Analytical Approaches to Non-Linear Value at Risk Simon Hubbert, Birkbeck College London

Overview Review Value at Risk approaches for linear portfolios. Consider the case for a portfolio of derivatives. Use Taylor approximations to derive closed form solutions. Based on: Non-linear Value at Risk : Britten-Jones and Schaeffer: European Finance Review 2. 1999.

Portfolio Monitoring Invest in n Risky Assets: Portfolio value today: Potential future loss/profit: where

Normal Value at Risk Q. How much are we likely to lose 100 % of the time over the future period? A. The number that satisfies: If then Typically or

Non-linear Portfolio Invest in n derivatives: Each is a non-linear function of and Potential future loss/profit: we cannot assume the are normally distributed. We need to approximate…

Simple Approximations 1st Order – Delta Approximation: 2nd Order – Gamma Approximation:

Coping with high dimensionality A large number of derivatives in the portfolio creates high computational demands. Eg. The covariance structure: requires numbers. We introduce a factor model: where .

Employing Delta Approximation With the Factor model we consider: The delta approximation then becomes: In vector notation:

Delta Normal VaR Suppose that: where Then.. Given a small we have:

Employing Gamma Approximation The Delta VaR is known to be a weak estimate (see BJ&S 1999). We turn to the gamma approximation: With a single factor this becomes:

Gamma approx cont’d How is distributed ? It is a quadratic: Complete the square - consider: Expand and match: and

Towards Gamma VaR If we assume that then: Furthermore: Use statistical tables to find

Gamma VaR Since We see that is equivalent to: Thus we can read off VaR estimates:

Gamma VaR: the multi-factor case BJ & S (1999) show that gamma VaR provides a much more accurate estimates than the delta approach applied to long options. We want to modify our analysis to cover multi-factor modelling: Idea: Use the approach used in the single factor case to develop a strategy.

Multi-factor Gamma approximation The approximate profit/loss is given by Where: :

Distribution of gamma approx Recall – Single factor case we considered: and found that . In the multi-factor case we set and analogously we can show: where,

Variable Transformations We assume that where is positive definite: and To make simplifications we set:

Towards Gamma approx With these transformations we can neatly write: One step further – spectral decomposition of B: Orthonormal matrix of Eigenvectors. Eigenvalues of B.

The Gamma Approximation One final transformation: Yields: A sum of squares of normal random variables each with unit variance, i.e., a sum of non-central chi-squared random variables.

Approximating the distribution What more can we say about ? We can write down analytic expressions for its moments: 1st 2nd 3rd

An idea.. The distribution of is not known. However.. We have expressions for the integer moments. Idea: Fit the moments to a more tractable distribution Hope for a good approximation to .

A candidate random variable Britten-Jones and Schaeffer (1999) consider a chi-squared random variable: where with p degrees of freedom. Such a random variable was proposed by Solomon and Stephens (1977) - showed that it can provide a good approximation to a sum of chi-square variables.

A distributional approximation The integer moments of the random variable are given by… where denotes the gamma function.

Moment matching We have analytic expressions for the integer moments of both : and Matching moments gives values for and

Gamma VaR Using the approximation We can read, from a table of , values such that We then set: For an appropriate confidence level .

Overview How to compute analytical non-linear VaR: Set up a factor model Employ first or second order Taylor approximations. Assume a distribution for the risk factors (eg, normal). Using the first order approximation with multi-factors Analytical solution – Delta VaR. Using the second order approximation with single factor Analytical solution – Gamma VaR Using the second order approximation with multi-factors Semi-analytical solution – Approximate Gamma VaR

Numerical Findings Numerical tests (BJ-S 1999) against Monte Carlo approach, suggest that: Delta approximations provide weak estimates of VaR. Gamma approximation (with a single factor) improves the VaR estimates – however a single factor assumption may not be realistic. Success of the approximate gamma VaR (with many factors) to VaR estimates is very dependent upon the curvature of the derivatives. Encouraging results are reported for portfolios of long European options.

Bibliography Britten-Jones, M and S. M. Schaeffer: (1999) Non-Linear Value at Risk Economic Finance Review 2: pp 161 – 187. Solomon, H and M. A. Stephens (1977) Distribution of a sum of weighted chi-square variables Journal of American Statistical Association 72: 881-885.