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

2. 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

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

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

5. 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…

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

7. 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

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

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

10. 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:

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

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

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

14. 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.

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

16. 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,

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

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

19. 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.

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

21. 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

22. 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.

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

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

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

26. 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

27. 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.

28. 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: