1. Value at Risk Chapter 20 Value at Risk part 1 資管所 陳竑廷

2. Agenda 20.1 The VaR measure 20.2 Historical simulation 20.3 Model-building approach 20.4 Linear model

3. The VaR measure 20.1 The VaR measure Value at Risk Provide a single number summarizing the total risk in a portfolio of financial assets. We are X percent certain that we will not lose more than V dollars in the next N days.

4. Example When N = 5, and X = 97, VaR is the third percentile of the distribution of change in the value of the portfolio over the next 5 days. ( 100-X ) % VaR

5. Advantages of VaR It captures an important aspect of riskin a single number It is easy to understand It asks the simple question: “How bad can things get?”

6. Parameters We are X percent certain that we will not lose more than V dollars in the next N days. –X The confidence interval –N The time horizon measured in days

7. Time Horizon In practice, set N =1, because there’s not enough data. The usual assumption:

8. Example Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume

9. Historical Simulation 20.2 Historical Simulation One of the popular way of estimate VaR Use past data in a vary direct way

10. When N = 1, X = 99 Step1 –Identify the market variables affecting the portfolio Step2 –Collect data on the movements in these market variables over the most recent 500 days Provide 500 alternative scenarios for what can happen between today and tomorrow

11. The fifth-worst daily change is the first percentile of the distribution

12. The Model-Building Approach 20.3 The Model-Building Approach Daily Volatilities –In option pricing we measure volatility “per year” –In VaR calculations we measure volatility “per day”

13. Single Asset Portfolio A consisting of $10 million in Microsoft Standard deviation of the return is 2% (daily) N = 10, X = 99 –N(-2.33) = 0.01 –1-day 99% ： 2.33 x ( 10,000,000 x 2% ) = $ 466,000 – 10-day 99% ：

14. Two Asset Portfolio B consisting of $10 million in Microsoft and $5 million in AT&T 1-day 99% ： 10-day 99% ：

15. The Linear Model 20.4 The Linear Model We assume The daily change in the value of a portfolio is linearly related to the daily returns from market variables The returns from the market variables are normally distributed

17. Linear Model and Options define

18. As an approximation Similarly when there are many underlying market variables where  i is the delta of the portfolio with respect to the i th asset

19. Example Consider an investment in options on Microsoft and AT&T. Suppose that S MS = 120, S A  = 30,  MS = 1000, and   = 1000

21. Thank you.