COMMON FALLACIES CONCERNING DURATION AND CONVEXITY
APPENDIX 2.1: OTHER FALLACIES CONCERNING DURATION AND CONVEXITY
Other fallacies found in the fixed-income literature regarding duration and convexity area are as follows:
1. Duration is based on the assumption of infinitesimal and parallel yield curve shifts, therefore, is not a useful risk measure when yield curve shifts are large and nonparallel.
2. Since duration is not derived using the framework of the modern port- folio theory, it does not relate risk to return.
3. Convexity is based on the assumption of large and parallel yield curve shifts, which imply the existence of arbitrage profits, and hence convexity is a theoretically invalid measure for interest rate risk analysis.
The first and third fallacies can be traced back to an influential critique of duration and convexity by Ingersoll, Skelton, and Weil (1978). The sec- ond fallacy resulted from comments of Sharpe (1983), which questioned whether duration is consistent with a risk-return equilibrium. A resolution to all of the above fallacies was provided by Nawalkha and Chambers (1999) and is outlined next.
The duration risk measure is consistent with a specific arbitrage-free term structure model of Heath, Jarrow, and Morton (HJM; 1992). This model is discussed in detail in the second part of the book. Under this model, only the “unexpected” portion of the yield curve shift remains par- allel. The expected portion of the yield curve shift is always nonparallel and is determined by the “forward rate drift restriction” imposed by the HJM model. Duration reflects the risk resulting only from the unexpected yield curve shifts, which are assumed to be parallel. However, when even the un- expected yield curve shifts are nonparallel, duration risk measure allows in- terest rate risk hedging against only the shifts in the height of the yield curve. Under this scenario, duration becomes a partial risk measure, and other higher order measures may be needed to hedge against shifts in the slope, curvature, and other higher order changes in the yield curve. Even as a partial risk measure, duration explains roughly 70 percent of the ex-post return differentials among bonds, and so remains the most important bond risk measure.
Further, the duration risk measure is consistent with Merton’s (1973a) intertemporal capital asset pricing model (ICAPM), and, hence, with continuous-time modern portfolio theory. Assuming that the entire invest- ment opportunity set is represented by the changes in the instantaneous short rate, a simplified form of the two-parameter ICAPM can be obtained
for securities subject to default risk (e.g., stocks and default-prone bonds).
However, the two-parameter ICAPM reduces to a single-factor model for all default-free securities. Interestingly, the appropriate equilibrium measure of the systematic risk of a default-free security is its durationand not its bond beta as derived by Alexander (1980); Boquist, Racette, and Schlarbaum (1975); Jarrow (1978); and Livingston (1978), under more restrictive as- sumptions. Intuitively, the above result obtains because under the two- parameter ICAPM, every default-free bond can serve as a hedge portfolio that is used to hedge against unexpected changes in the interest rates by risk- averse investors.
Finally, using the continuous-time HJM framework, the effect of con- vexity on the bond return can be shown to cancel out by a portion of the thetaof the bond. Bond theta measures the drift of the bond price due to the passage of time. Due to thisconvexity-theta trade-off,bond convexity is not priced under the single factor forward rate models of HJM. However, bond convexity may be priced under a two-factor HJM model that allows both level and slope shifts in the term structure of forward rates. The relation of convexity with slope shifts in the yield curve is the subject of Chapter 4.
NOTES
1. Duration was discovered more than half a century ago by Macaulay (1938) and Hicks (1939), and then rediscovered a number of times by researchers including Samuelson (1945) and Redington (1952).
2. Under discrete compounding, equation 2.15 leads to modified duration, which is different from duration. However, with continuous compounding, the defini- tion of duration given in equation 2.15 is identicalto the definition of duration given by equation 2.13.
3. Under nonparallel shifts in the yield curve, one can use the generalized duration vector models introduced in Chapter 5 or the key rate duration models intro- duced in Chapter 9.
4. Under discrete compounding, equation 2.18 leads to modified convexity, which is different from convexity. However, with continuous compounding, the defini- tion of convexity given in equation 2.18 is identicalto the definition of convex- ity given by equation 2.17.
5. See the following: Cole and Young (1995, p. 1); Fabozzi (1996, pp. 66, 73);
Fabozzi, Pitts, and Dattatreya (1995, pp. 97–98, p. 101, p. 109); Johnson (1990, p. 73); Kritzman (1992, p. 19); and Livingston (1990, p. 70).
6. See Nawalkha (1995) for analytical details of the pricing and duration of this callable bond—we are assuming (in his notation) that volatility of returns to the bond is V=0.01.
7. Since ∂lnP/∂y=(∂P/∂y)/Pis the slope, it follows from the definition of Dthat the slope equals −D.
Notes 43
8. Consider an initial price P* at yield y* and a new price Pat a new yield y.The vertical distance is lnP−lnP*=ln(P/P*)=ln(1+Rate of instantaneous return), as stated.
9. The slope or the first derivative of the plot at the initial yield y* equals [∂[(P− P*)/P*]/∂y]y=y*=[(∂P/∂y)y=y*]/P*= −D.The curvature (i.e., change in slope) or the second derivative of the plot at the initial yield y* equals [∂[∂[(P− P*)/P*]/∂y]/∂y]y=y*=[(∂2P/∂y2)y=y*]/P*=CON.
10. If Figure 2.5 were for a zero-coupon bond, the slopes and curvatures would be identical as initial yield changes.
44
3
Estimation of the Term Structure of Interest Rates
T
he duration model introduced in the previous chapter assumes infinitesi- mal and parallel shifts in a flat yield curve. In order to consider nonparal- lel shifts in a nonflat yield curve, we need to model the yields corresponding to different maturities. Theterm structure of interest ratesgives the relation- ship between the yield on an investment and the term to maturity of the in- vestment. This chapter focuses on how to estimate the default-free term structure of interest rates using cross-sectional U.S. Treasury bond data. The term structure obtained in this chapter will serve as an input in many chap- ters that follow, which introduce more complex risk measures for hedging against nonparallel term structure shifts (such as, M-absolute, M-square, duration vector, key rate durations, principal component durations). The term structure will also be an important input in various chapters in the sec- ond and third volumes of this book series. Since the valuation of default-free fixed-income securities and the derivatives based on these securities must fit an empirically observable term structure, estimation of the term structure using cross-sectional data is essential for the valuation process.The default-free term structure generally rises with maturity, because investors generally demand higher rates of interest on longer maturity in- vestments, both due to a preference for liquidity and as an aversion to inter- est rate risk. The term structure is typically measured using default-free, continuously compounded, annualized zero-coupon yields. Since coupon bonds are portfolios of zero-coupon bonds, the term structure can be used to value both coupon bonds and zero-coupon bonds. The term structure is not directly observable from the published coupon bond prices and yields.
Though default-free zero-coupon prices (such as the U.S. Treasury STRIPS) can be directly used for obtaining the term structure, the lack of liquidity in these markets, and the unavailability of a continuum of maturities, make the use of coupon bond prices necessary for obtaining more robust estimates of the term structure.
Bond Prices, Spot Rates, and Forward Rates 45
FIGURE 3.1 The Discount Function 0
1
Term
Discount factor
This chapter reviews three methods of term structure estimation: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. We consider various extensions to these methods based on error-weighing schemes that lead to more robust estimates of the term structure. Before introducing these methods however, we review some nota- tion and concepts.