Portfolio selection under VaR constraints - Springer Link

CMS 2: 123–138 (2005) DOI: 10.1007/s10287-004-0030-9

Portfolio selection under VaR constraints Kostas Giannopoulos1 , Ephraim Clark2 , Radu Tunaru3 1 United Arab Emirates University, UAEU, CBE, Po Box 17555, Al Ain, United Arab Emirates

(e-mail: [email protected])

2 Middlesex University, London 3 City University

Abstract. In this paper we show that by assuming a constant variance/covariance matrix over the holding period, the VaR limits can often be exceeded within the relevant horizon period. To minimize this risk, we formulate the problem in terms of portfolio selection and propose an innovative methodology using conditional VaR that minimizes the VaR at each point of the holding period. We rewrite the optimisation problem by taking into consideration the variability of risk on all assets eligible to be included in the portfolio. Keywords: VaR, portfolio selection, Monte-Carlo simulation, conditional heteroskedasticity JEL classification: C150, G110

1 Introduction Value at risk (VaR) is a popular, modern application of the Markowitz (1959) mean/variance risk management framework. It determines the maximum amount that a portfolio’s value could lose over a given period of time with a given probability, as a result of changes in market prices or rates of return. The concept of VaR is very appealing because it is consistent with the mean-variance paradigm and summarises in a single statistical measure all possible portfolio losses over a short period of time due to “normal” market movements. Thus, it allows investment managers to reconcile investment decisions based on a relatively long time horizon with risk averse investors that prefer to monitor their portfolio holdings on a more frequent basis and do not want to see their portfolio’s value drop below an acceptable level.

CMS Computational Management Science © Springer-Verlag 2005

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Most research on VaR takes the portfolio weights as given and concentrates on improving the accuracy of VaR estimates for the given portfolio. In this paper, we take a different tack and use VaR estimates to determine the portfolio weights. The intuition behind our approach is that in the presence of heteroskedasticity, VaR can vary within the horizon period. Thus, for example, a given VaR might be accurate for a ten day horizon, but the level might be breached several times between day 1 and day 10. The upshot of this is that during turbulent periods the historical volatility underestimates the actual risk while during quieter periods it overestimates it. The riskiness of the constant weighted portfolio becomes riskier on certain dates and, consequently, is exposed to higher losses. Clusters in asset volatility exacerbate the situation in so far as over the relevant investment horizon, the value of the portfolio may drop below acceptable levels set at the time of the portfolio formation. This has serious implications for investment decisions when additional constraints, such as asset liability management and VaR limits, are imposed. Higher levels of clustering and instability in the variance-covariance matrix increase the portfolio’s conditional risk and make it more likely that constraints will be violated. One way to overcome this inefficiency is to review portfolio weights on a more frequent basis. However, such a strategy is costly and will not necessarily result in a better investment performance. In this paper we propose an innovative way to solve the portfolio selection problem and at the same time control the portfolio’s potential losses. We rewrite the optimisation problem by taking into consideration the variability of risk on all those assets eligible to be included in the portfolio. In the Markowitz framework, expected returns, volatilities and correlations are used to build portfolios that minimize risk for a given level of return. It is based on the assumption that security returns are normal or log-normal i.i.d (independently, identically distributed). Under the normal/log-normal i.i.d. assumptions, means, variances and covariances are constant over time. Hence, once investors have chosen the “right” asset weights, the portfolio will remain efficient for the entire investment horizon. Another important corollary of the normal/log-normal i.i.d. hypothesis is that portfolio risk increases proportionally with the square root of time. The problem with the foregoing paradigm is that when the i.i.d. assumption is relaxed, the paradigm’s conclusions are no longer valid. In fact, a large and growing literature starting with Mandelbrot (1963) and Fama (1965) examines the distributional properties of asset price changes and rejects the normal/log-normal i.i.d. hypothesis.1 The implication for portfolio building and risk management is that changes in the means, variances or covariances of the individual assets upset the optimality of a given portfolio and require that it be rebalanced in order to re-establish optimality. Furthermore, the volatility of asset returns will no longer be proportional to the square root of time. The problem has not been neglected in the VaR literature. For example, Duffie and Pan (1997) in their overview of VaR emphasized that the i.i.d. assumption does not hold in the real world. Artzner et al. (1997, 1999) noted the undesirable proper1 See A. Bera and M. Higgins (1996) for a review of the literature.

Portfolio selection under VaR constraints

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ties of non-normal distributions for VaR such as non-additivity. Campbell-Pownall and Koedijk (1999) looked at the Asian equity markets and observed that during periods of financial turmoil, deviations from the mean-variance framework become more severe, resulting in periods with additional downside risk to investors. Nevertheless, most approaches to calculating VaR assume a joint normal/log-normal distribution of the underlying market parameters (Duffie and Pan (1997), Jorion (1996a), Pritsker (1997), Riskmetrics (1996) and Stambaugh (1996)). Bootstrapping and Monte Carlo simulations are used when the portfolio contains nonlinear instruments such as options (Jorion (1996b), Pritsker (1997), Riskmetrics (1996), Stublo Beder (1995) and Stambaugh (1996)). Efforts to overcome the difficulty have focused on developing methods that calculate the VaR more accurately using the ARCH/GARCH methodology.2 Hull and White (1998), for example, suggest adjusting historical data on each market variable to reflect the difference between the historical volatility of the market variable and its current volatility. Other approaches such as Rockafellar and Uryasev (2000), Andersen and Sornette (1999), Basak and Shapiro (1998), Emmer et al. (2000), Gourieroux et al. (2000); Puelz (1999) and Tasche (1999), seek to develop more efficient algorithims for portfolio optimizations that minimize VaR. What these approaches have in common is that they fix the holding period and assume that the variance/covariance matrix remains unchanged for its duration. However, the assumption that the matrix remains unchanged over the relevant holding period is equivalent to the assumption of i.i.d over the holding period. In this paper, we relax the assumption of an unchanged variance/covariance matrix over the holding period and show that when asset returns are not i.i.d., that is, when conditional variances and covariances vary through time, VaR can seriously mis-estimate the true riskiness of a portfolio. More precisely, we show that actual losses can be much higher than those predicted by VaR on any given day of the holding period even though it is accurate at the ending date. We then develop a method based on conditional variances and covariances, called C-VaR, that makes it possible to determine portfolio weights that respect the VaR limits at each point of the holding period as well as at the end. The rest of the paper is organized as follows. In Sect. 2, we review the links between VaR and modern portfolio theory. In Sect. 3, we specify how the presence of heteroskedasticity can cause VaR levels to fluctuate over the horizon period and in Sect. 4 we develop the methodology for simulating portfolio of two assets with heteroskedasticity. Section 5 presents the results of the simulations, which confirm the variability of VaR within the horizon period. The variability increases with the length of the period. We then use the methodology developed in the appendix to compute the portfolio that minimizes the variability of VaR. Section 6 concludes.

2 See Engle (1982) for the ARCH model and Bollerslev (1986) for the GARCH model.

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2 The link between VaR and modern portfolio theory The method for calculatingVaR depends both on the horizon chosen and on the kinds of assets in the portfolio. The method we consider here is suitable for portfolios consisting of stocks, bonds and currencies over a short horizon. If the portfolio contains non-linear positions (derivatives) either these positions must be linearised, e.g. by multiplying their delta by the volatility of the underlying asset, or a different VaR method may be employed, e.g. see Barone-Adesi et al. (1999). In statistical terms, the VaR is the lower sided confidence interval for the change in portfolio value over a specified time horizon. Thus, given a probability level v and a time horizon T , the portfolio VaR is : P rob VART,(1-v ) < PT − P0 ≤ (1 − v) (1.a) where PT is the value of the portfolio holdings at time T . The portfolio’s expected daily loss, also known as Daily-Earnings-at-Risk (DEaR), describes the magnitude of the daily losses on the portfolio for a given probability. For example the 99% probability DEaR for a portfolio with value P will be DEaR = P σP 2.33

(1.b)

where σp is the daily volatility (standard deviation) of portfolios returns and 2.33 is the number of standard deviations which gives the one-tailed probability of 99%. VaR uses a range of standard statistical methods to look at portfolio risk over time. The most popular method today is the "variance-covariance" VaR approach. It is so named because it is derived from the variance-covariance of the relevant underlying market rates of return. If W denotes the N-dimensional vector of current portfolio weights and is the variance-covariance matrix of their returns, the portfolio variance σp2 , is given by σP2 = W T W

(2)

Knowledge of the variance-covariance matrix of these variables for a given period of time implies knowledge of the variance or standard deviation of the portfolio over this period. The above approach to estimating portfolio risk is rooted historically in the pioneering work of Markowitz (1952), who laid down the cornerstone of modern portfolio theory. Markowitz was the first to show that the risk in a portfolio of securities is equal to the weighted second moments of the multivariate distribution of their returns3 . In Markowitz’s framework risk can be seen as the uncertainty that surrounds the future value of a portfolio as given by the spread of the probability 3 We use the term risk to refer to both variance and standard deviation of returns.


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0.5% 99.4

0.5% , E(P)=100

100.6

Fig. 1. p.d.f. of Portfolio exp. value

density function of the portfolio around its expected value. This is shown in Fig. 1 by the area on the far left and right of the expected value of the portfolio. Portfolios with high expected returns are attractive but, assuming market efficiency, are associated with higher risk. There is a direct link between VaR and modern portfolio theory. In the Markowitz portfolio theory, the risk of an asset mix is seen as the variability of actual return around its expected value, which is the centre of the distribution, at the end of the investment period. The area on the far left in Fig. 1 tells us that there is a 0.5% probability that at the end of the investment horizon, the portfolio with standard deviation of 20% faces a loss of 0.46% or more, which also represents the VaR of an equal length holding period4 5 . An investor who holds a portfolio with a distribution similar to that in Fig. 1 can expect to make a loss equal to or greater than the VaR value once out of two hundred days. That is, if the VaR measure is accurate, losses greater than the VaR value should occur on average 0.5% of the time. There are additional common properties between VaR and modern meanvariance portfolio theory. VaR summarises the amount of risk embedded in a portfolio as a single number and, like modern portfolio theory, takes into account the 4 Although upside risk is welcome by any investor, Markowitz treats upside and downside risk in the same manner. For risk management purposes, we are only concerned with potential losses, not gains. Given that on symmetrically distributed returns the DEaR is half the size of the risk in Markowitz definition, we will regard these two forms of risk as equivalent. Furthermore, DEaR and VaR are equally valid for calculating the potential for gain. 5 Of course this is valid only under the same assumptions on which Markowitz theory is based. These assumptions and their implications will be examined in the next paragraphs of this chapter.

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correlation between different types of risk. Since VaR complements itself with the associated probability that these losses will materialise, it gives a more objective assessment of the portfolio’s risk exposure. Furthermore, the VaR method enables senior corporate management to assess the magnitude of the risks involved, since the risk shown as the size of potential monetary losses can easily be compared with the portfolio’s expected return and with the firm’s own capital.

3 The uncertainty of VaR One of the primary limitations of the mean-variance portfolio selection model arises from the non-stationarity of security returns. Historical estimates of variances are simply static measures of the variability of returns around their mean. They fail, however, to utilise information about any dynamic properties that might be contained in the series, such as clustering effects and/or stochastic changes in means. The historical variance is easy to compute but relies upon strong assumptions regarding the process generating the underling series. The strongest is the assumption of homoskedasticity or constant variance. The GARCH methodology is an improvement in that it expresses the variance of returns as an autoregressive function of past squared errors. Hence, it represents a dynamic parameterisation by allowing the current variance to depend on the magnitude of the most recent errors and on its own lagged values. Among others, it allows the data to exhibit excess kurtosis, a phenomenon commonly observed in financial time series. Furthermore, it relaxes some of the restrictions imposed by the constant variance hypothesis and embodies the latter as a special case.6 However, as we mentioned above, even when the historical variance/covariance matrix is estimated with these more sophisticated methods, assuming that it will remain constant over the relevant holding period is equivalent to assuming i.i.d. over the holding period. Before analysing the market series and examining the impact that volatility instability can have on portfolio selection, we first set up an illustrative example showing what we mean by the unreliability of VaR measures and how this concept differs from VaR uncertainty due to Jorion’s (1996) estimation risk. Estimation risk in measuringVaR is a statistical problem due to sampling variation. The unreliability of VaR in this paper is due to the behaviour of actual stock returns over the holding period. Consider the returns of two hypothetical stocks, A and B, having equal historical standard deviations of 25% over a period of time, t = 1 to T . Let us assume that both series follow a GARCH(1,1) process as in Bollerslev (1986), over the same period. The conditional variances for each stock, as measured by the GARCH model, together with the historical (constant) variances are reproduced in Fig. 2. 6 The reasons why ARCH models are better estimators than historical and unconditional models have been well documented in the academic literature. For a synthesis of the ARCH modelling properties see Bollerslev et al. (1992).


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0.8

0.7

0.6

0.5 stock B

0.4 stock A

0.3

0.2

0.1 1988

1989

1990

1991

1992

1993

Fig. 2. Historical vs conditional volatilty

This graph illustrates how the frequency and the range of the conditional volatility of two stocks with the same unconditional volatility can differ over time. These conditional volatility graphs show that stock A’s volatility is more stable around the horizontal line representing the unconditional volatility while stock B’s volatility is more unstable and goes to extreme values. However, in the current framework of VaR calculations, both stocks get the same measure of risk, although it is clear that, over the whole time period, the risk is strikingly different. Whereas the conditional volatility for stock A is more stable over time than the conditional volatility for stock B, it is more often above stock B’s volatility. This is logical since both have the same unconditional volatility and stock A’s volatility has to make up for the short periods when stock B’s volatility jumps in value. The obvious conclusion from the foregoing example is that during turbulent periods the historical volatility underestimates the actual risk while during quieter periods it overestimates it. Hence, for the mean-variance investor seeking to optimise his portfolio’s risk/return trade off, changes in the portfolio variance-covariance matrix imply changes in the riskiness of the constant weighted portfolio: it becomes riskier on certain dates and, consequently, is exposed to higher losses. As a result of this volatility clustering, the value of the portfolio within the horizon period may drop below acceptable levels set at the time of the portfolio formation.

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This has serious implications for investment decisions when additional constraints, such as asset liability management and VaR limits, are imposed. Higher levels of clustering and instability in the variance-covariance matrix increase the portfolio’s conditional risk and make it more likely that constraints will be violated. One solution to the problem is to review portfolio weights on a more frequent basis. However, such a strategy is costly and will not necessarily result in a better investment performance. The solution we propose in this paper is to solve the portfolio selection problem and at the same time control the portfolio’s potential losses. We rewrite the optimisation problem by taking into consideration the variability of risk on all assets eligible to be included in the portfolio. In the rest of this analysis we will assume that the GARCH volatility estimators are the parametric estimators representing the true volatility. We should also mention that the two volatility estimators, the unconditional (historical) and the GARCH, share a common property. The average GARCH variance, E(ht ), is asymptotically equal to the historical estimate σ 2 .7 That is, the GARCH volatility, which is a function of past surprises, is mean reverting towards the historical volatility. Hence, the historical volatility is, asymptotically, the long run expectation of the GARCH volatility, E(ht ) E(ht )

=

σ2

(3)

Therefore, in the long run the two stocks, A and B, might have the same level of risk but they do not share the same degree of return uncertainty in each sub-period of the investment horizon. This phenomenon has been ignored so far in the VaR literature and it is of real interest to see what impact it may have on portfolio selection under VaR constraints. At this point we can ask, “if constant volatility is a biased estimate for the short term risk level associated with an investment,8 how can we rely exclusively on those historical estimates to compare and price different financial assets on a daily, weekly or even monthly basis?” Take, for example, the optimal portfolio selection problem. Because stocks A and B have an identical historical variance, the diversification mechanism, by its nature, will handle them as two equally risky investments and will offer both, other factors being equal, the same feasibility of being included in the optimal, constant-weighted, portfolio. Is it true that both stocks expose the investor to the same level of uncertainty? 4 Portfolio VaR and holding period heteroscedasticity 4.1 GARCH methodology used for simulation Let us now examine the implications that the failure to account for a variance/covariance matrix that varies through time might have on the portfolio’s daily 7 See, for example Greene (1997), Chapter 12. 8 By short term we refer to the time frequency over which returns are measured.


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Table 1. GARCH parameters used for simulation

δ ω α β

Asset A

Asset B

5.95238e-004 9.44822e-008 0.1 0.75

5.95238e-004 3.14941e-008 0.4 0.55

P&L account. Assume that the returns on assets A and B have the following characteristics: E(RA ) = E(RB ) (4) where RA , RB are the daily returns for assets A and B. Let us assume that the two series of return follow a similar GARCH (1,1) process, 2 + εt t = ht = ωY+ αεδt−1 + βht−1

(5.a) (5.b)

where εt is the residual return between the end of day t − 1 and the end of the day t and ht is the variance rate, that is the square of the volatility on day t. For simplicity we assumed that the price changes over the previous day are pure random walk with a constant drift, δ. To investigate the impact that conditional volatility may have on the portfolio’s VaR measured on a long horizon we use a Monte Carlo simulation. The simulation is done for two hypothetical assets A and B whose returns are described as in (5). The long run expectation (unconditional) for the mean and variance for asset A and B are equal. The way we set the parameters in (5) is described in the next paragraph. The values for all the parameters needed for simulation are given in Table 1. They are chosen on an ad-hoc basis for illustrative purposes in such a way to ensure the stability of the GARCH model. In essence, we set up E[ht ] = ω/ (1 − α − β) to be the same for both assets and the denominator to be positive. The values provided in Table 1 imply an expected return of 15% p.a. for each asset. Given that the long run expectation of the conditional volatility converges to a constant and is equal to ω E(ht ) = σ 2 = = 6.29882e − 007 (6) 1−α−β the above equality implies that the average conditional volatility between the two series is equivalent and is equal to 20% p.a. Despite that, in the long horizon like the end of the investment term, the volatility for the two stocks in the way their prices respond to everyday news is different. This response is measured by the two sets of the parameters ω, α, β. Stock B tends to respond more promptly to news but any changes in volatility has a short life. The two assets can be correlated in practice and the covariance may inherit all the problems associated with variances. In other words, the covariance must be modelled dynamically as well. We account for this by adding an extra GARCH(1,1) process for the covariance between assets A and B. Conditions ensuring the stability of the whole GARCH modelling are derived in the next section.

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4.2 Conditions ensuring GARCH stability The simulation is based on the following three GARCH processes, where the last is for the covariance between the two asset returns. 2 h1,t = w1 + a1 ε1,t−1 + b1 h1,t−1

(7)

2 + b2 h2,t−1 h2,t = w2 + a2 ε2,t−1 h12,t = w12 + a12 ε1,t−1 ε2,t−1 + b12 h12,t−1

(8) (9)

The variance-covariance matrix should be positive definite so we need to ensure h1,t h12,t > 0 , in other words that that h12,t h2,t h1,t h2,t − h212,t > 0

(10)

Using Eqs. (7–9) above gives 2 2 (w1 + a1 ε1,t−1 + b1 h1,t−1 )(w2 + a2 ε2,t−1 + b2 h2,t−1 )

(11)

−(w12 + a12 ε1,t−1 ε2,t−1 + b12 h12,t−1 ) > 0 2

It follows from the Cauchy-Buniakowski inequality that 2 2 (w1 + a1 ε1,t−1 + b1 h1,t−1 )(w2 + a2 ε2,t−1 + b2 h2,t−1 ) √ 2 √ ≥ w1 w2 + a1 a2 ε1,t−1 ε2,t−1 + b1 b2 h1,t−1 h2,t−1

(12)

So if we make a choice such that √ 2 √ w1 w2 + a1 a2 ε1,t−1 ε2,t−1 + b1 b2 h1,t−1 h2,t−1 > (w12 + a12 ε1,t−1 ε2,t−1 + b12 h12,t−1 )2 then the problem is solved. This condition is equivalent to √ √ w1 w2 + a1 a2 ε1,t−1 ε2,t−1 (13) + b1 b2 h1,t−1 h2,t−1 > w12 + a12 ε1,t−1 ε2,t−1 + b12 h12,t−1 √ because all the quantities are positive. We can choose w12 = w1 w2 and a12 = √ a1 a2 so the only thing left to satisfy is b1 b2 h1,t−1 h2,t−1 >b12 h12,t−1 which is √ h equivalent to bb121 b2 > √ 12,t−1 = corrt−1 . h1,t−1 h2,t−1 √ So, by choosing b12 = √b1 b2 (if we assume that perfect correlation is impossible in practice) or b12 = b1 b2 -0.0059 (if we allow perfect correlations of the two asset returns from time to time), we know that condition (13) is satisfied. 9 0.005 is used for a small quantity that still allows b > 0. In practice, depending on the choice of 12 b1 , b2 , any other small number can be used.


We must also take into account that w1 w2 = 1 − a 1 − b1 1 − a 2 − b2

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(14)

√ for i = 1, 2, 12. Therefore, because a12 = a1 a2 we must and that ai + bi < 1 √ √ ensure that a1 a2 + b1 b2 < 1 when we choose the parameters for simulation. 5 Asset management under uncertain VaR Now consider the situation where a manager must choose the portfolio of assets such that the uncertainty in the associated VaR measure is minimized. Depending on the correlations between the assets included in the portfolio, the weights chosen for the individual assets will determine the overall position for the portfolio. The argument detailed above can be imagined as the extreme case of portfolios with only one asset, first a portfolio consisting of asset A only and then a portfolio consisting only of asset B. In this section we are extending the argument about the uncertainty of the VaR measure to asset management. Moreover, in the appendix we describe a methodology that can simplify computationally the search for the right combination of assets in a portfolio. Table 2 reports the lower portfolio values at the 99% and 95% probabilities estimated with the conditional variance for portfolios A and B and with the unconditional variance under the “Historical” legend.10 As we can see, the values for the conditional variances are lower than those predicted by the unconditional VaR. Thus, the possible losses for each portfolio, calculated by subtracting the estimated conditional values in the table from 1000, are higher than those predicted by the unconditional VaR. Furthermore, this discrepancy becomes larger as the investment horizon increases. This finding shows that VaR estimates, as well as any other risk measure estimated by the Markowitz model, are inefficient and the inefficiency grows with the investment horizon. There is also a significant difference between the behaviour of the two portfolios. The losses for portfolio B are generally higher and the difference increases with the investment horizon. Table 3 shows the number of times that losses exceed those predicted by the unconditional VaR. Referred to as breaks, they are calculated for each (single asset) portfolio and probability level on daily basis. At the 95% probability on 10,000 simulation runs we expect the number of times the portfolio losses exceed the VaR estimate on each day to be approximately 500. Similarly, at 99% probability we expect the breaks to be around 100 each day. As we can see both at the 95% and 99% probability, the number of breaks are much higher than those predicted by the unconditional VaR model. Furthermore, as the investment horizon increases, the tendency for the unconditional VaR to underestimate the number of possible losses increases as well. 10 That is the one day DEAR multiplied by the square root of the time.

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K. Giannopoulos et al. Table 2. Lower daily values at 99% and 95% for portfolio A and B @99% DAY 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60

A 970.30 957.30 949.10 940.20 933.10 926.00 920.20 915.10 911.00 905.40 871.20 842.60 824.90 803.00 785.50

B 969.90 956.00 946.50 936.60 930.40 926.10 917.30 912.20 907.00 902.70 863.40 831.80 804.30 784.60 765.90

@95% Historical 971.20 959.70 950.90 943.70 937.30 931.70 926.50 921.70 917.30 913.10 880.60 857.10 838.20 822.20 808.30

A 979.00 970.20 964.50 959.30 954.60 950.50 946.30 943.70 939.70 936.10 909.80 890.60 873.20 860.30 848.50

B 979.20 970.70 965.30 959.80 955.40 950.90 948.70 944.30 939.60 936.30 908.50 885.90 868.80 856.70 840.60

Historical 979.80 971.80 965.80 960.80 956.50 952.70 949.20 946.00 943.00 940.20 918.90 904.00 892.30 882.80 874.70

Table 3. Break analysis

1 2 3 4 5 6 7 8 9 10 20 30 40 50 60

Breaks @99%

Breaks @95%

A 118 135 126 134 132 144 144 144 150 157 141 171 160 172 177

A 551 597 559 576 566 587 575 569 604 620 689 737 809 850 893

B 128 149 147 153 158 154 162 165 166 164 213 220 216 222 261

B 544 557 518 532 525 549 508 536 591 589 696 829 889 928 969

It is also interesting to note that the difference in behaviour between the portfolios detected in Table 2 is also reflected in Table 3. The breaks for portfolio B are generally higher than those of portfolio A. This is because portfolio B responds more promptly to the previous day’s “news”, thereby causing its volatility to rise to levels well above those of portfolio A. Table 4 shows the total number of breaks over the 60 day horizon for ten different combinations of portfolios A and B. Column 1 shows the proportion of A in the combined portfolio, columns 2 and 4 show the total number of breaks for the 95% and 99% confidence levels respectively and columns 3 and 5 show the proportion of total outcomes represented by the breaks. For example, there were


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Table 4. Portfolio of two assets Mixed portfolio breaks analysis Proportion of A 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Breaks at 95% No % 44,556 7.426% 40,823 6.804% 38,087 6.348% 36,104 6.017% 34,987 5.831% 34,915 5.819% 35,584 5.931% 37,155 6.193% 39,636 6.606% 42,910 7.152%

Breaks at 99% No % 9,595 1.599% 8,288 1.381% 7,308 1.218% 6,697 1.116% 6,486 1.081% 6,625 1.104% 7,020 1.170% 7,843 1.307% 9,060 1.510% 10,595 1.766%

NB: Column A displays the weight of asset A in the Portfolio. Of course, the weight of asset B is (1-A).

10,000 simulations and 60 days for a total of 600,000 outcomes. Thus, the 44,556 breaks at the 95% confidence level when the proportion of A is 100% represent 7.426% of the 600,000 outcomes. The first thing we notice is that all the portfolios have a higher number of breaks than what would be forecast by standard VaR. We can also see that the number of breaks decrease at first and then begin to increase. At the 95% confidence level the optimal portfolio is somewhere between A = 60% and A = 40%. At the 99% confidence level the optimal portfolio occurs somewhere between A = 70% and A = 50%. The common ground between the two confidence levels is between A = 60% and A = 50%. In the appendix we present a more efficient computational procedure for determining asset allocation. Using this methodology, we calculate the optimal portfolio at about 56%.

6 Conclusions In this paper, we have used GARCH modelling to show that simple VaR measures can seriously underestimate portfolio risk and that this underestimation grows with the horizon period. More effective risk management can be achieved if this risk can be reduced. We propose using conditional VaR to determine asset allocation that minimizes breaks in the unconditional VaR over the horizon period. Although we have illustrated the process for two asset portfolios, the same procedure can be easily expanded to a larger number of assets.

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Appendix: A more efficient computational procedure for asset allocation Here we show how to determine the proportion of different assets in an efficient portfolio. We choose the portfolio with only two assets A and B, in the proportion (w, 1− w), that is, w percent is invested in stock A and 1 − w percent in stock B. Thus, for simulation purposes, the value of the portfolio at time t and run n is P (t, n) = wSA (t, n) + (1 − w)SB (t, n)

(A1)

where t = 1, . . ., T , n = 1, . . ., N and w ∈ (0, 1). It is worth pointing out that P (0, n) = P0 is a known, fixed value for all simulations runs. If σa is the volatility per annum of the portfolio then the 99% value-at-risk is given by √ σa (A2) V aR(99%) = P0 × √ × 2.33 × T 252 Since there is no drift in the simulated prices for each of the two stocks, for all our simulations we check whether P (t, n) < P0 − V aR(99%).

(A3)

Suppose that out of N × T possible scenarios, with a momentarily fixed weight w, a number k(w) ≤ N T satisfies condition (9). Then if k(w) NT < 0.99 we reject the corresponding weight w. Otherwise we accept the portfolio with that combination. For the same simulation runs we calculate several portfolio combinations based on a grid of values for w. It is possible to decide automatically whether some weights from the grid will be accepted or not. It can be easily shown that condition (9) is equivalent to checking that V aR(99%) − SB (t, n) w< (A4.a) if SA (t, n) − SB (t, n) > 0 SA (t, n) − SB (t, n) and w>

V aR(99%) − SB (t, n) if SA (t, n) − SB (t, n) < 0 SA (t, n) − SB (t, n)

(A4.b)

Thus, if k + (w) is the number of times that condition (10.a) is satisfied and

k − (w) is the number of times that condition (10.b) is satisfied, we have the following decomposition

k(w) = k + (w) + k − (w)

(A5)


Now, for a given NT set of simulated runs, if

137 k + (w) NT

> 0.9911 then any other weight −

(w) w∗ ∈ (0, 1) satisfying w∗ < w will be accepted too. Similarly if k NT > 0.99 # # then any other weight w ∈ (0, 1) satisfying w > w will be accepted too. This is helpful to reduce the number of calculations that need to be made. Moreover, it can easily be proved that k + (w ∗ ) ≥ k + (w) for any w ∗ < w and that k − (w # ) ≥ k − (w) for any w # > w. These results imply that the computational algorithm can be made more efficient when the conditions (A4.a–b) are verified to decide what weights give the portfolio combinations that pass the VaR test. For example, one can start the procedure by simulating first N runs over T days and storing this data. Then the first weight providing the first portfolio that will be tested for VaR is w = 0.5. For this particular weight value k + (w) and k − (w) are calculated. Then if either k + (w) > 0.99N T or k − (w) > 0.99N T then the portfolio P = (0.5SA , 0.5SB ) passes the VaR test. If not there is still a chance that k(w) = k + (w) + k − (w) is greater than 0.99N T so again the corresponding portfolio is accepted. Having decided whether the portfolio with the weight w = 0.5 is good or not the searching procedure for other portfolios passing the VaR test can move in two directions, looking at weights smaller than 0.5 and weights greater than 0.5. Perhaps the best way of searching would be to consider next the extreme weights from the grid, for example 0.05 and 0.95. Then we know, without any further calculations and counting that k + (0.05) ≥ k + (0.5) and that k − (0.95) ≥ k − (0.5).

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