Value-At-Risk

1. Value-At-Risk By Stephen Lynagh Value-At-Risk

2. Outline Introduction What is Value-at-Risk Methods of Calculating VAR Critique of VAR Worst Case Scenario Analysis A VAR Exercise Value-At-Risk

3. Introduction • Risk management attempts to provide financial predictability for a company. • Every day firms face financial risks. Interest and exchange rate volatility, default on loans, and changes in credit rating are some examples. • These risks can be sorted into two categories – credit risk and market risk. 1) Credit risk: includes all risks associated with the credit of specific participants, such as potential default or changes in credit rating. Value-At-Risk

4. Introduction (Cont.) 2) Market risk: refers to risks affecting broad sectors of the economy, such as an increase in interest rates, currency devaluation, or a decline in commodities prices, like aluminum or oil. • Financial analysts use a number of innovations to calculate and hedge against these kinds of risk. • One innovation that has been receiving immense attention is value-at-risk. Value-At-Risk

5. What is Value-at-risk? • Value-at-risk (VAR) is a probabilistic measure of the range of values a firm’s portfolio could lose due to market volatility. • This volatility includes effects from changes in interest rates, exchange rates, commodities prices, and other general market risks. • In simple words, VAR is a statement of probable loss. AfshinMufti Value-At-Risk

6. Methods of calculating VAR • VAR can be calculated in many different ways. • As a result, firms using different calculating methods can arrive at different value-at-risk numbers for the same portfolio. • There are advantages and disadvantages in each method of calculating VAR and no one way is best. • So, when describing VAR, it is important to bear in mind the method of computation and the statistical significance of the result. • Regardless of the method of computation, VAR is a comprehensive measurement for an entire firm. Value-At-Risk

7. Methods of calculating VAR (Cont.) • In this case we will discuss three methods of calculating VAR: Correlation, Historical Simulation, and Monte Carlo Simulation. • Each method of calculating VAR is based on parameters derived from the historical price data over some past period of the assets in the portfolio. • The period can be as short as 100 days or as long as many years. The length of the period influences the model calculations. • Therefore, period is a consideration to make when estimating VAR. Value-At-Risk

8. Methods of calculating VAR (Cont.) • Each method values the portfolio in the next period. • To calculate VAR, risk managers compare the value of the portfolio in the future with the value of the portfolio today. • Each of the three methods estimates a value for the portfolio tomorrow. • The difference between the future value and the present value is the basis for the VAR. Because VAR is actually some value in the distribution of possible changes, it is the tail value of loss level where the tail is defined as a cut off at some confidence level, say 95%. Value-At-Risk

9. Methods of calculating VAR (Cont.) • Computing the VAR requires generating the distribution of outcomes of the risk portfolio out to some period in the future. • The parsimonious way to do this is to first obtain the distribution of possible outcomes for the underlying risk factors for the assets in the portfolio. • Risk factors are the primary sources of risk such as exchange rates, interest rates, etc. • Once the distribution of risk factors is obtained, it is easy to obtain the distribution of asset values since they are computed from the range of risk factors. Value-At-Risk

10. Methods of calculating VAR (Cont.) • For example, given a range of outcomes of the interest rate, it is possible to compute the range of values of a zero coupon bond. • Finally, aggregating the asset distributions using the weights of each asset in the portfolio delivers the portfolio distribution. Value-At-Risk

11. Methods of calculating VAR (Cont.) • Correlation Method: this method attempts to calculate the variance of the entire portfolio based on the variances of each asset in the portfolio and the relationships between risk factors. • To use this method to value VAR, a risk manager must perform the following steps: • Use the historical data to calculate the mean, variance, and correlations of each asset. • Assume each asset in the portfolio has a normally distributed return with its own mean and variance. Value-At-Risk

12. Methods of calculating VAR (Cont.) • Weight the assets in the portfolio with fractions so the weights add up to 1; for a single asset portfolio, the weight is one; for a two-asset portfolio, the weights can be any two numbers that add to 1 (0.3, 0.7; 0.41, 0.59; 1.5, -0.5). • The expected return of the portfolio is the weight of each asset multiplied by the return on each asset. • Therefore if the portfolio weight of the first asset is w, then the weight of the second asset will be (1-w). The expected return on the portfolio would be wr1+(1-w)r2. Here r stands for the return on the assets. Value-At-Risk

13. Methods of calculating VAR (Cont.) • The formula for the portfolio variance depends on the number of assets in the portfolio. • With two assets, the variance is the sum of the squared weights of each asset multiplied by the variance of each asset and twice the product of each weight and their covariance. • The formula mathematically represents this variance. Value-At-Risk

14. Methods of calculating VAR (Cont.) • Finally, assume the return on the portfolio is normally distributed, with mean and variance from the earlier calculations. Plot the distribution of expected returns. • The value of the portfolio at the chosen probability level is the VAR. • This value can also be calculated mathematically, given the mean and variance, or mean and standard deviation. • In a standard normal distribution, 95% of the values lie within 1.65 standard deviations from the mean. Value-At-Risk

15. Methods of calculating VAR (Cont.) • Therefore, for a VAR estimate to be accurate to the 95% confidence level, the VAR will lie on the distribution -1.65 standard deviations away from the expected return. • Similarly, 99% of the values in a standard normal distribution lie within 2.33 standard deviations from the mean, so to find the 99% confidence level, VAR is the value on the distribution 2.33 standard deviations on the left of the mean. Value-At-Risk

16. Methods of calculating VAR (Cont.) • The above picture of Standard Normal Distribution marks the lines where 95% of the distribution lies to the right of the line and where 99% of the distribution lies to the right of the line. Value-At-Risk

17. Methods of calculating VAR (Cont.) • Correlation Method Example: Suppose there is a two-asset portfolio. • One asset has an expected return of 20%, the other has an expected return of 12%. • The variance of the first asset’s return is 0.04, and the variance of the second asset’s return is 0.03. The covariance of the two assets is 0.02. • The two assets are equally weighted in the portfolio. • The expected return of the portfolio is: 0.5*0.2+0.5*0.12=0.16 16% Value-At-Risk

18. Methods of calculating VAR (Cont.) • The variance of the portfolio is: 0.52 *0.04+0.52*0.03+2(0.5)(0.5)0.02=0.0275 • The standard deviation of the portfolio is: √0.0275=0.1658 • The 5% tail on the left is: 1.65 * 0.1658 = 0.2736 away from the mean. • The 95% confidence level of VAR is: 0.16 - 0.2736 = -0.1136 there is a 5% chance the portfolio will lose more than 11.36%. • The 1% tail on the left is: 2.33 * 0.1658 = 0.3863 away from the mean. Value-At-Risk

19. Methods of calculating VAR (Cont.) • The 99% confidence level of VAR is: 0.16 - 0.3863 = -0.2263 there is a 1% chance the portfolio will lose more than 22.63%. • Picture (VAR estimates of return for Correlation Example) marks the lines where 95% of the distribution lies to the right of the line and where 99% of the distribution lies to the right of the line. Value-At-Risk