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Hedging and Value at Risk: A Semi-Parametric Approach

Zhiguang Cao Shanghai University of Finance and Economics Richard D.F. Harris University of Exeter Jian Shen University of Exeter Paper Number: 08/08 October 2008

Abstract The non-normality of financial asset returns has important implications for hedging. In particular, in contrast with the unambiguous effect that minimum-variance hedging has on the standard deviation, it can actually increase the negative skewness and kurtosis of hedge portfolio returns. Thus the reduction in Value at Risk (VaR) and Conditional Value at Risk (CVaR) that minimum-variance hedging provides can be significantly lower than the reduction in standard deviation. In this paper, we provide a new, semi-parametric method of estimating minimum-VaR and minimum-CVaR hedge ratios based on the Cornish-Fisher expansion. We apply the method to four equity index positions hedged with equity index futures. We find that the semiparametric approach is superior to the standard minimum-variance approach, and to the non-parametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently generates hedge portfolios that have lower VaR and CVaR. Keywords: Equity index futures; Hedging; Value at risk; Conditional value at risk; Skewness; Kurtosis; Cornish-Fisher expansion.

Address for correspondence: Professor Richard D. F. Harris, Xfi Centre for Finance and Investment, University of Exeter, Exeter EX4 4ST, UK. Tel: +44 (0) 1392 263215, Fax: +44 (0) 1392 263242 Email: [email protected].

1. Introduction

The motivation for minimum-variance hedging, and the benefits that it provides, are by now well established in the literature. However, minimum-variance hedging is optimal from a risk reduction perspective only when investors have quadratic utility, or when returns are drawn from a multivariate elliptical distribution. In practise, neither of these assumptions is likely to hold.1 When investors do not have quadratic utility and returns are not elliptically distributed, variance is no longer an appropriate measure of risk since it ignores the higher moments of the return distribution. This has led to the emergence of alternative measures of risk. Of these, perhaps the most widely used are Value at Risk (VaR) and Conditional Value at Risk (CVaR). The importance of VaR stems from its adoption by the Basle Committee on Banking Regulation for the assessment of the risk of the proprietary trading books of banks and its use in setting risk capital requirements (see Basle Committee, 1996; Jorion, 2007). Increasingly, VaR and CVaR are also now used in asset allocation (Huisman, Koedijk and Pownall, 1999; Campbell, Huisman and Koedijk, 2001; Alexander and Baptista, 2002, 2004) and portfolio performance measurement (Dowd, 1998; Alexander and Baptista, 2003).

In the context of hedging, Harris and Shen (2006) develop minimum-VaR and minimum-CVaR hedge ratios, estimated non-parametrically using historical simulation. The non-elliptical nature of returns is particularly important in the hedging context because while minimum-variance hedging unambiguously reduces portfolio variance, it can actually increase negative skewness and kurtosis, leading to portfolios that are riskier when measured by VaR or CVaR than when measured by variance. Harris and Shen show that that minimum-VaR and minimum-CVaR hedging generates modest out-of-sample improvements in the VaR and CVaR of the hedge portfolio, respectively, relative to minimum-variance hedging. However, a drawback of the non1

If returns are drawn from an elliptical distribution, the higher moments of returns are preserved when stocks are formed into portfolios. However, the evidence suggests that this is not the case (see, for example, Simkowitz and Beedles, 1978; Agarwal et al, 1989; Tang and Choi, 1998). A number of authors have also shown that investors have preferences over the higher moments of returns, which is inconsistent with quadratic utility (see, for example, Kraus and Litzenberger, 1976; Fang and Lai, 1997; Dittmar, 2002).

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parametric approach – in common with the historical simulation approach on which it is based – is its reliance on a large historical sample of data on spot and futures returns. This is unavoidable since by construction it is only possible to accurately measure the empirical frequency of a relatively rare event (such as those that are the focus of VaR and CVaR) by using a sample of data in which there are sufficient occurrences of such events. One is then left with a trade-off between using a large sample to improve the accuracy of the estimated minimum-VaR and minimum-CVaR hedge ratios, and a small sample to capture time-variation in the true hedge ratios.

In this paper, we develop an alternative, semi-parametric approach for estimating the minimum-VaR and minimum-CVaR hedge ratios that is much less dependent on historical data. The approach is based on the Cornish and Fisher (1937) expansion, which approximates the quantile of a standardised probability distribution by adjusting the corresponding quantile of the standard normal distribution using the higher moments of the distribution. The Cornish-Fisher expansion is already widely used to estimate the VaR and CVaR of long-only portfolios (see, for example, Zangari, 1996; Jorion, 2007) and so it is natural to extend its use to the case of hedging. Using the Cornish-Fisher expansion, we first derive an analytical expression for the first order conditions to the minimum-VaR and minimum-CVaR problems. We then solve these first order conditions numerically, using sample estimates of the variance, skewness and kurtosis of spot and futures returns in place of their population counterparts. Although, like the historical simulation approach of Harris and Shen (2006), our approach requires historical data, a much shorter sample is required because it does not rely solely on the tails of the empirical distribution.

We apply the semi-parametric approach to hedge different long positions in UK equity indices with index futures contracts daily data over the period 25/02/1994 to 19/08/2008. As a benchmark, we compare the semi-parametric approach with the nonparametric approach of Harris and Shen (2006). We find that the semi-parametric minimum-VaR and minimum-CVaR hedging offers significant improvements in VaR and CVaR reduction relative to minimum-variance hedging, and that in contrast with the non-parametric approach of Harris and Shen (2006), it works well with relatively short samples of historical data.

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The outline of this paper is as follows. In Section 2 we discuss the theoretical background to minimum-variance, minimum-VaR and minimum-CVaR hedging and introduce our semi-parametric approach. Section 3 discusses the data and methodology. Section 4 reports the empirical results. Section 5 concludes.

2. Theoretical Background

As in Harris and Shen (2006), suppose that there are two assets, Asset 1 and Asset 2, with per-period returns r1 and r2 , and that a short position in Asset 2 is used to hedge a long position in Asset 1. We assume that the mean return for both assets is zero.2 The hedge portfolio, given by a long position in Asset 1 and a fraction, h, of a short position in Asset 2, has a return equal to

rp = r1 − hr2

(1)

The variance of the hedge portfolio return is given by

σ p2 = σ 12 + h 2σ 22 − 2hρ1, 2σ 1σ 2

(2)

where σ 12 is the variance of r1 , σ 22 is the variance of r2 and ρ1, 2 is the correlation coefficient between r1 and r2 . The minimum-variance hedge ratio is the value of h that minimises (2), which is easily shown to be

hσ = ρ1,2

σ1 σ2

(3)

Minimum-variance hedging, while commonly used in practice, ignores the higher moments of the hedge portfolio return distribution, in particular its skewness and kurtosis. The skewness coefficient of the hedge portfolio is given by

2

This is a common assumption when dealing with daily financial asset returns, but could easily be relaxed to allow for a non-zero mean 4

E[rp3 ]

sp =

σ 3p

(4)

s σ 3 − 3hs aσ 12σ 2 + 3h 2 sbσ 1 σ 22 − h 3 s 2σ 23 = 1 1 (σ 12 + h 2σ 22 − 2hρ1, 2σ 1σ 2 ) 3 / 2

where the skewness and co-skewness coefficients of the two assets are defined by

s1 =

E[r13 ]

σ 13

, s2 =

E[r23 ]

σ 23

, sa =

E[r12 r2 ]

σ 12σ 2

, sb =

E[r1r22 ]

(5)

σ 1 σ 22

The kurtosis coefficient of the hedge portfolio is given by

E[rp4 ]

kp =

σ p4

(6)

k σ 4 − 4hk aσ 13σ 2 + 6h 2 k bσ 12σ 22 − 4h 3 k cσ 1 σ 23 + h 4 k 2σ 24 = 1 1 (σ 12 + h 2σ 22 − 2hρ1, 2σ 1σ 2 ) 2

where the kurtosis and co-kurtosis coefficients of the two assets are defined by

k1 =

E[r14 ]

σ 14

, k2 =

E[r24 ]

σ 24

, ka =

E[r13 r2 ]

σ 13σ 2

, kb =

E[r12 r22 ]

σ 12σ 22

, kc =

E[r1r23 ]

σ 1 σ 23

(7)

Value at Risk

One approach to measuring risk that implicitly incorporates the higher moments of portfolio returns is Value at Risk (VaR). VaR, which is defined as the largest loss on a portfolio that can be expected with a particular probability over a certain horizon. If returns are drawn from a location-scale family distribution, when the mean return is zero, the (1− α ) percent VaR of a portfolio can be written as

VaRp (1− α ) = −σ p q p (α )

(8)

5

where q p (α ) is the α percent quantile of the standardised distribution (i.e. zero mean

and unit variance) of hedge portfolio returns and σ p is the standard deviation of the hedge portfolio. When returns are normally distributed the VaR of a portfolio is simply a constant multiple of the standard deviation of portfolio returns adjusted by the portfolio mean return. Our analytical expression for the minimum-VaR hedge ratio is based on the Cornish-Fisher expansion, which approximates the quantile, q p (α ) using the higher moments of the distribution of hedge portfolio returns. In particular, using the skewness and kurtosis of the return distribution, the Cornish-Fisher expansion approximates q p (α ) by

q˜ p (α;sp ,k p ) = c(α ) +

1 1 c(α ) 2 −1]sp + [c(α ) 3 − 3c(α )](k p − 3) [ 6 24

1 − [2c(α ) 3 − 5c(α )]s 2p 36

(9)

where c(α ) is the α percent quantile of the standard normal distribution, and sp and

k p are given by equations (4) and (6) above, respectively. The Cornish-Fisher approximation for the VaR of the hedge portfolio is then given by VaRp (1 − α ) = −σ p q˜ p (α ;s p ,k p )

(10)

In order to derive the minimum-VaR hedge ratio, we differentiate (10) with respect to the hedge ratio, h, and set the first derivative equal to zero. This yields the following first order condition:

∂σ p (A1 + A2 s p + A3 k p + A4 s 2p ) ∂h ∂s ∂k ∂s + σ p (A2 p + A3 p + 2A4 s p p ) = 0 ∂h ∂h ∂h

6

(11)

where A1 = c(α ) − A4 = −

1 1 1 c(α ) 3 − 3c(α )], A2 = [c(α ) 2 −1] , A3 = [c(α ) 3 − 3c(α )] and [ 8 6 24

1 [2c(α )3 − 5c(α )]. We replace the population moments in (11) with their 36

sample estimates, and solve for the minimum-VaR hedge ratio, hVaR , numerically.

Conditional Value at Risk

A shortcoming of VaR is that it does not take into account the expected size of a loss in the event that this loss exceeds the VaR of the portfolio. An alternative measure of risk that addresses this shortcoming is Conditional Value at Risk (CVaR) or Expected Shortfall (see, for example, Tasche, 2002). The CVaR of a portfolio is given by

CVaRp (1− α ) =

1

1

∫ VaR(x)dx

α 1−α σ 1 = − p ∫ q xp dx α 1−α

(12)

where the second equality holds for any location-scale family distribution. Using the Cornish-Fisher expansion, this can be approximated as 1 CVaRp (1− α ) = −σ p (M1 + (M 2 −1)s p 6 1 1 + (M 3 − 3M1 )k p − (2M 3 − 5M1 )s2p ) 24 36

where M i=

(13)

1 c (α ) i ∫ x f ( x )dx and f (.) is the standard normal probability density

α

−∞

function. In order to derive the minimum-CVaR hedge ratio, we differentiate (13) with respect to the hedge ratio, h, and set the first derivative equal to zero. This yields the following first order condition:

∂σ p (B + B2 sp + B3 k p + B4 s 2p ) ∂h 1 ∂s ∂k ∂s + σ p (B2 p + B3 p + 2B4 s p p ) = 0 ∂h ∂h ∂h 7

(14)

where A4 = −

B1 = M1 −

1 [M 3 − 3M1] , 8

B2 =

1 [M 2 −1] , 6

B3 =

1 [M 3 − 3M1] 24

and

1 [2M 3 − 5M1 ]. We again replace the population moments in (14) with their 36

sample estimates, and solve for the minimum-CVaR hedge ratio, hCVaR , numerically.

3. Data and Methodology

Data

In the empirical analysis, we dynamically hedge the FTSE 100, FTSE 250, FTSE Small Cap and FTSE All Share indices using futures contracts on the FTSE 100 and FTSE 250 indices, considering all eight possible hedge positions. Daily data were obtained from DataStream for the period 25/02/1994 to 19/08/2008, which is the longest common sample available. We use the nearest futures contract to delivery, rolling over to the next nearest contract to delivery on the first day of the delivery month. We use the daily closing price for the spot indices and the corresponding daily settlement price for the futures contracts. We calculate log returns in order to estimate the minimum-variance, minimum-VaR and minimum-CVaR hedge ratios, but use simple returns to calculate the hedge portfolio return in each case. Panel A of Table 1 gives summary statistics of the three spot series and the two futures series, including the mean, standard deviation, skewness and kurtosis coefficients and the Bera-Jarque statistic. All of the series are both negatively skewed and leptokurtic and the BeraJarque statistic rejects the null hypothesis of normality for all six series at conventional significance levels. The largest departure from normality is for the FTSE SC, which has a skewness coefficient of -1.44 and kurtosis of 12.55. The last two columns report the 99% one-day VaR and CVaR of each currency. These are estimated using Equations (10) and (13), respectively. Panel B of Table 1 reports the correlations for each of the four spot series with each of the two futures series. For the FTSE 100 and FTSE 250 indices, the correlations are almost unity with their respective futures contracts. There is also a very high correlation between the FTSE All Share index and the FTSE 100 futures contract. The lowest correlation of 0.57 is between the FTSE SC index and the FTSE 100 futures contract.

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[Table 1]

Methodology

For a long position in each of the four indices, we use a short position in each of the two futures contracts, yielding eight hedge portfolios in total. We reserve the first 250 observations for initial estimation of the minimum-variance hedge ratio using Equation (7), the minimum-VaR hedge ratio using Equation (11) and the minimumCVaR hedge ratio using Equation (14). These are then used to construct the hedge portfolio for t=251,…500. We then re-compute the three hedge ratios using data for the period t=251,…500 and use these to construct the hedge portfolio for t=501,…750, and so on. This yields a total sample of 3500 out-of-sample hedge

portfolio returns for each of the three hedging strategies. As a benchmark, we also employ the historical simulation approach of Harris and Shen (2006), which estimates the minimum-VaR and minimum-CVaR hedge ratios by minimising the in-sample VaR and CVaR, respectively, at each step.

4. Empirical Results

Minimum-Variance Hedging

Table 2 reports the out-of-sample performance of the minimum-variance hedging strategy for the eight hedge portfolios. In each case, the table reports the average estimated hedge ratio, the change in the skewness and excess kurtosis coefficients of the hedge portfolio relative to the unhedged position, and the percentage change in standard deviation, 99% VaR and 99% CVaR. The table also gives the averages of each of these values. In all cases, the standard deviation is significantly reduced reflecting the relatively high correlations between the spot and futures series. The largest reductions are for the FTSE 100 index hedged with the FTSE 100 futures contract, the FTSE All Share index hedged with the FTSE 100 futures contract and the FTSE 250 index hedged with the FTSE 250 futures contract. In contrast with the diversification effects on standard deviation, in two of the eight cases, the skewness of the hedge portfolio is more negative than the skewness of the unhedged position, and 9

in five of the eight cases, the kurtosis of the hedge portfolio is greater than that of the unhedged position. The largest increase in kurtosis is associated with those spot and futures pairs that have the highest correlations. For the FTSE 250 index hedged with the FTSE 250 futures contract, minimum-variance hedging increases the kurtosis from 7.21 to 13.96. On average, negative skewness is reduced by 0.14, but kurtosis increases by 0.56. The consequence is that in terms of VaR and CVaR, the reduction in standard deviation that arises from hedging is offset by an increase in negative skewness and kurtosis in three of the eight cases. For these cases, the reduction in VaR and CVaR is less than the reduction in standard deviation, most notably in the case of the FTSE 250 index hedged with the FTSE 250 futures contract. On average, the reduction in VaR and CVaR relative to the unhedged position is 44.30% and 42.05%, respectively, compared to the reduction in standard deviation of 45.00%. These results are similar to those reported by Harris and Shen (2006), who analyse the risk reduction of cross-hedged currency portfolios.

[Table 2]

Non-Parametric Minimum-VaR and Minimum-CVaR Hedging

Panel A of Table 3 reports the results for the non-parametric minimum-VaR hedging strategy applied to the eight hedge portfolios, using the historical simulation approach of Harris and Shen (2006). It can be seen that the estimated minimum-VaR hedge ratios are less dispersed than the minimum-variance hedge ratios, and on average, marginally lower. Compared with minimum-variance hedging, minimum-VaR hedging yields a larger reduction in negative skewness in three of the eight cases, and a smaller increase in kurtosis in seven of the eight cases. In some cases, the differences are substantial. For example, for the FTSE Small Cap index hedged with the FTSE 100 futures contract, minimum-variance hedging reduces kurtosis relative to the unhedged position by 0.45, while minimum-VaR hedging reduces it by 4.76. On average, minimum-VaR hedging reduces negative skewness by 0.20 (which is marginally more than the reduction of 0.14 for minimum-variance hedging) and reduces kurtosis by 0.59 (rather than an increase of 0.56 for minimum-variance hedging). However, the improvement in skewness and kurtosis of minimum-VaR hedging relative to minimum-variance hedging is offset by the fact that the reduction 10

in standard deviation is lower. In particular, the average reduction in standard deviation for minimum-VaR hedging is 40.06 percent, versus 45.00 percent for minimum-variance hedging. As a consequence, the reduction in VaR (and also the reduction in CVaR) is generally lower for minimum-VaR hedging than it is for minimum-variance hedging. On average, the reduction in VaR is 42.69 percent for minimum-VaR hedging and 44.30 percent for minimum-variance hedging. For CVaR, the average reduction is 41.79 percent and 42.05 percent, respectively. This rather disappointing performance of the non-parametric minimum-VaR hedging is accounted for by the fact that the sample size used to estimate the minimum-VaR hedge ratios is very small (only 250 observations) and consequently provides insufficient information about the tails of the true distribution of hedge portfolio returns.

Panel B of Table 3 reports the analogous results for the minimum-CVaR hedge ratio. Here, the historical simulation approach works rather better. There is again a reduction in negative skewness and kurtosis for many of the portfolios. On average, the negative skewness of the hedge portfolio is reduced by 0.25 (versus 0.14 for minimum-variance hedging) and the kurtosis is reduced by 0.85 (versus an increase of 0.56 for minimum-variance hedging). Again, the reduction in standard deviation is less than for minimum-variance hedging in all cases, but in many cases this is not sufficient to outweigh the skewness and kurtosis effects. Consequently, in five of the eight cases, CVaR is reduced more by minimum-CVaR hedging than by minimumvariance hedging. On average CVaR is reduced by 44.08 percent with minimumCVaR hedging, compared to 42.05 percent with minimum-variance hedging.

[Table 3]

Semi-Parametric Minimum-VaR and Minimum-CVaR Hedging

Panel A of Table 4 reports the results for the semi-parametric minimum-VaR hedging strategy, using Equation (11) in Section 2, applied to the eight hedge portfolios. Compared with the non-parametric approach, the semi-parametric approach offers considerable improvement. In particular, relative to the non-parametric approach, the semi-parametric approach offers a greater reduction in negative skewness for all eight of the hedge portfolios and a greater reduction in kurtosis for seven of the eight 11

portfolios. In all cases, the reduction in negative skewness and kurtosis is greater than for minimum-variance hedging. The reduction in VaR is greater than that provided by the non-parametric approach in all cases, and lower than that provided by minimumvariance hedging in six out of eight cases. The average reduction in VaR is 45.58 percent, compared with 42.69 percent for non-parametric minimum-VaR hedging and 44.30 percent for minimum-variance hedging. The average reduction in CVaR is 45.31 percent, compared to 41.79 percent for non-parametric minimum-VaR hedging and 42.05 percent for minimum-variance hedging.

Panel B of Table 4 reports the results for the semi-parametric minimum-CVaR hedging strategy using Equation (14) in Section 2. The results are very similar to those for the minimum-VaR hedging. Negative skewness and kurtosis are reduced relative to the non-parametric minimum-CVaR hedge portfolios in almost all cases, and relative to the minimum-variance hedge portfolio in every case. The average reduction in negative skewness is 0.29 and the average reduction in kurtosis is 1.25, which is the highest among all five approaches considered. However, the reduction in standard deviation is the lowest of the five approaches, which to some extent counteracts the skewness and kurtosis effects. In terms of the reduction in CVaR, the semi-parametric minimum-CVaR approach is superior to the non-parametric minimum-CVaR approach for four of the eight hedge portfolios, and superior to the minimum-variance approach in all cases. However, the semi-parametric minimumVaR approach offers the best overall performance in terms of CVaR.

[Table 4]

5. Conclusion

In this paper, we provide a new, semi-parametric method of estimating minimum-VaR and minimum-CVaR hedge ratios based on the Cornish-Fisher expansion. We apply the method to four equity index positions hedged with equity index futures. We find that the semi-parametric approach works better than both the standard minimumvariance approach, and the non-parametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently also in VaR and CVaR. The superior performance of the 12

semi-parametric approach stems from the fact that it employs information from the entire distribution of hedge portfolio returns, rather than just from the left tail of the distribution. The semi-parametric approach is less dependent on historical data and is thus more likely to react to changes in the ‘true’ minimum-VaR and minimum-CVaR hedge ratios. This may make the new approach even more attractive to practitioners. There are a number of interesting avenues for future research. A natural extension would be to explicitly incorporate time-varying volatility into the semi-parametric approach. The simple form of the VaR and CVaR expressions that are provided by the Cornish-Fisher expansion are easily adaptable to the application of EWMA or GARCH-type models, as has been done in the case of long-only portfolios. This could be expected to offer further improvements in VaR and CVaR reduction.

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References

Alexander, G., and A. Baptista, 2002, “Economic Implications of Using a Mean-VaR Model for Portfolio Selection: A Comparison with Mean-Variance Analysis”, Journal of Economic Dynamics and Control 26, 1159-1193. Alexander, G., and A. Baptista, 2003, “Portfolio Performance Evaluation Using Valueat-Risk”, Journal of Portfolio Management 29, 93-102. Alexander, G., and A. Baptista, 2004, “A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model”, Management Science 50, 1261-1273. Basle Committee on Banking Supervision, 1996. “Overview of the Amendment to the Capital Accord to Incorporate Market Risks”, January. Campbell, R., R. Huisman and K. Koedijk, 2001, “Optimal Portfolio Selection in a Value-at-Risk Framework”, Journal of Banking and Finance 25, 1789-1804. Cornish, E., and R. Fisher, 1937, “Moments and Cumulants in the Specification of Distributions”, Revue de l'Institut International de Statistique 5, 307–320. Dittmar, R., 2002, “Non-Linear Pricing Kernels, Kurtosis Preference and the CrossSection of Equity Returns”, Journal of Finance 57, 369-403. Dowd, K., 1998, Beyond Value at Risk, John Wiley. Fang, H., and T-Y. Lai, 1997, “Co-Kurtosis and Capital Asset Pricing”, Financial Review 32, 293-307. Harris, Richard D. F., and J. Shen, (2006), “Hedging and Value at Risk”, Journal of Futures Markets 26, 369-390.. Huisman, R., K. Koedijk and R. Pownall, 1999, “Asset Allocation in a Value at Risk Framework”, Working paper, Erasmus University. Jorion, P., 2008, Value at Risk: The New Benchmark for Managing Financial Risk, McGraw Hill. Kraus, A., and R. Litzenberger, 1976, “Skewness Preference and the Valuation of Risky Assets”, Journal of Finance 31, 1085-1100. Simkowitz, M., and W. Beedles, 1978, “Diversification in a Three-Moment World”, Journal of Financial and Quantitative Analysis 13, 927-941. Tang, G., and D. Choi, 1998, “Impact of Diversification on the Distribution of Stock Returns: International Evidence”, Journal of Economics and Business 22, 119-127. Tasche, D., 2002, “Expected Shortfall and Beyond”, Journal of Banking and Finance 26, 1519-1533.

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Zangari, P., 1996, “A VaR Methodology for Portfolios that include Options”, RiskMetrics Monitor, First Quarter: 4–12.

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Table 1: Summary Statistics and Correlations Panel A: Summary Statistics Mean

SD

Spot FTSE 100 FTSE 250 FTSE SC FTSE All

0.01% 0.02% 0.01% 0.01%

1.07% 0.81% 0.58% 0.97%

-0.20 -0.48 -1.44 -0.28

Futures FTSE 100 FTSE 250

0.01% 0.02%

1.12% 0.83%

-0.13 -0.43

BeraJarque

99% VaR

99% CVaR

5.98 7.21 12.55 6.16

1418.40 2923.00 15648.00 1611.80

3.00% 2.62% 1.91% 2.82%

3.81% 3.16% 2.75% 3.54%

5.58 6.82

1058.10 2413.00

3.22% 2.61%

3.87% 3.20%

Skewness Kurtosis

Panel B: Correlations Futures FTSE 100 FTSE 250 Spot FTSE 100 FTSE 250 FTSE SC FTSE ALL

0.97 0.70 0.57 0.97

0.70 0.95 0.80 0.76

Notes: Panel A reports the mean, standard deviation, skewness coefficient, excess kurtosis coefficient, Bera-Jarque statistic, and 99% one-day VaR and CVaR for daily continuously compounded returns. Panel B reports the correlations of each of the four spot series with each of the two futures contracts. The sample period is 25/02/1994 to 19/08/2008. The Bera-Jarque statistic has a chi-squared distribution with two degrees of freedom under the null hypothesis that returns are normally distributed.

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Table 2: Minimum-Variance Hedging HR

∆Skew

∆Kurt

%∆SD

FTSE 100 FTSE 100 FTSE 250 FTSE 100 FTSE SC FTSE 100 FTSE ALL FTSE 100 FTSE 100 FTSE 250 FTSE 250 FTSE 250 FTSE SC FTSE 250 FTSE ALL FTSE 250

0.92 0.52 0.29 0.82 0.90 0.87 0.53 0.88

-0.24 0.13 0.13 -0.17 0.24 0.17 0.59 0.28

2.07 -0.18 -0.45 1.39 0.15 5.75 -4.31 0.09

-76.76 -29.72 -17.01 -74.57 -27.31 -64.14 -37.26 -33.24

-71.57 -31.84 -19.35 -70.86 -29.48 -49.82 -45.34 -36.17

-68.36 -31.41 -19.24 -68.53 -29.02 -37.27 -46.75 -35.85

Average

0.72

0.14

0.56

-45.00

-44.30

-42.05

Long

Short

%∆VaR %∆CVaR

Notes: The table reports the average hedge ratio, the change in the skewness and excess kurtosis coefficients and the percentage change in standard deviation, one-day 99% VaR and one-day 99% CVaR, relative to the long position, for each minimum-variance hedge portfolio. The results are based on a dynamic hedging strategy that uses a hedge ratio estimated every 250 days, using the previous 250 days observations.

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Table 3: Minimum-VaR and Minimum-CVaR Hedging (Non-Parametric) Panel A: Minimum-VaR Hedging HR

∆Skew

∆Kurt

%∆SD


0.86 0.59 0.47 0.79 0.78 0.69 0.55 0.74

-0.17 0.25 0.55 -0.20 0.19 0.11 0.59 0.25

1.33 -1.27 -4.76 1.47 0.02 3.44 -4.91 -0.02

-73.88 -25.95 -2.59 -72.94 -23.84 -55.27 -36.30 -29.68

-70.60 -31.43 -18.39 -68.94 -25.93 -48.35 -44.92 -33.00

-68.47 -31.72 -20.97 -66.52 -25.88 -40.89 -46.84 -33.05

Average

0.68

0.20

-0.59

-40.06

-42.69

-41.79

Long

Short

%∆VaR %∆CVaR

Panel B: Minimum-CVaR Hedging HR

∆Skew

∆Kurt

%∆SD


0.86 0.54 0.41 0.80 0.81 0.75 0.62 0.80

-0.13 0.23 0.46 -0.16 0.20 0.27 0.86 0.23

1.00 -1.16 -3.96 1.53 0.00 1.97 -6.08 -0.07

-74.69 -27.50 -7.20 -73.42 -25.22 -59.63 -35.38 -30.93

-71.99 -32.43 -20.94 -69.52 -27.87 -54.29 -47.75 -33.93

-70.13 -32.51 -23.00 -67.02 -27.99 -47.77 -50.16 -34.07

Average

0.70

0.25

-0.85

-41.74

-44.84

-44.08

Long

Short

%∆VaR %∆CVaR

Notes: The table reports the average hedge ratio, the change in the skewness and excess kurtosis coefficients and the percentage change in standard deviation, one-day 99% VaR and one-day 99% CVaR, relative to the long position, for each minimum-VaR hedge portfolio (Panel A) and each minimum-CVaR hedge portfolio (Panel B). The results are based on a dynamic hedging strategy that uses a hedge ratio estimated every 250 days, using the previous 250 days observations. The minimumVaR and minimum-CVaR hedge ratios are estimated using historical simulation.

Table 4: Minimum-VaR and Minimum-CVaR Hedging (Semi-Parametric) Panel A: Minimum-VaR Hedging HR

∆Skew

∆Kurt

%∆SD


0.84 0.63 0.47 0.78 0.82 0.74 0.62 0.80

-0.14 0.32 0.56 -0.16 0.23 0.28 0.82 0.28

0.93 -1.95 -4.74 1.29 -0.04 1.77 -5.89 -0.11

-74.02 -24.18 -4.33 -72.54 -27.38 -59.90 -35.44 -33.11

-71.09 -33.77 -20.38 -68.96 -30.95 -54.71 -47.20 -37.56

-69.31 -35.64 -23.37 -66.81 -31.42 -48.30 -49.47 -38.14

Average

0.71

0.27

-1.09

-41.36

-45.58

-45.31

Long

Short

%∆VaR %∆CVaR

Panel B: Minimum-CVaR Hedging HR

∆Skew

∆Kurt

%∆SD


0.82 0.66 0.50 0.77 0.79 0.72 0.63 0.78

-0.12 0.33 0.62 -0.15 0.21 0.29 0.86 0.26

0.68 -2.16 -5.14 1.26 -0.03 1.50 -6.05 -0.09

-72.68 -20.54 1.16 -71.52 -25.88 -58.58 -35.06 -31.79

-70.52 -32.10 -17.77 -68.04 -29.01 -54.24 -47.15 -35.89

-69.12 -34.59 -21.23 -65.94 -29.38 -48.44 -49.46 -36.35

Average

0.71

0.29

-1.25

-39.36

-44.34

-44.31

Long

Short

%∆VaR %∆CVaR

Notes: The table reports the average hedge ratio, the change in the skewness and excess kurtosis coefficients and the percentage change in standard deviation, one-day 99% VaR and one-day 99% CVaR, relative to the long position, for each minimum-VaR hedge portfolio (Panel A) and each minimum-CVaR hedge portfolio (Panel B). The results are based on a dynamic hedging strategy that uses a hedge ratio estimated every 250 days, using the previous 250 days observations. The minimumVaR and minimum-CVaR hedge ratios are estimated using Equations (11) and (14) that are based on the Cornish-Fisher expansion.