Hedging the interest-rate risk of mortgage-backed securities (MBSs henceforth) is an extremely difficult endeavor. The difficulty is due to the homeowner's prepayment option, which renders the duration and therefore the price sensitivity of MBSs to changes in interest rates highly uncertain. An intuitive understanding of this type of uncertainty can be gained by realizing that the value of a MBS is equal to the value of a riskless fully amortizing bond minus the value of a call option (i.e., the prepayment option). The two components of the MBSs react differently to changes in interest rates. When interest rates rise above the MBS coupon rate, the prepayment option becomes out of the money and the price of the MBS is a strictly convex function with respect to further increases in interest rates. When interest rates fall, however, the prepayment option becomes in the money and it prevents the value of the MBS from rising, a phenomenon that has been called negative convexity. To make matters worse, direct hedging instruments, such as futures on MBSs, are no longer available. The Government National Mortgage Association collateralized depository receipt (GNMA CDR) futures contract introduced by the Chicago Board of Trade (CBOT) in 1975 was withdrawn in 1987 due to investors' lack of interest. Johnson and McConnell (1989) find that the flexible delivery options of the contract reduced its hedging effectiveness. The fate of the mortgage-backed futures contract (MBF), introduced in 1989, was no different. After an initial period of success, trading volume declined continuously until the contract was withdrawn in 1992. According to Nothaft, Lekkas and Wang (1995) the failure was due to the contract's decreasing market liquidity and the existence of more liquid cross-hedging substitutes such as the Treasury-bond and the Treasury-note futures contracts. By most accounts, the 10-year Treasury-note futures contract is the most popular among MBS investors and mortgage originators. The popularity of this hedging instrument is due to the fact that, after allowing for prepayments, the average duration of a typical 30-year MBS is closer to the 10-year Treasury note than any other government fixed-income security for which a futures contract exists. Consequently, changes in MBS prices should be highly correlated with changes in 10-year Treasury-note prices. Evidence on the popularity of the T-note futures contract is provided by Fernald, Keane and Mosser (1994), who find that the hedging of MBSs with Treasury-note futures has essentially changed the short-run dynamics of the term structure of interest rates.
MBS risk hedging has been commonly based on (1) valuation methods and (2) regression methods. The former involves matching the effective duration of the MBS to that of a Treasury bond for which a futures contract exists. Subsequently, the hedger shorts a number of futures contracts equal to the ratio of the durations. Effective durations are derived from option-adjusted spread (OAS) models. Despite its wide acceptance in recent years, effective duration based on OAS suffers from some serious problems. These problems are linked to the sensitivity of duration estimates to the specific prepayment model used. To make matters worse, prepayment rates are extremely sensitive to changes in the term structure. Consequently, modeling prepayments requires explicit modeling of the term structure. Due to differences in prepayment and term-structure modeling, estimated duration measures vary widely across broker-dealer firms. Such wide variations have raised questions regarding the accuracy of OAS effective durations. Choi (1996) compares median broker OAS durations and empirical durations for MBS with nine different coupons for the period February 1992 to March 1994. He finds that the root-mean-square forecast error (RMSE) of OAS durations based on median broker forecasts is 1.13 when averaging across the nine coupons. The average RMSE based on empirical-duration forecasts was 0.83. Similarly, Breeden (1994) finds that effective-duration forecasts reported by major investment firms are inefficient in the sense that they do not take proper account of past information. Specifically, forecasts can be improved considerably by simply incorporating information from past empirical durations. This is an important issue, because inefficient duration estimates will inevitably lead to erroneous hedging decisions. A recent study by Goodman and Ho (1997) compares directly the hedging effectiveness of OAS effective-duration-based hedge ratios with that of empirical-duration-based hedge ratios. The authors find that the hedging performance of the OAS effective duration is worse than that of simple empirical durations.
Regression-based hedging strategies require neither prepayment modeling nor yield-curve forecasts. In most cases the hedge ratio is estimated from the regression [Delta][S.sub.t] = [Alpha] + [Beta] [Delta][F.sub.t] + [[Epsilon].sub.t] (1)
where [Delta] is the first-difference operator, [S.sub.t] is the price (or the logarithm of the price) of the MBS at time t, [F.sub.t] is the price (or the logarithm of the price) of the hedging instrument during period t, [[Epsilon].sub.t] is the random error term, assumed to be an i.i.d, process, and [Alpha] and [Beta] are fixed intercept and slope coefficients, respectively.(1) The minimum-variance hedge ratio is the estimated [Beta] which defines the number of dollars to go short in futures per dollar of investment in the spot (cash) market. If futures prices follow a martingale process, then [Beta] is also the expected-utility-maximizing hedge ratio for an agent with quadratic utility function [e.g., Anderson and Danthine (1981), Ederington (1979), Johnson (1960) and Malliaris and Urrutia (1991), among others].
The simplicity of the regression approach is undoubtedly the major factor in its wide usage in both academic research and practical decision making. However, this may be one case where the price of simplicity is higher than commonly thought. Economic analysis and intuition suggest that the prices of the spot asset and the futures contract are jointly (simultaneously) determined (see e.g. Stein 1961). Consequently, the estimation of (1) is subject to simultaneity bias, i.e., the estimated hedge ratio [Beta] will be upward biased and inconsistent. Furthermore, if both the spot price and the futures price have unit roots in their univariate representations (i.e., they follow martingale processes) but some linear combination is stationary, the regression should include an error correction term (see Engle and Granger 1987). Omitting this term will lead to downward-biased estimates of [Beta] [e.g., Brenner and Kroner (1995) and Kroner and Sultan (1993)]. Since the two biases work in opposite directions, one may hope that they will, on average, cancel each other, even though such a happy coincidence cannot be assured. Even if the biases cancel out, however, there still remains a problem which, in our view, is much more serious, namely, the assumption that the volatility as well as the covariance-correlation of spot and futures prices is constant over time. This assumption runs contrary to the findings of several studies, e.g., Anderson (1985), Malliaris and Urrutia (1991), Kroner and Sultan (1993) and Park and Bera (1987), among others. Significant variation in variances and covariances strongly suggests the use of time-varying hedge ratios. Investors that relied on constant, regression-based hedge ratios suffered huge losses in the period 1985-1986. During that period the fall in interest rates did not produce price increases in MBSs, due to the prepayment option becoming in the money. At the same time Treasury-note futures prices rose, thus producing large losses for short cross-hedgers (see Batlin 1987). Avoiding or, at least, limiting losses during such periods of falling interest rates requires adjusting hedge ratios downward. Several recent studies find that time-varying hedge ratios lead to larger risk reduction than constant hedge ratios for such diverse assets as commodities (Baille and Myers 1991), Treasury bonds (Cecchetti, Cumby and Figlewski 1988), bankers' acceptances (Gagnon and Lypny 1995), foreign currency (Kroner and Sultan 1993) and stock index futures (Park and Switzer 1995).
Despite the growing evidence that dynamic hedging is superior to traditional regression-based static hedging, no effort has been devoted to ascertaining the relative importance and feasibility of dynamic hedging strategies for MBSs. This stands in sharp contrast to the growing importance of the MBS market. In this particular market, dynamic hedging techniques are likely to be more appropriate because of (1) the prepayment option of MBSs and (2) the lack of direct hedging instruments, which necessitates the use of cross-hedging. These two factors imply that the correlation-covariance of the spot instrument and the hedging instrument will be subject to time variation.(2)
This article focuses on regression-based hedging techniques, and it proposes a dynamic hedging model for Government National Association mortgage-backed securities (GNMA MBSs). This model is free of the drawbacks associated with the static regression hedging strategies currently used. The simultaneity bias of the regression approach is dealt with by modeling the distribution of price changes of GNMA MBSs and 10-year Treasury-note futures as a system of simultaneous equations. Error correction (EC) terms from cointegrating relationships, when present, are included in the conditional mean equations of the two markets. The dynamic variance-covariance structure of the two markets is modeled via a version of the bivariate GARCH model suggested by Engle and Kroner (1995). This formulation of the variance-covariance matrix, known as the BEKK model, assures that the variance-covariance matrix is positive semidefinite for all time periods. It also economizes on the number of parameters that need to be estimated.(3) The dynamic error-correction GARCH model (EC GARCH) is estimated using daily data on six different coupon GNMA MBSs. Dynamic cross-hedge ratios are obtained from the time-varying variance-covariance matrix using the 10-year Treasury-note futures contract as the hedging instrument. The hedging effectiveness of the dynamic hedge ratios are contrasted with the effectiveness of regression-based hedge ratios both within and out of sample. Dynamic and static hedge ratios are evaluated in terms of overall risk reduction, as well as expected utility maximization. There is overwhelming evidence that dynamic hedge ratios are superior to static ones even when transaction costs, due to rebalancing, are incorporated into the analysis. This conclusion holds for all six different coupon GNMA MBSs under investigation.
The rest of this paper is organized as follows: The next section describes the dynamic hedging model. The section after describes the data and presents several preliminary findings. Then we present the main findings and compare the hedging performance of the dynamic hedge ratios with that of the static hedge ratios. A final section summarizes the principal findings.
Dynamic Hedging and the EC GARCH Model
The return on a hedged portfolio of MBSs with a short position of [h.sub.t-1] dollars' worth of futures for every dollar's worth of the MBS held at time t - 1 is equal to
[r.sub.h,t] = [r.sub.s,t] - [h.sub.t-1][r.sub.f,t] (2)
where [r.sub.h,t] is the hedged return, [r.sub.s,t] is the return on the spot asset, [h.sub.t-1] the hedge ratio as of t - 1, and [r.sub.f,t] is the return on the futures contract.(4) Following Kroner and Sultan (1993), we assume that the representative investor has a time-separable quadratic utility function. Consequently, his optimal hedge ratio, [h.sub.t-1], will be such that expected utility as of time t - 1 will be maximized, or
[Mathematical Expression Omitted] (3)
where [E.sub.t-1] is the conditional expectation as of time t - 1, U([center dot]) is the utility from holding the hedged portfolio, [Theta] is the coefficient of risk aversion, and [Mathematical Expression Omitted] is the conditional variance of the hedged return. The first-order condition for a maximum yields the following optimal hedge ratio:
[Mathematical Expression Omitted] (4)
As can be seen, the optimal hedge ratio is composed of two distinct components. The first component is the conditional-variance-minimizing hedge ratio. The second component is the speculative demand for futures contracts. The expected-utility-maximizing hedge ratio coincides with the variance-minimizing ratio if either the coefficient of risk aversion is extremely high. or the expected percentage change in the futures price is zero.(5) In most applications the speculative component is insignificant because futures prices follow driftless martingale processes, especially in high frequencies (e.g., Baille and Myers 1991, Gagnon and Lypny 1995, and Kroner and Sultan 1993). Consequently, the second component of the optimal hedge ratio vanishes, and the expected-utility-maximizing ratio coincides with the variance-minimizing hedge ratio.
Estimation of dynamic hedge ratios based on (4) can be obtained once the conditional joint distribution of spot and futures returns is fully specified. The model that is used for this purpose is described by the following set of equations:
[r.sub.s,t] = [[Beta].sub.s,0] + [[Beta].sub.s,1] E[C.sub.t-1] + [[Epsilon].sub.s,t] (5)
[r.sub.f,t] = [[Beta].sub.f,0] + [[Beta].sub.f,1] E[C.sub.t-1] + [[Epsilon].sub.f,t] (6)
[[Epsilon].sub.t][where][[Omega].sub.t-1] [similar to] N(O, [H.sub.t]) (7)
[H.sub.t] = C[prime]C + A[prime][[Epsilon].sub.t-1][[Epsilon][prime].sub.t-1] A + B[prime][H.sub.t-1]B (8)
where [[Omega].sub.t-1] is the information set at time t - 1, N([center dot]) is the bivariate conditional normal density, [[Epsilon].sub.t] is the error vector with zero mean, and [H.sub.t] the variance-covariance matrix. A and B are diagonal matrices for the slope coefficients, and C is the vector of the intercept coefficients.(6) The specification of the variance-covariance matrix given in (8) implies that the individual variances are functions of their own past squared residuals and past variances, whereas the covariance is a function of past cross residuals and its own lagged value. As mentioned earlier, this particular parameterization of [H.sub.t] assures positive semidefiniteness for all t.(7) The term E[C.sub.t] in the conditional mean equations (7) and (8) is the error correction term, which ensures that deviations from long-term equilibrium are transitory.(8)
The model outlined above is estimated for six different coupon GNMA MBSs, using in all cases the 10-year Treasury-note futures contract as the cross-hedging instrument. The log likelihood function is maximized using the numerical algorithm of Berndt, Hall and Hausman (1974).
Data and Preliminary Statistics
Daily prices for 30-year fixed-rate GNMA securities with coupons 7.5%, 8%, 8.5%, 9%, 9.5% and 10% were obtained from Smith Barney, Inc. They represent dealer-quoted bid prices for forward delivery on a to-be-announced basis.(9) Data used for the futures market are the daily settlement prices for the 10-year Treasury-note futures contracts obtained from Tick Data, Inc. The delivery months for the 10-year Treasury-note futures contract are March, June, September and December. On any given trading day there are prices for all four outstanding contracts. A time series of non-overlapping observations is created by using the most actively traded contract, thereby avoiding problems related to thin trading. The same approach is followed by Koutmos and Tucker (1996). Other related studies roll over to the next nearest contract three weeks before maturity. Since the major consideration here is biases due to thin trading, it is better to explicitly use the most actively traded contact. The two approaches are roughly equivalent, however, since the trading volume of the nearby contract declines two to three weeks before expiration. Also, yield to maturity and duration for the 10-year T-note are obtained from DRI. The sample extends from July 22, 1992 to August 7, 1995 for a total of 749 daily prices for each series.
The time-series properties of the various coupon 30-year GNMA MBSs and the 10-year Treasury futures are important in choosing the best model specification. The prices of most financial assets, or their logarithmic transformations, have been found to be non-stationary, i.e., they have a unit root in their univariate representation. Table 1, panel A, reports the results from unit-root tests based on the Phillips-Perron procedure (see Phillips and Perron 1988) for the logarithms of the prices of the MBSs and the Treasury-note futures contract.(10) The calculated unit-root statistics are in all instances [TABULAR DATA FOR TABLE 1 OMITTED] below their critical value.(11) Consequently, the logarithms of the prices have a unit root, and first-differencing is sufficient to induce stationarity. As expected, the PP tests reject the hypothesis of a unit root in the first logarithmic differences (returns). The hypothesis of cointegration (i.e., that spot and futures prices share a common stochastic trend) is tested using the procedure suggested by Engle and Granger (1987). The procedure is essentially a test for a unit root applied to the residuals of a regression involving the variables to be tested for a common stochastic trend. The notion is that if the linear combination implied by the estimated regression is stationary, then the variables are cointegrated. As can be seen from Table 1, panel A, the estimated Engle-Granger (EG) statistic suggests that the log of the price of the 10-year Treasury futures is cointegrated with the log of the price of the GNMAs across all coupons. Thus, the inclusion of an EC term in the conditional mean equations is justified.
Table 1, panel B, reports several descriptive statistics for the percentage logarithmic differences of the prices (referred to as returns henceforth) of the GNMA securities and the 10-year Treasury futures. The estimated skewness and excess kurtosis, as well as the Kolmogorov-Smirnov (D) statistic, are statistically significant, indicating departures from normality in all instances. Non-normality can be caused by temporal dependencies in the return series, especially second moment temporal dependencies. The presence of such dependencies is tested by means of the Ljung-Box (LB) statistic. For the returns, the hypothesis that all autocorrelations up to the 10th lag are jointly zero is rejected for coupons 8%, 8.5%, 9% and 9.5% but retained for coupons 7.5% and 10% as well as the futures returns. Thus, the evidence on first moment dependencies is rather mixed. The hypothesis that the autocorrelations for the squared returns up to the 10th lag are jointly zero is rejected across all GNMA coupons but retained for the Treasury-note futures. It is interesting to note that the LB statistics are much higher for the squared returns. This provides indirect evidence of time-varying second moments and justification for the subsequent use of the bivariate GARCH specification for the variance-covariance matrix.
Hedging Performance and Comparisons
The bivariate error correction EC GARCH model, described by Equations (5) - (8), is estimated using the 10-year Treasury-note futures returns along with each one of the six GNMA-MBS returns. Thus, a total of six bivariate models are estimated. Maximum-likelihood estimates are reported in Table 2, panel A. The EC terms are insignificant with minor exceptions. They are retained, however, because they improve out-of-sample hedging effectiveness. The fixed coefficients that describe the variance-covariance matrix [H.sub.t], are significant at any level of significance. Therefore, conditional variances and covariances can be predicted on the basis of past innovations (errors) and past variances and covariances. This constitutes indirect evidence that time-varying hedge ratios that exploit the dynamic relationship of MBS and Treasury-note futures can provide more effective hedging than the static ratios commonly used. Park and Bera (1987) find that allowing for conditional heteroskedasticity improves the confidence intervals of static hedge ratios for GNMA MBSs. The improvement is greater in cross-hedging situations.
Correct specification of the time-varying variance-covariance matrix is of paramount importance, given that the hedge ratios are based on it. For the variances to be well specified, the squared standardized residuals have to be uncorrelated. Likewise, correct specification of the covariance requires that the cross products of the standardized residuals be uncorrelated.(12) To test both hypotheses, the LB statistic is used with ten lags (two weeks). The estimated values of the LB statistics are below their critical values for both the squared standardized residuals and their cross products (see Table 2, panel B). Consequently, the model captures successfully the dynamic nature of the variance-covariance structure of MBSs and the 10-year Treasury-note futures.
Even though the LB statistics show no evidence of misspecification in the variance-covariance structure, they do not provide any indication as to how well the models capture the impact of positive and negative innovations on volatility. For this purpose the set of diagnostics proposed by Engle and Ng (1993) are used. These diagnostics are based on the simple intuition that if the volatility process is correctly specified, then the squared standardized residuals should not be predictable on the basis of observed variables. The Engle-Ng estimated diagnostics are reported in Table 3. With no exception, all these statistics turn out to be statistically insignificant, lending support to the hypothesis that additional information such as the size and sign of past [TABULAR DATA FOR TABLE 2 OMITTED] [TABULAR DATA FOR TABLE 3 OMITTED] innovations cannot be used to improve one-step-ahead forecasts of the variance-covariance matrix. The hypothesis of conditional bivariate normality of the normalized error vector is tested using the multivariate normality test proposed by Bera and John (1983). The estimated values of these test statistics reject the normality assumption. This implies that the estimated coefficients will still be unbiased but the standard errors are probably understated. All the parameters describing the variance-covariance matrix, however, remain significant at the 1% level. Thus the results are robust to departures from normality. In addition, the usefulness of the model lies mostly in its ability to reduce risk. Its performance in reducing risk is tested extensively below.
Table 2, panel C, reports likelihood ratio test statistics for various restrictions imposed on the coefficients of the full model. The restriction that the EC term is insignificant is retained. The only exception is the 10% coupon. The hypothesis that the parameters describing the time-varying component of the variance-covariance matrix are jointly zero, i.e., A = B = 0, is rejected at any level of significance and across all coupons. This model is referred to as the error-correction ordinary least-squares (EC OLS) model. Rejection of this model provides strong evidence that the second moments of GNMA MBSs are time-varying. Finally, the classical OLS model is considered by testing the restriction that the error correction term is zero. This restriction is rejected only for the 10% coupon. Even though the EC OLS and the OLS models are rejected, their hedging effectiveness is contrasted to that of the full (EC GARCH) model within sample and out of sample. The reason is that the full model will probably require frequent adjustment of the hedge ratio. If transaction costs are assumed to be present, such frequent adjustment may not be economically feasible, so that a simpler model may be preferable.
Within-Sample and Out-of-Sample Hedging Performance
The time-varying optimal hedge ratios given in (4) are calculated using the estimated variance-covariance matrix based on the unrestricted bivariate EC GARCH model and a few restricted versions, including the simple regression. In addition, hedge ratios based on empirical durations are used for comparing the findings in this paper directly with those of Goodman and Ho (1997). The evidence concerning stationarity of dynamic hedge ratios based on bivariate GARCH models is mixed. Depending on the asset in question, hedge ratios may follow random walks [e.g., Baille and Myers (1991) for commodities] or some stationary process [e.g., Kroner and Sultan (1993) for exchange rates]. Using the Phillips-Perron test, we reject the hypothesis that within-sample hedge ratios for GNMA MBSs based on the unrestricted model have a unit root. This holds true across all coupons (see Table 4). It is interesting, however, to note that the same test (not reported) fails to reject the hypothesis that out-of-sample hedge ratios follow random walks.
The maxima, minima and averages of within-sample estimated hedge ratios along with the standard deviations are reported in Table 4 for all six coupons. As can be seen, the range of possible values is quite substantial. Minimum hedge ratios can even be negative for higher coupons. Even though this may appear strange at first, it makes perfect sense if we recall the negative convexity feature of MBSs. The negative convexity implies that as interest rates are falling the hedge ratio should be reduced, and if the fall continues the hedger should even go long in futures. Another interesting finding is that there is an inverse relationship between coupon rates and average hedge ratios. This makes intuitive sense, because the higher the coupon, the lower the duration and hence the lower the price sensitivity of the MBS. Thus the average hedge ratio is lower.
Hedging effectiveness is evaluated in terms of variance reduction of the hedged portfolio using hedge ratios based on the full model, various restricted versions and empirical duration. Both within-sample and out-of-sample evaluations are performed. Within-sample evaluations are based on hedge ratios implied by each particular model estimated over the entire sample period. The results are reported in Table 5. Panel A reports the calculated variances based on the full model, three of its restricted (nested) versions and empirical duration. The dynamic hedge ratios based on the EC GARCH and GARCH models produce the most significant reductions in variance. Panel B reports percentage reductions in variance using hedges based on the EC GARCH model vis-a-vis hedges based on the GARCH model, the EC OLS model, the standard OLS model, the empirical duration and finally the unhedged position. Compared to the unhedged position, the bivariate EC GARCH model produces on the average a 62.87% reduction in variance, with the minimum reduction being 36.30% (for the 10% coupon) and the maximum reduction being 76.89% (for the 7.5% coupon).(13) Compared to the static OLS hedging, the average percentage variance reduction across coupons is 18.93%, with the maximum reduction being 31.17% (for the 8.5% coupon) and the minimum reduction being 7.25% (for the 10.0% coupon). [TABULAR DATA FOR TABLE 4 OMITTED] [TABULAR DATA FOR TABLE 5 OMITTED] Interestingly, the empirical-duration-based hedging provides better protection than the static OLS hedging. This is probably due to the fact that the empirical duration uses information related to the duration of the 10-year T-note.(14) It should be noted that relaxing the assumption that the conditional mean in the futures return is zero and using the expected-utility-maximizing hedge ratio as stated in Equation (4) does not change the results. For example, Table 5, panel C reports within-sample variances using the full hedge ratio. As can be seen, the estimated variances of the hedged portfolios remain almost unchanged. Also, the ordering is exactly the same. Thus, the speculative component of the hedge ratio is not important.
Historical (ex post) performance, however, is not the safest way to evaluate the usefulness of hedging models. Out-of-sample evaluation is an alternative and, perhaps, a more meaningful way of judging the validity and usefulness of a hedging model. To this end, all models are re-estimated on the basis of the first 649 observations, thus creating a holdout sample of 100 observations. At each time period the hedge ratios are calculated and the models are re-estimated by adding one more observation. This process is repeated until the holdout sample is depleted. Figure 1 presents a graphical representation of the estimated out-of-sample cross-hedge ratios based on the EC GARCH and the traditional OLS models. The EC-GARCH-based hedge ratios are clearly time-varying, while the OLS-based hedge ratios are nearly constant. In addition, the inverse relationship between coupon size and average hedge ratios is apparent in Figure 1. Table 6, panel A, reports out-of-sample variances of hedged portfolios. The estimation is based on the initial sample and the updating procedure described above. The picture that emerges is very much the same as within sample. The smallest variance is achieved when cross-hedge ratios from the full EC GARCH model are used. The average percentage reduction in variance is slightly smaller in the out-of-sample experiment. Specifically, the average reduction is 58.91% when the alternative is the unhedged position, 7.28% when the alternative is the traditional OLS, 7.96% when the alternative is the EC OLS and 6.93% when the alternative is the empirical duration.
[TABULAR DATA FOR TABLE 6 OMITTED]
Without considering transaction costs, it appears that the superior hedging performance of the dynamic hedge ratios within sample and out of sample more than justifies the costs arising from using a more complex model. Moreover, this superior performance is consistent across all coupons. Studies dealing with other financial assets find smaller improvements in performance from using dynamic hedge ratios. For example, Kroner and Sultan (1993), using five different exchange rates, find an average improvement of 2.5% in terms of variance reduction over and above the reduction achieved by the traditional OLS hedge. It is reasonable to expect that in situations of direct hedging the correlation between the spot and the futures will be very stable because of non-arbitrage pricing relationships. In situations of cross-hedging, however, no such relationships exist. Consequently the correlation between the spot asset and the hedging instrument exhibits substantial time variation. Under these circumstances, hedge ratios that allow, correlations and variances to change in the light of new information provide better protection.
A question that naturally arises is whether the dynamic hedging approach suggested here compares favorably to hedging techniques based on OAS effective duration. An obvious difficulty in carrying out such a comparison is the diversity of effective-duration models used by broker-dealer firms and analytics vendors and the lack of any baseline model. In the words of Davidson, Kulason and Herskovitz (1995):
The major drawback to the OAS approach is that it is basically a black box into which an investor puts assumptions and out of which comes risk and return measures. The prepayment function and term structure models embedded in OAS models are generally proprietary, precluding the possibility of an investor inspecting these key aspects of the model.
A meaningful comparison could be attempted however, in the light of the recent findings of Goodman and Ho (1997). These authors base their OAS effective durations on a prepayment model that uses consensus prepayment projections from prepayment models of eleven dealers. Subsequently they test the hedging effectiveness of OAS effective duration versus empirical durations. Their findings present the most serious challenge to the view that effective duration provides better risk assessment for MBSs. Specifically, it is found that empirical-duration-based hedge ratios reduce the variance of the hedged returns by 28.65% (for FNMA MBS) and 36.28% (for GNMA MBS) over and above to the reduction achieved by the use of OAS effective duration. Using empirical-duration hedge ratios as a baseline, it can be seen that dynamic hedging based on the bivariate EC GARCH model reduces the variance of hedged returns by an additional 14.15% (within sample) and 6.93% (out of sample) compared to hedged ratios based on empirical durations. Since the time period examined by Goodman and Ho (1997) largely overlaps the period examined in this study, it can be said that the performance of EC-GARCH-based hedge ratios compares favorably to that of OAS effective-duration-based hedge ratios.(15)
Measuring Economic Significance
While the empirical findings illustrate very clearly the superiority of the dynamic hedge ratios, it is still necessary to investigate whether the percentage risk reduction is economically meaningful. Following Kroner and Sultan (1993), we consider the risk reduction to be economically significant if it increases the value of the utility function, net of transaction cost. For the mean-variance hedger the objective is to maximize the expected utility function of the form [Mathematical Expression Omitted], where E([r.sub.h]) and [Mathematical Expression Omitted] are the expected return and the variance of the return of the hedged portfolio and [Theta] is the coefficient of risk aversion. Assuming [Theta] equals 4 and the expected return on the hedged portfolio is zero, the utility from hedging is [Mathematical Expression Omitted], where -y is the percentage reduction in expected return due to transaction costs.(16) A typical round trip (one buy and one sell) transaction costs $10-$15 for an institutional investor and $25-$50 for a retail investor, implying a minimum percentage transaction cost of 0.001% and a maximum of 0.005%. Thus, if the hedger invested in the classical hedge (OLS) portfolio for the 7.5% coupon GNMA MBS, he would have had an average utility of U([r.sub.h]) = -4(0.0295) = -0.118 per day. Had he invested in the dynamic hedge, his average utility would have been U([r.sub.h]) = -y - 4(0.0268) = -y - 0.1072. That is, the investor's utility increases by -y + 0.0108 if he uses dynamic hedging, implying that the dynamic hedging strategy will be preferred over the static strategy whenever y [less than] 0.0108. Thus, given the assumed transaction costs, the dynamic hedge would result in an economic improvement for both institutional and retail investors with a mean-variance utility function and degree of risk aversion [Theta] = 4.
The preceding analysis probably understates the economic significance of the dynamic model, because it assumes that the investor rebalances his portfolio at the beginning of every day. In practice, a mean-variance investor will choose to rebalance only if his utility from rebalancing exceeds that [TABULAR DATA FOR TABLE 7 OMITTED] [TABULAR DATA FOR TABLE 8 OMITTED] from not rebalancing. Several within-sample and out-of-sample simulations were conducted, and the expected total utility along with the number of times the hedge would have been rebalanced is reported in Tables 7 and 8. The estimated total expected utilities show clearly the economic superiority of the dynamic hedges across all coupons.(17)
This paper has shown that hedge ratios based on the time-varying joint distribution of GNMA MBSs and 10-year Treasury futures offer superior hedging to traditional hedge ratios. The improvement over and above the static hedging is 18.9% (within sample) and 7.9% (out of sample), averaging over six different coupon GNMA MBSs. These improvements are both statistically and economically significant. Assuming realistic transaction costs, it was shown, within a mean-variance framework, that economic gains are substantial irrespective of the frequency of updating the hedge. These results hold across six different coupon GNMA MBSs.
The dynamic hedge ratios vary substantially over the sample period, in some cases turning even negative presumably due to the negative convexity of MBS. There is an inverse relationship between average hedge ratios and coupon sizes. The same inverse relationship is observed in the size of variance reduction and coupon size. The implication is that the 10-year Treasury futures contract is more effective in hedging MBSs with low coupons. This is a direct result of the fact that the lower the coupon of the MBS, the lower is the value of the prepayment option and the higher the correlation with the 10-year Treasury futures contract.
The first author gratefully acknowledges research support from Fairfield University. We thank Michael D. Youngblood for providing the data. We also thank William Goetzmann, the co-editor, and three anonymous reviewers for valuable comments and suggestions. The views expressed in this paper are those of the authors alone.
1 Using first differences in the regression implies that the underlying distribution is normal, whereas using first logarithmic differences implies that the underlying distribution is lognormal.
2 Tong (1996) finds that in the case of foreign-exchange risk hedging, dynamic hedging is not substantially better than static hedging. He attributes this to the rather stable relationship between the cash asset and the direct hedging instrument. He goes on to speculate that dynamic hedging might be more advantageous in situations of cross-hedging.
3 There are several alternative parameterizations of the multivariate GARCH model (see Bera and Higgins 1993, Engle and Kroner 1995). The vector model used by Bollerslev, Engle and Wooldridge (1988) is one of the most commonly used parameterizations. However, the restrictions for positive definiteness of the variance-covariance matrix are very difficult to impose during estimation (see Engle and Kroner 1995).
4 The conditional mean and the conditional variance of the hedged rerum can then be written as [Mathematical Expression Omitted], [Mathematical Expression Omitted], where, [Mathematical Expression Omitted], [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the conditional variances for the hedged return, the spot return and the futures return, respectively, and [[Sigma].sub.s,f,t] is the conditional covariance between spot and futures returns.
5 See Kroner and Sultan (1993), Cecchetti, Cumby and Figlewski (1988), Baille and Myers (1991), Park and Bera (1987) and Ederington (1979), among others.
6 The matrices A and B are taken to be diagonal because likelihood-ratio tests fail to reject the hypothesis that the off-diagonal elements are zero.
7 The matrices in (8) are defined as follows:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
8 The error correction E[C.sub.t] is the residual from the regression [s.sub.t] = [[Delta].sub.0] - [[Delta].sub.1][f.sub.t] + E[C.sub.t], where [s.sub.t] and [f.sub.t] are the natural logarithms of the spot and the futures price, respectively. According to Engle and Granger (1987), if two variables are cointegrated, then their short-term movements should be influenced by the deviations from the cointegrating relationship.
9 The to-be-announced market (TBA) is used by mortgage originators with a portfolio of mortgages that have not yet been pooled.
10 The Phillips-Perron (PP) tests are based on the regression [y.sub.t] = [Mu] + [Beta](t - T/2) + [Alpha][y.sub.t-1] + [u.sub.t-1], where [y.sub.t] is the series to be tested for a unit root, and T denotes the number of observations. The null hypothesis is that [Rho] = 1, and the stationary alternative is that [Rho] [less than] 1. The 5% critical values are -1.95 if the regression is estimated without constant and time trend, -2.86 if the regression is estimated without time trend and -3.41 if the regression is estimated with constant and time trend.
11 Here, and in all subsequent hypotheses testing, the significance level utilized is the 5% level for simplicity and uniformity.
12 It should be noted that the correlations implied by Equation (8) are also time-varying. A constant-correlation version along the lines of Kroner and Sultan (1993) and Park and Switzer (1995) was also estimated. However, in all cases the hypothesis of constant correlation was rejected across all coupons. Such rejections are highly likely in situations of cross-hedging as opposed to direct hedging.
13 It may appear counterintuitive that the GARCH model performs slightly better than the EC GARCH within sample, given that the inclusion of the EC term improves the value of the likelihood somewhat. The paradox is easily resolved when we realize that the variance of the hedged portfolio, Var([r.sub.h]), is calculated subsequent to the estimation of the time-varying hedge ratio and it is not the same as the variance of the error term, Var([[Epsilon].sub.s,t]), in the conditional-mean equation.
14 Empirical durations are calculated from the regression [Delta][S.sub.t] = a + b [Delta]y + e, where [S.sub.t] is the log of the price of the MBS, a and b constant parameters and e the error term. The empirical duration is the estimate of b. The empirical-duration-based hedge ratio is simply the ratio of b to the duration of the 10-year T-note. Note that these hedge ratios are not the same as the OLS-based hedge ratios, because the information used is different. Specifically, they utilize information on the duration of the 10-year T-note and the covariance between changes in the MBS price and changes in the 10-year yield. Since the duration of the 10-year T-note changes as the yield changes, these hedge ratios are time-varying both within and out of sample.
15 Goodman and Ho (1997) examine FNMA and GNMA MBSs over the period September 12, 1994 to May 29, 1996.
16 This value for [Theta] is in line with most empirical studies in the literature [see Kroner and Sultan (1993) and references therein].
17 The results using empirical duration (not reported) are very close to those produced by simple OLS.
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