Traditional risk measures cannot be compared across the different activities of the firm. For example, even though they both measure first-order changes in the portfolio as a result of movements in the underlying, the duration of a bond portfolio (which is measured in months or years) cannot be compared with the delta of an options portfolio. For comparative purposes the duration or delta can be multiplied by a notional amount and a variation in the underlying risk factor (interest rates or share price). Nevertheless it is difficult to assess which activities are taking the most risk using these sensitivity-based measures, and to allocate capital accordingly. One of the main advantages of VaR is that it takes into account the volatilities and correlations of risk factors, so it is comparable across different asset classes.

Sensitivity-based risk measures have a number of limitations. They only make sense within each trading unit, and they cannot be compared across different activities to see which area has the most risk. They cannot be aggregated to give an overall exposure across all products and currencies. They do not indicate how much could be lost, either in normal circumstances or under extreme events.

9.2.2 VaR-Based Trading Limits

Long-term capital allocation between different asset classes and different activities is normally addressed using risk-adjusted performance measures (§7.2.3). On the other hand, short-term trading limits are based only on risk and not on returns: in the short term returns can be assumed to be nearly zero. Sensitivity-based trading limits have been used for some time, but more recently there have been considerable internal and regulatory forces to implement trading limits that are based on VaR. VaR provides a risk measure that focuses on the profit and loss from different activities in the firm taken together. It is therefore natural that a firm would seek to use VaR in a unified framework for allocating capital, not just to satisfy regulatory purposes, but also to allocate capital among the different activities of the firm.

However, it is a very difficult task to replace the traditional sensitivity-based trading limits with trading limits based on VaR. Not only are traders thinking in terms of sensitivities, they have the software to calculate immediately the impact of a proposed trade on their limit. In contrast, real-time VaR systems for large complex portfolios are simply not available, unless many approximations are made (§9.4.3). Therefore, if trading limits are based on VaR, the trader will need to use both VaR- and sensitivity-based systems, at least initially, so that he knows what are the implications of a trade from all perspectives.

Real-time VaR systems for large complex portfolios are simply not available, unless many approximations are made

9.2.3 Alternatives to VaR

As a measure of risk, VaR has some limitations. First, like volatility, it is affected by good risk as well as bad risk. Following Dembo and Freeman

Market Models Table 9.2: Good risk, bad risk, VaR and volatility

Return: Probability of that Probability of that

return in portfolio A return in portfolio

0.6 0.5-0.4-

Figure 9.2 Which is more risky?

(2001), a simple example to illustrate this is given. Consider the distributions of returns for two portfolios A and that are shown in Table 9.2 and Figure 9.2. Portfolio A has a greater chance of positive returns, and this is reflected in the fact that the expected return for portfolio A is larger than it is for portfolio B. However, if one only looks at the standard risk measures, these tell the same story for both portfolios. That is, the volatility (standard deviation), the 5% VaR and the 1% VaR are identical for both portfolios. Even though much of the variation in the return to portfolio A is concentrated on the up side, so that A has more good risk than B, this is not revealed by looking at the volatility or the VaR of these portfolios.

Downside risk measures are based on the returns that fall short of a benchmark return. For example, the semi-variance operator introduced by Markovitz (1959) measures the variance of all returns that are less than the expected return:

SV = £((min(0, R - E(R)))2).

(9.2)

More generally, E(R) may be replaced by any benchmark return that can be time-varying or fixed. Dembo and Freeman (2001) advocate the use of regret as a measure of downside risk. The regret operator is defined as

Regret = -£(min(0, R - B)). (9.3)

This has the same form as the pay-off to a put option with strike equal to the benchmark return. Regret therefore has the intuitive interpretation of an insurance cost - the cost of insuring the downside risk of a portfolio. Regret Regret will easily will easily distinguish good risk from bad risk. For example, if the distinguish good risk benchmark return - 0 then the regret of portfolio A in the example above is from bad risk 0.9, whereas the regret of portfolio is 2.275.

Artzner et al. (1997) have used an axiomatic approach to the problem of defining a satisfactory risk measure. They set out certain attributes that one should reasonably require of any risk measure, and call risk measures that satisfy these axioms coherent. A coherent risk measure p assigns to each loss X a risk measure p(X) such that the following conditions hold.

1. Risk is monotonic: if X Y then p(X) p(F).

2. Risk is homogeneous: p(tX) = tp(X) for / > 0.

3. Risk-free condition: p(X+nr) - p(X) - n, where r is the risk-free rate.

4. Risk is sub-additive: p(X+ Y) < p(X) + p(F).

These attributes guarantee that the risk function is convex; this corresponds to risk aversion (§7.2.4).10 The risk-free condition ensures that if an amount n of the riskless asset is added to the position then the risk measure will be reduced by n; thus capital requirements will be reduced accordingly. The last attribute is very important: it ensures that the total risk is no more than the sum of the risks of individual positions and without this there would be no incentive to diversify portfolios.

VaR is not a coherent risk measure because it does not necessarily satisfy axiom 4. The violation of axiom 4 has serious consequences for risk management, since axiom 4 allows decentralized calculation of the risks from different positions in the firm and the sum of these individual risk measures will be conservative (over-) estimate of the total risk. In a structure of limits and sub-limits for different activities or individual traders, axiom 4 implies that the respect of sub-limits will guarantee the respect of global limits. Therefore one of the great disadvantages of VaR is that it is not sub-additive.

Artzner et al. (1999) have introduced a new risk measure called conditional VaR that is a coherent risk measure, and has a simple relation to the ordinary VaR. Ordinary VaR corresponds to a lower percentile of the theoretical P&L distribution, a threshold level of loss that cuts off the lower tail of the

That is, p(0"+(l - 0 < <PW + (1 - 0 ( -

One of the great disadvantages of VaR is that it is not sub-additive