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

nSS ,...,1

n

iii tStV

1

)()(

n

iii SV

1

.,...,1)()( nitSttSS iii

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

VaR

VaRVPr

),(~ 2NV );(1 VaR

33.2)01.0(1 65.1)05.0(1

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…

)(),...,( 11 nn SgSg

ig iS .t

n

iii gV

1

ig

Simple Approximations

1st Order – Delta Approximation:

2nd Order – Gamma Approximation:

ii

iiii S

S

gt

t

ggg

22

2

2

1i

i

ii

i

iiii S

S

gS

S

gt

t

ggg

22

2

2

1i

i

ii S

S

gg

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 .

),cov( ji gg )1( nji 2

2n

kikiii fbfbaS 11 nk

Employing Delta Approximation

With the Factor model we consider:

The delta approximation then becomes:

In vector notation:

),...,(),...,,...,( 111 knk ffgffg

V

n

ii

1

ig

k

jj

j

ii ff

gt

t

g

1

n

ij

k

j

n

i j

ii

ii f

f

gt

t

g

1 1 1

k

jjjt f

1

fV Tt

j

k

j

n

i j

iit ff

g

1 1

Delta Normal VaR

Suppose that:

where

Then..

Given a small we have:

),0(~ ANf ),cov( jiij ffA

),(~ ANfV Tt

Tt

.)(1 AVaR T

t

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:

fV t 21

2

2

2

1f

f

gn

i

ii

22

1f

Gamma approx cont’d

How is distributed ?

It is a quadratic:

Complete the square - consider:

Expand and match:

and

V

22

1ffV t

22

1efV t

2

2

1

tt

e

Towards Gamma VaR

If we assume that then:

Furthermore:

Use statistical tables to find

),0(~ 2Nf

1,~e

Nef

1,~2

22

eef

Z

)(z

)(Pr zZ

Gamma VaR

Since

We see that is equivalent to:

Thus we can read off VaR estimates:

2/2

2

tVefZ

)(Pr zZ

2)(

2

1Pr zV t

2)(2

1 zVaR t

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:

:

fV Tt

n

i

k

r

k

ss

sr

iri f

ff

gf

1 1 1

2

2

1

k

rs

k

s

n

i sr

iir f

ff

gf

1 1 1

2

2

1 s

k

r

k

srsr ff

1 12

1

kkR

n

i sr

iirs ff

g

1

2

ff T 2

1

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,

2)( ef 1e

1e

,2

1 11 ffVT

t

1

2

1 Ttt

Variable Transformations

We assume that

where is positive definite:

and

To make simplifications we set:

and

ANf ,0~ A

)()( 2/12/1 AAA T )()( 2/12/11 AAA T

)( 12/1 fAy )()( 2/12/1 AAB T

Towards Gamma approx

With these transformations we can neatly write:

One step further – spectral decomposition of B:

Orthonormal matrix of Eigenvectors.

Eigenvalues of B.

yByV Tt

2

1

Tk CdiagCB ),...,( 1

kkRC

k ,...,1

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.

yCz T

k

jjjt zV

1

2

2

1

),(~ 1k

T ICN

Approximating the distribution

What more can we say about ?

We can write down analytic expressions for its moments:

1st

2nd

3rd

V

)(2

11 ATrm t

AATrmm T 2212 )(

2

1

)(3)()(3 32121

313 AAATrmmmmm T

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 .

V

V

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.

kpaY 2~ p

A distributional approximation

The integer moments of the random variable

are given by…

where denotes the gamma function.

kpaY

...3,2,1)2/(

)2/(2

rp

prka krr

r

)(

Moment matching

We have analytic expressions for the integer moments of both

:

and

:

Matching moments gives values for and

V ,...,, 321 mmm

kpaY ,...,, 321

pa, .k

Gamma VaR

Using the approximation

We can read, from a table of , values such that

We then set:

For an appropriate confidence level .

VY

2 )(y

)(Pr yY

VaRy )(

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.