spot curve is also shifted, although not in the same “parallel” manner. Therefore, the change in the bond price is not exactly the same as it would be if its own yield-to-maturity changed by the same amount as the change in the par curve. In general, the modified duration and effective duration on a traditional option-free bond are not identical. The difference narrows when the yield curve is flatter, the time-to-maturity is shorter, and the bond is priced closer to par value (so that the difference between the coupon rate and the yield-to-maturity is smaller). 224 Fixed Income Analysis The modified duration and effective duration on an option-free bond are identical only in the rare circumstance of an absolutely flat yield curve. 3.3. Key Rate Duration Above, the effective duration for a sample callable bond was calculated as: 102.890738 − 99.050120 EffDur = = 7.6006 2 × 0.0025 × 101.060489 This duration measure indicates the bond’s sensitivity to the benchmark yield curve assuming that all yields change by the same amount. “Key rate” duration provides further insight into a bond’s sensitivity to changes in the benchmark yield curve. A key rate duration (or partial duration) is a measure of a bond’s sensitivity to a change in the benchmark yield curve at a specific maturity segment. In contrast to effective duration, key rate durations help identify “shaping risk” for a bond—that is, a bond’s sensitivity to changes in the shape of the bench- mark yield curve (e.g., the yield curve becoming steeper or flatter). The previous illustration of effective duration assumed a parallel shift of 25 bps at all maturities. However, the analyst may want to know how the price of the callable bond is expected to change if benchmark rates at short maturities (say up to 2 years) shifted up by 25 bps but longer maturity benchmark rates remained unchanged. This scenario would rep- resent a flattening of the yield curve, given that the yield curve is upward sloping. Using key rate durations, the expected price change would be approximately equal to minus the key rate duration for the short maturity segment times the 0.0025 interest rate shift at that segment. Of course, for parallel shifts in the benchmark yield curve, key rate durations will indicate the same interest rate sensitivity as effective duration. 3.4. Properties of Bond Duration The Macaulay and modified yield duration statistics for a traditional fixed-rate bond are func- tions of the input variables: the coupon rate or payment per period, the yield-to-maturity per period, the number of periods to maturity (as of the beginning of the period), and the fraction of the period that has gone by. The properties of bond duration are obtained by changing one of these variables while holding the others constant. Because duration is the basic measure of interest rate risk on a fixed-rate bond, these properties are important to understand. The closed-form formula for Macaulay duration, presented as Equation 4 and again here, is useful in demonstrating the characteristics of the bond duration statistic. 1 + r 1 + r + N × ( c − r )   MacDur =  − [ ]  − (t T )   r c × (1 + r ) − 1 + r   N   The same characteristics hold for modified duration. Consider first the fraction of the period T). Macaulay and modified durations depend on the day-count basis used that has gone by (t/T to obtain the yield-to-maturity. The duration of a bond that uses the actual/actual method to Chapter 5 Understanding Fixed-Income Risk and Return 225 count days is slightly different from that of an otherwise comparable bond that uses the 30/360 method. The key point is that for a constant yield-to-maturity (rr), the expression in braces is unchanged as time passes during the period. Therefore, the Macaulay duration decreases smoothly as t goes from t = 0 to t = T, which creates a “sawtooth” pattern. This pattern for a typical fixed-rate bond is illustrated in Exhibit 6. EXHIBIT 6 Macaulay Duration between Coupon Payments with a Constant Yield-to-Maturity Macaulay Duration Time-to-Maturity Coupon Payment Dates As times passes during the coupon period (moving from right to left in the diagram), the Macaulay duration declines smoothly and then jumps upward after the coupon is paid. The characteristics of bond duration related to changes in the coupon rate, the yield-to-maturity, and the time-to-maturity are illustrated in Exhibit 7. EXHIBIT 7 Properties of the Macaulay Yield Duration Macaulay Zero-Coupon Bond Duration Discount Bond l+r Perpetuity r Premium Bond Time-to-Maturity t/T = 0, thus not displaying the sawtooth pattern between coupon payments. The relationship between the Macaulay duration and the time-to-maturity for a zero-coupon bond is the 45-degree line: MacDur = N when c = 0 (and t/T = 0). Therefore, the Macaulay duration of a zero-coupon bond is its time-to- maturity. 226 Fixed Income Analysis A perpetuityy or perpetual bond, which also is called a consol, is a bond that does not mature. There is no principal to redeem. The investor receives a fixed coupon pay- ment forever, unless the bond is callable. Non-callable perpetuities are rare, but they have an interesting Macaulay duration: MacDur = (1 + r)/r as N approaches infinity. In effect, the second expression within the braces approaches zero as the number of periods to maturity increases because N in the numerator is a coefficient but N in the denominator is an exponent and the denominator increases faster than the numerator as N grows larger. Typical fixed-rate coupon bonds with a stated maturity date are portrayed in Exhibit 7 as the premium and discount bonds. The usual pattern is that longer times-to-maturity correspond to higher Macaulay duration statistics. This pattern always holds for bonds trading at par value or at a premium above par. In Equation 4, the second expression within the braces is a positive number for premium and par bonds. The numerator is positive because the coupon rate (cc) is greater than or equal to the yield-to-maturity (r), whereas the denominator is always positive. Therefore, the Macaulay duration is always less than (1 + r)/r, and it approaches that threshold from below as the time-to-maturity increases. The curious result displayed in Exhibit 7 is in the pattern for discount bonds. General- ly, the Macaulay duration increases for a longer time-to-maturity. But at some point when the time-to-maturity is high enough, the Macaulay duration exceeds (1 + r)/r, reaches a maximum, and then approaches the threshold from above. In Equation 4, such a pattern develops when the number of periods (N) is large and the coupon rate (c) is below the yield-to-maturity (r). Then the numerator of the second expression within the braces can become negative. The implication is that on long-term discount bonds, the interest rate risk can actually be less than on a shorter-term bond, which explains why the word “generally” is needed in describing the maturity effect for the relationship between bond prices and yields-to-maturity. Generally, for the same coupon rate, a longer-term bond has a greater percentage price change than a shorter-term bond when their yields-to-maturity change by the same amount. The exception is when the longer-term bond actually has a lower duration statistic. Coupon rates and yields-to-maturity are both inversely related to the Macaulay duration. In Exhibit 7, for the same time-to-maturity and yield-to-maturity, the Macaulay duration is higher for a zero-coupon bond than for a low-coupon bond trading at a discount. Also, the low-coupon bond trading at a discount has a higher duration than a high-coupon bond trading at a premium. Therefore, all else being equal, a lower-coupon bond has a higher du- ration and more interest rate risk than a higher-coupon bond. The same pattern holds for the yield-to-maturity. A higher yield-to-maturity reduces the weighted average of the time to re- ceipt of cash flow. More weight is on the cash flows received in the near term, and less weight is on the cash flows received in the more-distant future periods if those cash flows are discounted at a higher rate. In summary, the Macaulay and modified duration statistics for a fixed-rate bond depend primarily on the coupon rate, yield-to-maturity, and time-to-maturity. A higher coupon rate or a higher yield-to-maturity reduces the duration measures. A longer time-to-maturity usually leads to a higher duration. It alwayss does so for a bond priced at a premium or at par value. But if the bond is priced at a discount, a longer time-to-maturity mightt lead to a lower duration. This situation only occurs if the coupon rate is low (but not zero) relative to the yield and the time-to-maturity is long. Chapter 5 Understanding Fixed-Income Risk and Return 227 EXAMPLE 10 A hedge fund specializes in investments in emerging market sovereign debt. The fund manager believes that the implied default probabilities are too high, which means that the bonds are viewed as “cheap” and the credit spreads are too high. The hedge fund plans to take a position on one of these available bonds. Bond Time-to-Maturity Coupon Rate Price Yield-to-Maturity (A) 10 years 10% 58.075279 20% (B) 20 years 10% 51.304203 20% (C) 30 years 10% 50.210636 20% The coupon payments are annual. The yields-to-maturity are effective annual rates. The prices are per 100 of par value. 1. Compute the approximate modified duration of each of the three bonds using a 1 bp change in the yield-to-maturity and keeping precision to six decimals (because approximate duration statistics are very sensitive to rounding). 2. Which of the three bonds is expected to have the highest percentage price increase if the yield-to-maturity on each decreases by the same amount—for instance, by 10 bps from 20% to 19.90%? Solution to 1: Bond A: PV V0 = 58.075279 PV V+ = 58.047598 10 10 110 + +⋯+ = 58.047598 (1.2001)1 (1.2001)2 (1.2001)10 PV−= 58.102981 10 10 110 1+ 2 +⋯+ = 58.102981 (1.1999 ) (1.1999 ) (1.1999 )10 The approximate modified duration of Bond A is 4.768. 58.102981 − 58.047598 ApproxModDur = = 4.768 2 × 0.0001 × 58.075279 Bond B: PV V0 = 51.304203 PV V+ = 51.277694 228 Fixed Income Analysis 10 10 110 1+ 2 +⋯+ = 51.277694 (1.2001) (1.2001) (1.2001)20 PV− = 51.330737 10 10 110 1+ 2 +⋯+ = 51.330737 (1.1999 ) (1.1999 ) (1.1999 )20 The approximate modified duration of Bond B is 5.169. 51.330737 − 51.277694 ApproxModDur = = 5.169 2 × 0.0001 × 51.304203 Bond C: PV V0 = 50.210636 PV V+ = 50.185228 10 10 110 1+ 2 +⋯+ = 50.185228 (1.2001) (1.2001) (1.2001)30 PV− = 50.236070 10 10 110 1+ 2 +⋯+ = 50.236070 (1.1999 ) (1.1999 ) (1.1999 )30 The approximate modified duration of Bond C is 5.063. 50.236070 − 50.185228 ApproxModDur = = 5.063 2 × 0.0001 × 50.210636 Solution to 2: Despite the significant differences in times-to-maturity (10, 20, and 30 years), the approximate modified durations on the three bonds are fairly similar (4.768, 5.169, and 5.063). Because the yields-to-maturity are so high, the additional time to receipt of interest and principal payments on the 20- and 30-year bonds has low weight. Nevertheless, Bond B, with 20 years to maturity, has the highest mod- ified duration. If the yield-to-maturity on each is decreased by the same amount— for instance, by 10 bps, from 20% to 19.90%—Bond B would be expected to have the highest percentage price increase because it has the highest modified duration. This example illustrates the relationship between the Macaulay duration and the time-to-maturity on discount bonds in Exhibit 7. The 20-year bond has a higher duration than the 30-year bond. Chapter 5 Understanding Fixed-Income Risk and Return 229 Callable bonds require the use of effective duration because Macaulay and modified yield duration statistics are not relevant. The yield-to-maturity for callable bonds is not well defined because future cash flows are uncertain. Exhibit 8 illustrates the impact of the change in the benchmark yield curve (∆Curve) on the price of a callable bond price compared with that on a comparable non-callable bond. The two bonds have the same credit risk, coupon rate, payment frequency, and time-to-maturity. The vertical axis is the bond price. The horizontal axis is a particular benchmark yield—for instance, a point on the par curve for government bonds. EXHIBIT 8 Interest Rate Risk Characteristics of a Callable Bond Price Value of the Embedded Call Option PV– PV0 Non-Callable Bond PV+ Callable Bond Δ Curve Δ Curve Benchmark Yield As shown in Exhibit 8, the price of the non-callable bond is always greater than that of the callable bond with otherwise identical features. The difference is the value of the embedded call option. Recall that the call option is an option to the issuer and not the holder of the bond. When interest rates are high compared with the coupon rate, the value of the call option is low. When rates are low, the value of the call option is much greater because the issuer is more likely to exercise the option to refinance the debt at a lower cost of funds. The investor bears the “call risk” because if the bond is called, the investor must reinvest the proceeds at a lower interest rate. Exhibit 8 shows the inputs for calculating the effective duration of the callable bond. The entire benchmark curve is raised and lowered by the same amount, ∆Curve. The key point is that when benchmark yields are high, the effective durations of the callable and non-callable bonds are very similar. Although the exhibit does not illustrate it, the slopes of the lines tan- gent to the price–yield curve are about the same in such a situation. But when interest rates are low, the effective duration of the callable bond is lower than that of the otherwise com- parable non-callable bond. That is because the callable bond price does not increase as much when benchmark yields fall. The slope of the line tangent to the price–yield curve would be flatter. The presence of the call option limits price appreciation. Therefore, an embedded call option reduces the effective duration of the bond, especially when interest rates are falling and the bond is more likely to be called. The lower effective duration can also be interpreted as a shorter expected life—the weighted average of time to receipt of cash flow is reduced. Exhibit 9 considers another embedded option—a put option. 230 Fixed Income Analysis EXHIBIT 9 Interest Rate Risk Characteristics of a Putable Bond Price Putable Bond Value of the Embedded Put Option PV– PV0 PV+ Non-Putable Bond Δ Curve Δ Curve Benchmark Yield A putable bond allows the investor to sell the bond back to the issuer prior to maturity, usually at par value, which protects the investor from higher benchmark yields or credit spreads that otherwise would drive the bond to a discounted price. Therefore, the price of a putable bond is always higher than that of an otherwise comparable non-putable bond. The price difference is the value of the embedded put option. An embedded put option reduces the effective duration of the bond, especially when rates are rising. If interest rates are low compared with the coupon rate, the value of the put option is low and the impact of a change in the benchmark yield on the bond’s price is very similar to the impact on the price of a non-putable bond. But when benchmark interest rates rise, the put option becomes more valuable to the investor. The ability to sell the bond at par value limits the price depreciation as rates rise. In summary, the presence of an embedded option reduces the sensitivity of the bond price to changes in the benchmark yield curve, assuming no change in credit risk. 3.5. Duration of a Bond Portfolio Similar to equities, bonds are typically held in a portfolio. There are two ways to calculate the duration of a bond portfolio: (1) the weighted average of time to receipt of the aggregatee cash flows, and (2) the weighted average of the individual bond durations that comprise the portfo- lio. The first method is the theoretically correct approach, but it is difficult to use in practice. The second method is commonly used by fixed-income portfolio managers, but it has its own limitations. The differences in these two methods to compute portfolio duration can be exam- ined with a numerical example. Suppose an investor holds the following portfolio of two zero-coupon bonds: Macaulay Modified Market Bond Maturity Price Yield Duration Duration Par Value Value Weight (X) 1 year 98.00 2.0408% 1 0.980 10,000,000 9,800,000 0.50 (Y) 30 years 9.80 8.0503% 30 27.765 100,000,000 9,800,000 0.50 Chapter 5 Understanding Fixed-Income Risk and Return 231 The prices are per 100 of par value. The yields-to-maturity are effective annual rates. The total market value for the portfolio is 19,600,000. The portfolio is evenly weighted in terms of mar- ket value between the two bonds. The first approach views the portfolio as a series of aggregated cash flows. Its cash flow yieldd is 7.8611%. A cash flow yield is the internal rate of return on a series of cash flows, usu- ally used on a complex security such as a mortgage-backed bond (using projected cash flows based on a model of prepayments as a result of refinancing) or a portfolio of fixed-rate bonds. It is the solution for r in the following equation. 10,000,000 0 0 100,000,000 19,600,000 = + 2 +⋯+ 29 + , r = 0.078611 (1 + r )1 (1 + r ) (1 + r ) (1 + r )30 The Macaulay duration of the portfolio in this approach is the weighted average of the times to receipt of aggregated cash flow. The cash flow yield is used to obtain the weights. This calculation is similar to Equation 1, and the portfolio duration is 16.2825.  1 × 10,000,000 + 30 × 100,000,000   (1.078611)1 (1.078611)30  MacDur =  = 16.2825  10,000,000 100,000,000  1+  (1.078611) (1.078611) 30  There are just two future cash flows in the portfolio—the redemption of principal on the two zero-coupon bonds. In more complex portfolios, a series of coupon and principal pay- ments may occur on some dates, with an aggregated cash flow composed of coupon interest on some bonds and principal on those that mature. The modified duration of the portfolio is the Macaulay duration divided by one plus the cash flow yield per period (here, the periodicity is 1). 16.2825 ModDur = = 15.0958 1.078611 The modified duration for the portfolio is 15.0958. That statistic indicates the percentage change in the market value given a change in the cash flow yield. If the cash flow yield increases or decreases by 100 bps, the market value of the portfolio is expected to decrease or increase by about 15.0958%. Although this approach is theoretically correct, it is difficult to use in practice. First, the cash flow yield is not commonly calculated for bond portfolios. Second, the amount and tim- ing of future coupon and principal payments are uncertain if the portfolio contains callable or putable bonds or floating-rate notes. Third, interest rate risk is usually expressed as a change in benchmark interest rates, not as a change in the cash flow yield. Fourth, the change in the cash flow yield is not necessarily the same amount as the change in the yields-to-maturity on the individual bonds. For instance, if the yields-to-maturity on the two zero-coupon bonds in this portfolio both increase or decrease by 10 bps, the cash flow yield increases or decreases by only 9.52 bps. In practice, the second approach to portfolio duration is commonly used. The Macaulay and modified durations for the portfolio are calculated as the weighted average of the statistics for the individual bonds. The shares of overall portfolio market value are the weights. This weighted average is an approximation of the theoretically correct portfolio duration, which is obtained using the first approach. This approximation becomes more accurate when the 232 Fixed Income Analysis differences in the yields-to-maturity on the bonds in the portfolio are smaller. When the yield curve is flat, the two approaches produce the same portfolio duration. Given the equal “50/50” weights in this simple numerical example, this version of port- folio duration is easily computed. Average Macaulay duration = (1 × 0.50) + (30 × 0.50) = 15.50 Average modified duration = (0.980 × 0.50) + (27.765 × 0.50) = 14.3725 Note that 0.980 = 1/1.020404 and 27.765 = 30/1.080503. An advantage of the second ap- proach is that callable bonds, putable bonds, and floating-rate notes can be included in the weighted average using the effective durations for these securities. The main advantage to the second approach is that it is easily used as a measure of inter- est rate risk. For instance, if the yields-to-maturity on the bonds in the portfolio increase by 100 bps, the estimated drop in the portfolio value is 14.3725%. However, this advantage also indicates a limitation: This measure of portfolio duration implicitly assumes a parallel shiftt in the yield curve. A parallel yield curve shift implies that all rates change by the same amount in the same direction. In reality, interest rate changes frequently result in a steeper or flatter yield curve. Yield volatility is discussed later in this chapter. EXAMPLE 11 An investment fund owns the following portfolio of three fixed-rate government bonds: Bond A Bond B Bond C Par value EUR25,000,000 EUR25,000,000 EUR50,000,000 Coupon rate 9% 11% 8% Time-to-maturity 6 years 8 years 12 years Yield-to-maturity 9.10% 9.38% 9.62% Market value EUR24,886,343 EUR27,243,887 EUR44,306,787 Macaulay duration 4.761 5.633 7.652 The total market value of the portfolio is EUR96,437,017. Each bond is on a cou- pon date so that there is no accrued interest. The market values are the full prices given the par value. Coupons are paid semiannually. The yields-to-maturity are stated on a semiannual bond basis, meaning an annual rate for a periodicity of 2. The Macaulay durations are annualized. 1. Calculate the average (annual) modified duration for the portfolio using the shares of market value as the weights. Chapter 5 Understanding Fixed-Income Risk and Return 233 2. Estimate the percentage loss in the portfolio’s market value if the (annual) yield-to-maturity on each bond goes up by 20 bps. Solution to 1: The average (annual) modified duration for the portfolio is 6.0495.      4.761 24,886,343   5.633 27,243,887   × + × + 0.0910 96,437,017   0.0938 96,437,017   1 +   1 +  2 2    7.652 44,306,787   × = 6.0495 0.0962 96,437,017   1 +  2 Note that the annual modified duration for each bond is the annual Macaulay du- ration, which is given, divided by one plus the yield-to-maturity per semiannual period. Solution to 2: The estimated decline in market value if each yield rises by 20 bps is 1.21%: −6.0495 × 0.0020 = −0.0121. 3.6. Money Duration of a Bond and the Price Value of a Basis Point Modified duration is a measure of the percentage price changee of a bond given a change in its yield-to-maturity. A related statistic is money duration. The money duration of a bond is a measure of the price changee in units of the currency in which the bond is denominated. The money duration can be stated per 100 of par value or in terms of the actual position size of the bond in the portfolio. In the United States, money duration is commonly called “dollar duration.” Money duration (MoneyDur) is calculated as the annual modified duration times the full VFulll) of the bond, including accrued interest. price (PV VFulll MoneyDur = AnnModDur × PV (10) The estimated change in the bond price in currency units is calculated using Equation 11, which is very similar to Equation 6. The difference is that for a given change in the annu- al yield-to-maturity (∆Yield), modified duration estimates the percentage price change and money duration estimates the change in currency units. VFull ≈ −MoneyDur × ∆Yield ∆PV (11) For a theoretical example of money duration, consider the 6% semiannual coupon payment bond that matures on 14 February 2027 and is priced to yield 6.00% for set- tlement on 11 April 2019. The full price of the bond is 100.940423 per 100 of par value, and the annual modified duration is 6.1268. Suppose that a Nairobi-based life 234 Fixed Income Analysis insurance company has a position in the bond for a par value of KES100,000,000. The market value of the investment is KES100,940,423. The money duration of this bond is KES618,441,784 (= 6.1268 × KES100,940,423). Therefore, if the yield-to-maturity rises by 100 bps—from 6.00% to 7.00%—the expected loss is approximately KES6,184,418 (= KES618,441,784 × 0.0100). On a percentage basis, that expected loss is approximately 6.1268%. The “convexity adjustment” introduced in the next section makes these esti- mates more accurate. Another version of money duration is the price value of a basis pointt (PVBP) for the bond. The PVBP is an estimate of the change in the full price given a 1 bp change in the yield-to-maturity. The PVBP can be calculated using a formula similar to that for the approx- imate modified duration. Equation 12 is the formula for the PVBP. PVBP = ( PV− ) − ( PV+ ) (12) 2 PV− and PV V+ are the full prices calculated by decreasing and increasing the yield-to-maturity by 1 bp. The PVBP is also called the “PV01,” standing for the “price value of an 01”or “pres- ent value of an 01,” where “01” means 1 bp. In the United States, it is commonly called the “DV01,” or the “dollar value of a 01.” A related statistic, sometimes called a “basis point value” (or BPV), is the money duration times 0.0001 (1 bp). For a numerical example of the PVBP calculation, consider the 2.875% semiannual cou- pon paymen