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Class 23 - Chap 10 1 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 2 Financial Institution Dealer Trading Book Investor Banking Book “Tradable” assets/liabilities •Short horizon investments Investment assets/liabilities • Long horizon investments • Liquid securities • Illiquid securities • Long and short positions in: • Usually consist of: • • • • • • Bonds Commodities FX Futures/Options Equity Securities Options Securitizations • CMO • RMBS • • • • 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 3 4 FIs need an answer to the following question to understand their exposure to market risk 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 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 5 Three Main Measurement Methods 1. RiskMetrics (variance/covariance) 2. Historical (Back Simulation) 3. Monte Carlo Simulation 6 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 7 RiskMetrics begins by measuring the FI’s Daily Earnings at Risk (DEAR) DEAR = Total position Value X Extreme Loss Per Unit Example: if a financial institution has a DEAR of $2 mill at For the most paraverage, this Howhas do awe calculate the 95% level – on that bank 5% chance of is easy to calculate this piece? losing $2 mill or more tomorrow We are going to do this for three markets 1. Fixed Income 2. Foreign Exchange 3. Equity The analysis is shown for the 1day horizon but it can be generalized to any horizon 8 Dear for Fixed Income Portfolio 9 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 Yield Baathis so we need some historical data We are going to use value at risk toonfind 9.5 for the yield on BBB rated bonds – Federal Reserve 8.5 7.5 6.5 5.5 Change in Baa Yield 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 10 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? Changes in Baa Yield Frequency 80 6 60 σ = 0.069795% 4 40 20 2 0 0 Mean = -0.00034% Standard Deviation = 0.069795% 5% 1.6449 STDevs -0.00034% 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) Question: So how many standard deviations from the mean will 5% occur on the distribution above? 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. Changes in Baa Yield Frequency 80 6 60 4 40 20 2 0 0 Mean = -0.00034 Standard Deviation = 0.069795 σ = 0.069795 5% 1.6449 STDevs -0.00034 Find the change in interest rates under a really “bad case” scenario X z 0.00034% 1.6449(0.06975%) 0.114466% Based on historical data – the change in interest rates will exceed 0.1145% only 5% of the time 12 Step #2 Calculate the daily earnings at risk DEAR a) Calculate the value of the bond position under the current YTM 13.5% V b) Calculate the value of the bond position under the new YTM V c) 10,631,483 395,094.92 26 (1.135) 10,631,483 384,874.78 26 (1.135 0.001145) DEAR equals the difference or potential loss in value DEAR $384,874.78 $395,094.92 $10,220.15 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 13 Dear for Foreign Exchange (FX) Portfolio 14 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 $/€ 15 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 -0.06 300 45 250 40 35 200 30 150 25 20 100 15 50 10 5 0 0 -5 Frequency Frequency Step 1 Find the extreme change in FX rates $/€ Changes in $/€ FX rate Changes -0.04 -0.02 in 0$/€ FX 0.02 rate 0.04 0.06 0.08 45 40 35 30 25 20 15 10 5 0 5% -1.6449 Will the FI lose money if this goes up or down? X z 0.00 1.6449(.01) 0.016449 Based on historical data – the decrease in exchange rates will exceed -0.016449 only 5% of the time Mean = 0.00 Standard Dev. = .01 16 Step #2 Calculate the daily earnings at risk DEAR a) Calculate the dollar value of the euro position at the current FX rate 1.36 $/€ V = (€1,400,000)(1.36$/€) = $1,904,000 a) Calculate the dollar value of the euro position at the extreme FX rate 1.344 $/€ V = (€1,400,000)(1.36-.016449$/€) = $1,880,971 b) DEAR equals the difference or potential loss in value 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. 17 Dear for Equity Portfolio 18 Suppose an FI holds an equity portfolio in their trading book with market value of $500,000. 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 Daily S&P 500 Returns Daily-0.05 S&P 500 Returns -0.10 0.00 0.05 -0.15 20 20 5% 15 15 10 -1.6449 0.12 0.10 0.09 0.07 0.06 0.04 0.03 0.01 0 0.00 0 -0.02 0 -0.03 5 -0.09 50 X z E[ RP ] z 2 2 2 2 E[ RP ] R f E[ RM ] R f Extreme Return R f E[ RM 5 10 -0.05 100 25 -0.06 150 Frequency 200 -0.11 Frequency 250 0.15 25 -0.08 300 0.10 ] R Z f 2 2 0.00001 1.3(.00023 0.00001) 1.6449 1.32 0.01582 0.0003 1.64490.02183 0.0362 Mean = -0.00023 Standard dev = 0.0158 Based on historical data – there is a 5% chance that losses on the portfolio will exceed -0.0362 tomorrow due to market exposure 19 Step #2 Calculate the daily earnings at risk DEAR a) 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 20 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: 2 DEARFI2 DEARFX DEARE2 DEARP DEAR DEAR 2 FI , FX DEARFI DEARFX DEAR 2 FI , E DEARFI 2 FX , E FX E E 21 Following our example suppose the following correlation matrix for S&P returns, changes in FX rates and changes in Baa bond yields Fixed income FX Market Returns Fixed income 1 0.011877 0.269737 FX Equity portfolio 1 0.202521 1 Portfolio DEAR: 10,220.15 2 23,028.60 2 18,073.20 2 DEARP 2(0.01188)10,220.1523,028.60 2(0.2025)23,028.60 18,073.20 $35,145.76 2(0.2697)10,220.1518,073.20 22 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 23 JP Morgan holds: a) 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%. b) (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 24 Market risk: ◦ A bank’s risk of experiencing losses (on their trading book) due to market exposure. Measurement ◦ RiskMetrics 25 Appendix 26 Historical Back Simulation 27 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 28 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. Find the 5% DEAR for the fixed income portfolio • We want to find the cutoff value where 5% of all observations fall below • Procedure 1. Collect historical changes in interests rates 4 year (1000 observations is a good number) • 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 2. 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 3. 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 Observations Change in Yield 45 -56185.6 46 -56185.6 47 -56185.6 48 -56185.6 49 -56185.6 50 -56185.6 51 -56185.6 52 -56185.6 53 -56185.6 54 -56185.6 55 -55157.9 56 -55157.9 57 -55157.9 29 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 Obs 45.00 46.00 47.00 48.00 49.00 50.00 51.00 52.00 53.00 54.00 55.00 56.00 Bonds -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -56185.63 -55157.94 -55157.94 FX -23800.00 -23660.00 -23520.00 -23520.00 -23380.00 -23240.00 -22960.00 -22680.00 -22540.00 -22540.00 -22400.00 -22400.00 Equity -56385.64 -56358.65 -56250.42 -56118.47 -55837.17 -55021.25 -53718.18 -53211.42 -52851.18 -51687.01 -51646.37 -50888.61 Total -68336.54 -67306.65 -67047.75 -66199.20 -66150.30 -65822.59 -65795.34 -65332.19 -65017.40 -64389.43 -63928.60 -63542.26 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 30 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 31 Monte Carlo Simulation 32 We will do this for the 2 asset case only – things get a little more complicated for more than 2 assets Procedure: 1. 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 33 We will do this for the 2 asset case only – things get a little more complicated for more than 2 assets Procedure: 1. Generate 2 standard normal variables you can do this in excel using the following command =NORMINV(RAND(), 0, 1) 2. Calculate the correlation between changes in asset prices or returns 1. 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 3. Transform variables using estimated parameters: X1 1 1 z1 X 2 2 2 z1 z2 1 2 4. Repeat for as many simulations as you want 5. Calculate the simulated price and the change in value 6. Calculate the DEAR using the simulated data 34 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 … mean st dev FI -0.0002 0.0698 FX 0.0000 0.0100 Correlation 0.0130 Pull 5,000 draws from the standard normal distribution Convert the draws to draws from a bivariate normal X1 1 1 z1 X 2 2 2 z1 z2 1 2 X1 0.0002 .0698(1.529) X 2 0 .01(0.1419) 1.529 0.013 (0.1419) 1 0.0132 35 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 obs 247 248 249 250 251 252 253 254 FI -55,454.6 -55,449.8 -55,437.8 -55,432.8 -55,430.1 -55,412.6 -55,399.2 -55,390.1 FX -22,672.5 -22,628.4 -22,620.6 -22,470.3 -22,458.5 -22,453.8 -22,444.9 -22,435.6 total -78,127.1 -78,078.2 -78,058.4 -77,903.1 -77,888.6 -77,866.4 -77,844.1 -77,825.7 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 36 Basel II Standardized Approach 37