ABSOLUTE VERSUS DURATION

MEASURING TERM STRUCTURE SHIFTS

M- ABSOLUTE VERSUS DURATION

Recall that duration is defined as the weighted average 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:

Duration was defined in Chapter 2 using the bond’s yield to maturity. Du- ration can be defined more generally using the entire term structure of in- terest rates, with the following weights in equation 4.14:

where, CF_tis the cash flow occurring at time t. Duration computed using the yield to maturity is often known as the Macaulay duration, while dura- tion computed using the entire term structure of interest rates, as in equa- tions 4.14 and 4.15, is known as the Fisher and Weil (1971) duration. For (4.15)

w CF

e

P CF

t e

t f s ds

t y t t

= t

∫











 =

 



( ) / ( )⋅

0

//P

(4.14) D t w_t

t t t t_N

= ×

=

∑

= 1

f s ds t t t

t '( ) . . / . / .

0

2 3

0 065 0 016 2 0 003 3 0

∫

^{= × + × − × + 00004 4}

0 065 0 008 0 001 0 0001

4

2 3

×

= × + × − × +

t

t t t

/

. . . . ××t⁴

f t f t f t

t t

'( ) ( ) ( )

( . . . .

= +

= + × − × +

∆

0 06 0 02 0 003 ² 0 00004 0 005 0 004 0 065 0 016 0 003

× 3 + − ×

= + × −

t t

t

) ( . . )

. . . ×× +t² 0 0004. ×t³

M-Absolute versus Duration 85

brevity, we will refer to both duration definitions as simply “duration,”

though in this chapter we will be using the latter definition.

Duration gives the planning horizon, at which the future value of a bond or a bond portfolio remains immunized from an instantaneous, paral- lel shift in the term structure of interest rates. By setting a bond portfolio’s duration to the desired planning horizon, the portfolio’s future value is im- munized against parallel term structure shifts. 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 the bond’s (or the bond portfolio’s) price:

Unlike duration, the M-absolute measure is specific to a given planning horizon. The M-absolute risk measure selects the bond that minimizes the M-absolute of the bond portfolio. For the special case, when planning hori- zon is equal to zero, the M-absolute converges to the duration of the bond.

To get more insight regarding the M-absolute risk measure, consider the lower bound on the change in the target future value of a bond portfolio

∆V_H=V_H'− V_H,given as follows:²

where V_H is the target future value of the bond portfolio at the planning horizon H,given as:

and V_H' is the realized future value of the bond portfolio at time H,given an instantaneous change in the forward rates from f(t) to f'|(t), and V₀ is the current price of the bond portfolio.

Equation 4.17 puts a lower bound on the change in the target future value of the bond portfolio, which is a function of a constant K₃ and the portfolio’s M-absolute. The term K₃ depends on the term structure move- ments and gives the maximum absolute deviation of the term structure of

V_H =V⁰×^exp_

∫

^0Hf s ds

( )

^_

(4.17)

∆V

V^H K M

H

≥ − 3× A

(4.16) M^A t H w_t

t t t tN

= − ×

=

∑

= 1

the initial forward rates from the term structure of the new forward rates.

Mathematically, K₃can be defined as follows:

for all tsuch that, 0≤t≤t_N.

The term K₃is outside of the control of a portfolio manager. A portfo- lio manager can control the portfolio’s M-absolute, however, by selecting a particular bond portfolio. 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, which implies that only this bond is completely immune from interest rate risk. An implicit condition required for the inequality 4.17 to hold is that the bond portfolio does not contain any short positions(see Appendix 4.1). The immunization objective of the M-absolute model is to select a bond portfolio that mini- mizes the portfolio’s M-absolute. We call this objective the M-absolute im- munization approach.

Both the duration model and the M-absolute model are single risk- measure models. An important difference between them arises from the nature of the stochastic processes assumed for the term structure move- ments. The difference between duration and M-absolute can be illustrated using two cases.

Case 1: The term structure of instantaneous forward rates experiences an instantaneous, infinitesimal, and parallel shift (i.e., slope, curva- ture, and other higher order shifts are not allowed). In this case, the 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 risk but not to a com- plete elimination of immunization risk except in certain trivial situ- ations.³ Hence, the performance of the duration model would be superior to that of the M-absolute model under the case of small parallel shifts.

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 im- munizing 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 (4.18) K₃=^Max

(

K₁^, K₂

)

^, where K^, ₁^≤∆f t

^{( )}

^≤K₂

M-Absolute versus Duration 87

may result in a “stochastic process risk” for the duration model.

The effects of the stochastic process risk are especially high for a

“barbell” portfolio as compared to a “bullet” portfolio.

Although the use of M-absolute does not entirely eliminate the risk of small and parallel interest rate shifts, it does offer enhanced protection against nonparallel shifts. Equation 4.17 gives the lower bound on the tar- get future value of the bond portfolio as a product of its M-absolute and the parameter K₃, which gives the maximum absolute deviation of the term structure of new forward rates from the term structure of initial forward rates. In general, the value of K₃depends on the shifts in the height, slope, curvature, and other relevant higher order term structure shape parameters.

The essential difference between the duration model and the M-absolute model can be summarized as follows.

The duration model completely immunizes against the height shifts but ignores the impact of slope, curvature, and other higher order term struc- ture shifts on the future target value of a bond portfolio. This characteristic allows the traditional duration model to be neutral toward selecting a bar- bell or a bullet portfolio. In contrast, the M-absolute model immunizes only partially against the height shifts, but it also reduces the immunization risk caused by the shifts in the slope, curvature, and all other term structure shape parameters by selecting a bond portfolio with cash flows clustered around its planning horizon date.

The relative desirability of the duration model or the M-absolute model depends on the nature of term structure shifts expected. If height shifts completely dominate the slope, curvature, and other higher order term structure shifts, then the duration model will outperform the M-absolute model. If, however, slope, curvature, and other higher order shifts are rela- tively significant—in comparison with the height shifts—then the M-absolute model may outperform the traditional duration model.

Example 4.4 The M-absolute of a bond portfolio is computed identically to the duration of a bond portfolio, except that the longevity of each cash flow is reduced by Hand then its absolute value is taken. For example, con- sider a bond portfolio Aconsisting 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_A =

(

^50%×^{2 0. years}

)

⁺

(

^{50%×3 0. years}

)

^{=2 5. yearss}

The M-absolute of this portfolio, however, would depend upon the in- vestor’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,

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,

Note that both bond portfolios have equal durations and, based upon duration alone, would appear to be equally risky. Portfolio A,however, of- fers generally superior immunization because its cash flows are closer to the horizon and therefore are less subjected to the effects of large and nonpar- allel term structure shifts. This difference in riskiness is captured by the M- absolute risk measures of the two portfolios. Note that the M-absolute of portfolio B is greater than the M-absolute of portfolio A:

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 Ais equal to 0.5 years for any value of investment horizon Hfrom two to three years. Ultimately, the usefulness of M-absolute must be re- solved empirically.

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, 31 annual coupon bonds are constructed with seven different maturi- ties (1, 2, 3, . . . , 7 years) and five different coupon values (6, 8, 10, 12, and 14 percent) for each maturity.⁴

For December 31, 1951, two different bond portfolios are con- structed corresponding to the duration strategy and the M-absolute strat- egy. Under the duration strategy, an infinite number of portfolios exist that would set the portfolio duration equal to the investment horizon H.To de- termine a unique portfolio, the following quadratic objective function is minimized:⁵

M^A_B=

(

^50%×^{1 0 2 5.} − ^.

)

⁺

(

^{50%×4 0 2 5. − .}

)

^{=1 5. years}

D_B=

(

^50%×^{1 0. years}

)

⁺

(

^{50%×4 0. years}

)

^{=2 5. yearss}

M_A^A=

(

^50%×^{2 0 2 5.} − ^.

)

⁺

(

^{50%×3 0 2 5. − .}

)

^{=0 5. years}

M-Absolute versus Duration 89

subject to:

where p_igives the weight of the ith bond in the bond portfolio, and D_ide- fines the duration of the ith bond. The objective function of the M-absolute strategy is to minimize the portfolio’s M-absolute:

subject to:

where M^A_idefines the M-absolute of the ith bond.

The planning horizon, H,is assumed to equal four years. The two port- folios are rebalanced on December 31 of each of the next three years (i.e., 1952, 1953, 1954) when annual coupons are received. At the end of the four- year horizon (i.e., December 31, 1955), the returns of the two bond portfo- lios are compared with the return on a hypothetical four-year zero-coupon bond (computed at the beginning of the planning horizon). The differences between the actual values and the target value are defined as deviations in the interest rate, risk-hedging performance. The immunization procedure is re- peated over 32 overlapping four-year periods: 1951 to 1955, 1952 to 1956, . . . , 1982 to 1986. Because interest rate volatility in the 1950s and 1960s was lower than in the 1970s and 1980s, and to test the robustness of these

p

p i J

i i

J

i

∑

= ⁼

≥ =

1

0 1 2

1

for all . . .

, , , ,

(4.20) Min p M_i ^A_i

i J

∑

=



 



1

p D H

p

p i

i i

i J

i i

J

i

=

=

∑

∑

=

=

≥ =

1

1

0 1 2

1

for all .

, , , . . , J

(4.19) Min p_i

i J

2

=1



∑

 



models against the possible nonstationarities in the stochastic processes for the term structure, results for these two periods are analyzed separately.

Table 4.1 reports the sum of absolute deviations of the M-absolute hedging strategy as a percentage of the sum of absolute deviations of the du- ration strategy, for the two separate time periods. The M-absolute strategy reduces the immunization risk inherent in the duration model by more than half in both 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, and other higher order term structure shape parameters.