An Evaluation of Contingent Immunization

Transcription

1 An Evaluation of Contingent Immunization Antonio Díaz María de la O González Eliseo Navarro Frank S. Skinner* Departamento de Análisis Económico y Finanzas Universidad de Castilla-La Mancha Facultad de CC Económicas y Empresariales de Albacete, Plaza de la Universidad, 1, Albacete (Spain) Tel: Antonio.Diaz@uclm.es MariaO.Gonzalez@uclm.es Eliseo.Navarro@uclm.es *Frank S. Skinner (Corresponding author) University of Surrey School of Management Guildford, Surrey,United Kingdom Tel: +44 (0) Fax: +44 (0) F.Skinner@surrey.ac.uk JEL Classification: C61, E43, G11, G12 We acknowledge the financial support provided by Junta de Comunidades de Castilla-La Mancha grants PAI and PCI and Ministerio de Educación y Ciencia grant SEJ C02-01 partially supported by FEDER funds. Anyway any error is entirely our own.

2 An Evaluation of Contingent Immunization Abstract This paper tests the effectiveness of contingent immunization, a stop loss strategy that allows portfolio managers to take advantage of their ability to forecast interest rate movements as long as their forecasts are successful, but switches to a pure immunization strategy should the stop loss limit be encountered. This study uses actual daily transactions in the Spanish Treasury market covering the period and uses performance measures that accounts for skewness and kurtosis as well as mean variance. The main result of this paper is that contingent immunization provides excellent performance despite its simplicity. Keywords: Immunization; Contingent Immunization; Active portfolio management JEL Classification: C61, E43, G11, G12 1

3 1. Introduction The aim of this research is to test the effectiveness of contingent immunization techniques. The pioneer works in this field, Leibowitz and Weinberger (1981, 1982 and 1983), developed contingent immunization as a midpoint in a risk-return framework between pure immunization and active bond management strategies. Contingent immunization is a stop loss strategy that allows portfolio managers to take advantage of their ability to forecast interest rate movements as long as their forecasts are successful, but switches to a pure immunization strategy should the stop loss limit be encountered. Specifically, contingent immunization consists of forming a bond portfolio with a duration larger or smaller than the investor s planning period depending on interest rate expectations. If the investor thinks that interest rates are going to rise more than the market expects she would buy a bond portfolio with a duration smaller that her planning period and vice versa. However, if interest rates move opposite to the investor s expectations and the portfolio value falls to a given stop loss limit then she would immunize and guarantee this lower limit for the eventual portfolio return. This strategy gives contingent immunization an option like feature: 1 limiting losses but preserving upside potential if interest rates movements are favourable. Therefore contingent immunization strategies represent an attempt to capture positive (or avoid negative) skewness. Moreover, investors can achieve these attractive option-like distributional characteristics synthetically by trading in government bonds without the use of costly and illiquid options. 2

4 While prior work is supportive of contingent immunization, the accuracy of the results is affected by the assumption that portfolio weights are adjusted periodically rather than when a payment is made from the underlying portfolio. For instance, Fooladi and Roberts (1992) assume semi-annual rebalancing while Soto (2001, 2004) assumes quarterly adjustments. Late rebalancing can lead to poor results because immunization can be applied late after the stop loss limited is violated. Late rebalancing therefore can have an important impact on the assessment of contingent immunization as a viable strategy especially if interest rates fluctuate sharply as in the case of the Spanish market during the period This paper makes a number of contributions. First this paper makes a high computational effort in measuring the holding period returns of all strategies as realistic and exact as possible by rebalancing the portfolio each time payments are due instead of periodically and checking the portfolio value every day to determine whether the stop loss rule should be implemented. Consequently this paper makes the most accurate assessment of contingent immunization to appear in the literature so far. Second, we borrow from the recent hedge fund literature to include the mean, variance, skewness and kurtosis in our assessment of performance. This is especially important for contingent immunization because, as mentioned earlier, the stop loss limit inherent in contingent immunization can be seen as a deliberate attempt by investors to capture moments of the distribution other than mean and variance. Moreover we examine the performance of not only a variety of contingent immunization strategies, but also classical immunization, active bond and passive equity strategies. Third, we employ an extensive data set of actual daily transactions in the Spanish Treasury market covering a ten-year period from January 4, 1993 to January 3,

5 We find that contingent immunization strategies provide excellent results, as these strategies are able to capture upside potential while the stop loss limit is by and large effective in preventing large losses. We find that by adjusting performance measures for skewness and kurtosis the relative ranking of contingent immunization strategies do improve suggesting that contingent immunization strategies do improve the distributional properties of holding period returns. Moreover these attractive results are achieved without the need for interest rate derivatives that are often illiquid and require complex valuation methods. This paper is structured as follows. First, we describe the data. Then, we determine the structure of the portfolios and propose a model to implement alternative contingent immunization as well as active and pure immunization strategies. Third we introduce traditional and innovative portfolio performance measures, specifically the Shape ratio and adjusted Sharpe ratio that account for mean variance, and the modified Sharpe ratio that accounts for four moments of the distribution of holding period returns. Then we present and comment on the results and finally summarize the main conclusions. 2. Data The data set consists of mean daily bonds, bills and repo prices of actual transactions in the Spanish public debt market over the period from January 4, 1993 until January 3, This data is provided by Banco de España and is comprised of daily information on more than 66 different bonds 2 during the whole sample period as well as bills and one-week repo market transactions. 4

6 To understand the behaviour of Spanish interest rates during the sample period we first estimate the Spanish term structure of interest rates every day for the entire sample period. 3 The summary statistics of monthly changes of interest rates are given in Table 1. The levels of one-month, one-year and ten-year interest rates are represented in Figure 1. Clearly there was a dramatic decrease in interest rates and a twist in the yield curve during the sample period especially during the first half. Moreover, these dramatic changes in interest rates are reflected in the distributional characteristics of changes in interest rates where, especially in the first half, interest rate changes are characterised by negative skewness and large excessive kurtosis. As expected, short rates have a greater volatility than long-term interest rates. Moreover we conduct a factor analysis of the yield curve. 4 The first three factors can be identified (as usual) as parallel, slope and curvature changes of the yield curve accounting for 77.94, and 4.68% of the total variance respectively. Compared with other countries (see Barret, Gosnell and Heuson 2004 and Driessen, Melenberg and Nijman 2003) parallel shifts seem to explain less of the behaviour of the term structure so the risk of failure of immunization strategies due to twists of the yield curve is greater than in other default free bond markets. 5 < Insert Table 1 about here > < Insert Figure 1 about here > 3. Portfolio design We specify a three-year planning period and divide the sample into twenty-nine three-year overlapping periods. Each period starts at quarterly intervals on the first trading day of January, April, July and October from January 1993 to January The 5

7 opportunity set consists of six Treasury bonds with the highest liquidity selected from all the Spanish government bonds outstanding. 6 Two bonds must have a remaining maturity shorter than three years, two bonds must have a maturity longer than three years and the final two bonds must have a maturity close to three and ten years. The initial portfolio is rebalanced each time coupons are paid either by reinvesting these payments among the existing bonds in the portfolio or in new bonds with a duration to maintain the given strategy. This is an important innovation as prior empirical work, for example Fooladi and Roberts (1992) and Soto (2001, 2004), periodically adjust the portfolios irrespective of the pattern of underlying bond payments. Our procedure can have an important impact on the accuracy of the holding period returns especially since interest rates fluctuate sharply in the Spanish market during the period Díaz, Merrick and Navarro (2006) and Sarig and Warga (1989) note that bonds become less liquid as they approach maturity. Therefore if we cannot find transactions for bonds with a suitable maturity to replace maturing bonds we use Treasury bills. In case neither bills nor bonds are found to replace maturing bonds we use one-week repos. This means that the portfolio must be rebalanced each week until the end of the holding period. In any event we use the daily mean of actual transaction prices of bonds, bills and repos. 4. Methodology We develop a portfolio selection model to conduct out of sample contingent and pure immunization strategies that allows for different degrees of risk and expectations 6

8 about interest rate movements. These contingent immunization strategies are modelled through the following quadratic programming set-up: N 2 Min ω j (1) j= 1 subject to H N j= 1 ω D j j = dif N j= 1 ω j = 1 0 ω j 1 where N is the number of bonds available, ω j is the weight of bond j in the portfolio, H is the investor horizon, dif is the difference between the investor horizon and the m portfolio duration (H-D), and D = t C P ( t ) j i = ij ij ij P 1 0 j is the Fisher and Weil duration of bond j. 7 In the duration term C ij is the cash flow i generated by the bond j (C ij > 0) due at time t ij, P 0 (t ij ) is the present value of a unit zero coupon bond with maturity at time t ij and P j is the bond price. The objective function is set in order to choose, among those portfolios that meet the constraints, that one with the maximum dispersion. This portfolio is the one with the maximum degree of diversification and the lowest idiosyncratic risk. The variable dif determines the degree of activeness of the strategy. We have a three-year planning horizon and we consider six values for dif for the active strategies, -1.5, -1, -0.5, 0.5, 1, 1.5 years, that corresponds to starting durations of 4.5, 4.0, 3.5, 2.5, 2.0 and 1.5 years. The first three would be applied if the investor expects a fall in interest rates below forward rates and the last three if the investor expects a rise in interest rates above forward rates. The larger the absolute value of dif the more active portfolio management 7

9 and thus, the greater the investor s expected return but also the greater the risk. Finally the last set of constrains indicate that short sales are not allowed in the bond market. We proceed to run our contingent immunization strategies in the following way. Let R H be the default-free zero coupon rate at the beginning of the holding period with maturity of three years. This is the target return of pure immunization strategies. Let V 0 be the initial and V H be the final portfolio value. When following a contingent immunization strategy we allow the manager to undertake active bond portfolio management but we require that losses derived from unexpected interest rate movements to be within a given limit. In our study, we consider three different minimum returns: 50, 100 and 150 basis points below the target return, that is, below R H. In the first case, for instance, the minimum final portfolio value we would demand is V * H = V 0 (1 + R H ) H. To guarantee this minimum final portfolio value we check every day the current portfolio value in the following way. Assume that t days after the initial investment, portfolio value is V t. We calculate, according to the interest rates outstanding at t, the current portfolio value that we need to guarantee, at the end of the holding period, an amount equal to V H * if we immunize our portfolio at t. This value is given by: V * t * VH = (1 + R ) H t H t (2) * where R H-t is the time t zero coupon rate with term to maturity equal to H-t. If V t > V t we continue with the active portfolio management. On the other hand, if V t V * t we would proceed to immunize our bond portfolio making its duration equal to the remaining time left in the three-year planning period, that is, applying the quadratic programming model again but replacing the first constrain to make dif equal to zero. 8

10 Note that if the portfolio value reaches this limit (V * t ) and so the portfolio is immunized, we consolidate losses, not being able to take advantage of favourable future interest rates moves. However recall that the aim of contingent immunization is to eliminate the possibility of larger losses. In order to compare contingent immunization with active management as well as with pure immunization we also obtain the results of these two extreme strategies. In the case of active management, we apply the portfolio selection model, described in equation [1], but we do not require any lower limit for the final portfolio value. That is, we do not immunize our portfolio even if we are incurring large losses. We go on with the active management until the end of the holding period as if the investor is still waiting for a final interest rate movement according to their initial expectations. We do this by maintaining portfolio duration larger or smaller than the holding period. In the case of pure immunization, we apply our portfolio selection model (equation [1]) but make dif equal to zero (D=H). That is we proceed to immunize the portfolio over the whole holding period. Since we have six active strategies with durations of 4.5, 4.0, 3.5, 2.5, 2.0 and 1.5, and three levels of the stop loss at 50, 100 and 150 basis points, we have in total 18 different contingent immunization strategies. Counting the six active strategies and the pure three-year immunization strategy we examine 25 strategies in all. To achieve these objectives we have to overcome two difficulties. First, we have to determine the exact moment when immunization should be activated (the moment when V t V * t ). We check if the portfolio value has hit the stop limit every day even though this requires a high computational effort. Second, in the case of contingent immunization with dif larger than zero, it is sometimes impossible to maintain the original strategy as we approach the end of the 9

11 three-year planning horizon. For instance if dif=0.5 with a holding period of three years we must decide what to do when we approach two and a half years after the beginning of the holding period as it is no longer possible to keep the portfolio s duration 0.5 years below the remaining holding period. In this case, we immunize the portfolio until the end of the holding period provided it was not immunized before due to adverse changes in interest rates. 8 We should note that there are two factors that can cause contingent immunization to fail in its objective of guaranteeing a minimum return. First immunization can fail because our immunization strategies are based on Fisher-Weil duration and Fisher-Weil duration requires parallel shocks in the term structure. We know from Table 1 and Figure 1 that the Spanish term structure experienced non-parallel shifts during the sample period. Second, it is possible that a very sharp and sudden interest rate movement could cause the portfolio value to fall well below the limit before we can immunize Results For each strategy we calculate the difference between the three-year zero coupon bond yield available at the beginning of each strategy (benchmark return) and the realized return. Table 2 provides a description of the main statistics of the benchmark return and its three-year changes over the 29 holding periods. The volatility of the target return is much higher during the first six years, which roughly corresponds to the convergence period of the Spanish currency to the Euro. Therefore we split the sample period into two sub-periods to see if a difference in the behaviour of interest rates during these two periods influences the effectiveness of the different strategies

12 < Insert Table 2 about here > Evidently changes in the three-year interest rates are dramatic over the sample period with a striking maximum variation of 736 basis points. 11 Overall interest rates are declining during the sample period and this decrease is only interrupted in two intervals, during June 1994-September 1995 and June 1999-January This is a very interesting scenario for active management. If an investor forecasts these movements of interest rates she can obtain an extraordinary return by investing in long-term bonds. However, if an investor follows the wrong strategy (in this case investing in short term bonds) the aim of contingent immunization is to limit the losses to the stop limit. Table 3 describes the results obtained for each strategy. In columns 2 to 4 we show the mean of the differences between the actual return obtained when following contingent immunization and the three-year benchmark return of the pure immunization strategy over the 29 holding periods. As expected investing in long term bonds (dif = H- D < 0) yield excellent results because interest rates are decreasing during the sample period. An active bond manager forecasting these interest rates movements would enjoy a very significant extra annual return, the greater the difference between duration and investor horizon the greater the return. Thus, for dif = years the extra annual return is between 80 and 83 basis points while for dif = -1.5 years this reward is between 166 and 198 basis points. < Insert Table 3 about here > However if the investor follows the wrong strategy and invests in short term bonds the average losses are cut off at the stop limit. This lower limit guarantees a minimum return of 50, 100 or 150 basis points below the target return. 12 Table 3 shows that if this minimum return is fixed at 50 basis points below the three year zero coupon 11

13 rate, the mean return when following the wrong strategy is between 24 and 34 basis points less than the target return. With a 150 basis point stop limit the average loss ranges between 42 and 76 basis points. We conclude that, on average, contingent immunization is successful. It is also interesting to see that pure immunization (dif = 0) yields an average return of 19 basis points over the target return. This is a typical result since fixed coupon bonds have positive convexity so that a pure immunization strategy obtains the promised yield as a minimum or a higher amount (for details see Skinner, 2005). However we also observe a great deal of variation where the results from the pure immunization strategy can be 64 basis points below and 83 basis points above the benchmark return. This highlights the limitations of the pure immunization strategy and suggests the importance of developing more accurate immunization techniques. 13 These two extreme cases correspond to the holding periods from October 1993 to October 1996 and from October 1999 to October Figure 1 shows that during the first period an exceptional twist of the entire yield curve occurs where the yield curve moves from a very steep downward slope to an increasing curve at the end of this period. During the second period the yield curve flattened to become, eventually, upward sloping. This behaviour of the yield curve can explain the relative failure of immunization techniques based Fisher-Weil duration as Fisher-Weil duration assumes parallel shifts in the yield curve. When examining the range of results of contingent immunization we can see that the stop limit is also trespassed. In fact, the maximum loss is nearly 50 basis points below the stop limit in the three cases, specifically for a duration difference of for a 50 and 100 basis point stop limit and for a duration difference of +1.0 and a 50 basis 12

14 point stop limit. However, the violation of the lower limit is only serious in the case of very active strategies when the duration difference is large. Moreover when we look at the maximum values we can see the attractiveness of contingent immunization. Investors following an active portfolio management can obtain nearly an annual 5 % extra yield. Figures 2(a) to 2(d) clearly illustrate these results. They show the return above the three-year benchmark rate for contingent immunization with a stop loss limit of 50 basis points for each of the 29 holding periods. The vertical axis represents the actual return above the three-year zero coupon rate available at the beginning of each holding period and the horizontal axis represents the unexpected change in the three year rate during the holding period. The unexpected changes are estimated as the three-year zero coupon bond rate at the end of each holding period minus the corresponding forward rate outstanding at the beginning of the each holding period. To help interpret these figures the solid horizontal line represents the performance of a perfectly immunized portfolio where the return is zero basis points above the target three-year zero-coupon rate. These figures illustrate the different patterns of the extra return for each strategy. Figure 2(a) shows the most active strategy where the investor expectations are correct. Generally this strategy achieves very good results but there are some poor results even when interest rates unexpectedly decrease. This is because we compute unexpected changes in interest rates at the end of the holding period. However if during the investor s planning period interest rates increase and the stop loss limit is reached we immunize the portfolio thereby consolidating losses irrespective of later interest rate movements. If we maintain the active strategy instead of immunizing it is possible that some of these losses would have been recovered. This is in fact the price of 13

15 contingent immunization. Figure 2(b) is similar but with a lower degree of activeness (dif = -0.5) the results are less striking. < Insert Figure 2 about here > Figures 2(c) and 2(d) provide the results of an active strategy where the investor s expectations are incorrect. We can see that in the case of a very active and risky strategy (dif = +1.5) the losses are limited. It is obvious that the stop loss limit (50 basis points) is sometimes trespassed but this infringement is not too damaging. Figure 3 reports the results for pure immunization and illustrates that for this sample period, even the most passive strategy sometimes failed. Although the extra return is close to zero in most periods sometimes the difference between the target and actual return is 50 basis points or more. Since Fisher-Weil duration is used to implement all strategies and Fisher-Weil duration assumes parallel term structure changes, it is evident that changes in the term structure during the sample period are much more complex and this probably damages the effectiveness of both contingent and pure immunization strategies. < Insert Figure 3 about here > Returning to Table 3 we compare the results of contingent immunization with those obtained by an active strategy without a stop loss limit. The greatest loss from the active strategy is when the investor chose a duration difference of This loss, 103 basis points below the target return, is in contrast to the loss from the contingent immunization strategies, which are 30, 51 and 70 basis points depending on the specific strategy. Moreover the maximum loss from the pure active strategy is 343 basis points. This is a much larger loss than any of the three contingent immunization strategies considered in this study (97, 149 and 190 basis points). These results illustrate the 14

16 effectiveness of the stop loss limit of contingent immunization. Moreover the possibilities of extraordinary gains are very similar in both contingent immunization and active strategies. Another noteworthy outcome is the minimum return obtained by active management when following the strategy dif = In this case, the worst result for the active strategy is +9 basis points over the target return whereas the worst result when following the contingent strategy is -144 basis points. This result again highlights the price of contingent immunization as once the stop loss limit is reached losses are consolidated and the strategy is then unable to take advantage of later favourable interest rate movements. 6. Performance evaluation Until now we evaluate active, contingent immunization and pure immunization strategies by examining the potential gains and losses relative to the three-year zerocoupon yield. The next step is to examine the performance of these strategies. However the stop loss limit inherent in the contingent immunization strategy impose a floor on potential losses and represents an attempt by the investor to transform the distribution of returns to enhance positive skewness or at least avoid negative skewness. Therefore our performance measure should account for more than simply mean variance, but also skewness and kurtosis. Table 4 measures the mean, standard deviation, skewness and excessive kurtosis (compared to the normal distribution) of the 29 holding period returns for the twentyfive bond strategies and for a buy and hold strategy for investing in the IBEX 35 index. The IBEX 35 is a value-weighted index of the largest and most liquid 35 stocks on the 15

17 Madrid stock market. 14 We expect that investors have a preference for the mean and positive skewness and wish to avoid standard deviation and excessive (positive) kurtosis. Hass (2007) finds that if investors have non-increasing absolute risk aversion then excessive kurtosis is unattractive. <<Table 4 about here>> First, reading up the columns we see that as duration increases the mean and standard deviation of holding period returns generally increase. In other words as the degree of activeness of the bond strategy increases, risk, as measured by the standard deviation, is rewarded by extra return. Moreover as we read along the row we see that as we relax the stop loss limit the mean and standard deviation generally increases for strategies that hope for decreases in interest rates (dif < 0) but the mean and variance generally decreases for strategies that hope for increases in interest rates (dif > 0). As Table 1 reports that interest rates generally decrease during the sample period it is no surprise that the strategies that hope for interest rate decreases enjoyed higher mean holding period returns, but at a cost of higher standard deviation. Overall the empirical results conform very well to the mean variance framework. Looking at other moments of the distribution we see that as duration increases positive skewness usually decreases. In other words, another cost of increasing duration risk is to reduce positive skewness as well as increase standard deviation. Reading along the rows we are unable to determine if there is any systematic tendency of skewness or excessive kurtosis to change as we relax the stop loss limit. Finally we note that when compared to the buy and hold equity strategy as reported in the second row of Table 4, the duration strategies have far lower expected returns, but far more 16

18 attractive distributional characteristics with a much lower standard deviation, higher positive skewness and lower kurtosis. 15 While these results are interesting it is difficult to reach any general conclusion regarding the performance of the strategies based on examining each of the four moments of the distribution one by one. What is needed is a performance statistic that accounts for all four moments of the distribution. Fortunately the hedge fund literature suggests just such a measure. The hedge fund literature (see Kouwenberg, 2003; Gregoriou and Gueyie, 2003 and Favre and Galeano, 2002) proposes adjustments to the Sharpe ratio to account for non-normality in the return distribution through use of the value at risk (VAR) technology. VAR measures the expected loss that can occur on a portfolio within a given time interval. Portfolio managers are to set the likelihood that such a loss would occur to some small value like 5 or 1%, so this technique focuses exclusively on minimising tail risk. The corresponding adjusted Sharpe ratio ASR is as follows. ASR R R VAR p f = (3) Note that R p is the average holding period return, R f is the three-month Spanish interest rate and VAR is a measure of risk. Therefore ASR is a reward to risk ratio like the Sharpe ratio. However the ASR still assumes that holding period returns are normally distributed since VAR is measured as VAR = Nσ (4) 17

19 where σ is the standard deviation of the holding period return and N is the number of standard deviations associated with a given level of probability assuming that holding period returns are normally distributed. 16 For example N = 2.33 represents 2.33 standard deviations from the mean (representing the full range of outcomes that can occur 99% of the time assuming a normal distribution). Nevertheless the above does focus on what we are really interested in, tail risk, and it can be extended to other moments of the distribution. This modified Sharpe ratio MSR that adjusts the ASR to include the impact of skewness and excessive kurtosis of the holding period return is shown below. MSR R R MVAR p f = (5) Now MVAR is measured as follows (see Favre and Galeano, 2002) = MVAR N ( N 1) S ( N 3 N ) K (2N 5 N ) S σ (6) Notice that MVAR is simply VAR with N adjusted by the terms in brackets. The terms S and K refer to the skewness and kurtosis. S measures deviations in the symmetry from the normal distribution and K measures deviations in the peakness (and by implication, the fatness of tails) from the normal distribution. Note that if S and K were zero, MVAR becomes VAR. Table 5 reports the Sharpe, M 2, ASR and MSR ratios. The Sharpe ratio is the traditional reward to risk ratio where the numerator is the return of the strategy at hand above the three-month Spanish interest rate and the denominator is the standard deviation of the strategy s holding period return. The M 2 adjusts the Sharpe ratio for the differences in standard deviation between the IBEX 35 and the bond portfolio strategy 18

20 at hand so that we can clearly see if the equity strategy outperformed the bond strategy. Specifically if M 2 is positive, then the bond strategy is superior but if the M 2 is negative, the equity strategy is superior in the mean variance sense. If the combination of skewness and excessive kurtosis improves performance then the MSR ratio will be greater than the ASR ratio. The Sharpe ratios confirm that our empirical results conform well to mean variance theory. Specifically, as duration increases, Sharpe ratios generally increase, and as the stop limit increases the Sharpe ratios generally improve as long as duration is above the three-year benchmark, but the Sharpe ratios generally decrease with the stop limit for strategies where the duration is less than the three years. We noted earlier that this later result is due to the generally decreasing interest rates of the sample period. The M 2 ratio clearly shows that high duration strategies performed better than the generic buy and hold equity strategy. The interesting question is does the attractive distributional characteristics of the bond strategies change our perception of performance? A comparison of the ASR and MSR ratios emphatically suggests that it does since for every strategy the MSR ratio is higher than the corresponding ASR ratio. Moreover when comparing the MSR ratios for bond and equity strategies we again find that high duration strategies outperform the generic buy and hold equity strategy, but now the duration threshold where bond strategies outperform equity is lower. 17 Specifically notice that all mean variance performance measures, Sharpe, M 2 and ASR, suggests that the buy and hold equity strategy is superior to the bond strategies with a dif of 0.5 but now the MSR ratio suggests that these strategies are superior to the buy and hold equity strategies. 19

21 Additionally when the buy and hold equity strategy is superior to the contingent immunization strategies, the evident dominance of the equity strategy is much reduced when the additional distributional characteristics of skewness and kurtosis are incorporated into our measure of performance. Clearly the distributional characteristics of bond holding period returns are important as not only can performance be seen as better in absolute terms but the relative ranking of what is the best strategy can change once the impact of skewness and kurtosis is recognised. Finally we note that for strategies where the investor s expectations are correct and the stop loss limit is not usually encountered, all measures of performance agree that contingent immunization strategies generally improves as the stop loss limit is raised but none of these contingent immunization strategies outperforms the active bond strategy. Moreover for strategies where the investor s expectations are incorrect and the stop loss limit is usually encountered, all measures of that contingent immunization strategies generally improves as the stop loss limit is lowered but only one of these contingent immunization strategies fails to outperform the active bond strategy. Clearly contingent immunization does affect performance in the way it is supposed to and acts as a midpoint in the risk return framework between pure immunization and active management strategies even when we generalize risk to include skewness and kurtosis. 7. Conclusions This paper is the first to expend a great computational effort to mimic the behaviour of actual contingent immunization strategies by rebalancing portfolios every time a cash payment is made from the underlying portfolio and testing every day whether the stop loss limit is violated. Therefore this paper makes the most accurate 20

22 comparison amongst contingent immunization, active management and pure immunization strategies to appear in the literature to date. The main conclusion is that contingent immunization does indeed provide a mid point between pure immunization strategies and active bond portfolio management. As claimed by the earlier studies, contingent immunization allows investors to carry on active management but the stop loss limit is effective in limiting losses derived from failures in predicting future interest rates. Just as important however is that contingent immunization adjusts the distribution of holding period returns. Clearly the distributional characteristics of bond holding period returns are important as not only can performance be seen as better but the relative ranking of what is the best strategy can change once the impact of skewness and kurtosis is recognised. We conclude that once should recognize the impact of skewness and excessive kurtosis when measuring the performance of contingent immunization strategies. One drawback of these strategies is inherited from the effectiveness of immunization itself to guarantee a target return due to non-parallel shifts in the yield curve. Additionally contingent immunization strategies do experience violations of the stop loss limit although these violations are limited in both frequency and size. On the other hand contingent immunization strategies do limit the potential loss from an active bond portfolio strategy. Overall contingent immunization provides, on average, excellent results. It is important to note that these strategies are very simple to implement and monitor. They provide a very flexible instrument to adjust the degree of risk assumed by 21

23 the investor and, at the same time, they give the investor much of the upside potential available from the more risky active management strategies. Moreover contingent immunization achieves an attractive distribution of returns without the need for complex valuation models and hedging strategies requiring the use of often illiquid interest rate derivatives. 22

24 Reference Barrett, W., Brian, T., Gosnell F. and Heuson J., "Term-Structure Factor Shifts and Economic News, Financial Analysts Journal, Vol. 69, n.5, Sept, Bierwag, G., Duration Analysis. Managing Interest Rate Risk. Ballinger Publishing Company, Bierwag, G., Kaufman, G. G. and Toevs, A., Recent Developments in Bond Portfolio Immunization Strategies. Innovations in Bond Portfolio Management. JAI Press, Buhler, W., Uhrig-Homburg, M. Walter U. and Weber, T., An Empirical Comparison Of Forward-Rate And Spot-Rate Models For Valuing Interest-Rate Options, Journal of Finance, Vol. 54 m. 1, Feb, Díaz, A., Merrick, J. J. and Navarro, E., Spanish Treasure bond market liquidity and volatility pre- and post- European Monetary Union. Journal of Banking and Finance, Vol. 30, n. 4. April, Driessen, J., Melenberg B. and Nijman T., Common Factors in International Bond Returns. Journal of International Money and Finance, Vol. 22 n. Oct, Elton, E., Gruber, M and Michaelly, The structure of spot rates and immunization. The Journal of Finance Vol. 45, Fabozzi, F., Bond Markets, Analysis and Strategies. Prentice-Hall International Editions, Favre, L. and J. Galeano, Mean modified Value-at-Risk Optimization with Hedge Funds. Journal of Alternative Investments Vol. 5,

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26 Sarig, O. and Warga, A., Bond price data and bond market liquidity. Journal of Financial and Quantitative Analysis, vol. 24, n. 3. September, Skinner, F., Pricing and Hedging Interest and Credit Risk Sensitive Instruments. Elsevier Finance. Soto, G. M., Immunization Derived from a Polynomial Duration Vector in the Spanish Bond Market. Journal of Banking and Finance, vol. 25, n. 6, Soto, G. M., Generalized M-Vector Models for Hedging Interest Rate Risk. Journal of Banking and Finance, vol 27, n. 8, Soto, G. M., Duration Models and IRR Management. A Question of Dimensions. Journal of Banking and Finance, vol. 28, n. 5, Vasicek, O. and Fong, H Term Structure Modelling Using Exponential Splines. The Journal of Finance. Vol. 37, n. 2, pp

27 Table 1. Summary statistics of monthly changes of interest rates from the estimated Spanish term structure. Jan 93-Jan 03 Jan 93-Dec 98 Jan 99- Jan 03 1month 1 year 10 year 1month 1 year 10 year 1month 1 year 10 year Mean (%) Stand.Dev Median Maximum Minimum Skewness Kurtosis

28 Table 2. Summary statistics of target returns (three-year zero coupon bond yields). Level (%) Three year changes Jan93-Jan03 Jan93-Dec98 Jan98-Jan03 Jan93-Jan03 Jan93-Dec98 Jan98-Jan03 Mean St. Dev Median Maximum Mínimum Skewness Kurtosis

29 Table 3. Summary Results of Contingent Immunization Strategies for the period January 1993 to January Mean Minimum Maximum H D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) (years) Act Act Act N/A N/A N/A

30 Table 4. Distributional characteristics of the strategies for the period January 1993 to January Moment Mean Standard Deviation Skewness Kurtosis IBEX H D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) (years) Act Act Act Act

31 Table 5. Performance of the strategies for the period January 1993 to January Moment Sharpe M 2 ASR MSR IBEX N/A H D Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) Lower bound for losses (b.p.) (years) Act Act Act Act

32 Figure 1. Level of one-month, one-year and ten-year interest rates % interest rates /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/2002 Date 1 month interest rate 1 year interest rate 10 years interest rate

33 Figure 2. Results of Contingent Immunization 1(a) dif = Stop loss: 50 basis points Actual-target return (b.p.) Unexpected interest rate change (b.p.) 2(b) dif = Stop loss: 50 basis points ctual-target return (b.p.) Unexpected interest rate change (b.p.) 32

34 Figure 2 (continued). Results of Contingent Immunization 2(c) dif = Stop loss: 50 basis points Actual-Target return (b.p.) Unexpected interest rate change (b.p.) 2(d) dif = Stop loss: 50 basis points Actual-Target return (b.p.) Unexpected interest rate change (b.p.) 33

35 Figure 3. Pure immunization strategy Actual-Target return (b.p.) Unexpected interest rate change (b.p.) 1 In particularly it could be compared to a synthetic call option. 2 See Of the 66 instruments, at least 29 and at most 33 are outstanding at any point in time during the sample period. 3 Daily estimates were made using Vasicek and Fong (1982) methodology applied to mean daily prices of all bond and bills traded each day. 4 Monthly changes in one, three, six and twelve-month interest rates and two year to ten-year interest rates were used as inputs to undertake factor analysis. 5 Immunization risk is the risk of obtaining a lower return than target at the end of the investor s holding period due to term structure movements different from those assumed (usually parallel shifts of the term structure). 6 Liquidity is measured by its trading volume. For a more detailed analysis of the liquidity of the Spanish Treasury bond market and other related institutional issues see, for instance, Díaz, Merrick and Navarro (2006). 7 Our version of Fisher & Weil s duration assumes parallel shifts in the term structure, but the term structure itself can be of any shape. 8 Another possibility would be to invest in the shortest asset available, for instance in one week repo. However this strategy would incur very heavy transaction costs making this strategy extremely costly and unrealistic. 9 One could avoid this problem by checking portfolio value continuously but this would be unfeasible in practice. 10 Although the new European currency came into effect the first of January 1999, the convergence process was effectively complete prior to that date. A common measure of convergence is the difference between the Spanish forward rates and the forward rates of other European economies. This difference narrowed to virtually zero by January 1998 so we use this date to subdivide our sample. 11 The maximum variation of 736 basis points occurred during the three-year holding period from April 1995 to April Recall that the target return is the three-year zero-coupon bond yield at the beginning of each holding period. 13 These results are in accordance with other studies where immunization has been tested using one-factor models of the term structure. For this reason, some authors suggest the use of multifactor term structure 34

36 models for immunization. See for instance Elton, Gruber and Michaely (1990) and Reitano (1992) who suggest the use of a duration vector, or Navarro and Nave (1997) and Soto (2001, 2003) for its application to the Spanish Treasury market. These sorts of models usually imply further constrains for the portfolio holdings adding complexity to the portfolio selection model. 14 The holding period returns for the IBEX 35 is derived from total returns that assumes that all dividends are re-invested in the stocks just as we assume all coupons are re-invested in the bond portfolio strategies. The IBEX 35 holding period returns are calculated over precisely the same dates as the bond portfolio management strategies. 15 We split the sample into two parts to see whether this conclusion is robust to the different interest rate environments as reported earlier in Table 1 and Figure 1. The conclusions we reach with respect to the bond strategies are the same. However during the first period when interest rates were sharply decreasing the equity market performed better with an extraordinarily high mean of 34% and higher positive skewness and lower excessive kurtosis that the bond strategies. Again however the standard deviation of the equity strategy was far higher than all the bond strategies. For the sake of brevity we chose not to report these results but they are available from the authors upon request. 16 Strictly speaking this is a striped down version of VAR as usually VAR = V p σnt 0.05 where V p is the value of the portfolio and T is the time in days it takes to windup a position. However as shown in Favre and Galeano (2002) the value of the portfolio V p appears in the numerator of the adjusted Sharpe ratio and so cancels out when the full VAR expression is included in the denominator. Therefore we neglect the term V p as it will cancel out anyway in the ASR. Also, the square route of T is neglected by Favre and Galeano (2002). In essence they assume that the position can be liquidated in one day so the daily earnings at risk are the same as the value at risk. 17 As discussed in note 15 we split the sample into two parts to see whether this conclusion is robust to the different interest rate environments as reported earlier in Table 1 and Figure1. The sub period results are somewhat different. During the first period when interest rates were sharply decreasing, low duration portfolios with tight stop loss limits performed better when we include the impact of skewness and excessive kurtosis than the more active strategies. Also the equity market performed better that the bond strategies. Nevertheless the conclusion is the same. Accounting for the additional distributional characteristics of skewness and kurtosis can change the relative ranking of which strategy performs best. For the sake of brevity we chose not to report these results but they are available from the authors upon request. 35