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Market Risk

Market Risk. Class 23 - Chap 10. Lecture Outline. Purpose: to understand what market risk is and how it is measured Brief introduction to market risk Measurement methods: RiskMetrics Historical Back Simulation Monte Carlo Simulation. What is Market Risk?. Financial Institution. Dealer

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Market Risk

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  1. Market Risk Class 23 - Chap 10

  2. Lecture Outline Purpose: to understand what market risk is and how it is measured • Brief introduction to market risk • Measurement methods: • RiskMetrics • Historical Back Simulation • Monte Carlo Simulation

  3. What is Market Risk? Financial Institution Dealer Trading Book Investor Banking Book “Tradable” assets/liabilities • Short horizon investments • Liquid securities • Long and short positions in: • Bonds • Commodities • FX Futures/Options • Equity Securities • Options • Securitizations • CMO • RMBS Investment assets/liabilities • Long horizon investments • Illiquid securities • Usually consist of: • Consumer loans • Commercial Loans • Retail Loans • Branches Market Risk is the risk associated with daily fluctuations in the price of actively traded assets, liabilities and derivatives - i.e. the risk of losses in value on an FIs trading book

  4. Is Market Risk Really a Problem?

  5. Thinking About Market Risk • FIs need an answer to the following question to understand their exposure to market risk • Value at risk (VaR) is an essential tool used in answering this question • What horizon? • Regulators usually consider “tradable” assets/liabilities as those held for horizons less than 1 year – these assets/liabilities are included in the trading book and the VaR • FIs usually consider “tradable” assets/liabilities as those held for a much shorter horizon How much money will the firm lose on its trading portfolio if the market has a really bad day, month, year …? How do we define a bad day

  6. Methods for Measuring Market Risk Three Main Measurement Methods • RiskMetrics (variance/covariance) • Historical (Back Simulation) • Monte Carlo Simulation

  7. RiskMetrics – history • Developed by JPMorgan in 1994 • The object was to produce a single number that summarized the firms market exposure across all markets in which it traded • In 1994 JPMorgan had 120 independent units trading: • Fixed income • Foreign Exchange • Commodities • Derivatives • Emerging Markets Securities • Proprietary assets • 2008 JPMorgan held a trading portfolio of $460 billion – typical value for a major money center bank

  8. RiskMetrics – Methodology • RiskMetrics begins by measuring the FI’s Daily Earnings at Risk (DEAR) • We are going to do this for three markets • Fixed Income • Foreign Exchange • Equity • The analysis is shown for the 1day horizon but it can be generalized to any horizon Example: if a financial institution has a DEAR of $2 mill at the 95% level – on average, that bank has a 5% chance of losing $2 mill or more tomorrow Total position Value Extreme Loss Per Unit X DEAR = For the most par this is easy to calculate How do we calculate this piece?

  9. RiskMetrics: Dear for Fixed Income Portfolio

  10. RiskMetrics – Fixed Income Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity of the bonds is 13.5%. Find the 95% DEAR of the portfolio. Step 1 find the extreme change in interest rates We are going to use value at risk to find this so we need some historical data for the yield on BBB rated bonds – Federal Reserve

  11. RiskMetrics – Fixed Income Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity of the bonds is 13.5%. Find the 95% DEAR of the portfolio. What is the change in bond YTM that I would expect to be exceeded only 5% of the time Why is it positive? From the tables, 5% of the area under the curve is to the right of 1.6449 on the standard normal distribution. So, We know that 5% occurs 1.6449 standard deviations away from the mean (on any normal distribution) σ = 0.069795 5% 1.6449 STDevs Mean = -0.00034 Standard Deviation = 0.069795 -0.00034 Question: So how many standard deviations from the mean will 5% occur on the distribution above?

  12. RiskMetrics – Fixed Income Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity of the bonds is 13.5%. Find the 95% DEAR of the portfolio. σ = 0.069795 5% 1.6449 STDevs Mean = -0.00034 Standard Deviation = 0.069795 -0.00034 Find the change in interest rates under a really “bad case” scenario Based on historical data – the change in interest rates will exceed 0.1145% only 5% of the time

  13. RiskMetrics – Fixed Income Step #2 Calculate the daily earnings at risk DEAR • Calculate the value of the bond position under the current YTM 13.5% • Calculate the value of the bond position under the new YTM • DEAR equals the difference or potential loss in value Based on historical data – There is a 5% chance that the FI’s daily losses on their fixed income portfolio will exceed $10,220.15

  14. RiskMetrics: Dear for Foreign Exchange (FX) Portfolio

  15. RiskMetrics – Foreign Exchange (FX) Suppose an FI has a position in €1.4 million on their trading book currently the FX rate is 1.36 $/ € find the 95% daily earnings at risk (DEAR) for the companies FX portfolio Step 1 Find the extreme change in FX rates $/€

  16. RiskMetrics – Foreign Exchange (FX) Suppose an FI has a position in €1.4 million on their trading book currently the FX rate is 1.36 $/ € find the 95% daily earnings at risk (DEAR) for the companies FX portfolio Step 1 Find the extreme change in FX rates $/€ Will the FI lose money if this goes up or down? 5% -1.6449 Mean = 0.00 Standard Dev. = .01 Based on historical data – the decrease in interest rates will exceed -0.016449 only 5% of the time

  17. RiskMetrics – Foreign Exchange (FX) Step #2 Calculate the daily earnings at risk DEAR • Calculate the dollar value of the euro position at the current FX rate 1.36 $/€ • Calculate the dollar value of the euro position at the extreme FX rate 1.344 $/€ • DEAR equals the difference or potential loss in value V = (€1,400,000)(1.36$/€) = $1,904,000 V = (€1,400,000)(1.36-.016449$/€) = $1,880,971 V = $1,880,971 – 1,904,000 = – $23,028.60 Based on historical data – There is a 5% chance that the FI’s daily losses on its FX portfolio will exceed $23,928.06.

  18. RiskMetrics: Dear for Equity Portfolio

  19. RiskMetrics –Equity Suppose an FI has a $500,000 equity position in their trading portfolio. The portfolio has a market beta of 1.3 the daily risk free rate is currently .001%. Calculate the 95% DEAR on the FIs equity trading portfolio Step 1 Find the extreme Market Return 5% -1.6449 Mean = -0.00023 Standard dev = 0.0158 Based on historical data – there is a 5% chance that the portfolio return will exceed -0.0362 tomorrow

  20. RiskMetrics – Equity Step #2 Calculate the daily earnings at risk DEAR • Calculate the 95% DEAR – the extreme portfolio return times the total equity position DEAR = –0.0362($500,000) = –18,073.20 Based on historical data – There is a 5% chance that the FI’s daily losses on its equity portfolio will exceed $18,073.20

  21. RiskMetrics – Aggregate risk measures • The last step is to put it all together • We cannot just add them up because that ignores diversification • we need to account for how bonds, currency and stocks are related (correlated) Portfolio DEAR:

  22. RiskMetrics – Aggregate risk measures • Following our example suppose the following correlation matrix for S&P returns, changes in FX rates and changes in Baa bond yields Portfolio DEAR:

  23. RiskMetrics – Application • FI’s usually calculate their DEAR and work to reduce portfolio risk when these DEARs are violated • We have done some pretty simple DEARs but in reality banks trade in many different markets • In 2008 Citigroup’s DEAR calculation required updating 250,000 correlation and variance parameters

  24. JP Morgan holds: • A BBB rated bond portfolio with $12M in face value that it plans to hold for less than 1 month. The portfolio has an average time to maturity of 7.5 years, aggregate semiannual coupon of 8.3% and average YTM of 9.2%. • (ii) A $360.5M position in their equity trading portfolio. The portfolio has a market beta of .73 and the daily risk free rate is currently 0.003%. Find JP Morgan’s 99 % DEAR if the mean and standard deviation of daily changes in YTM for BBB rated bonds is -0.0005 and 0.039 respectively over the last year. The daily mean and standard deviation of market returns is 0.00046 and 0.012 over the last year. The correlation between changes in YTM and market returns is 0.24

  25. Lecture Summary • Market risk: • A bank’s risk of experiencing losses (on their trading book) due to market exposure. • Measurement • RiskMetrics

  26. Appendix

  27. Historical Back Simulation

  28. Historical Back Simulation • The biggest drawbacks of the RiskMetrics approach is that: • It assumes a normal distribution • This may not always be appropriate – for example options have a minimum negative return but unlimited positive return • Correlation must be calculated • The biggest change with historical back simulation is that it: • Does not assume any distribution. It uses the empirical distribution to find the daily earnings at risk (DEAR) • Do not need to calculate correlations and variances when aggregating risks • Basic Idea • We are going to use historical observations to simulate potential scenarios or outcomes for tomorrow

  29. Historical Back Simulation – Fixed Income Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $1,631,483 with an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity of the bonds is 13.5%. Find the 95% DEAR of the portfolio. • We always calculate the change in value in relation to a change in the market (interest rates, market return, FX rate) • The value of the portfolio could be affected by other factors (liquidity) but we just want to measure the exposure to market risk Find the 5% DEAR for the fixed income portfolio • We want to find the cutoff value where 5% of all observations fall below • Procedure • Collect historical changes in interests rates 4 year (1000 observations is a good number) • Calculate the change in value for each observation ie if the interest rate is at 13.5% calculate: ΔV = P(13.5+ΔI) – P(13.5%) for each value of ΔI • Sort values from largest to smallest loss. Find the 5% VaR i.e. 95% of all observations fall below this value VaR(.95) = (1003)(0.05) = 50.15 We used 1003 historical observations

  30. Historical Back Simulation – Aggregate • We can do the same thing for the foreign currency position and the equity position • Finally, to aggregate the risk we just sum up the change in value across all portfolios and sort the total Aggregating each day and then calculating the VaR accounts for the correlation. That is, the interactions between assets is taken into account when we create the full portfolio of bonds stocks and currency

  31. Historical Back Simulation – Disadvantages • Back simulation relies on prior data • Because it uses historical data, there are relatively few observations. This decreases the accuracy (statistical precision) of the estimate • We can use more observations but the further back we go the less relevant those observations become as potential outcomes for tomorrow • We can try to weight prior observations less i.e. give them a lower probability of occurring • The other solution is just to make up numbers. However, we want to do that in a reasonable way → Monte Carlo simulation • We are going to generate observations such that the probability that they occur tomorrow is the same as the probability that they have occurred in the past

  32. Monte Carlo Simulation

  33. Monte Carlo Simulation – Intuition • We will do this for the 2 asset case only – things get a little more complicated for more than 2 assets • Procedure: • Generate 2 standard normal variables you can do this in excel using the following command =NORMINV(RAND(), 0, 1) Transform the uniform variable into a standard normal Generates a uniform random variable between 0-1

  34. Monte Carlo Simulation – Process • We will do this for the 2 asset case only – things get a little more complicated for more than 2 assets • Procedure: • Generate 2 standard normal variables you can do this in excel using the following command =NORMINV(RAND(), 0, 1) • Calculate the correlation between changes in asset prices or returns • MCS assumes a distribution (multivariate normal)so we want to makes sure the variables we are modeling are normally distributed – prices and values are non-normal • Transform variables using estimated parameters: • Repeat for as many simulations as you want • Calculate the simulated price and the change in value • Calculate the DEAR using the simulated data

  35. Monte Carlo Simulation – Example Example: Excel Spread Sheet We can estimate the mean, standard deviation and correlation of the change in FX and equity values calculated above. • Note: with Monte Carlo simulation you could simulate anything prices, changes in returns, FX rates, interest rates … • Pull 5,000 draws from the standard normal distribution • Convert the draws to draws from a bivariate normal

  36. Monte Carlo Simulation – Example • Using simulated values calculate the change in value of the portfolio • Now we just repeat the procedure for back simulation • Calculate the change in value of the portfolios • Sort the values from smallest to largest • Calculate the 5% DEAR – (5000)(.05) = 250th observation • If markets experience a really bad day, the FI will lose: • $55,432.80 on their fixed income portfolio • $22,470.30 on their currency portfolio • 77,903.10 combine

  37. Basel II Standardized Approach

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