VaR is today the most widely used quantitative analysis tool for financial risk. It describes the maximal portfolio loss for a given confidence level over a specified trading horizon. Mathematically speaking, VaR characterizes the tail (extreme quantile) of the portfolio return distribution.7Despite a number of pitfalls (see later), VaR has introduced a new dimension of risk analysis into the financial com- munity. The magic of VaR is that it introduces a uniform measuring system for the various instruments in a portfolio by providing a method for comparing risk across security and asset classes (prior to VaR, an investor was unable to quantify the risk of a combination of a $1 million position in US equity and a $1 million position in short duration German Government Bonds). The two important parameters that a risk manager has to define for VAR are the time period and the confidence level (e.g. 99% for a one-day horizon). VaR can be calculated on a variety of different aggregation levels within the portfolio (fund-wide risk attribu- tion, asset classes, sectors, instruments, trading managers, geographic regions).
In most respects VaR is a natural progression from MPT, but there are also important differences:
■ MPT interprets risk in terms of the standard deviation of returns, while VaR describes risk as the maximum likely loss.
■ The variance–covariance approach of MPT is just one of several methods to calculate VaR (others are the Monte Carlo method and the historical method).
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■ VaR can be applied to a range of different risks beyond market risk, while MPT is limited to market risk. The concept of VaR can even be used for firm- wide risk management.
■ VaR offers a wider frame for the incorporation of more complex statistical methods such as non-normal returns.
While VaR is conceptually and intuitively quite simple, the technical details of its calculation can be rather involved depending on the heterogeneity of the portfolio and the distributional assumptions made. Detailing the technical aspects of portfo- lio VaR calculation is beyond the scope of this book8 but a short overview is provided in the following.
The three main elements of standard VaR calculation are:
■ mapping the portfolio positions to risk factors
■ calculating the risk factor covariance matrix
■ determining the VaR method and calculating VaR.
Mapping risk factors
Risk factor mapping is the decomposition of individual securities into components over which the risk manager has control (the exposure) and exogenous influences he cannot control (the risk factors). A risk factor is a variable which affects directly the future value of a security (e.g. interest rates, equity market valuations, FX rates etc.). The prices of thousands of worldwide available financial securities are influenced by common risk factors.
The dependency of a security on risk factors (the security’s ‘sensitivities’) is expressed by a ‘pricing function’ which can depend on the VaR method employed (see below). Note that the concept of risk factors and sensitivities is not new. Sensitivities have been used for the analysis of yield curve risks as well as in option theory for many years (examples are the ‘value of a basis point move’ or the ‘Greeks’).
VaR calculations should account for every important risk factor in the portfo- lio, which can be quite numerous (>1,000). The following provides a broad categorization.9
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■ Equity indices: The risk of an equity position can be separated into two components, the sensitivity to one or several relevant market/sector indices and an idiosyncratic residual risk. With a sufficient number of stocks in the portfolio, the latter is diversified away. Commonly employed equity risk factors are broad market or specific sector risks, expressed as ‘sensitivities’, or in the language of the ‘Capital Asset Pricing Model’, ‘betas’.
■ Yield and credit spread curves: Fixed income risk arises from potential moves in interest rate term structures (yield curves) and credit ratings. Each currency has its own yield curve. Credit risk is expressed in terms of the spread to a ‘risk-free’ (i.e.
not defaultable) reference bond or probability of default. A portfolio exposure to yield and spread curve risk can be described by certain representative maturity points on the yield and credit spread curves (e.g. one-month, six-month, one-year, two-year, five-year, ten-year, 15-year and 30-year maturities).10
■ Foreign exchange rates: Foreign exchange spot rates are risk factors in portfolios that hold positions in currencies different from the investor’s home currency.
For positions in FX forwards and futures, the two underlying yield curves define additional risk factors. The interest rate parity theorem provides the exact relationship between the spot rates, the forward rates and the two interest rates.
■ Commodities: Spot commodity positions are exposed to single risk factors, the respective commodity prices. Similar to yield curves defining risk factors related to interest rates, commodity Forward and Futures contracts define a term structure for each commodity which is determined by the costs of carry (mostly related to interest rates) and the ‘convenience yield’.
One of the most common approaches to mapping positions to risk factors is via the use of a security’s cash flow. This method assumes that all instruments can be repre- sented by defined cash flows to the investor at present or specified times in the future (note that this assumption is invalid for options). The cash flows are attributed to spe- cific risk factors and discounted to present value at the prevailing interest rates. A simple example is given by the cash flows of a (default free) bond with yearly coupon payments and principal payback in ten years. The risk factors are represented by the interest rates at various maturities on the yield curve. The present value of the cash flow to the investor in a given year (coupon payments in years one to ten and
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principal payback in year ten) is uniquely determined by the x-year interest rate. Cash flows that fall in between two maturity points are split between the two nearby points. A second commonly used approach is the sensitivity approach, which is based on directly determining the portfolio’s sensitivity to changes in each risk factor.11 Calculation of covariance matrix
The next step in the calculation of VaR consists of estimating volatilities and correla- tions of the risk factors. Making the standard assumption that security returns are multivariate normally distributed, the correlation matrix (together with the vector of expected returns) uniquely determines the distribution of portfolio returns, and therefore portfolio VaR. There are various methods for calculating volatility (vari- ances) and correlations (covariances). The following approaches are most common (the RiskMetrics Technical document (1996) provides a discussion of the details):12
■ Equally weighted moving average of squared returns (variance) and cross- products of returns (covariance).
■ Exponential moving average of squared returns and cross-products of returns with specified decay parameter. This is the method chosen by RiskMetrics and many others. The decay parameter most often chosen is 0.94 for daily
observations and 0.97 for monthly observations.
■ The family of ARCH models: ARCH (autoregressive conditional
heteroskedastic) and GARCH (General ARCH) models were developed in the early 1980s and have become a starting point for sophisticated volatility and correlation forecasting models.13The exponential moving average model can be seen as a special case of a GARCH model.
Calculation of VaR
After attributing cash flows to risk factors and calculating the risk factors’ covari- ance matrix, VaR can be calculated using several methods which differ mainly in respect to two factors.
1 Assumptions regarding security valuation as a function of risk factors: A determination must be made as to how to model the sensitivity of security
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prices to changes in risk factors. The industry distinguishes between local valuation and full valuation methods.
2 The distributional assumptions made: The industry distinguishes between parametric (mostly normal) distributions versus historical distributions.
The three most popular VaR methods are:
■ Parametric approach: The idea behind parametric VAR models is to approximate the pricing function of each instrument (i.e. the relationship between each instrument and the risk factors), in a way that an analytical formula for VaR can be obtained. The simplest parametric approach is the ‘delta method’, also referred to as ‘variance–covariance-based VaR’.14Assuming normal return distributions and a linearrelationship between risk factors and the securities in the portfolio, the returns of the portfolio themselves are conditionally normally distributed (here
‘conditional’ refers to the standard deviation of this distribution changing over time depending on the most recent volatility). The covariance matrix of risk factors determines the standard deviation of portfolio returns. The method possesses the charm of simplicity, but is severely flawed in circumstances where the assumption of linearity does not hold. An enhancement of the delta method is the ‘delta–gamma’ method, where second-order approximations are added to the security pricing function. Both the delta and delta–gamma methods are ‘local’
valuation methods, as only the mark to market values of the portfolio positions enter into the valuation of the portfolio (and no other price scenarios).
■ Monte Carlo simulations:15The Monte Carlo method is a ‘full valuation’
method based on simulating the behaviour of the underlying risk factors through a large number of draws produced by a random generator. Using given pricing functions, the values of the portfolio positions are calculated from the simulated values of risk factors. A possible non-linear relationship between security and risk factors is fully accounted for. The positions in the portfolio are fully revalued under each of the random scenarios. Every random draw of risk factor values thus leads to a new portfolio valuation. A high number of iterations (several thousand) provides a simulated return distribution of the portfolio, from which the VaR value can be determined.
(The number of iterations necessary can be somewhat reduced by
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‘deterministic sampling’, i.e. pseudo-random selection of values to ensure that the full spectrum of relevant scenarios is selected.) The underlying distribution of the randomly generated risk factor values can theoretically be chosen freely, but in most cases the simulation draws from a normal distribution. The
random values are transformed such that the empirical correlations between the risk factors are recognized (mathematically, this is achieved through a
‘Cholesky decomposition’ of the correlation matrix). The use of non-normal distributions can quickly lead to complex mathematical problems.16
■ Historical simulation: Instead of simulating return distributions, the distribution is determined by looking into the past. The historical method is also a full valuation method and relies on the (unconditional) historical distribution of returns by applying past asset returns to the present holdings in the portfolio. The values of the portfolio positions are fully evaluated for each historical return by the use of the specified pricing functions. This method has the advantage that no explicit assumptions about the underlying return distribution have to be made.
The problem of the method is that it relies on historical price behaviour, which might no longer be relevant in current market environments.
It was mentioned that the full security re-valuation employed by the Monte Carlo as well as the historical method requires specific valuation models or pricing functions.
These models have to be correctly adjusted, which happens by calibrating the model parameters to the market prices of specific benchmark instruments. Further, as long as normally distributed draws are used, the Monte Carlo method does not address the issue of non-normally distributed asset returns. The dependence and correlation struc- ture of non-normal multivariate distributions is still the subject of intense research.
Stretching out beyond the normal distribution very quickly leads to intractable mathe- matical problems, as non-normal multivariate distributions are much more difficult to deal with (see the section on Extreme Value Theory later in this chapter).17
Because of complex and non-linear (option) positions that are present in most AIS portfolios, the variance-based method can be quite misleading and should be avoided. Historical simulations are attractive as they make no distributional assumptions, but the results depend strongly on the historical period chosen for the analysis. The Monte Carlo approach is usually the most reliable method, but is at the same time computationally the most intense.
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