The VaR framework for risk management

Transcription

1 The VaR framework for risk management May 24, 2001 Page 1 of 20

2 Overview Systemic risk in the market Risk management using margins Exploring the concepts of VaR Some examples of VaR for derivatives portfolios Page 2 of 20

3 1. Systemic risks in the market Page 3 of 20

4 Risks in the equity market 1. Price risk 2. Counterparty risk 3. Liquidity risk 4. Operational risk Operational risk is not quantifiable. Of the first three, liquidity risk is difficult to measure. Page 4 of 20

5 The link between price risk and counterparty risk In a transaction between L and S, the mutual credit risk is a central barrier to obtaining soundly functioning securities markets. When prices fall, L has an incentive to default. When prices rise, S has an incentive to default. This is the mechanism for contagion a domino effect in defaults in the market when one economic agent fails. Default can take place when: 1. There is an unexpectedly large movement upward or downward in prices. 2. Prices move down or up consistently over a period of time. A clearing corporation (CC) has a neat solution this problem: Novation. Legally, L buys from the CC, S sells to the CC. L CC S Page 5 of 20 The CC protects itself using a system of margins.

6 2. Risk management using margins Page 6 of 20

7 Risk management at the CC The CC recovers losses (and pays out profits) on a daily basis. This is the mark to market margin. The mark-to-market margin breaks a large multi-day loss into a series of small one-day losses, and reduces the profits from defaulting. Once this is done, the gains from declaring bankruptcy are limited to the losses made on one day. The one day losses that can be made on any given day is collected upfront, and is called initial margin. The initial margin should substantially covers the one day risk. Page 7 of 20

8 Multi day losses vs. one day losses 2 0 Page 8 of Time series of profits/losses

9 The role for VaR at the futures clearing corporation The initial margin should be larger than the one-day loss on most days. Futures exchanges abroad use crude thumb-rules. In modern parlance, initial margin is a Value at Risk (VaR) on a one day horizon. If the one day rupee profit on a position is x f(x), then the VaR v at a 95% level is: v f(x)dx = 0.05 Page 9 of 20 In India, the L. C. Gupta Committee has recommended that initial margin at the futures clearing corporation should use VaR at a 99% level on a one day horizon.

10 3. Exploring the concepts of VaR Page 10 of 20

11 VaR: Definition The VaR is the loss that will be exceeded with a known probability, by a 1. Well defined portfolio: If an investor holds a portfolio, there is no sense in calculating the VaR for any single asset. The VaR of a portfolio is not the same as the sum of the VaR of individual assets. 2. Over a defined period of time: The VaR of a portfolio is the loss expected over a given investment horizon. Most investors calculate a VaR over a day. Corporations with assets that do not have a daily mark to market valuation technology would calculate their VaR over a longer horizon, say, a month. 3. For a defined risk level: The VaR of a portfolio is the loss expected with a given level of uncertainty. A 95% VaR is the minimum level of loss you would expect to make on 5 days out of a 100 day investment period. Page 11 of 20

12 Interpreting the VaR Therefore, if you have a portfolio with an initial investment of Rs. 1 million, and you want to hold it for 100 days. If you calculate that the 95%, one day VaR is Rs. 15,000, then 5 out of 100 days, you can expect to make losses of at least Rs. 15,000. The 100% VaR for any portfolio of risky assets over any time horizon is. Page 12 of 20

13 A simple example of calculating the 95% VaR An investor puts Rs. 1,000,000 into Nifty on 16 th June Date Profits/Losses Date Profits/Losses (Rs.) (Rs.) Page 13 of 20

14 How to calculate the 95% one day VaR Sort the list of the profits and losses in descending order Page 14 of 20 The 95% VaR is the loss that can be exceeded 5% of the times in this sorted list, which is between a loss of Rs. 22,984 and Rs. 21,428. The 99% VaR is the loss that can be exceeded 1% of the times in this list, which is a loss of greater than Rs. 58,580.

15 VaR estimation In general, we look at the distribution of the portfolio to calculate the VaR. This would require a very large sample of data about the portfolio, or the constituents of the portfolio. For linear products, with certain pdfs, VaR is known in closed form. In general, specifically for portfolios containing options, VaR requires simulation: 1. Model the joint distribution of the underlying prices, 2. Simulate T realisations from this process, 3. At each draw, apply theoretical models to reprice the portfolio, 4. Obtain T realisations of the one day change, 5. Sort these values, and read off the 1% order statistic. Page 15 of 20

16 Complications in estimating VaR Changing variance of the assets Non normal distributions. Very large number of assets, which means 1. Larger complications in modelling the distributions 2. Larger number of draws from the distribution Page 16 of 20

17 VaR estimation in real-time Trading takes place at high speed intra-day. Electronic trading implies that positions change rapidly. NSE was the 5th largest exchange in the world in 1999 by number of trades. The target load is 1000 trades per second. Each trade updates a client portfolio; this means 2000 VaR calculations per second. In 1997, our first estimates suggested that doing real-time VaR by Monte Carlo would require a Cray. Page 17 of 20

18 4. Some examples of VaR for derivative portfolios on Nifty Page 18 of 20

19 Pure futures portfolios: 99% one day VaR We invested Rs into a simple 1 month Nifty futures position. The VaR depends purely on the volatility of Nifty: 99% one day VaR = σ Nifty 1000 = Rs If we invest Rs. 100,000 into the same portfolio as above, the VaR gets scaled linearly to Rs. 3, Now we invest Rs into one long 1 month and one short 1 month futures. 99% one day VaR = σ Nifty σ Nifty 1000 = Rs. 0. We invest Rs. 10,000 into 10 long 1 month futures and -11 short Nifty. Both the futures and the spot are driven purely by Nifty volatility. 99% one day VaR = σ Nifty 10, σ Nifty 10, 000 = Rs. 0. Page 19 of 20

20 Some options portfolios:99% one day VaR We calculate each of the VaRs in these examples using a full Monte Carlo simulation with Nifty value at investment being taken as We invest Rs in a deep in the money call (X = 500), which has a little more than a month to expire. 99% one day VaR = Rs Note that this is like the VaR on a one month futures contract. We have a portfolio where we are long Nifty spot, long a put option (X=2000) and short a call option (X=2000). Both options have a year to expire. 99% one day VaR = Rs (Put call parity) Page 20 of 20