1. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management The Oxford Guide to Financial Modeling Thomas S. Y. Ho and Sang Bin Lee Copyright © 2004 by Thomas Ho and Sang Bin Lee. All rights reserved.

2. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 2 15.1 Risk Measurement -Value at Risk (VaR) Definition: a measure of potential loss at a level (99% or 95% confidence level) over a time horizon

3. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 3 15.2 Market Risk Market risk is the losses that arise from the mark to market of the trading securities “Prices” for tradable securities of a portfolio are marked to market that are often derived from the fair values of the valuation models

4. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 4 15.3 VaR for single securities Definition : is called the critical value which determines the one-tail confidence level of standard normal distribution. Time factor is defined as where t is the time horizon in measuring the VaR. Volatility is the standard deviation of the stock measured in $ over one year.

5. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 5 15.3 VaR for single securities - portfolio return distribution - the price of the bond - the critical value for a particular interval of a normal distribution

6. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 6 - the VaR of the bond - the bond price uncertain value is a multivariate normal distribution 15.3 VaR for single securities

7. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 7 15.3 Delta Normal Methodology (2) VaR for a Portfolio (I) - The portfolio value - The portfolio uncertain value - The VaR of the portfolio

8. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 8 15.3 Portfolio VaR VaR for a Portfolio (II) - Component VaR

9. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 9 15.3 Three Stocks Case A Numerical Example - Calculating the VaR of a portfolio of three different stocks (GM, WMT, and IBM) - Calculating the daily rates of return and the variance-covariance matrix

10. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 10 15.3 Correlations of the Stock Returns Calculating the daily rates of return and the variance- covariance matrix

11. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 11 15.3 VaR Derivations The detailed derivation of the individual VaR as well as the portfolio VaR is given as follows. Where,

12. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 12 15.3 VaR Derivation

13. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 13 15.3 Component VaR

14. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 14 15.3 VaR Calculation VaR calculation output by Delta-Normal Method

15. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 15 15.4 Historical Simulation Methodology The Historical Simulation methodology

16. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 16 15.4 Historical Returns Historical Return data set

17. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 17 15.4 VaR Calculation 5-day VaRGMWMTIBMTotal Weight1/3 1 Individual stock VaR4.32923.22644.036211.5917 Portfolio VaR---8.1626 Beta1.06310.84181.0950- Beta*Weight0.35440.28060.36501 Component VaR2.89252.29062.97958.1626 Portfolio Effects1.43670.93581.05673.4291 VaR calculation output by Historical Simulation Method

18. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 18 15.5 Monte Carlo Simulation Methodology Random numbers generated from Multi Normal Distribution

19. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 19 15.5 Simulations based on the Correlation Matrix The variance-covariance matrix of stock returns generated by Monte Carlo simulation

20. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 20 15.5 VaR Calculation VaR calculation output by Monte Carlo Simulation Method

21. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 21 15.6 Extreme Value Theory multiply historical returns by – 1 to convert them into positive values. choose a threshold (): a parametric distribution of the tail beyond the threshold. The ratio: count how many observations are beyond the threshold in the actual data and divide it by the total observation. Parameters () estimation VaR calculation

22. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 22 15.6 Extreme Value Theory - Historical return data vs. Standard Normal Distribution - cumulative distribution functions

23. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 23 15.6 calculate the VaR by the extreme value theory - The formula to calculate the VaR based on the Extreme Value Theory - probability density function

24. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 24 15.7 Credit Risk Definition the loss of principal or interest or any promised payments from the borrow for bonds or loans of any securities

25. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 25 15.7 Credit Risk and Market Risk Model VaR of a Bond - Firm value process (Merton) Integrating Credit Risk and Market Risk in a Portfolio Context (I) - Firm value process (Merton, Longstaff and Schwarzt) - interest rate model (Hull and White)

26. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 26 15.7 Portfolio Credit and Market Risk Integrating Credit Risk and Market Risk in a Portfolio Context - The stock risk

27. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 27 15.7 Portfolio Credit Risk Specify a set of macro-economic factors that would affect the credit risk of the firms. Define the default index by measuring default rate. The macro economic factors are used as the independent variables to explain the default rate. Measure the rating migrations against the speculative default rates: change of the speculative default rate determines the change in the rating migrations. The simulations can then be used to simulate the change in value of a bond portfolio.

28. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 28 15.7 Credit VaR - a Numerical Example by CreditMetrics Initial ratin g Rating at year-end (%) AAAAAABBBBBBCCCDefault AAA90.818.330.680.060.120.00 AA0.7090.657.790.640.060.140.020.00 A0.092.2791.055.520.740.260.010.06 BBB0.020.335.9586.935.301.171.120.18 BB0.030.140.677.7380.538.841.001.06 B0.000.110.240.436.4883.464.075.20 CCC0.220.000.221.302.3811.2464.8619.79 - One-year Transition Matrix

29. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 29 Category1234 AAA3.604.174.735.12 AA3.654.224.785.17 A3.724.324.935.32 BBB4.104.675.255.63 BB5.556.026.787.27 B6.057.028.038.52 CCC15.0515.0214.0313.52 Bond numberCredit GradeFace valueMaturityCoupon Rate Recovery Rate 1A1005zero0.60 2BBB10050.060.55 3BB10050.030.40 - Bond Data set 15.7 cont.. - Example one -year forward zero curves by crediting rating category

30. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 30 GradeAAAAAABBBBBBCCCDefault Price95.624195.45694.95993.92888.765485.095171.920840 Profit/Loss6.85876.69096.19365.16260-3.6703-16.8446-48.7654 Probability0.030.140.677.7380.538.841.001.06 Cumulative Probab ility 100.0099.9799.8398.1691.4310.902.061.06 15.7 cont.. - 1year forward bond Price, Profit/Loss and Probability for the BB-grade bond

31. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 31 15.7 cont.. - Cumulative Probability of BB-Grade Bond’s Profit/Loss

32. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 32 - Estimate the correlation matrix or variance-covariance matrix among the bond returns 15.7 cont.. - Z-threshold

33. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 33 15.7 cont.. - Bond Portfolio Default Risk Distribution

34. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 34 15.8 Risk Reporting Aggregation of Risks to Equity ($mil.) (the VaR Table)

35. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 35 15.9 Risk Monitoring Back testing

36. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 36 15.10 Risk Measurement Strategic Risk Management –Smith and Smithson (1998) determines the economic factors affecting the equity value of a firm –Hedging against these economic factors is strategic risk management Business Process –Build a model of the firm as a system of processes –Manage the processes by monitoring and controlling the risks in each phase

37. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management 37 15.10 Risk Measurement (2) Investment Cycle