Figure 9.1 The P&L density and value-at-risk.

Table 9.1: VaR tables

Holding period, h Significance level, a

0.01 0.05 0.1

2 <- VaR increases

Prob(AP, < -VaRa „) = a (9.1)

is a mathematical statement that is equivalent to saying that the 100a% / -day VaR measurement x is the lower a quantile of the unrealized P&L distribution, as depicted in Figure 9.1.

Table 9.1 shows how VaR varies with the choice of significance level and holding period. A single VaR model will generate a whole table of VaR estimates, that will increase as the significance level increases (that is, as a decreases). From Figure 9.1 it is evident that the percentile that cuts off 100a% of the area under the density will move to the left as the area a decreases. The VaR will also increase as the holding period increases because the uncertainty in P&L will generally increase with holding period. In some cases the square-root-of-time rule is employed (§3.3) so that the / -day VaR is simply taken as Jh times the 1-day VaR. If the square-root-of-time rule is not invoked then the VaR measure is simply the lower percentile of the historical / -day P&L distribution or, when a covariance matrix is used, it is the VaR based on the / -day covariance matrix.6

6 When a covariance matrix is used in the VaR model, a 1-day covariance matrix is transformed to an A-day covariance matrix by multiplying each element in the matrix by h. In linear portfolios this is equivalent to multiplying the 1-day VaR by vti: but it is not so for option portfolios, where the square-root-of-time rule should not be used. More details are given in the help sheets for the VaR models on the CD.

]00a% h-day VaR measurement x is the lower a quantile of the unrealized P&L distribution

The regulatory MRR is a multiple of the average of the last 60 days 1% 10-day VaR estimates when netted across the whole firm, or the previous days VaR estimate, whichever is greater (§9.5.1). Normally VaR models produce more realistic risk capital measurements than those obtained using the standardized approach. In fact for well-diversified or well-hedged portfolios capital requirement savings may exceed 50%. VaR models have many advantages and considerable potential for internal and external risk management and control, but there are also a number of disadvantages with their use, and these are discussed in §9.2.

9.1.3 Basel 2 Proposals

There are number of problems with the current framework for measuring risk capital. For example, the measurement of MRRs is quite complex, whereas the measurement of credit risk is still quite crude and operational risk is totally ignored. And even MRRs only focus on a few short-term risks with a single criterion of limited usefulness: market risks in the banking book are ignored. There is also no proper integration of the separate measures of market and credit risk: the two measures are simply summed, as if they were perfectly correlated (§9.3.5). But, if anything, market and credit risks will be negatively correlated. For example, with a simple instrument such as a swap, one is either owed money and therefore subject to credit risk, or one owes money and therefore subject to market risk.

There is also no proper integration of the separate measures of market and credit risk: the two measures are simply summed, as if they were perfectly correlated

In 1999 and 2001 the Basel Committee prepared three consultative papers that aimed to address some of these problems. The intention of Basel 2 is to widen the scope of its regulatory framework to cope with more sophisticated institutions and products and to cover a broader range of risks. In particular, for the first time operational risk will be included in risk capital requirements. More advanced models for measuring CRRs will be introduced, that allow better credit quality differentiation and risk mitigation techniques. Another proposal that will have a great impact is that market risk capital will be required to cover positions in the banking book. Current expectations are that the Basel 2 Amendment will be implemented in stages, starting in 2005. More details can be found on www.bis.org, www.isda.org and www.bba.org.uk.

9.2 Advantages and Limitations of Value-at-Risk

Since regulators have imposed minimum capital requirements to cover market risks that are based on internal models, VaR has become the ubiquitous measure of risk. This has many advantages. VaR can be used to compare the market risks of all types of activities in the firm, and it provides a single measure that is easily understood by senior management. The VaR concept can be extended to other types of risk, notably credit risk and operational risk. It takes into account the correlations and cross-hedging between various asset

categories or risk factors, and it can be calculated according to a number of different methods. VaR may also take account of specific risks by including individual equities among risk factors or including spread risk for bonds. And it may be calculated separately by building block, although without assessed correlations between building blocks, VaR measurements are simply added.

There are, however, many disadvantages with the use of VaR. It does not distinguish between the different liquidities of market positions, in fact it only captures short-term risks in normal market circumstances. VaR models may be based on unwarranted assumptions, and some risks such as repo costs are ignored. The implementation costs of a fully integrated VaR system can be There is a danger that huge and there is a danger that VaR calculations may be seen as a substitute VaR calculations may be for good risk management. Furthermore, in the course of this chapter we shall seen as a substitute for see tat yaR measures are very imprecise, because they depend on many good risk management assumpti0ns about model parameters that may be very difficult to either support or contradict.

9.2.1 Comparison with Traditional Risk Measures

The traditional measures of risk for a fixed income portfolio that can be represented by a cash-flow map, such as a portfolio of bonds or loans, are based on sensitivity to movements in a yield curve. For example, the standard duration measure is a maturity-weighted average of the present values of all cash flows.7 Another traditional measure of yield curve risk is the present value of a basis point move (PVBP), the change in present value of cash flows if the yield curve is shifted up by 1 basis point.8

The traditional risk measures of an equity portfolio are based on the sensitivities to the risk factors, that is the portfolio betas.9 However, in §8.1 the market risk of an equity portfolio was attributed to three sources: the variances (and covariances) of the underlying risk factors, the sensitivities to these risk factors, and the specific variance or residual risk. Therefore the traditional beta risk measure relates only to the undiversifiable part of the risk: the risk that cannot be hedged away by holding a large and diversified portfolio. The beta ignores the risk arising from movements in the underlying risk factors and the specific risks of a portfolio. One of the major advantages of VaR is that it does not ignore these other two sources of risk.

7 The basic measure of duration is T(PV, x M/SiPV,). It is a measure of interest rate risk because if interest rates rise the present value of cash flows will decrease, but the income from reinvestment will increase, and the duration marks the break-even point in time where the capital lost from lower cash flows has just been recovered by the increased reinvestment income.

8 There are 104 basis points in 100%, thus to convert a cash flow to PVBP terms, a simple approximation is to multiply the cash flow amount by / and then divide the result by 104. To see this note that the interest rate sensitivity is -dIn P/dr. For a single cash flow of $1 at time I, P = e~r! so -din P/dr = t.

9 And risk measures in option portfolios have also been based purely on sensitivities. The main risk factors in options portfolios are the price and the volatility of the underlying; the main sensitivities to these risk factors, delta, gamma and vega, are defined in §2.3.3.