A Simplified Approach to the Conditional Estimation of Value at Risk (VAR)

A Simplified Approach to the Conditional Estimation of Value at Risk (VAR) by Giovanni Barone-Adesi(*) Faculty of Business University of Alberta and Center for Mathematical Trading and Finance, City University and Kostas Giannopoulos(**) University of Westminster (*) 3-20H Faculty of Business, Edmonton, Alberta, Canada T6G 2R6 (**)32-38 Wells st, London W1P 4DJ, UK

Abstract Emerging risk-management techniques use Value at Risk (VAR) to assess the market risk of a portfolio. We propose a relative simple method to estimate VAR conditionally to reflect new information about the volatility of securities held in a portfolio with changing weights. While portfolio holdings might aim at diversifying risk, this risk is subject to continuos changes. The GARCH methodology allows us to estimate past and current and predict future risk levels of our current position. The use of historical returns of portfolio components and current weights can produce accurate estimates of current risk for a portfolio of traded securities. Information on the time series properties of the returns of the portfolio components is transformed into a conditional estimate of the current portfolio volatility with no need of using complex time series procedures. Stress testing and correlation stability are discussed in this framework. 1

1. Introduction Emerging risk-management techniques use Value at Risk (VAR) to assess the market risk of a portfolio. We propose a relative simple method to estimate VAR conditionally to reflect new information about the volatility of held securities in a portfolio with changing weights. A risk management system comprises an information system, a risk measurement system and a capital allocation system. The information system is necessary to collect the necessary information flow, the risk measurement system obtains an assessment of risk from the collected information and the capital allocation system maximises risk adjusted profitability and controls the overall firm risk. The different components of risk that need to be monitored include market risk, counteparty risk, liquidity and operation risk. For traded securities information about the first three components of risk can be gathered from market data. However, the possibility of marking to market traded contracts limits the relevance of counteparty risk for them, leaving market and liquidity risk as main concerns. The VAR approach attempts to address both of these concerns simultaneously by multiplying the number of days deemed to be necessary to close out a portfolio by the largest daily loss expected on a given day at some level of probability. Often operationally the largest daily loss expected in one month, a measure of market risk, is multiplied by ten days, a measure of the average time necessary to close out a large portfolio. A criticism of VAR is that in some recent crises, such as the MBO and the Latin American ones, sudden drops in liquidity made previous estimates of close-out periods illusory. This criticism however reflects more the difficulty of producing reliable estimates of future liquidity in newer markets rather than an intrinsic weakness of the VAR approach. The other input necessary to the estimation of VAR is the largest daily loss expected in one month, usually referred as DEAR (daily earnings at risk). Under the approximation of normality DEAR equals 1.65 portfolio volatility, because the cumulative normal at that point, N~(-1.65) = 0.05 = 1/20, the number of trading days in a typical month. Therefore up to date volatility estimates are crucial to the implementation of 2

the VAR approach. We estimate the conditional volatility of a portfolio with changing weights and we discuss how to apply stress analysis to our estimates to allow for some robust inferences. Our approach allows also for a quantitative evaluation of correlation risk. 2. Volatility The estimation of VAR and DEAR requires an estimation of portfolio volatility. Unfortunately the historical volatility on a bank portfolio is an ill suited measure of its current volatility because investment weights may change rapidly and even individual securities volatility may shift over time. More over the composition of the volatilities of individual components into a portfolio volatility requires the knowledge of the correlation matrix of the returns of the different components. This correlation matrix is also possibly subject to shifts over time. Even if the correlation matrix were constant the effort required to estimate it in a multivariate time series framework would pose a computational challenge. A simple procedure to overcome the difficulties of inferring current portfolio volatility from past data may rely on the knowledge of current portfolio weights and historical returns of the portfolio components to construct the hypothetical return series the portfolio would have earned if it had been kept constant at its current weights in the past. Securities with strong non-stationarities like options may be included by substituting them with the products of their current delta multiplied by the volatility of their notional underlying assets. The resulting time series of portfolio returns is then analysed to identify the best fitting time series model. Accurate point estimates of current volatility are then produced and VAR is computed from them. 3. A simplified way to compute the portfolio s risk and return Let R t be the Nx1 vector (R 1,t, R 2,t,..,R n,t ) where R i,t is the return on the ith asset over the period (t-1,t) and W be the Nx1 vector of the portfolio weights over the same period. The historical returns of our current portfolio holdings are given by: 3

Y t = W T R t (1) In investment management if W represents actual investment holdings the series Y can be seen as the historical path of the portfolio returns. If however W represents an investment holding under consideration, Y describes the behaviour of this hypothetical portfolio over the past. Following Markowitz (1956) the portfolio s risk and return trade-off can be expressed in terms of the statistical moments of the multivariate distribution of the weighted investments as: E(Y t ) = E(W T R) = m (2.1) var(y t ) = W T W = 2 (2.2) where is the unconditional variance-covariance matrix of the returns of the N assets. A simplified way to find the portfolio s risk and return characteristics is by estimating the first two moments of Y. E(Y) = m (3.1) var(y) = E[Y - E(Y)] 2 = 2 (3.2) Hence if historical returns are known the portfolios mean and variance can be found as in (3.1), (3.2). This is easier than (2.1), (2.2) and still yields the same results. The method in (3.1), (3.2) can easily be deployed in risk management to compute the value at risk at any given time t. However, 2 will only characterise current volatility if W has not changed. If positions are being modified, the series of past returns, Y, 4

needs to be reconstructed and 2, the volatility of the new position need to be reestimated as in (3.2). 4. Time varying risk The portfolios risk and return estimates given by (2.1), (2.2) or (3.1), (3.2) rely upon a very strong assumption; that the series of returns, Y, is stationary. That means that both m and 2 do not change over the measurement period. Several studies have concluded that asset variances and covariances are not constant but change over time, e.g. Christie 1982. This problem, of using historical estimates of asset means and variances in VAR analysis, is well known to market practitioners. As a result a number of methods have been proposed to overcome the non stationarity problem and to estimate in the best possible way current variances and covariances. Perhaps today the most popular method is the exponential smoothing (ES) proposed by JP Morgan. A more sophisticate approach can be found in the GARCH methodology based on the work of Engle (1982) and Bollerslev (1986). Both approaches use past information in a more efficient way to compute current variances. The GARCH methodology seems to be superior but the ES is computationally easier 1. Because of the huge dimensions that a variance-covariance matrix may have both methods seek first to partition this matrix into (N-1)N/2 off-diagonal elements and then to capture the joint dynamics of the second moments for each possible pair-wise combinations of investment holdings. The volatility of current investment holdings is then computed as in (2.2). The problems both methods face stem from the way they partition the variance-covariance matrix. That is because unless certain preconditions are satisfied there is no guarantee that the resulting variance-covariance matrix comes from a NxN multivariate distribution. Hence the portfolio variance estimates are very likely to be biased. 1 For a comparison of the two models see Giannopoulos and Eales (1996). 5

5. Our approach to conditional VAR In this study we are going to adopt a simplified approach to compute a portfolio s VAR which aims to overcome both of the above problems, the non-stationarity and dimensionality, and will still provide us with unbiased estimates of portfolio s volatility. We believe that past returns contain all the necessary information about the current portfolio s risk return trade-off. Thus, we can obtain volatility estimates for the portfolio by studying directly its own past returns rather the returns of its components. If for example the volatility is constant it can then be estimated as in (3.2) and it will match the one computed using equation (2.2). However, it is very likely that the volatility of most individual assets included in the portfolio does change over the time, particularly if returns are measured over high frequency, i.e. daily. If that is the case then why should we believe that portfolio Y' s volatility is constant? If the constant volatility hypothesis is rejected estimates computed by (2.2) or (3.2) cease to be reliable. As we have seen above, one possible way to compute the volatility of the portfolio Y as time varying is to update the variance covariance matrix as soon as new prices are available and compute portfolio volatility as in (2.2) for that period. This approach is however problematic for the two reasons mentioned earlier. We can however compute portfolio Y' s volatility as time varying by treating past returns as time series on their own. This approach has many advantages. It is simple, easy to compute and overcomes the dimensionality and bias problems that arise from the NxN covariance matrix being estimated. On the other hand the portfolio s past returns contain all the necessary information about the dynamics that govern the aggregate current investment holdings and we should really make the best use of these information 2. For example it might be possible to capture the time path of portfolio 2 Markowitz (1956) incorporates equation (2.2) in the objective function of his portfolio selection problem because his aim was to find the optimal vector of weights W. However if W is known a priory then the portfolio s (unconditional) volatility can be computed more easily as in (3.2). 6

volatility using a GARCH model. This hypothesis is based on the fact that most of the high frequency security returns have been found to contain volatility clusters. 6. Empirical investigation To illustrate our procedure we collected daily data for assets with different risk exposure and we constructed three hypothetical portfolios as in equation (1). We then employed GARCH methodology and stress analysis on the portfolio return to study its riskiness. Our data set consist of the following daily data series: futures on bonds (LIFE) : Equities (cash) : Italian, German, Long Gilt FTSE100, S&P500 Commodities futures (IPE and LCE): Brent crude oil, Cocoa, Copper, Aluminium high grade We generated constant weighted portfolios for the period 1 November 1991 until 15 November 1994. The futures contracts have been rolled to create a single series. Missing observations and bank holidays have been set equal to a smoothed value 3. When a futures contract was rolled to the next one the first observation was considered as missing and so was set equal to its smoothed value. The three portfolios we constructed had the following weights: 3 We used a sophisticated approach to smooth each series. First the downhill simplex algorithm was used to find the optimal smoothing coefficients for a variety of smoothing specifications. Then we selected the smoothing model that minimised the Schwarz criterion. 7

Table 1 Portfolio composition Portfolio Bonds Equity Commodities A Italian 40% German 30% L Gilt 30% B Italian 30% German 20% L Gilt 20% C Italian 15% German 15% L Gilt 10% S&P500 15% FTSE100 15% S&P500 15% FTSE100 15% Oil 10% Cocoa 8% Copper 6% Alumin 6% Portfolio A contains only bond futures which are believed to be less volatile than the other two types of assets. On the other end portfolio C is invested 40% in bond futures, 30% in commodity futures and 30% in equities. The descriptive statistics of the three portfolios are reported on table 2. Although portfolio C contain more risky assets it is less volatile than A or B because it is more diversified. Table 2 Descriptive Statistics portfolio mean (p.a.) standard deviation (p.a.) A 1.29% 6.59% B 2.08% 6.36% C 3.41% 5.45% For a portfolio diversified across a wide range of assets the non-constant volatility hypothesis is an open issue. The LM test can be used to verify whether there are any GARCH effects. The test consist on regressing the squared residuals of an autoregressive process against their own lagged values. The test has been carried out on our portfolios and the results are reported in table 3. 8

Table 3 LM test for ARCH portfolio: A B C LM test(5) 110.83* 76.75* 18.59* * significant at 99% or above The statistic for lag order of five which is distributed as a chi-squared with five degrees of freedom is significant at 99% or above for each portfolio. However, as the portfolio becomes more diversified the statistic decreases. Perhaps in a much more widely diversified portfolio the null hypothesis of non-arch might not be rejected. Obviously if the null hypothesis is accepted then the portfolio s volatility is constant and could be estimated as in (3.2). We tested each portfolio for a number of GARCH parameterisations and found that one GARCH specification best fits in all three portfolios. This is as follows: Y t = t t ~ NI (0,h t ) (4.1) h t = ( t-1 + ) 2 + h t-1 (4.2) Table 4 Parameter estimates of equation (4) portfolio: A B C 0.070 (4.85) 0.888 (51.70) -2.396 (6.02) 0.048 (4.18) 0.933 (71.45) -1.783 (2.51) 0.035 (2.83) 0.944 (44.57) -2.522 (3.44) in parenthesis are White t statistics The parameter estimates with t-statistics are reported in table 4. All the coefficients are highly significant confirming that the portfolio returns follow a GARCH process. The coefficient measures the impact of last period s squared innovation, t, on to- 9

day s variance and it is positive and significant; In addition, in each series 0< 2 1 which indicates that the conditional variance is time stationary. Moreover, the constant volatility model for the current portfolio holdings, which is a special case of + =0, can be rejected. The coefficient captures any asymmetries in volatility that might exist. In every portfolio this coefficient is significant and negative indicating that volatility tends to be higher when portfolio values are falling. Correct model specification requires that diagnostic tests be carried out on the fitted residual,. Table 5 contains estimates of the regression: 2 t a b h (5) t with heteroskedasticity-consistent, White (1980), t- statistics given in parentheses. As Pagan and Ullah (1988) shows, if the forecasts are unbiased then a=0 and b=1. These hypotheses cannot be rejected at the 95% confidence level. The uncentered coefficient of determination in (5), R², measures the fraction of the total variation of everyday returns explained by the estimated conditional variance, which is known one day in advance. This coefficient has a value between 35% and 38%, meaning that our model on average can predict more than one third of next day s squared price movement. Table 5 Diagnostic test on fitted residuals Portfolio a b R 2 A -1.077 (0.73) B -0.716 (0.43) C -1.076 (0.61) 1.086 (0.75) 1.055 (0.40) 1.099 (0.52) 0.379 0.347 0.367 In parenthesis, below estimates, a and b, are: first row the t-statistics for testing a=0, second row the t- statistic for the hypothesis b=1. Last row reports the uncentered R 2. 10

All diagnostic test results are very satisfactory and allow us to conclude that the implemented GARCH parameterisation, although it has been very general and simple, has removed the GARCH effects from the portfolio. Figures 1 to 3 illustrates how the daily annualised standard deviation of the three portfolio over the tested period behave over time. The upper line shows the volatility of an undiversified portfolio; thus the volatility the same portfolios would have if all the pair-wise correlation coefficients of the assets invested were 1.0. The undiversified portfolio s volatility is simply the weighted average of the conditional volatilities of each asset included in the portfolio. Fig 1 Portfolio A 0.150 diversified vs non-diversified volatility 0.125 non-diversified 0.100 0.075 0.050 diversified 0.025 92 93 94 11

Fig 2 Portfolio B 0.144 diversified vs non-diversified volatility non-diversified 0.128 0.112 0.096 0.080 0.064 0.048 diversified 0.032 92 93 94 Fig 3 Portfolio C 0.160 diversified vs non-diversified volatility 0.144 0.128 non-diversified 0.112 0.096 0.080 0.064 diversified 0.048 0.032 92 93 94 12

The range over which the volatility for each portfolio oscillates are reported in table 6. Because portfolio s C is diversified across a wider range of assets its volatility oscillates between annual standard deviations of 3.65% and 7.29%. This range is less wide than the ones for portfolios, A and B. Table 6 Portfolio volatility range diversified undiversified portfolio minimum maximum minimum maximum A 0.0293 0.1188 0.0392 0.1353 B 0.0377 0.1114 0.0585 0.1344 C 0.0365 0.0729 0.8839 0.1576 The ranges of volatility in table 6 are the ones that would have been observed had the portfolio weights at dummy been effective over the whole tested period. There are three useful products of our methodology. The first one is a simple and accurate measure for the volatility of the current portfolio from which an accurate assessment of current risk can be made. This is achieved without using computationally intense multivariate methodologies. The second one is the possibility of comparing a series of volatility patterns similar to figures 1 to 3 with the historical volatility pattern of the actual portfolio with its changing weights. This comparison allows for an evaluation of the managers ability to time volatility. Timing volatility is an important component of performance especially if expected security returns are not positively related to current volatility levels. Finally, the possibility of using the GARCH residuals on the current portfolio weights allows for the implementation of meaningful stress testing procedures. We will focus on stress testing and the evaluation of correlation risk because of their importance in risk management models. 13

7. Stress analysis The innovations affecting the volatility of the portfolios are exhibited in figures 4,5 and 6. It is apparent that the distribution of the innovations is not normal with values reaching up to six standard deviations for the least diversified portfolio, A. Negative innovations are more modest, ranging up to four. Worst case scenarios for stress analysis may be build applying the largest outliers in the innovation series to the current GARCH parameters. This exercise simulates the effect of the largest historical shock on the current market conditions. Thus, to stress our portfolios it is not necessary to choose between the largest shocks for the different securities because the most interesting shocks are a direct by-product of the GARCH estimation of portfolio volatility. Fig 4 Portfolio A 6 Stress analysis 4 2 0-2 -4-6 92 93 94 14

Fig 5 Portfolio B 6 Stress analysis 4 2 0-2 -4 92 93 94 Fig 6 Portfolio C 5 Stress analysis 4 3 2 1 0-1 -2-3 92 93 94 15

8. Correlation stability and diversification benefits Conditional VAR models which use the quadratic equation (2.2) to update portfolio volatility, e.g. Riskmetrics, need first to estimate all the possible pairwise covariances. In a widely diversified portfolio, e.g. containing 100 assets, they are 4950 conditional covariances and 100 variances to be estimated. Furthermore, any model used to update the covariances must keep the multivariate features of the joint distribution. With a large matrix like that it is unlikely to get unbiased estimates 4 for all the 4950 covariances and at the same time to guarantee that the joint multivariate distribution still holds. Obviously errors in covariances as well in variances will affect the accuracy our portfolio s VAR estimate and lead to the wrong risk management decisions. Our approach estimates conditionally the volatility of only one, univariate, time series, the portfolio s return, overcoming all of the above problems. Hence, it does not require the variance-covariance matrix, it can be computed easily and it can handle an unlimited number of assets. On the other hand it measures in full the changes in assets variances and covariances. Another appealing property of our approach is to disclose the impact that the overall changes in covariances have on the portfolio volatility. It can tell us in what proportion an increase/decrease in the portfolios VAR is due to changes in asset variances or correlations. We will refer to this type of analysis as correlation stability. It is known that each correlation coefficient is subject to changes at any time. However, changes across the correlation matrix might not be correlated and therefore their impact on the overall portfolio risk may be diminished. Our conditional VAR approach allows to attribute any changes in the portfolio s conditional volatility to two main components; changes in asset volatilities and changes in asset correlations. If h t 4 The Pagan Ullah test can also be applied to measure the goodness of fit of a conditional covariance model. This stands on regressing the cross product of the two residual series against a constant and the covariance estimates. The unbiasedness hypothesis requires the constant to be zero and the slope one. The uncentered coefficient of determination of the regression tells us the forecasting power of the model. Unfortunately, even with daily observations, for most financial time series the coefficient of determination tends to be very low, pointing to the great difficulty of getting good covariance estimates. 16

st is the portfolio s conditional variance, as estimated in (4.2), its time varying volatility is t = h t. This is the volatility estimate of a diversified portfolio at period t. By setting equal to 1.0 all the pairwise correlations coefficients in each period, the portfolio s volatility becomes the weighted volatility of its asset components. Conditional volatilities of the individual asset components can be obtained by fitting a GARCH type model for each return series. We note the volatility of this undiversified portfolio as s t. The quantity (1- t ) tells us in what proportion the portfolio volatility has been diversified away because of non perfect correlations. If that quantity does not change significantly over time then the overall effect of time varying correlations is invariant and we have correlation stability. Figures 7, 8 and 9 show how the correlation stability improves for more diversified portfolios. Portfolio A, containing only bonds is subject to greater correlation risk because of the tendency of bonds to fall in step in the presence of large market moves. This effect can also be observed in the greater excursion of the volatility of portfolio A in table 6. Risk managers who rely on the average standard deviations in table 2 will be surprised by the extreme values of volatility over our bond portfolio may produce in a crash. Our conditional volatility estimates provide early warnings about this risk increase and therefore are a useful supplement to existing risk management systems. 17

Fig 7 Portfolio A volatility ratio 1.00 diversified over undiversified portfolio 0.75 0.50 0.25 0.00 92 93 94 Fig 8 Portfolio B volatility ratio 1.00 diversified over undiversified portfolio 0.75 0.50 0.25 0.00 92 93 94 18

Fig 9 Portfolio C volatility ratio 1.00 diversified over undiversified portfolio 0.75 0.50 0.25 0.00 92 93 94 9. Conclusions While portfolio holdings might aim at diversifying risk, this risk is subject to continuous changes. The GARCH methodology allows us to estimate past and current and predict future risk levels of our current position. The use of historical returns of portfolio components and current weights can produce accurate estimates of current risk for a portfolio of traded securities. Information on the time series properties of the returns of the portfolio components is transformed into a conditional estimate of the current portfolio volatility with no need of using complex multivariate time series procedures. Our approach leads to a simple formulation of stress analysis and correlation risk. 19

A software for the calculations used in this paper is available. For more information contact the authors at gbarone@gpu.srv.ualberta.ca (Barone-Adesi) or 100331.154@compuserve.com (Giannopoulos). References Giovanni Barone-Adesi (1994) ALM in Banks, working paper, University of Alberta. Bollerslev T (1986), "Generalised Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31, 307-28. Christie A (1982), The stochastic behaviour of Common stock Variance: Value, Leverage and Interest Rate Effects, Journal of Financial Economics, 10, 407-432. Engle R (1982), "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance in the U.K. inflation", Econometrica, 50, 987-1008. Giannopoulos K and B Eales (1996), Educated Estimates, Futures and Options World, April, 45-47. Mandelbrot B (1963), "The Variation of Certain Speculative Prices", Journal of Business, 36, 394-419. JP Morgan (1995) Risk Metrics Pagan A and A Ullah (1988), "The Econometric Analysis of Models with Risk Terms", Journal of Applied Econometrics, 3, 87-105. 20