Joel R. Barber

Mark L. Copper

Insurance Markets and Companies

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“The measurement and control of interest rate risk”AUTHORSJoel R. BarberMark L. CopperARTICLE INFOJoel R. Barber and Mark L. Copper (2010). The measurement and control ofinterest rate risk. Insurance Markets and Companies, 1(2)RELEASED ON Tuesday, 07 September 2010JOURNAL "Insurance Markets and Companies"FOUNDER LLC “Consulting Publishing Company “Business Perspectives”NUMBER OF REFERENCES0NUMBER OF FIGURES0NUMBER OF TABLES0© The author(s) 2021. This publication is an open access article.businessperspectives.orgInsurance Markets and Companies: Analyses and Actuarial Computations, Volume 1, Issue 2, 2010  7Joel R. Barber (USA), Mark L. Copper (USA) The measurement and control of interest rate risk Abstract This paper develops a general measure of interest rate risk in an infinite factor model. We show that interest rate risk measures developed by Fong and Vasicek (M2), by Bowden (Direction X), and by Nawalkha and Chambers (|M|) are special cases derived by placing different restrictions on term structure changes. Futher, we extend Bowden's meas-urement to the case of an arbitrary time horizon. This extension has theoretical advantages when long-term interest rate volatility is small and has computational advantages inherent in positive quadratic forms. Keywords: immunization, duration, interest rate risk, term structure of interest rates. Introduction  Traditional immunization theory (Redington, 1952; Fisher and Weil, 1971) assumes that a single factor determines the evolution of interest rates. Fong and Vasicek (1984) examined immunization risk in an infinite factor model. They considered the effect of an arbitrary change in the instantaneous forward rates, subject to a restriction on the maximum slope of the forward rate curve. They found that the classically immunized portfolio has a negative lower bound that depends upon the magnitude of the rate change and a portfolio dispersion meas-ured, which they call M-squared (M2). Fong and Vasicek’s work has received consider-able attention. Two promising variations on the same theme have appeared. Barber and Copper (1998) and Bowden (1997) obtained a lower bound based upon a restriction on the root-mean square of the interest rate innovation. Nawalkha and Chambers (1996) obtained a lower bound based upon a restriction on the maximum forward rate change. The contribution of this paper is to show how the ideas are related, and how they may all be devel-oped off the same platform. In the process, we gen-eralize Fong and Vasicek’s M2 risk measure, and recast and extend Bowden's risk measure. In fact, both Nawalkha and Chamber’s risk measure and Bowden’s are seen to be special members of a large family of possible risk measures. 1. Interest rate risk We will examine the interest rate risk of an account A consisting of a schedule of future cash flows. Let a(t) denote the cumulative cash flow into account A from time 0 to a future time t  0. Although cash flows typically are discrete, we will allow cash to continuously flow in or out of A. A discrete cash flow  at time t0 means the cumulative cash flow will have discrete jump at time t0 of . For any two successive times, t1 and t2, ([t1, t2]) will be the                                                       Joel R. Barber, Mark L. Copper, 2010. measure of net cash flows credited into A from time t1 to t2. Any cash flows occurring precisely at t1 and t2 are also included1. Next, let P(t1, t2) be a function that provides the value at time t1 of a dollar promised at time t2. If t = 0 is the present, P(0, t2) would be the present value of a dollar promised at t2, while P(T, t2) is the value at horizon date T of a dollar promised at t2. Now, let V(T) denote the value of A at time T, based upon the current term structure (spot yield curve) observed at time 0. In other words, V(T) is the future value of the portfolio at the horizon T assuming that term structure does not change. Our goal is to examine how the value at the hori-zon date changes when the term structure shifts in some manner. The future value of account A under the current term structure is determined by the following Stieltjes integral: tdtTPTV a, .     (1) If the set of cash flows is a discrete set, say C1,..., CN occurring at times t1,..., tN, then (1) can be ex-pressed in the usual form as: Niii tTPCTV1, . If the cash flow stream is continuous, then d (t) = '(t)dt and dtttTPTV , . In this case '(t) is usually called the continuous cash flow stream. The advantages of the Stieltjes integral are: (1) the integrals can easily be manipulated using well-known formulas from the calculus, and (2) the integral form of the present value allows us to examine both the continuous and discrete (and mixtures of both) cash flow streams at the same time2.                                                      1 Technically, we are assuming that  be of finite variation in finite intervals of time so that  denotes the corresponding Lebesgue-Stieltjes measure (cf. Hewitt and Stromberg, 1965). 2 Bowden (1997), for example, separately derives his interest rate risk measure for continuous and discrete cash flow streams. Insurance Markets and Companies: Analyses and Actuarial Computations, Volume 1, Issue 2, 2010  8We assume there exists an instantaneous forward rate function r that explains the time value of money: 2121 ,logttduurttP .     (2) The forward rates are determined from the cur-rent term structure. Since our goal here is to ana-lyze how the future value of A might change, we write V(r, T) to indicate the dependence of V on the particular array of bond prices given by r. In fact, let us consider the transition from one pricing re-gime r to another r + h. Here h is the function, whose value at t is simply the difference in regimes as it pertains to the time t forward rate. If  is a parameter that ranges in value from 0 to 1, then V(r + h, T) ranges from the value of A under the old regime to the new. Under the assumption that h is “small” in an appropriate sense, we can safely use the derivative: 0,, ThrVTrVDh     (3) to estimate the change V(r, T) to V(r + h,T). In fact, this derivative, when h  -1 or 1 and V(r, T) is scaled to 1, is the standard duration. Following Fong and Vasicek, Nawalkha and Chambers, and Bow-den, we relax the requirement that h is known ex-plicitly and consider what then can be said regard-ing the interest rate risk of A. Even with h un-known, we will follow Fabozzi (1996) and refer to (3) as the dollar duration of account A. Note that the immunization condition requires that the dol-lar duration equals zero. Incorporating the forward rate (2) and differentiat-ing under the integral sign, we can formulate the dollar duration as: tdtTduPuhtT, . Furthermore, by letting (T, t) denote the T value of all cash flows attributed to account A up to and including time t and by integrating by parts, we may express this same dollar duration as1: Th dtthTVdttTthtrVD ,, .   (4) 2. A Fong-Vasicek-type inequality In this section, we derive the analog of the Fong and Vasicek (1984) inequality appropriate to our con-text. In our notation, their restriction on changes                                                      1 It is necessary to place some restriction on h; say, that h be inte-grable. in the forward rate curve means that the magnitude of the slope of the change in forward rate curve |h'(t)| is bounded by positive number K: |h'(t)| <  for all t. It will help our exposition to assume that the hori-zon T is fixed and to write (t) = (T, t) and dTPd , . Integrating the right hand side of (4) by parts, leads to the expression: dtssdtthTata dttbTtsTdsthtTbT sdTsbhsdsTThbTTa. Now assume that |h'(t)| <  for some fixed con-stant  and all t. Furthermore, assume that  > 0. Under these assumptions, the expression we have just obtained will always dominate the quantity sdsbTsKsdTsKbTba2sdTsbhsdsTThbTTa. Unfortunately, this expression still depends on h. However, if we assume that account A is immu-nized to time horizon T in the sense that 0tdTt , then, by virtue of the mean value theorem, h drops out and sdTsKTrVDbah223, . Finally, by using – h as well as h and letting the limits of integration include all cash flows, we obtain sdsTPTshTrVDh ,23,2.   (5) The integral in this expression evaluates to Fong and Vasicek’s M2 whenever cash flows are finite in number and discrete. The extra factor of 3 re-sults from the linearization implicit in differentia-tion and the resultant neglect of the convexity of present value of positive cash flows. Nevertheless the use of this inequality in controlling interest rate is exactly the same as detailed in (Fong and Vasicek, 1984): If one knows a bound for h' or, better, a confidence interval for that bound, then the interest rate risk inherent in using account A to fund a single liability at time T is given by M2. Insurance Markets and Companies: Analyses and Actuarial Computations, Volume 1, Issue 2, 2010  93. Bowden-type inequalities Let us now return to the expression for dollar dura-tion (4). Just as we used (T, t) to represent the time T value of all cash flows attributed to account A up to and including time t, now let tT ,  denote the time T value of all cash flows attributed thereafter, TVtTtT ,, . Thus, for example, if T is taken before any cash flow, then dollar duration is compactly expressed dttTth , . Continuing with this assumption, we have, by Holder’s inequality: qqpph dttTdtthTrVD11,, ,      (6) whenever p and q are nonnegative extended real numbers, whose reciprocals total to one. In particular, if we specialize to the case p = 2, then (6) becomes the Cauchy inequality, 2122,, dttThTrVD h . Bowden has observed that since Cauchy's inequal-ity becomes an equality when ,Th ,  that ,T  represents the “direction” of interest rate shift to which the T-value of account A is most sen-sitive (Bowden’s (1997), “Direction X”). 4. Inequalities for arbitrary time horizon The logic of Fong and Vasicek’s derivation of their inequality underlines the importance of being able to measure interest rate risk relative to horizons other than the present. This suggests a generalization of Bowden’s inequalities. If we place no restriction on T and work directly from (4), then (6) generalizes to pph dtthTrVD1,  .,,11qTqqT qdttTdttT    (7) Now, if we take the extreme case q = 1 in (7) we obtain1 sdsTPTshTrVDh ,, ,   (8) an inequality quite similar to the Fong and Vasicek inequality, except that the second moment M2 is re-placed by the first moment of T-valued cash flows and the maximum of h' is replaced by the maximum of h itself. This inequality generalizes that obtained by Nawalkha and Chambers in (1996). In fact, the manager of account A who feels that in-terest rate risk arises from term structure shifts that are "close" to parallel might consider this rough and ready strategy to control the interest rate risk result-ing from funding a single liability with a variety of instruments with different maturities: Minimize both the horizon “duration” (8) and the horizon “convex-ity” (5). The importance of minimizing “duration” increases with the magnitude of the interest shift expected while the importance of minimizing “con-vexity” increases with the magnitude of the “twist” in the term structure anticipated. Bowden’s version of assumption (7) has computa-tional tractability. In fact, if we take p = 2 in (7), we may express it as TrVDh ,  212,,,22vdudvTPuTPvuTvuhHere we have used the notation x+ as shorthand for max(0, x). This inequality shows that, with 2h  fixed, the risk of the portfolio is given by a positive quadratic form evaluated at the portfolio vector. Such forms have several attractive properties. Here we mention the minimum value of such a form constrained to a linear manifold may be found by solving the appropriate linear equations using La-grange multiplier (Fleming, 1977). Conclusion We show that interest rate risk measures developed by Fong and Vasicek, by Bowden, and by Nawalkha and Chambers are special cases derived by placing different restrictions on term structure changes. Futher, we extend Bowden's measurement to the case of an arbitrary time horizon. This extension has theoretical advantages when long term interest rate volatility is small and has computational advantages inherent in positive quadratic forms. References1 1. Barber, J.R. and M.L. Copper. A minimax risk strategy for portfolio immunization. Insurance: Mathematics and Economics, 592, 1998. 2. Bowden, R.J. Generalising Interest Rate Duration with Directional Derivatives: Direction X and Applications. Management Science, May 1997.                                                      1 The measure | | is defined in the same way as  except that the signed net cash flows for each date are replaced by the absolute value of the net cash flows. In particular, | | =  whenever all cash flows are positive. Insurance Markets and Companies: Analyses and Actuarial Computations, Volume 1, Issue 2, 2010  103. Fisher, L., and R.L. Weil. Coping with the Risk of Interest Rate Fluctuations: Returns to Bondholders from Na-ive and Optimal Strategies. Journal of Business, October, 1971. 4. Fabozzi, F.J. Bond Markets, Analysis and Strategies, 3rd ed. Prentice Hall, 1996. 5. Fleming, W. Functions of Several Variables. Springer-Verlag, 1977. 6. Fong, H.G. and O.A. Vasicek. A Risk Minimizing Strategy for Portfolio Immunization. Journal of Finance, Dec 1984. 7. Hewitt, E. and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, 1965. 8. Macaulay, F.R. Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the U.S. since 1856. National Bureau of Economic Research, 1938. 9. Nawalkha, S.K. and D.R. Chambers. An improved immunization strategy: M-absolute. Financial Analysts Jour-nal, Sept/Oct 1996. 10. Redington, F.M, Review of the principle of life-office valuations. Journal of the Institute of Actuaries 78, 1952. 

The measurement and control of interest rate risk

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