What is Value-at-Risk, and Is It Appropriate for Property/Liability Insurers? Neil D. Pearson Associate Professor of Finance University of Illinois at Urbana-Champaign April 1999

What is Value-at-Risk?

If You Like the Normal Distribution

What is Value-at-Risk? Notation:  V change in portfolio value f(  V) density of  V x specified probability, e.g Value-at-risk (VaR) satisfies or

What is Value-at-Risk? Value-at-risk (for a probability of x percent) is the x percent critical value If you like the normal distribution, it is proportional to the portfolio standard deviation

What is Value-at-Risk? Value-at-risk is something you already understand Value-at-risk is a particular way of summarizing the probability distribution of changes in portfolio value The language of Value-at-Risk eases communication

If Value-at-Risk Isn’t New, Why Is It So Fashionable? It provides some information about a firm’s risks It is a simple, aggregate measure of risk It is easy to understand “Value” and “risk” are business words

Basic Value-at-Risk Methodologies 3 methodologies: –Historical simulation –Variance-covariance/Delta-Normal/Analytic –Monte Carlo simulation Illustrate these using a forward contract –current date is 20 May 1996 –in 91 days (19 August) receive 10 million pay $15 million

First step: Identify market factors Market factors: S, r GBP, and r USD

Historical Simulation Start with current situation: –date: 20 May 1996 –portfolio: 1 forward contract –market factors: S=1.54, r GBP =6.06%, and r USD =5.47% Obtain values of market factors over last N days Use changes in market factors to: – simulate values of market factors on 21 May –compute mark-to-market values of forward contract on 21 May –compute hypothetical profit/loss

Historical simulation: P/L

Repeat N times

Sort

Variance-covariance method

Portfolio standard deviation Portfolio variance X i = dollar investment in i-th instrument  i = standard deviation of returns of i-th instrument  ij = correlation coefficient

Risk mapping: Main idea

The option price change resulting from a change in the oil price is: In this sense the option “acts like”  barrels of oil The option is “mapped” to  barrels of oil

Risk mapping: Interpret forward as portfolio of standardized positions Change in m-t-m value of forward: Find a portfolio of simpler (“standardized”) instruments that has same risk as the forward contract “Same risk” means same factor sensitivities etc.

Risk mapping: Interpret forward as portfolio of standardized positions Let V = X 1 + X 2 + X 3 denote value of portfolio of standardized instruments –each standardized instrument depends on only 1 factor Change in V is Choose X 1, X 2, X 3 so that:

Choice of X 1, X 2, X 3 Recall that the m-t-m value of the forward is This implies

Compute variance of portfolio of standardized instruments Variance of portfolio of standardized instruments: where and  USD is the standard deviation of % changes in the $ interest rate.

Compute value-at-risk Portfolio standard deviation Value-at-risk

Variance-covariance method

Monte Carlo simulation Like historical simulation Use psuedo-random changes in the factors rather than actual past changes Psuedo-random changes in the factors are drawn from an assumed multivariate distribution

What Is VaR, Again One need not focus on change in portfolio value over the next day, month, or quarter Instead, one could estimate the distribution of: –cash flow –net income –surplus –or almost anything else one cares about VaR (broadly defined)  DFA

Is VaR Appropriate for Property/Liability Insurers? Do property/liability insurers have investment portfolios? Do they care about the possible future values of things like: –Cash flow? –Net income? –Surplus?

What is Value-at-Risk?

Limitations of VaR VaR  DFA –it is a particular, limited summary of the distribution VaR is an estimate of the x percent critical value –based on various assumptions –sampling variation VaR doesn’t indicate what circumstances will lead to the loss –2 portfolios with opposite interest rate exposure could have same VaR