## Value At Risk

1.3.1 The Origin and Development of VaR In the late 1970s and 1980s, a number of major financial institutions started work on internal models to measure and aggregate risks across the institution as a whole. They started work on these models in the first instance for their own internal risk management purposes as firms became more complex, 12This problem is especially acute for gamma risk. As one risk manager noted On most option desks, gamma is a local measure designed for very small moves up...

## Normal Var And

The normal or Gaussian distribution was briefly introduced in Chapter 3. It is very widely used, and has plausibility in many contexts because of the central limit theorem. Loosely speaking, this theorem says that if we have a random variable from an unknown but well-behaved distribution, then the means of samples drawn from that distribution are asymptotically i.e., in the limit normally distributed. Consequently, the normal distribution is often used when we are concerned about the...

## T62 Uses Of Monte Carlo Simulation

We might use Monte Carlo for any of four different purposes. The first is to price a derivative. To do so, we run a sample path of S in a risk-neutral world i.e., we run a sample path taking the mean expected return to be the risk-free return r instead of h , and calculate the payoff from the derivative e.g., so the payoff from a standard Black-Scholes call with strike price X would be max ST - X, 0 . We do this a large number M of times, calculate the sample mean payoff to our derivative, and...

## Estimating Liquidityadjusted Var And Etl 821 A Transactions Cost Approach

One approach is to adjust VaR and ETL for liquidity effects through an examination of the impact of transactions costs on our P L. Given our earlier discussion of liquidity, we can plausibly assume that transactions costs rise with the size of the transaction relative to the market size for the instrument concerned i.e., because of adverse market reactions due to limited liquidity and with the bid-ask spread. We can also assume that transactions costs fall with the length of time taken to...

## Benefits And Difficulties Of Stress Testing

Stress testing ST is ideal for showing up the vulnerability of our portfolio and of our VaR calculations to otherwise hidden risks or sources of error. Schachter 1998 suggests five different ways in which ST can provide valuable information to risk managers that may not otherwise be available to them Since stress events are unlikely, the chances are that the data used to estimate VaR or ETL will not reveal much about stress events. The short holding period often used for VaR will often be too...

## Of Tail Losses

1We should keep in mind that the critical value associated with the null hypothesis will depend on the alternative hypothesis e.g., whether the alternative hypothesis is that the 'true' prob-value is different from, or greater than, or less than, the prob-value under the null hypothesis . However, in what follows we will assume that the alternative hypothesis is the last of these, namely, that the 'true' prob-value is less than the null-hypothesis prob-value. where n is the number of P L...

## Negative Contribution To

We turn now to consider the component VaR, CVaR, and begin by considering the properties that we want CVaR to satisfy. The two main properties we want are Incrementality. We want the component VaR to be, at least to a first order of approximation, equal to the IVaR the increase or decrease in VaR experienced when the relevant component is added to or deleted from the portfolio. Additivity. We want the arithmetic sum of component VaRs to be equal to the VaR of the total portfolio. This ensures...

## A52 Deltagamma Approaches

A5.2.1 The Delta-Gamma Approximation An obvious alternative is to take a second-order approach to accommodate non-linearity by taking a second-order Taylor series approximation rather than a first-order one. This second-order approximation is usually known in finance as a delta-gamma approximation, and taking such an approximation for a standard European call option gives us the following This second-order approximation takes account of the gamma risk that the delta-normal approach ignores cf....

## A32 Mapping Positions And Var Estimation

We now consider how to map and estimate VaR for specific types of position. Naturally, there is a huge variety of different financial instruments, but the task of mapping them and estimating their VaRs can be simplified tremendously by recognising that most instruments can be decomposed into a small number of more basic, primitive instruments. Instead of trying to map and estimate VaR for each specific type of instrument, all we need to do is break down each instrument into its constituent...

## Estimating Liquidity At Risk

We turn now to liquidity at risk or LaR , sometimes also known as cash flow at risk or CFaR . LaR or CFaR relates to the risk attached to prospective cash flows over a defined horizon period, and can be defined in terms analogous to the VaR.9 Thus, the LaR is the maximum likely cash outflow over the horizon period at a specified confidence level for example, the 1-day LaR at the 95 confidence level is the maximum likely cash outflow over the next day, at the 95 confidence level, and so forth. A...

## Extreme Value Distributions

If we are concerned about VaRs and ETLs at extreme confidence levels, the best approach is to use an EV distribution see Tool No. 5 Extreme Value VaR and ETL . In a nutshell, EV theory tells us that we should use these and only these to estimate VaRs or ETLs at extreme confidence levels. 5.4.1 The Generalised Extreme Value Distribution EV theory offers a choice of two approaches. The first of these is to work with a generalised extreme value GEV distribution. The choice of this distribution can...

## The Lognormal Distribution

Another popular alternative is to assume that geometric returns are normally distributed. As explained in Chapter 3, this is tantamount to assuming that the value of our portfolio at the end of our holding period is lognormally distributed. Hence, this case is often referred to as lognormal. The pdf of the end-period value of our portfolio is illustrated in Figure 5.5. The value of the portfolio is always positive, and the pdf has a distinctive long tail on its right-hand side. Figure 5.5 A...

## Estimation Of Historical Simulation Var And

Having obtained our historical simulation P L data, we can estimate VaR by plotting the P L or L P on a simple histogram and then reading off the VaR from the histogram. To illustrate, suppose we have 100 observations in our HS P L series and we plot the L P histogram shown in Figure 4.1. If we choose our VaR confidence level to be, say, 95 , our VaR is given by the x-value that cuts off the upper 5 of very high losses from the rest of the distribution. Given 100 observations, we can take this...

## Weighted Historical Simulation

One of the most important features of traditional HS is the way it weights past observations. Recall that Rit is the return on asset i in period t, and we are implementing HS using the past n observations. An observation Rit_j will therefore belong to our data set if j takes any of the values 1, , n, where j is the age of the observation e.g., so j 1 indicates that the observation is 1-day old, and so on . If we construct a new HS P L series, P Lt, each day, our observation Rit_ j will first...

## Estimating Confidence Intervals For Historical Simulation Var And

The methods considered so far are good for giving point estimates of VaR or ETL, but they don't give us any indication of the precision of these estimates or any indication of VaR or ETL confidence intervals. However, there are a number of methods to get around this limitation and produce confidence intervals for our risk estimates. 4.3.1 A Quantile Standard Error Approach to the Estimation of Confidence Intervals for HS VaR and ETL One solution is to apply the theory of quantile standard...

## The Meanvariance Framework For Measuring Financial Risk

The traditional solution to this problem is to assume a mean-variance framework we model financial risk in terms of the mean and variance or standard deviation, the square root of the variance of P L or returns . As a convenient although oversimplified starting point, we can regard this framework as underpinned by the assumption that daily P L or returns obeys a normal distribution.1 A random variable X is normally distributed with mean and variance a2 or standard deviation a if the probability...

## Estimating Parametric

We can also estimate VaR using parametric approaches, the distinguishing feature of which is that they require us to specify explicitly the statistical distribution from which our data observations are drawn. We can think of parametric approaches as fitting curves through the data and then reading off the VaR from the fitted curve. In making use of a parametric approach, we therefore need to take account of both the statistical distribution and the type of data to which it applies. We now...

## Estimating Historical Simulation

The simplest way to estimate VaR is by means of historical simulation HS , which estimates VaR by means of ordered L P observations. Suppose we have 100 L P observations and are interested in the VaR at the 95 confidence level. Since the confidence level implies a 5 tail, we know that there are five observations in the tail, and we can take the VaR to be the sixth highest L P observation. Assuming that our data are in L P form, we can therefore estimate the VaR on a spreadsheet by ordering our...

## Choosing 1 Or 2 Tail

1 The Risk Measurement Revolution 1 1.1 Contributory Factors 1 1.1.1 A Volatile Environment 1 1.1.2 Growth in Trading Activity 2 1.1.3 Advances in Information Technology 2 1.2 Risk Measurement Before VaR 3 1.2.5 Derivatives Risk Measures 6 1.3.1 The Origin and Development of VaR 7 1.3.2 Attractions of VaR 10 1.3.3 Criticisms of VaR 11 1.4 Recommended Reading 12 2 Measures of Financial Risk 13 2.1 The Mean-Variance Framework for Measuring Financial Risk 13 2.1.1 The Normality Assumption 13 2.1.2...