Backtesting Derivative Portfolios with FHS

Filtered historical simulation provides the general framework to our backtests of portfolios of derivative securities held by a large sample of financial institutions. We allow for stochastic volatility and exchange rates. Correlations are preserved implicitly by our simulation procedure. Options are repriced at each node. Overall results support the adequacy of our framework, but our VaR numbers are too high for swap portfolios at long horizons and too low for options and futures portfolios at short horizons. Filtered Historical Simulation 2 We backtest a large sample of LIFFE derivative portfolios held by financial institutions. Our backtests are based on a new generation of VaR models, filtered historical simulation (FHS). FHS overcomes some shortcomings of the traditional bootstrapping approach. Historical simulation (bootstrapping) consists of generating scenarios, based on historical price changes, for all the variables in the portfolio. Since the estimated VaR is based on the empirical distribution of asset returns it reflects a more realistic picture of the portfolio’s risk. Unfortunately this methodology presents a number of disadvantages. To overcome some of them Barone-Adesi, Bourgoin and Giannopoulos (1998) and Barone-Adesi, Giannopoulos and Vosper (1999) introduce filtered historical simulation (FHS hereafter). They take into account the changes in past and current volatilities of historical returns and make the least number of assumptions about the statistical properties of future price changes. We backtest our FHS model for Value at Risk on three types of portfolios invested over a period of two years. The first set of backtests consists of actual portfolios of LIFFE financial futures and options contracts traded on LIFFE. In the second set of backtests we examine the suitability of the FHS model on interest rate swaps. Finally, we backtest a set of portfolios obtained mixing LIFFE interest rate futures and options with interest rate swaps. We go beyond the criteria of the BIS recommendations by evaluating daily risk at four different confidence levels and five different trading horizons for a large number of portfolios of derivative securities. In the first section we review backtesting methodologies. In the second section we report results for LIFFE portfolios. We run two sets of backtests: in the first backtest we keep constant implied volatilities and FX rates, while in our second backtest we simulate implied volatilities. Our results show that fixed implied volatility performs better at short VaR horizons, while at longer ones (5 to 10 days) our stochastic implied volatility performs better. In the third section we investigate the performance of FHS on books of interest rate swaps. We compare each book’s daily values with the FHS lower forecasted value. For each book we produce two types of forecast; an aggregate market value risk expressed in GBP and a set of currency components (plain vanilla swaps in USD, JPY, DEM, and GBP are used). We find that our methodology is too conservative for swap portfolios at longer horizons. * USI and City Business School, University of Westminster and London Clearing House respectively. 1 For the first set of backtests, a total of 75,835 daily portfolios; the second and the third set use 75,985 daily portfolios. Filtered Historical Simulation 3 In the final section we investigate the performance of the FHS model on diversified portfolios across interest rate swaps and futures and options traded on LIFFE. This section shares the same data with the separate LIFFE and swap backtests while restricting the number of portfolios to 20 among the largest members on LIFFE. By adding to each LIFFE portfolio one of the four swap books used in the swap backtest we form 20 combined (combo) portfolios. Our analysis is based on two sets of criteria: statistical and economic. The former examine the frequency and the pattern of losses exceeding the VaR predicted by FHS (breaks); the latter examine the implications of these breaks in economic terms, with reference to the total VaR allocated. Overall our findings support the validity of FHS as a risk measurement tool. 2 Four SWAP books consisting of 500 Swaps each were formed for the SWAP backtest. Details of the portfolios are available from the authors. Filtered Historical Simulation 4 1. Overview of VaR models. VaR models play a core role in the risk management of today’s financial institutions. A number of VaR models are in use. All of them have the same aim, to measure the size of possible future losses at a predetermined level of probability. There are a variety of approaches used by VaR models to estimate the potential losses. Models differ in fact in the way they calculate the density function of future profits and losses of current positions, as well as the assumptions they rely on. Although VaR analysis has been used since early 1980’s by some departments of few large financial institutions, it wasn’t until the middle 1990’s that it became widely accepted by banks and also imposed by the regulators. The cornerstone behind this wide acceptance was RiskMetrics, a linear VaR model based on the variance-covariance of past security returns, introduced by JP Morgan in 1993. The variance-covariance approach to calculate risk can be traced back to the early days of Markowitz’s (1959) Modern Portfolio Theory, which is now common knowledge among risk managers. Linear VaR models, however, impose strong assumptions about the underlying data. For example, the density function of daily returns follows a theoretical distribution (usually normal) and has constant mean and variance. The empirical evidence about the distributional properties of speculative price changes provides evidence against these assumptions, e.g. Kendall (1953) and Mandelbrot (1963). Risk managers have also seen their daily portfolio’s profits and losses to be much larger than those predicted by the normal distribution. Embrechts et al. (1997) and Longin (2000) propose the use of extreme value theory to overcome the last problem. Unfortunately their approach is unsuitable in general for portfolios of derivatives, where losses may be limited by contractual terms. As an example, the maximum loss on an option spread is bounded by the difference between the two strike prices. The RiskMetrics VaR method has two additional major limitations. It linearises derivative positions and it does not take into account expiring contracts. These shortcomings may result in large biases, particularly for longer VaR horizons and for portfolios weighed with short out-of-the money options. To overcome problems of linearising derivative positions and to account for expiring contracts, risk managers look at simulation techniques. Pathways are simulated for scenarios for linear positions, interest rate factors and currency exchange rate and are then used to value all positions for each scenario. The VaR is estimated from the distribution (e.g. 1 percentile) of the simulated portfolio values. Monte-Carlo simulation is widely used by financial institutions around the globe. Nevertheless, this method can attract severe criticisms. First, the generation of the scenarios is based on random numbers drawn from a theoretical distribution, often normal. Such a distribution not only does not conform to the empirical distribution of most asset returns, but it also limits the losses to around three or four standard deviations when a very large number of simulation runs is carried out. Second, to maintain the multivariate properties of the risk factors when generating scenarios, historical correlations are used; during market crises, when most correlations tend to increase rapidly, a Monte Carlo system is likely to underestimate the possible losses. Third, Monte Carlo simulation tends to be slow, because a large number of scenarios have to be generated. Recognising the fact that most asset returns cannot be described by a theoretical distribution, an increasing number of financial institutions are using historical simulation. Here, each historical obser3 The RiskMetrics VaR approach recognises the fact that variances and covariances are changing over time and uses a simple method (exponential smoothing) to capture these changes. There is some contradiction however in its constant volatility assumption for multiperiod VaR (i.e. the last trading day’s volatility is scaled by time). 4 The extent to which these assumptions are violated depend on the frequency of the data. Daily data, which are of interest in risk management, tend to deviate to a great extend from normality. 5 Jamshidian and Zhu (1997) proposed a method that limits the number of portfolio valuations. Filtered Historical Simulation 5 vation forms a possible scenario, see Butler and Schachter (1998). A number of scenarios are generated and in each of them all current positions are priced. The resulting portfolio distribution is more realistic since it is based on the empirical distribution of risk factors. Historical simulation has still some serious drawbacks. Long time series of data are required to include extreme market conditions. The fact that asset risks are changing all the time is ignored. Historical returns are used as if they were i.i.d. random numbers. A consequence of this usage is that during highly volatile market conditions historical simulation underestimates risk, as documented by van den Goorbergh and Vlaar (1999) and Vlaar (2000), who support the use of GARCH volatilities. Furthermore, historical simulation uses constant implied volatility to price the options under each scenario. Some positions, which may appear well hedged under the constant implied volatility hypothesis, may become very risky under a more realistic scenario. It is hard to determine the extent of this problem, as sensitivity analysis is difficult with historical simulation. To remedy the above problems Barone-Adesi, Bourgoin and Giannopoulos (1998) and Barone-Adesi, Giannopoulos and Vosper (1999), extending the work of Barone-Adesi and Giannopoulos (1996), have suggested to draw random standardised returns from the portfolio’s historical sample. After scaling these standardised historical returns by the current volatility, they use them as innovations in a conditional variance equation for generating scenarios for both future portfolio variance and price (level). This method generates the complete distribution of the current portfolio’s profits and losses including the effects of volatility changes, overcoming an additional limitation of most current VaR models. A comparison of the approach of Barone-Adesi et al. with alternative risk measurement methods is in Pritsker (2000). 1.1 Filtered historical Simulation To overcome the shortcomings of historical simulation it is necessary to filter historical returns; that is to adjust them to reflect current information about security risk. Our complete filtering methodology is discussed in Barone-Adesi, Giannopoulos and Vosper (1999). A brief synopsis is presented below. Simulating a Single Pathway Many pathways of prices are simulated for each contract (futures or interest rate) in our dataset over several holding periods. In our backtests we use 5000 simulation pathways over 10 days. Our algorithm can be described by starting with the simulation of a single pathway for a single contract. From this we generalise to the simulation of many pathways for many contracts and their aggregation into portfolio pathways. The set of portfolio pathways for each day in the holding period defines 10 empirical distributions i.e. over holding periods from 1 to 10 days. The portfolio composition is held constant over each holding period. Our methodology is non-parametric in the sense that simulations do not rely on any theoretical distribution on the data as we start from the historical distribution of the return series. We use two years of earlier data to calibrate our GARCH models, (Bollerslev 1986), for asset returns and to build the data bases necessary to our simulation. By calibrating GARCH models to the historical data we form residual returns from the returns series. Residual returns are then filtered to become identically and independently distributed, removing serial correlation and volatility clusters. As the computation of the i.i.d. residual returns involves the calibration of the appropriate GARCH model, the overall approach 6 The portfolio’s standardised returns must be i.i.d. A mean equation is fitted when appropriate to achieve that. 7 For example one futures may be modelled by a GARCH (1,1) with no AR or MA terms, another one by an AGARCH with an MA term; we examine a variety of processes to attempt to fit the appropriate Filtered Historical Simulation 6 can be described as semi-parametric. For example, assuming a GARCH (1,1) process with both moving average (θ) and autoregressive ( μ ) terms, our estimates of the residuals εt and the variance ht are: rt = μ rt −1 + θ ε t −1 + εt εt ~ N(0, ht) (1) ht = ω + α(εt -1 γ) + βht-1 (2) To bring residuals close to a stationary i.i.d. distribution, so that they are suitable for historical simulation, we divide the residual ε t by the corresponding daily volatility estimate: