572 MA Introduction to Mathematical Finance
Lecture #10
Paolo Guasoni
[e-mail]
Boston University
c 2004 by Paolo Guasoni p. 1/22
Measuring Risk
So far we have evaluated future payoffs from the standpoint of expected utility. This approach is well-suited to problems involving individual choice, but it is not feasible for other applications, such as risk management, which involve assessing the impact of unfavorable events only. The main question is: which properties are diserable for a good measure of risk? To answer this question, we need to look at potential applications.
c 2004 by Paolo Guasoni p. 2/22
Motivation
Capital allocation for banks and insurance companies: when taking on additional risks, financial institutions need to evaluate their impact on the solvability of the firm. Internal risk-management systems usually require to set the probability of extreme losses (such as those leading to bankrupcy) below certain levels. Capital requirements for banks: regulators want to set rules for capital requirements so as to ensure the stability of the entire financial system. Rules should be clear and not allow for loopholes. Calculating risk may be technical, but the result should be understandable to a person without technical prerequisites.
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Risk Properties
Risk should be...
...translation invariant. If you add a certain amoutn to a position, its risk is reduced by the same amount. ...linear: if you double a position, its risk should also double. ...monotonic: if a payoff is always higher than another, the risk of the former should be lower than that of the latter. ...subadditive: if you combine two positions, their combined risk should be lower than the sum of their risks.
These axioms have been formulated by Artzner, Delbaen, Eber and Heath (1999). A risk measure with this set of properties is called coherent.
c 2004 by Paolo Guasoni p. 4/22
Value at Risk
Value at Risk was introduced by JP Morgan in 1996 as a tool for measuring market risk. The Value at Risk of a certain position is the answer to the following question: what is the minimum size of a loss that will be exceeded or met no more than x% of the time? Equivalently, the question may be reformulated as follows: what is the minimum fixed amount to be added, so that the resulting position will remain solvable with 1 - x% probability?
c 2004 by Paolo Guasoni p. 5/22
Value-at-Risk as a Quantile
Given a random variable X , the left quantile at level is defined as: q (X) = inf{x : P (X x) } Then the Value at Risk of X is defined as: V @R (X) = -q (X) In other words, Value at Risk is just minus the -quantile of X .
c 2004 by Paolo Guasoni p. 6/22
Value at Risk as a Risk Measure
Value at Risk is...
...measured in currency units. In fact q (X + c) = q (X) + c. ...linear. In fact, q (cX) = cq (X) ...monotonic. In fact, if X Y then q (X) q (Y ). ...not subadditive.
The last pitfall is closely related to a more evident one: Value at Risk in general does not account for the size of the losses above the threshold.
c 2004 by Paolo Guasoni p. 7/22
Value at Risk is not subadditive
Take two independent random variables X and Y with the following common distribution. P (X = 0) = P (Y = 0) = 1 - /2, P (x = -1) = P (Y = -1) = /2. From the definition, we have that V @R (X) = V @R (Y ) = 0. If we consider Z = (X + Y )/2, we obtain that V @R (Z) = 1, and hence V @R (X + Y ) = 2 > 0 = V @R (X) + V @R (Y ).
c 2004 by Paolo Guasoni p. 8/22
Value at Risk and diversification
The morale is that V@R ignores the worst cases, as long as they do not exceed a certain probability. Diversification, as beneficial as it is, may turn large losses with a small probability (which are invisible to V@R) into moderate losses with a larger probability (which are visible to V@R). Minimizing V@R may mean concentrating as much downside as possible on a few cases, only to sweep them under the rug of a small probability. Despite this pitfall, V@R continues to be used widely, mainly for inertia, and it is important to understand how to calculate it.
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Historical V@R
Suppose we have n IID observations x1 , . . . , xn from an unknown distribution. The natural estimate for q (X) is given by the corresponding quantile of the empirical distribution. This provides an estimate for V@R called historical V@R. For example, to estimate 1% V@R from 100o observations, one takes minus the tenth smallest. This approach is very general, but is not always feasible.
c 2004 by Paolo Guasoni p. 10/22
Gaussian V@R
If X N (, ), then q (X) = + q In particular, V @R1% (X) 2.33 - . This means that one can simply estimate and , thus obtaining an estimate for V@R. The advantage of this approach is that it only requires means and covariances. For example, if we want to calculate the V@R of some linear combination of X , Y , and Z , we only need the means, the and covariance matrix. But this estimate will be reasonable only if we believe that the sample is Gaussian.
X-
.
c 2004 by Paolo Guasoni p. 11/22
Expected Shortfall
A risk measure which is related to V@R, but that overcomes its pitfalls, is the Expected Shortfall, defined as follows: ES (X) = - inf{E [X|A] : P (A) > } When X has a continuous distribution, then: ES (X) = -E [X|X q (X)] In other words, ES is the expected loss, conditioned on a loss higher than V@R. ES is always greater than V@R: ES (X) = -E [X|X q (X)] -E [q (X)|X q (X)] = = -q (X) = V @R (X) ES coincides with V @R if and only if the conditional loss is always equal to V @R .
c 2004 by Paolo Guasoni p. 12/22
Expected Shortfall as a Risk Measure
Expected Shortfall is...
...measured in currency units. ES (X + c) = ES (X) + c. ...linear. In fact, ES (cX) = ESq (X) ...monotonic. In fact, if X Y then ES (X) ES (Y ). ...subadditive. Exercise.
Therefore ES is superior to V@R as a measure of risk, because it accounts for both probability and the size of losses.
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Example
Take two independent random variables X and Y with the following common distribution. P (X = 0) = P (Y = 0) = 1 - /2, P (x = -1) = P (Y = -1) = /2. Now we have that ES (X) = ES (Y ) = -1. If we consider Z = (X + Y )/2, we obtain that ES (Z) = 1/2, and hence ES (X + Y ) = 1 < 2 = ES (X) + ES (Y ). Expected shortfall now encourages diversification.
c 2004 by Paolo Guasoni p. 14/22
Relationship between ES and V@R
If X has a continuous distribution, there is a simple relationship between expected shortfall and V@R: 1 ES (X) = Proof: Exercise.
0
V @R (X)d
In practice, ES is equal to the averaged value of all values of V@R with level from 0 to . Estimating ES must not be much harder than estimating V@R.
c 2004 by Paolo Guasoni p. 15/22
Historical ES
Suppose we have n IID observations x1 , . . . , xn from an unknown distribution. The natural estimate for ES(X) is given by the corresponding quantity of the empirical distribution. This is historical ES. For example, to estimate 1% V@R from 1000 observations, one takes minus the average among the tenth smallest. As for V@R, this approach is very general, but is not always feasible.
c 2004 by Paolo Guasoni p. 16/22
Gaussian V@R
If X N (, ), then q (X) = + q
X-
.
Denote by and respectively the distriibution function and the density of a standard normal. We have that V @R (X) = --1 () - . It follows that ES (X) =
(-1 ())
- . Exercise.
In particular, ES1% (X) 2.66 - . For 0, we have that V @R (X) ES (X). Exercise.
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The Cornish-Fisher Expansion
The Cornish-Fisher expansion expresses the quantile of certain distributions in terms of their moments, and the quantiles of the standard normal (henceforth z ). The approximation for a normalized distribution (mean 0, variance 1) involving the first four moments is:
2 3 z - 1...