Interest Rate Risk Modeling The Fixed Income Valuation Course Sanjay K. Nawalkha Gloria M. Soto Natalia A. Beliaeva
Interest t Rate Risk Modeling : The Fixed Income Valuation Course. Sanjay K. Nawalkha, Gloria M. Soto, Natalia K. Beliaeva, 2005, Wiley Finance. Chapter 4: M-Absolute and M-Square Risk Measures es Goals: Introduction to the two risk measures: M-Absolute and M-Square. Reconcile the two different views for measuring fixedincome securities risk. 2
Chapter 4: M-Absolute and M-Square Risk Measures Measuring Term Structure Shifts M-Absolute versus Duration M-Square versus Convexity Closed-Form Solutions for M-Square and M- Absolute 3
Chapter 4: M-Absolute and M-Square Risk Measures Measuring Term Structure Shifts M-Absolute versus Duration M-Square versus Convexity Closed-Form Solutions for M-Square and M- Absolute 4
Measuring Term Structure Shifts Reconsider the price of a bond with periodic coupon C paid k times a year, and face value F: C C C C F P = + + +... + + (4.1) e yt1 e yt2 e yt3 e ytn e ytn To derive general interest rate risk measures that allow large and non-parallel shifts in the term structure of interest rates the assumption of a single yield must be relaxed. 5
Measuring Term Structure Shifts As shown in chapter 3, the term structure of interest rates can be defined in two equivalent ways - the term structure of zero-coupon yields and the term structure of instantaneous forward rates. The following two sections consider the shifts in these two term structures of interest rates and show an equivalence relationship between them. 6
Measuring Term Structure Shifts Shifts in the Term Structure of Zero-Coupon Yields Shifts in Term Structure of Instantaneous Forward Rates 7
Measuring Term Structure Shifts Shifts in the Term Structure of Zero-Coupon Yields Shifts in Term Structure of Instantaneous Forward Rates 8
Shifts in the Term Structure of Zero-Coupon Yields The term structure of zero-coupon yields can be applied to price the bond given in equation (4.1) as follows: C C C C F = + + +... + + (4.2) e P y( t1) t1 y( t2) t2 y( t3) t3 y( tn) tn y( tn) tn e e e e where each cash flow is discounted by the zero-coupon yield y(t) corresponding to its maturity t. 9
Shifts in the Term Structure of Zero-Coupon Yields We assume a simple polynomial form for the term structure of zero-coupon yields, as follows: () 2 3 y t = A0 + A1 t + A2 t + A3 t +... + (4.3) where parameters A 0, A 1, A 2, and A 3, are the height, slope, curvature, and the rate of change of curvature (and so on) of the term structure. Though about 5 to 6 terms may be needed for adequately capturing the shape of the term structure of zero-coupon yields, the height, slope, and curvature parameters are the most important. 10
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.1 Example 4.1 Assume the following term structure shape: 2 3 () 0 1 2 3 y t = A + A t + A t + A t +... + = 0.06 + 0.01 t 0.001 t + 0.0001 t 2 3 Using this, the different maturity yields can be given as: y 2 3 ( ) 2 3 () 0 = 0.06 + 0.01 0 0.001 0 + 0.0001 0 = 0.06 = 6.00% = y 1 = 0.06 + 0.01 1 0.001 1 + 0.0001 1 = 0.0691= 6.91% A 0 11
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.1 Similarly, y y ( ) y( ) ( ) y( ) 2 =... = 7.68%, 3 =... = 8.37%, 4 =... = 9.04%, 5 =... = 9.75%. The above assumed parameters define a rising shape for the term structure of zero-coupon yields. By changing the values of the shape parameters, one can define different types of shapes (e.g., rising, falling, humped, etc.) for the term structure. t 12
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.1 We can price a $1,000 face value, 5-year, 10% annual coupon bond, by substituting these yields in the equation: 100 100 100 100 100 1000 P = + + + + + e e e e e e 0.0691 2 0.0768 3 0.0837 4 0.0904 5 0.0975 5 0.0975 = $1002 $1,002.1111 Now, let the term structure experience a non-infinitesimal and a non-parallel shift given as: ' ' ' 2 ' 3 ( ) = 0 + 1 + 2 + 3 +... + (4.4) y' t A A t A t A t... (4.4) 0 1 2 3 where, A 0 = A 0 + ΔA 0, A 1 = A 1 + ΔA 1, A 2 = A 2 + ΔA 2, A 3 = A 3 + ΔA 3, 13
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.1 Equation (4.4) can be rewritten as follows: y'( t) = y( t) + Δ y( t) (4.5) where, Δ y( () t = Δ A + ΔA t + ΔA t + ΔA t +... + (4.6) 2 3 0 1 2 3 Equation (4.6) defines the shift in the term structure of zero-coupon yields as a function of the changes in height, slope, curvature, and other parameters. 14
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 Reconsider the initial term structure of zero coupon yields as defined in Example 4.1. Assume that the central bank increases the short rate (y(0) may proxy for the short rate) by fifty basis points: ΔA 0 = 0.005; This increase in the short rate signals a slowing down of the economy to the bond traders. 15
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 Expecting slower growth and hence, a lower inflation in the future, the bond traders start buying the medium term bonds (for simplicity, in our case, one year is a short term, and three to five years is a medium term). This leads to a negative slope shift of twenty basis points per year: ΔA 1 = -0.002. 002 The changes in curvature and other higher-order parameters are zero. 16
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 The shift in the term structure is given as follows (see equation (4.6)): Δ yt () = 0.005005 0.002002 t The new term structure is given as follows: () y' t = yt ( ) +Δyt ( ) = + + + 2 3 (0.0606 0.0101 t 0.001001 t 0.00010001 t ) (0.005 005 0.002002 t) = 0.065 + 0.008 t 0.001 t + 0.0001 t 2 3 17
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 Substituting different maturity values in the above equations, the new zero-coupon yields and the changes in zero coupon yields can be easily computed. These are given as follows: y'(0) = 0.065, 065 Δ y'(0) = 0.005, 005 y'(1) = 0.0721, 0721 Δ y'(1) = 0.003, y'(2) = 0.0778, Δ y'(2) = 0.001, y'(3) = 0.0827, 0827 Δ y'(3) = 0.001, 001 y'(4) = 0.0874, 0874 Δ y'(4) = 0.003, y'(5) = 0.0925, Δ y'(5) = 0.005 18
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 The new bond price consistent with the new term structure of zero coupon yields equals $1019.84. Hence, the bond price change equals (the original bond price in Example 4.1 was $1002.11): 11) Δ P = $1019.84 $1002.11 11 = $17.7373 Though the central bank increased the instantaneous short rate by fifty basis points, the 5-year bond gained in value by approximately $17.73. 73 This happened because the long rates fell even as the short rates rose, leading to an overall increase in the bond price. 19
Shifts in the Term Structure of Zero-Coupon Yields-Example 4.2 This example demonstrates the inadequacy of using the simple duration model, which would have falsely predicted a decrease in the 5-year bond's price, given an increase in the short rate (since it would assume a parallel increase in all yields). The percentage increase in the bond price is: Δ PP= 17.73 1002.11= 1.769% 20
Measuring Term Structure Shifts Shifts in the Term Structure of Zero-Coupon Yields Shifts in Term Structure of Instantaneous Forward Rates 21
Shifts in Term Structure of Instantaneous Forward Rates In many instances, it is easier to work with instantaneous forward rates as certain interest rate risk measures and fixed income derivatives are easier to model using forward rates. Though any shift in the term structure of zero coupon yields corresponds to a unique shift in the term structure of instantaneous forward rates, the instantaneous forward rates are a lot more volatile than the zerocoupon yields, especially at the long end of the maturity spectrum. 22
Shifts in Term Structure of Instantaneous Forward Rates The relationship between the term structure of zero- coupon yields and the term structure of instantaneous forward rates can be given as follows: t yt () t= fsds ( ) (4.7) 0 where y(t) is the zero-coupon-yield for term t, and f(t) is the instantaneous forward rate for term t (which is the same as the forward rate that can be locked-in at time zero for an infinitesimally small interval t to t + dt) 23
Shifts in Term Structure of Instantaneous Forward Rates It is also possible to obtain the term structure of instantaneous forward rates, given the term structure of zero-coupon yields, by taking the derivative of both sides of equation (4.7) as follows: ft () = t yt ()/ t + yt () (48) (4.8) If the term structure of zero-coupon yields is rising (falling), then y(t)/ t>0 (<0), and instantaneous forward rates will be higher (lower) than zero-coupon yields. 24
Shifts in Term Structure of Instantaneous Forward Rates Equation (4.7) can be used to express the bond price in equation (4.2) as follows: P C C C C F = 1 + 2 + 3 +... + N + N (4.8) f( s) ds f( s) ds f( s) ds f( s) ds f( s) ds 0 0 0 0 0 e e e e e To compute the bond price, we need an expression for the forward rates. 25
Shifts in Term Structure of Instantaneous Forward Rates Substituting g( (4.3) into (4.8), and taking the derivative of y(t) with respect to t, the instantaneous forward rate can be given as: () 2 f t = t( A + 2A t + 3 A t +... + ) or 1 2 3 + ( A + A t + A t + A t +... + ) (4.10) 2 3 0 1 2 3 ( ) 2 3 f t = A + 2A t + 3A t + 4 A t +... + (4.11) 0 1 2 3 Substituting the forward rates given in (4.11) in (4.9) and computing the respective integrals, the bond price can be obtained using the term structure of instantaneous forward rates. 26
Shifts in Term Structure of Instantaneous Forward Rates Both the term structure of zero-coupon yields and the term structure of instantaneous forward rates have the same height, but the term structure of forward rates has twice the slope, and thrice the curvature (and four times the rate of change of curvature, and so on) of the term structure of zero coupon yields. This makes the term structure of forward rates more volatile, especially for longer maturities. 27
Shifts in Term Structure of Instantaneous Forward Rates Using g( (4.11), the shift in the term structure of instantaneous forward rates can be given as follows: Δ ft ( ) = Δ A+ 2ΔA t+ 3ΔA t + 4 ΔA t +... + (4.12) 2 3 0 1 2 3 where the new term structure is given as: f '( t) = f( t) + Δ f( t) (4.13) 28
Shifts in Term Structure of Instantaneous Forward Rates-Example 4.3 Example 4.3 This example demonstrates how instantaneous forward rates are used in practice. In this example we price the $1,000 face value, 5-year, 10% annual coupon bond using the instantaneous forward rates with the same term structure parameter values given in Example 4.1. Then, we obtain the new price of this bond using the instantaneous forward rates corresponding to the new parameter values given in Example 4.2. 29
Shifts in Term Structure of Instantaneous Forward Rates-Example 4.3 Obviously, we expect to find the same prices as in Example 4.1 and Example 4.2, which used zero-coupon yields. Substituting the term structure parameter values from Example 4.1 into (4.11): 2 3 ( ) = 0.0606 + 0.0202 0.003003 + 0.00040004 f t t t t t fsds t t t t 0 2 3 4 ( ) = 0.06 + 0.02 / 2 0.003 / 3 + 0.0004 / 4 = 0.0606 t + 0.01 01 t 0.001001 t + 0.00010001 t 2 3 4 30
Shifts in Term Structure of Instantaneous Forward Rates-Example 4.3 Using the above equation, the following five integrals can be computed: 1 2 3 4 5 fsds ( ), fsds ( ), fsds ( ), fsds ( ), fsds ( ) 0 0 0 0 0 Substituting the results in equation (4.9): $100 $100 $100 $100 $100 $1000 P = + + + + + = e 0.0691 e 0.1536 e 0.2511 e 0.3616 e 0.4875 e 0.4875 $1002.11 The bond price is the same as Example pe 4.1. 31
Shifts in Term Structure of Instantaneous Forward Rates-Example 4.3 Similarly, we can obtain the new bond price using the new instantaneous forward rates. Substituting the values of ΔA 0 = 0.0050, ΔA 1 = -0.0020, ΔA 2 = 0.0, and ΔA 3 = 00 0.0, from Example 42i 4.2 into (412) (4.12) we get: Δ ft () = 0.005005 0.004004 t The new term structure of instantaneous forward rates is given as: f'( t) = ft ( ) +Δft ( ) = + + + 2 3 (0.06 0.02 t 0.003 t 0.0004 t ) (0.005 0.004 t) = 0.065 + 0.016 t 0.003 t + 0.0004 t 2 3 32
Shifts in Term Structure of Instantaneous Forward Rates-Example 4.3 The above equation can again be used to compute the integral: t f s ds t t t t 0 2 3 4 '( ) = 0.065 + 0.016 / 2 0.003 / 3 + 0.0004 / 4 = 0.065 t + 0.008 t 0.001 t + 0.0001 t 2 3 4 Using the above equation, the five integrals can be computed for t = 1, 2,3,4, and 5. Substituting these integrals in the bond price equation, the new price of the bond can be obtained as $1019.84, which h is identical to the new price given in Example 4.2 using zero-coupon yields. 33
Chapter 4: M-Absolute and M-Square Risk Measures Measuring Term Structure Shifts M-Absolute versus Duration M-Square versus Convexity Closed-Form Solutions for M-Square and M- Absolute 34
M-Absolute versus Duration Recall that duration is defined as the weighted-average g of the maturities of the cash flows of a bond, where weights are the present values of the cash flows, given as proportions of the bond s price: D t= t N t= t = 1 t w t (4.14) Duration can be defined more generally using the entire term structure of interest rates, with the following weights in the above equation: CF CF w t t t = / P t = / P (4.15) y( t) t f( s) ds e 0 e 35
M-Absolute versus Duration Duration computed using the yield-to- maturity is often known as the Macaulay duration, while duration computed using the entire term structure of interest rates (as in equations and ) is known as the Fisher and Weil [1971] duration. For brevity, we will refer to both duration definitions as simply duration duration, though in this chapter we will be using the latter definition. 36
M-Absolute versus Duration Duration: the planning horizon, at which the future value of a bond or a bond portfolio remains immunized from an instantaneous, parallel shift in the term structure of interest t rates. By setting a bond portfolio s duration to the desired planning horizon, the portfolio s future value is immunized against parallel term structure shifts. 37
M-Absolute versus Duration The M-absolute risk measure is defined as the weighted average of the absolute differences between cash flow maturities and the planning horizon, where weights are the present values of the bond s (or a bond portfolio s) cash flows, given as proportions of bond s (or the bond portfolio s) price: t= t N A M = t H w t t= t 1 (4.16) Unlike duration, the M-absolute measure is specific to a given planning horizon. The M-absolute risk measure selects ects the bond that minimizes the M-absolute of the bond portfolio. 38
M-Absolute versus Duration Consider the lower bound on the change in the target future value of a bond portfolio (no short position, proof is in Appendix A): ΔV V V H H K A 3 M (4.17) where V H is the target future value of the bond portfolio at the planning horizon H, given as: V 0 exp H H = V f ( s) ds 0 Set V H as the realized future value of the bond portfolio at time H, given an instantaneous change in the forward rates, and V 0 is the current price of the bond portfolio. 39
M-Absolute versus Duration In (4.17), K 3 depends on the term structure movements and gives the maximum absolute deviation of the term structure of the initial forward rates from the term structure t of the new forward rates. Mathematically, ti ( ) ( ) K = Max K, K, where, K Δ f t K, (4.18) 3 1 2 1 2 for all t such that, 0 t t N. 40
M-Absolute versus Duration A portfolio manager can control the portfolio s M- absolute but not K 3. The smaller the magnitude of M-absolute, the lower the immunization risk exposure of the bond portfolio. Only a zero-coupon bond maturing at horizon H has zero M-absolute. The immunization objective of the M-absolute model is to select a bond portfolio that minimizes the portfolio s M- absolute. We call this objective the M-absolute immunization approach. 41
M-Absolute versus Duration An important difference between duration and M- absolute arises from the nature of the stochastic processes assumed for the term structure movements. The difference can be illustrated t using two cases. Case 1. The term structure of instantaneous forward rates experiences an instantaneous, infinitesimal, and parallel shift (i.e., slope, curvature, and other higher order shifts are not allowed). 42
M-Absolute versus Duration Under this case, the duration model leads to a perfect immunization performance (with duration equal to the planning horizon date). In contrast, the M-absolute model leads to a reduction in immunization i risk but not to a complete elimination of immunization risk except in certain trivial situations. Hence, the performance of the duration model would be superior to that of the M- absolute model under the case of small parallel shifts. (4.17) gives the maximum absolute deviation of the term structure of new forward rates from the term structure of initial forward rates. 43
M-Absolute versus Duration Case 2. The term structure of instantaneous forward rates experiences a general shift in the height, slope, curvature, and other higher order term structure shape parameters, possibly including large shifts. Because the traditional duration model focuses on immunizing against small and parallel shifts in the term structure of instantaneous forward rates, the presence of shifts in the slope, curvature, and other higher order term structure shape parameters may result in a stochastic process risk for the duration model. 44
M-Absolute versus Duration Summary: The duration model completely immunizes against the height ht shifts but ignores the impact of slope, curvature, and other higher order term structure shifts on the future target value of a bond portfolio. The relative desirability of the duration model or the M- absolute model depends on the nature of term structure shifts expected. 45
M-Absolute versus Duration-Example 4.4 The M-absolute of a bond portfolio is computed identically to the duration of a bond portfolio, except that t the longevity of each cash flow is reduced by H and then its absolute value is taken. Example 4.44 Consider a bond portfolio A consisting of equal investments in two zero-coupon bonds maturing in two years and three years, respectively. The duration of this portfolio would be equal to 2.5 years; that is, ( ) ( ) D = 50% 2.0 years + 50% 3.0 years = 2.5 years. A 46
M-Absolute versus Duration-Example 4.4 The M-absolute of this portfolio, however, would depend upon the investor s time horizon. For an investor with a time horizon of 2.5 years, the portfolio M-absolute would be equal to 0.5; that is, ( ) ( ) M = 50% 2.0 2.5 + 50% 3.0 2.5 = 0.5 years. A A Now, consider bond portfolio B, consisting of equal investments in two zero-coupon bonds maturing in one year and four years, respectively. The duration of this portfolio would also be equal to 2.5 years; that is, ( ) ( ) D = 50% 1.0 years + 50% 4.0 years = 2.5 years B 47
M-Absolute versus Duration-Example 4.4 Note that both bond portfolios have equal durations and, based upon duration alone, would appear to be equally risky. Portfolio A, however, offers generally superior immunization because its cash flows are closer to the horizon and therefore are less subjected to the effects of large and non-parallel term structure shifts. 48
M-Absolute versus Duration-Example 4.4 This difference in risk is captured by the M-absolute risk measures of the two portfolios. Note that t the M-absolute of portfolio B is greater than the M-absolute of portfolio A: ( ) ( ) M = 50% 1.0 2.5 + 50% 4.0 2.5 = 1.5 years. A B Because M-absolute is a single-risk-measure model, it does not generally provide perfect interest rate risk protection. For example, the M-absolute of portfolio A is equal to 0.5 years for any value of investment e horizon o H from two to three years. Ultimately, the usefulness of M- absolute must be resolved empirically. 49
M-Absolute versus Duration-Example 4.4 Nawalkha and Chambers [1996] test the M-absolute risk measure against the duration risk measure using McCulloch s term structure data over the observation period 1951 through 1986. On December 31 of each year, thirty-one annual coupon bonds are constructed with seven different maturities (1, 2, 3,, 7 years) and five different coupon values (6%, 8%, 10%, 12%, 14%) for each maturity. 50
M-Absolute versus Duration-Example 4.4 For December 31, 1951, two different bond portfolios are constructed t corresponding to the duration strategy t and the M-absolute strategy. Under the duration strategy, an infinite number of portfolios exist that would set the portfolio duration equal to the investment horizon H. 51
M-Absolute versus Duration-Example 4.4 In order to determine a unique portfolio, the following quadratic objective function is minimized: i i J 2 Min pi (4.19) i = 1 where J J i i i i K i= 1 i= 1 p D = H, p =1, p 0, for all i = 1,2,,, J where p i gives the weight of the ith bond in the bond portfolio, and D i defines the duration of the ith bond. 52
M-Absolute versus Duration-Example 4.4 The objective function of the M-Absolute strategy is to minimize i i the portfolio s M-absolute: J A Min pm i i (4.20) i = 1 st.. J i =11 p =1, p 0, for all i = 1,2, K, J i i where M A i defines the M-absolute of the ith bond. 53
M-Absolute versus Duration-Example 4.4 The planning horizon, H, is assumed to equal four years. The two portfolios are rebalanced on December 31 of each of the next three years when annual coupons are received. At the end of the four-year horizon, the returns of the two bond portfolios are compared with the return on a hypothetical four-year zero-coupon bond (computed at the beginning of the planning horizon). The immunization procedure is repeated over 32 overlapping four-year periods: 1951 55, 1952 56,, 1982 86. 86 54
M-Absolute versus Duration-Example 4.4 55
M-Absolute versus Duration-Example 4.4 Table 4.1 reports the sum of absolute deviations. The M-absolute strategy reduces the immunization risk inherent in the duration model by more than half in both the time periods. This finding implies that the changes in the height of the term structure of instantaneous forward rates must be accompanied by significant changes in the slope, curvature, u and other higher order term structure u shape parameters. 56
Chapter 4: M-Absolute and M-Square Risk Measures Measuring Term Structure Shifts M-Absolute versus Duration M-Square versus Convexity Closed-Form Solutions for M-Square and M- Absolute 57
M-Square versus Convexity Recall that convexity is defined as the weighted-average of fthe maturity-squares of fthe cash hflows of a bond, where weights are the present values of the cash flows, given as proportions of the bond s price: t= t N 2 CON = t wt t= t 1 (4.21) The weights are defined using the entire term structure of interest rates as in (4.15). Convexity measures the gain in a bond s (or a bond portfolio s) o o value due to the second order effect of a large and parallel shift in the term structure of interest rates. 58
M-Square versus Convexity The M-square risk measure is defined as the weighted average of fthe squared ddifferences between cash hflow maturities and the planning horizon, where weights are the present values of the bond s (or a bond portfolio s) cash flows, given as proportions of bond s (or the bond portfolio s) price: 2 t= t N 2 t= t M = ( t H ) wt (4.22) 1 Unlike convexity, the M-square measure is specific to a given planning horizon. 59
M-Square versus Convexity The M-square model selects the bond portfolio that minimizes i i the M-square of fthe bond portfolio, subject tto the duration constraint (i.e., duration = planning horizon H). For the special case, when H=0, the M-square converges to the convexity of the bond. To get more insight regarding the M-square risk measure, consider the following inequality (no short position). Δ V H V H 1 D H Δf H K M 2 2 ( ) ( ) 4 (4.23) where V H is the target future value of the bond portfolio at the planning horizon H. 60
M-Square versus Convexity The M-square model is based upon two risk measures. If the portfolio is immunized with respect to duration (i.e., D = H), then (4.23) puts a lower bound on the target future value. The term K 4 is outside of the control of a portfolio manager. Mathematically, K 4 can be defined as follows: K4 [ Δf( t)]/ t, (4.24) for all t such that, 0 t t N. 61
M-Square versus Convexity The smaller the magnitude of M-square, the lower the risk exposure of the bond portfolio. Only a zero-coupon bond maturing at horizon H has a zero M-square, which implies that only this bond is completely immune from interest rate risk. The immunization objective of the M-square model is to select a bond portfolio that minimizes the portfolio s M- square, subject to the duration constraint D = H. 62
M-Square versus Convexity A linear relationship exists between M-square and convexity, given as follows: M 2 = CON 2 D H + H 2 (4.25) If duration is kept constant, then M-square is an increasing function of convexity. This leads to the well-known convexity-m-square paradox: as shown in (3.18) in chapter 3, higher convexity is beneficial since it leads to higher returns. On the other hand, (4.23) suggests that M-square should be minimized, in order to minimize immunization risk. 63
M-Square versus Convexity This convexity-m-square paradox can be resolved by noting that t the convexity view assumes parallel l term structure shifts, while the M-square view assumes non- parallel term structure shifts. Which view is valid, depends upon the extent of the violation of the parallel term structure shift assumption. Actually, the convexity view is not consistent with bond market equilibrium, while the M-square view is consistent with equilibrium conditions, as it requires no specific assumptions regarding the shape of these shifts. 64
M-Square versus Convexity In the following subsection we provide a unified framework, which h allows both the convexity view and the M-square view as special cases of the general framework. Then we show the empirical relationship between bond convexity (which is linear in M-square) and ex ante bond returns. We also investigate whether higher-convexity portfolios lead to higher immunization risk. 65
M-Square versus Convexity Resolving the Convexity/M-square Paradox Convexity, M-Square, and Ex-Ante Returns Convexity, M-Square, and Immunization Risk 66
M-Square versus Convexity Resolving the Convexity/M-square Paradox Convexity, M-Square, and Ex-Ante Returns Convexity, M-Square, and Immunization Risk 67
Resolving the Convexity/M-square Paradox Fong and Fabozzi [1985] and Lacey and Nawalkha [1993] suggest an alternative ti two-term t Taylor-seriesexpansion approach to the M-square model, which leads to a generalized framework for resolving the convexity/m-square paradox. Consider a bond portfolio at time t = 0 that offers the amount C t at time t=t 1 1, t 2,...,t, N. The return R(H) on this portfolio between t=0 and t=h (an investment horizon) can be given as: 68
Resolving the Convexity/M-square Paradox ( ) R H = [ V ' V ] H V 0 0 (4.26) V H is the realized future value of the portfolio at the planning horizon H after the term structure of forward rates shifts to f (t), and V 0 is the current value of the portfolio using the current term structure f(t). Using a two-term Taylor series expansion, (4.26) can be simplified as: 2 ( ) ( ) [ ] R H = R F H + γ 1 D H + γ 2 M + ε (4.27) 69
Resolving the Convexity/M-square Paradox R F (H) is the riskless-return on any default-free zero- coupon bond with maturity H given as: H R F ( H) = exp f ( s) ds 1 (4.28) 0 The duration coefficient in (4.27) is defined as follows: γ 1 = Δ fh ( ) (1 + RF ( H )) (4.29) 70
Resolving the Convexity/M-square Paradox And the M-square coefficient in (4.27) is defined as a difference of two effects as follows: γ 2 = CE RE (4.30) 1 2 CE = Convexity Effect = ( 1 + R ( H) )( Δ f ( H) ) F 2 (4.31) ( ft ()) 1 RE = Risk Effect = ( 1 + R ( )) Δ F H (4.32) 2 t t= H 71
Resolving the Convexity/M-square Paradox CE is positive for any term structure shift such that an increase in convexity (or increase in M-square) enhances return regardless of the direction of the shift. This demonstrates the traditional view of convexity. RE can be either positive or negative, depending upon whether the instantaneous forward rate at the planning horizon H experiences a positive or a negative slope shift. 72
Resolving the Convexity/M-square Paradox A positive slope shift will decrease the value of the M- square coefficient i (see (4.30)) such that t a higher-m-square portfolio (or higher convexity portfolio) will result in a decline in portfolio return. A negative slope shift will increase the value of the M- square coefficient such that a higher-m-square portfolio (or a higher convexity portfolio) will result in an increase in portfolio return. The convexity view assumes an insignificant risk effect (i.e., parallel shifts) such that only the convexity effect matters. Conversely, the equilibrium-consistent i i M- square view assumes an insignificant convexity effect such that only the risk effect matters. 73
Resolving the Convexity/M-square Paradox- Example 4.5 Example 4.5 Reconsider the $1,000 face value, 5-year, 10% annual coupon bond that was priced in Example 4.3. In this example, the initial term structure of instantaneous forward rates is given as: 2 3 () = 0.06 + 0.02 0.003 + 0.0004 f t t t t The instantaneous shift in the forward rates and the new term structure of forward rates were given in that example as: Δ ft ( ) = 0.005 0.004 t f '( t) = 0.065 + 0.016 t 0.003 t + 0.0004 t 2 3 74
Resolving the Convexity/M-square Paradox- Example 4.5 Consider the instantaneous return on this bond (at H = 0) using equation. Substituting H = 0, (4.27) simplifies to the following equation: V0' V0 R( 0 ) = = D Δ f(0) + CON [ CE RE] + ε V V 0 The convexity effect and the risk effect are given as follows: 1 2 1 2 CE = Convexity Effect = ( Δ f ( 0) ) = ( 0.005 ) = 0.0000125 2 2 1 ( Δft ()) 1 RE = Risk Effect = = [ 0.004 ] = 0.002 2 t 2 t = 0 75
Resolving the Convexity/M-square Paradox- Example 4.5 The M-square (or convexity) coefficient is equal to CE RE = 0.0020125. The risk effect completely dominates the convexity effect in this example. This could be because that the slope change is high (negative 40 basis points per year). However, even if slope change were only 1 basis point per year, the RE would still be equal to (1/2) [0.0001] 0001] = 0.00005, which is four times the convexity effect of 0.00001250000125 produced by 50 basis points shift in the height of the term structure. 76
Resolving the Convexity/M-square Paradox- Example 4.5 It suggests that even very small changes in the slope (or curvature, etc.) of the term structure of forward rates can violate the assumption of parallel term structure shifts sufficiently, such that the risk effect dominates the convexity effect. From Example 4.3, the instantaneous return on the bond can be given as: R V ' V 1019.84 1002.11 17.73 0 = 1.769% V = 1002.11 = 1002.11 = 0 0 ( ) 0 77
Resolving the Convexity/M-square Paradox- Example 4.5 Approximating this return using the two-term term Taylor series expansion given above we get: V0' V0 R( 0) = = D 0.005 + CON 0.0020125 + ε V 0 1 100 2 100 3 100 4 100 5 100 5 1000 + + + + + 0.0691 0.1536 0.2511 0.3616 0.4875 0.4875 D = e e e e e e = 4.146 1002.11 2 2 2 2 2 2 1 100 2 100 3 100 4 100 5 100 5 1000 + + + + + 0.0691 0.1536 0.2511 0.3616 0.4875 0.4875 CON = e e e e e e $1002.1111 = 19.1 78
Resolving the Convexity/M-square Paradox- Example 4.5 Substituting duration and convexity in the Taylor series expansion above, we get: R R V ' V 0 = = 4.146 0.005 + 19.1 0.0020125 + ε V ( ) 0 0 ( ) 0 0 1.771% The approximation of 1.771% is extremely close to the actual return of 1.769% (the difference equals 0.002%). The approximation is good because it considers the risk effect consistent with the M-square view. 79
Resolving the Convexity/M-square Paradox- Example 4.5 Suppose, we assumed perfect parallel shifts consistent with the convexity view, instead. Then the risk effect would be assumed to be zero, and the approximation would be given as follows: R R V ' V 0 = = 4.146 0.005 + 19.1 0.0000125 + ε V ( ) 0 0 ( ) 0 0 2.049% The return of 2.049% is consistent t with the convexity view outlined in the previous chapter, and is very different from the actual return of 1.769% (the difference equals -3.818%). 80
M-Square versus Convexity Resolving the Convexity/M-square Paradox Convexity, M-Square, and Ex-Ante Returns Convexity, M-Square, and Immunization Risk 81
Convexity, M-Square, and Ex-Ante Returns Lacey and Nawalkha [1993] test a modified version of (4.27) in order to empirically distinguish between the risk effect (caused by slope changes) and the convexity effect (caused by second-order effect of height changes). The equation tested is obtained from (4.27) by substituting the linear relationship between M-square and convexity (given in 4.25) as follows: RH ( ) R( H) = β + β D+ γ CON+ ε (4.33) where F 0 1 2 β = γ H + γ H and β = γ 2 γ H (4.34) 2 0 1 2 1 1 2 82
Convexity, M-Square, and Ex-Ante Returns An ex-ante version of (4.33) is tested with CRSP government bond data using pooled cross-sectional sectional time-series regressions of two-month excess holding period returns of U.S. Treasury bonds on their duration and convexity measures over the period January 1976 through November 1987. The results of these tests over the whole sample period as well as over selected five-year periods are reported in Table 4.2. 83
Convexity, M-Square, and Ex-Ante Returns Table 4.2 The convexity coefficient is negative in all eight sub-periods, and is negative and statistically ti ti significant over two of these periods. High positive- convexity is not associated with positive excess returns, a conclusion that rejects the convexity view. 84
Convexity, M-Square, and Ex-Ante Returns However, the statistically significant negative values of the convexity coefficient over two sub-periods provide some evidence that convexity is priced, and that increasing the level of positive convexity reduces the ex- ante excess return on a bond portfolio. The negative relationship between the convexity exposure and excess holding-period bond returns is not inconsistent with the results of Fama [1984], which imply a positive slope, but a negative curvature for the term structure of excess holding-period returns. 85
M-Square versus Convexity Resolving the Convexity/M-square Paradox Convexity, M-Square, and Ex-Ante Returns Convexity, M-Square, and Immunization Risk 86
Convexity, M-Square, and Immunization Risk This section shows that holding duration constant, and increasing the absolute size of convexity (or M-square) leads to higher immunization risk for bond portfolios. The excess returns are defined as the portfolio s duration- immunized return less the riskless return. Table 4.3 reports the standard deviation of the excess holding period returns for portfolios with different levels of convexity exposures, for the full sample period and the two sub-periods 1976 through 1981 and 1982 through 1987. Figure 4.1 demonstrates these standard deviations graphically. 87
Convexity, M-Square, and Immunization Risk Table 4.3 A clear relationship is shown between portfolio convexity and immunization risk. These results demonstrate that the magnitude of convexity exposure increases immunization risk. 88
Convexity, M-Square, and Immunization Risk Figure 4.1 Relationship between convexity and risk 89
Chapter 4: M-Absolute and M-Square Risk Measures Measuring Term Structure Shifts M-Absolute versus Duration M-Square versus Convexity Closed-Form Solutions for M-Square and M- Absolute 90
Closed-Form Solutions for M-Square and M- Absolute In this section we present closed-form solutions of M- square and M-absolute. The closed-formulas are valid both at coupon payment dates and between coupon payment dates. The expression for M-square of a security between cash flow payment dates, with cash flows paid k times a year at regular intervals, is given as: M = N ( ) i ( j s ) 2 j s H CF 2 j = 1 2 N e CF j ( ) i j s j 1 e = j / k (4.35) 91
Closed-Form Solutions for M-Square and M- Absolute where i =y/k is the continuously-compoundedcompounded yield of the bond divided by k, CF i is the ith the cash flow payment (i = 1,2,,N), N is the total number of cash flows, s is the time elapsed since the date of last cash flow payment in the units of time interval between coupon payment dates, and H is the planning horizon in the units of time interval between coupon payment dates. At the coupon payment dates, s =0 0. 92
Closed-Form Solutions for M-Square and M- Absolute The M-square formula in (4.35) applies to all fixed income securities with fixed regular cash flows, such as bonds, annuities, and perpetuities. The division of the bracketed expressions by k 2 on the right hand side of (4.35) gives the M-square value in annualized units. Even though the derivation of M- square risk measure uses the whole forward rate curve, the formula given above uses the continuouslycompounded yield per period of the security. It has been shown that the error by using a single yield versus the whole yield curve in calculating the M-square is insignificant for portfolio immunization and other hedging applications. i 93
Closed-Form Solutions for M-Square and M- Absolute Simplifying (4.35) gives: ( ) 2 N j s CF N j CF j i( j s) i( j s) j= 1 e j= 1 e N ( j scf N 2 ) j CFj 2 M = 2 H / k (4.36) i( j s) i( j s) j= 1 e j= 1e N N 2 CFj CFj + H i( j s) i( j s) j= 1e j= 1e Alternatively, 2 2 M = CON 2 D H + H (4.37) 94
Closed-Form Solutions for M-Square and M- Absolute (4.37) expresses M-square in terms of duration and convexity. Hence, substituting the closed-form solutions of D and CON obtained in chapter 2 into (4.37) gives the closed-form formula of M-square. Since chapter 2 obtained the formulas of D and CON, both between coupon payment dates and at coupon payment tdates, for three types of securities (i.e., regular bonds, annuities, and perpetuities), appropriate substitutions of D and CON in (4.37) lead to the corresponding M-square formulas. 95
Closed-Form Solutions for M-Square and M- Absolute The M-absolute in annualized unit is given in the summation form as: M A = N j s H CF i ( j s ) j = 1 N e CF j i ( j s ) j = 1 e j / k (4.38) 96
Closed-Form Solutions for M-Square and M- Absolute A closed-form formula for M-absolute between cash flow payment dates is given as follows: A M = (4.39) e L CF 1 1 I D H I H s k e 1 e 1 ( e 1) e e P i 1 B ( ) A + 2 + + / i Li i is ( L s ) i P = price of the underlying security with the accrued interest (the security may be a regular fixed-coupon bond, an annuity, or a perpetuity), D = duration of the underlying security, CF 1 = first cash flow of the underlying security, 97
Closed-Form Solutions for M-Square and M- Absolute L = INT(H+s), INT(x) = integer function defined as the closest integer less than or equal to x, and IA and IB are indicator functions defined as: I A 1 if H + s < N = 1 if H + s N I B 1 if 1< H + s < N = 0 else By substituting the appropriate formulas into (4.39), the M-absolute of a regular bond, or an annuity, or a perpetuity, can be calculated, easily. 98
Closed-Form Solutions for M-Square and M- Absolute-Example 4.6 Example 4.6 Consider twelve bonds, all of which have a $1,000 face value and a 10% annual coupon rate, but different maturities ranging from 1 year to 3.75 years in increments of 0.25 years. Assume that the yields to maturity of all bonds are identical and equal 5% and the planning horizon is set to 2 years. Table 4.44 gives the M- square and M-absolute of the nine bonds obtained by applying the closed-form solutions presented above. All the figures are expressed in years. 99
Closed-Form Solutions for M-Square and M- Absolute-Example 4.6 Table 4.44 M-square and M-absolute for all twelve bonds 100
Interest Rate Risk Modeling The Fixed Income Valuation Course Sanjay K. Nawalkha Gloria M. Soto Natalia A. Beliaeva