Sensitivity with Respect to the Yield Curve: Duration in a Stochastic Setting

Abstract

Bond duration in its basic deterministic meaning form is a concept well understood. Its meaning in the context of a yield curve on a stochastic path is less well developed. In this paper we extend the basic idea to a stochastic setting. More precisely, we introduce the concept of stochastic duration as a Malliavin derivative in the direction of a stochastic yield surface modeled by the Musiela equation. Further, using this concept we also propose a mathematical framework for the construction of immunization strategies (or delta hedges) of portfolios of interest rate securities with respect to the fluctuation of the whole yield surface.

Appendix: Macaulay Duration and Portfolio Immunization

1.1 A.1 Discrete Case

In Macaulay’s original concept duration was the weighted average by present value of the number of periods to maturity for a series of cash flows, typically those of interest and principal payments for a bond, normalized by the total present value [32]. For notation, let V be the present value (or price) of the bond, r>0 be the (constant) rate of interest, and n be the number of periods to maturity. The expression

is the closed form for the present value of an annuity in arrears for n periods at rate r, reflecting the typical payment scheme of a bond, e.g. a United States Treasury bond. Therefore the Macaulay duration d _Mac has the following definition for equally spaced cash flows of size C and return of principal P:

$$ d_{\mathrm{Mac}}:=\frac{C\sum_{k=1}^{n}k(1+r)^{-k}+nP(1+r)^{-n}}{C\sum_{k=1}^{n}(1+r)^{-k}+P(1+r)^{-n}} $$

or

(A.1)

In the simple case of single cash flow—a zero coupon bond—Macaulay duration reduces to the number of periods n to that payment, justifying the name.

Soon, however, practitioners began preferring a version of duration as the simple negative of the derivative of V with respect to r, dropping the factor (1+r). This version became known as the modified duration d _mod, with this definition:

(A.2)

Such redefinition provides the relationship

$$ d_{\mathrm{Mac}}=(1+r) \, d_{\text{mod}}\text{,} $$

so that the modified duration of a zero coupon bond is (1+r)n.

In ordinary parlance, either form of duration is stated as a positive number, e.g., “The duration of this bond is ten years”, as indicated. A rationale exists, however, for stating the duration as a negative number, reflecting the inverse relationship between changes in the level of interest and changes in price. Such versions, inverting the minus signs of (A.1) and (A.2), more typically appear in Taylor series expansions of bond price, and in more developed mathematical expositions. The latter approach is assumed in this paper.

1.2 A.2 Continuous Case

The continuous case is a straightforward extension of the discrete case. Let C, as previously, be the cash flow assigned to a single period, but consider it divided equally into j parts flowing at the ends of j equally spaced sub-periods. As well, consider the interest rate r as that assigned to the entire period, but let it be divided by j providing a sub-rate for compounding across the sub-periods. Then term of (A.1) then becomes

So, if

then (A.1) and (A.2), respectively, become

and

in the latter case because lim _j→∞(1+r/j)=1. So

$$ \widehat{d}_{\mathrm{Mac}}=\widehat{d}_{\text{mod}}\text{,} $$

(A.3)

justifying the use of the combined name continuous duration for both versions. As in the case of discrete Macaulay duration, in the simple case of a zero coupon bond continuous duration reduces to the number of periods n to that payment.

An alternative description of this result is that the modified duration is a continuous approximation to the Macaulay duration, or conversely, the Macaulay duration is a discrete approximation to the modified duration. As n→∞ with rn constant the two definitions merge.

It is stated without proof that the other common form of annuity timing, payments in advance, i.e., at the beginnings of the compounding periods rather than at the ends, results in the same continuous forms of (A.3).

1.3 A.3 Portfolio Immunization

An active part of portfolio management is the targeting of a specific duration. For example, a pension fund manager may wish to have a value certain at some future time t=T, starting at t=0 now. Consider two portfolios A and B, with respective durations d _A and d _B , and present values (prices) of v _A and v _B . If these portfolios are combined, then the new portfolio A+B has duration

$$ d_{A+B}=\frac{v_{A}}{v_{A}+v_{B}}d_{A}+\frac{v_{B}}{v_{A}+v_{B}}d_{B}. $$

If A be the portfolio to be immunized to desired duration d _A+B , then one can solve for v _B knowing all other quantities. Specifically,

$$ v_{B}=\frac{d_{A+B}-d_{A}}{d_{B}-d_{A+B}}\, v_{A}, $$

which may be positive or negative. If negative one can interpret the result as an amount proportioned to portfolio B to be sold from portfolio A to achieve the objective, or alternatively, the amount to sell short of portfolio B.

Bond immunization is a very big business. In recent years Japanese banking interests have been heavy buyers of 30-year United States Treasury Bond strips—having a duration of 30 years—in order to extend the durations of portfolios. The activity has been so significant as to keep the longest-term yields below those of somewhat shorter-term yields for extended periods of time, even in strongly positive yield curve environments otherwise.