How does the choice of Value-at-Risk estimator influence asset allocation decisions?

Abstract

Considering the growing need for managing financial risk, Value-at-Risk (VaR) prediction and portfolio optimisation with a focus on VaR have taken up an important role in banking and finance. Motivated by recent results showing that the choice of VaR estimator does not crucially influence decision-making in certain practical applications (e.g. in investment rankings), this study analyses the important question of how asset allocation decisions are affected when alternative VaR estimation methodologies are used. Focusing on the most popular, successful and conceptually different conditional VaR estimation techniques (i.e. historical simulation, peak over threshold method and quantile regression) and the flexible portfolio model of Campbell et al. [J. Banking Finance. 2001, 25(9), 1789–1804], we show in an empirical example and in a simulation study that these methods tend to deliver similar asset weights. In other words, optimal portfolio allocations appear to be not very sensitive to the choice of VaR estimator. This finding, which is robust in a variety of distributional environments and pre-whitening settings, supports the notion that, depending on the specific application, simple standard methods (i.e. historical simulation) used by many commercial banks do not necessarily have to be replaced by more complex approaches (based on, e.g. extreme value theory).

Keywords:

• Value-at-Risk
• Portfolio optimisation
• Shortfall constraints
• Simulation

JEL Classification:

• G10
• G11
• G17

Acknowledgements

We are grateful for the valuable suggestions of two anonymous reviewers which have significantly improved the quality of our manuscript. We also thank the participants of the 15th Colloquium on Financial Markets in Cologne for their comments on an earlier draft.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Interestingly, this strong focus on the VaR continues even though it is well known that it suffers from a variety of theoretical shortcomings (see Artzner et al. 1999, Yamai and Yoshiba 2005).

2 Note that Zakamouline (2011) and Ornelas et al. (2012) challenge the finding that other risk measures produce rankings similar to the standard deviation. They do not focus on similarities between different VaR estimators.

3 Basically we can estimate portfolio VaR by (i) aggregating loss data and proceed with a univariate forecast model for the aggregate or by (ii) starting with disaggregate data and use multivariate structural VaR models. However, we follow Kuester et al. (2006) by restricting our attention to the univariate case because Berkowitz and O’Brien (2002) show that complicated structural models cannot outperform simple univariate models and that the latter model class tends to yield less conservative VaR forecasts and hence is cheaper to implement.

4 The appendix of our study reports the results of additional unconditional estimators such as the Student-t method and the unconditional counterparts to our conditional techniques.

5 We do not wish to enter the debate on finding the most appropriate specifications.

6 Note that we may also calculate the true optimum weights under the true model and compare them with the estimated weights. However, since our focus is on a comparison of estimates, we leave this exercise for future research.

7 Readers with background knowledge on the VaR estimators and/or the portfolio selection model can skip those sections and proceed directly to section 4.

8 As highlighted by Bollerslev et al. (1992) and Hansen and Lunde (2005) such GARCH models of low order typically capture empirical volatility dynamics quite well.

9 Switching to a Student-t (ST) distribution yields , where is the c-quantile of a Student-t distribution with degrees of freedom (see Jorion 1996, Alexander 2008). This distribution can be used to capture fat-tails by setting a low value for (e.g. ; see Campbell et al. 2001, Bao et al. 2006).

10 An alternative would be the historical simulation variant of Boudoukh et al. (1998) which places more weight on recent returns to better represent the risk of today’s portfolio. A more recently proposed bootstrap-based extension can be found in Su (2014).

11 A comprehensive overview of extreme value theory is provided by Embrechts et al. (1997), while Christoffersen (2003) gives a highly accessible and streamlined account.

12 They also show that the mean squared error for quantile estimates based on the POT method are far less sensitive to the choice of threshold than other methods like the Hill estimator for the tail index.

13 Depending on the algorithm used in the optimisation (see Chen and Wei 2005, Chen 2007) and the specific way of obtaining quantiles in historical simulation (see Choudhry 2013), a quantile regression result can be different from its direct theoretical counterpart historical simulation.

14 An alternative optimisation routine is the interior point algorithm of Koenker and Park (1996).

15 For other models of this kind, see, for example, Roy (1952), Leibowitz and Kogelman (1991), Lucas and Klaassen (1998) and Jansen et al. (2000).

16 Indeed, S(p) collapses to a multiple of the Sharpe index, if the portfolio returns are normally distributed and the risk-free rate is zero.

17 W(0) is just a constant which does not affect the portfolio optimisation. Also note that is the negative value of the q(c, p) definitions in section 2.

18 For other types of reward-to-VaR performance measures, see Alexander and Baptista (2003).

19 Thus, the two-fund separation of Tobin (1958) holds like in the mean–variance framework.

20 It is linked to a distribution through the quantile definition. However, no assumptions about the specific form of this distribution are required.

21 The Datastream codes for these series are SP500COMP, BMUS10Y and USGBILL3.

22 Using this sample allows a direct comparison to Campbell et al. (2001). In section 7, we illustrate the results for a more recent sample ranging from January 2000 to December 2015.

23 Note that choosing an unnecessarily high level of confidence (such as 99.9%) would lead to a false sense of risk management as the losses will rarely exceed that level.

24 A setting which reflects the current regime of very low interest rates is analysed in section 7.

25 In the appendix, we also see that some methods deliver identical weights due to their methodological closeness. The weights from the unconditional normal and Student-t approaches are the same. In addition, there are only small deviations of the unconditional historical simulation and the unconditional quantile regression.

26 This is in line with the observation made by Campbell et al. (2001) for unconditional historical simulation.

27 The appendix shows that such a result also occurs among non-normal unconditional estimators. The somewhat higher weights for the Student-t approach reduce when obtaining by fitting the Student-t distribution to the data via maximum likelihood instead of using .

28 We have also used the same VaR level (the 95% conditional historical simulation VaR) for all of our methods. However, we find that our main results on differences in asset weights, which we present in the remainder of the article, qualitatively hold.

29 Of course a higher VaR can be obtained by borrowing money at the risk-free rate (resulting in more money being invested in the risky portfolio of stocks and bonds), i.e. moving along the capital market line to the right.

30 For an excellent review of models with similar features (such as Christodoulakis and Satchell 2002, Tse and Tsui 2002) and other multivariate GARCH models, see Bauwens et al. (2006).

31 We use the Akaike information criterion (AIC) to identify the optimal values of M, N, , , and and validate the model with the Q-statistic for linear and squared residuals (as in Block et al. 2015). We cannot reject the hypothesis of lack of autocorrelation in the innovation process and its square.

32 Unlike the values suggest, the constants of the GARCH models are not zero. They are positive with .

33 We have also implemented an asymmetric model (as in Ang and Chen 2002) capturing evidence on higher asset correlations in periods of extreme downside moves. Our findings are qualitatively similar to the baseline setting.

34 and control skewness and kurtosis but do not give the actual values of these moments (see Jondeau and Rockinger 2003). Thus, table 4 should not be interpreted as modeling rather low skewness and kurtosis.

35 In setting E, we limit ourselves to positive skewness because investors typically have a preference for assets with positive skewness (see Harvey and Siddique 2000, Lucey et al. 2006). However, note that changing the signs of the skewness parameters does not affect our main conclusions.

36 We do not plot the distributions for the cash weights because they are the residual component directly determined by the stock and bond weights.

37 It is worth noting that the mean weights of the normal VaR are closer to the mean weights of the conditional methods than the mean weights of the non-normal unconditional methods reported in the appendix.

38 This also holds for the additional unconditional estimators in the appendix. When repeating our calculations with the unconditional historical simulation as the benchmark, we can also confirm that differences among non-normal unconditional approaches are rather small.

39 Because our results are similar to those of sections 4 and 6, we concentrate on a summary of our research design.

40 A comprehensive overview of other volatility structures can be found in Hansen and Lunde (2005). For other types of filters apart from GARCH models, see Duffie and Pan (1997).

41 Bao et al. (2007) provide an excellent overview of much more alternative distributional assumptions.

42 We also experimented with gold which is a typical portfolio component because of its interesting diversification and hedging properties (see Lucey et al. 2006, Lucey and Li 2015).

43 Kuester et al. (2006) provide an interesting extension using mixtures of random variables following a generalised error distribution (GED) which provides better VaR forecasts than approaches using normal mixtures.

44 It does not satisfy the sub-additivity property. That is, if two assets are combined for risk management purposes, the VaR can go up rather than down. Thus, it is not in line with the idea that diversification helps reduce risks. When we aggregate two risks, the total of the risk measures corresponding to the risks should either decrease or stay the same.

Additional information

Funding

We are indebted to the Fritz Thyssen Stiftung [grant number 20.15.0.079 WW] and the Wissenschaftsförderung der Sparkassen Finanzgruppe e.V. for generous financial support.