Market Risk Market Risk Market risk is the uncertainty resulting from changes in market prices. It can be measured over periods as short as one day. Market Risk Measurement Calculating Market Risk Exposure Generally concerned with estimated potential loss under adverse circumstances JPM RiskMetrics (or variance/covariance approach) Historic or Back Simulation Monte Carlo Simulation JP Morgan RiskMetrics Model Determine the daily earnings at risk (DEAR): DEAR = dollar value of position × price sensitivity × potential adverse move in yield Or DEAR = Dollar market value of position × Price volatility For fixed income securities: Daily price volatility=[D/(1+R)] × adverse daily yield move Assume that changes in the daily yield are normally distributed, 90% (95%, 99%)of the time the changes in the daily yield will be within 1.65 (1.96, 2.33) standard deviations of the mean. Example 101: calculate DEAR for a position of $1 million of 7year zerocoupon bonds with a yield of 7.243%. Assume that the mean of standard deviation of the daily yield change is 0 and 10 basis points, respectively. Given 90% confidence interval: Market value of position = $1,000,000/(1+7.243%)^7=$612,900 Price volatility = [D/(1+R)] (Potential adverse change in yield) = (7/1.07243) (1.65x0.0010) = 1.077% DEAR = Market value of position (Price volatility) = ($612,900) (.01077) = $6,600 Question: Calculate DEARs given 95% and 99% confidence interval. To calculate the potential loss for more than one day: Market value at risk (VAR) = DEAR × N Example: For a fiveday period VAR = $6,600 × 5 = $14,758 For foreign exchange & equities: 
In the case of foreign exchange, DEAR is computed in the same fashion we employed for interest rate risk.

Example
Export Bank has a trading position in Japanese Yen and Swiss Francs. At the close of business on February 4, the bank had ¥300,000,000 and Swf10,000,000. The exchange rates for the most recent six days are given below:
Exchange Rates per U.S. Dollar at the Close of Business
2/4 2/3 2/2 2/1 1/29 1/28
Japanese Yen 112.13 112.84 112.14 115.05 116.35 116.32
Swiss Francs 1.4140 1.4175 1.4133 1.4217 1.4157 1.4123
a. Calculate the foreign exchange (FX) position in dollar equivalents using the FX rates on February 4.
Japanese Yen: ¥300,000,000/¥112.13 = $2,675,465.98
Swiss Francs: Swf10,000,000/Swf1.414 = $7,072,135.78
b. Calculate the volatility of change in exchange rates for each currency over the fiveday period.
Day Japanese Yen: Swiss Franc
2/4 0.62921% 0.24691% % Change = (Rate_{t}/Rate_{t1})  1 * 100
2/3 0.62422% 0.29718%
2/2 2.52934% 0.59084%
2/1 1.11732% 0.42382%
1/29 0.02579% 0.24074%
0.01205 0.00428
c. Determine the bank’s DEAR for both currencies by using 90% confidence level.
Japanese Yen: $2,675,465.98 * 1.65*0.01205 = $53,194.95
Swiss Francs: $7,072,135.78 * 1.65*0.00428 = $49,943.42
For equities, if the portfolio is well diversified then
DEAR = dollar value of position × βx1.65x _{M}.
In order to aggregate the DEARs from individual exposures, we cannot simply sum up individual DEARs. Instead, it requires the correlation matrix.
DEAR portfolio = [DEAR_{a}^{2} + DEAR_{b}^{2} + DEAR_{c}^{2} + 2_{ab} × DEAR_{a} × DEAR_{b} + 2_{ac} × DEAR_{a} × DEAR_{c} + 2_{bc} × DEAR_{b} × DEAR_{c}]^{1/2}
Calculate the DEAR for the following portfolio with and without the correlation coefficients.
Estimated
Assets DEAR _{S,FX} _{S,B} _{FX,B}
Stocks (S) $300,000 0.10 0.75 0.20
Foreign Exchange (FX) $200,000
Bonds (B) $250,000
What is the amount of risk reduction resulting from the lack of perfect positive correlation between the various assets groups?
The DEAR for a portfolio with perfect correlation would be $750,000. Therefore the risk reduction is $750,000  $559,464 = $190,536.
Basic idea: Revalue portfolio based on actual prices (returns) on the assets that existed yesterday, the day before, etc. (usually previous 500 days). Then calculate 5% worstcase (25^{th} lowest value of 500 days) outcomes Does not require normal distribution of returns Does not need correlations or standard deviations of individual asset returns 
500 observations is not very many from statistical standpoint

Increasing number of observations by going back further in time is not desirable.
To overcome problem of limited number of observations Employ historic covariance matrix and random number generator to generate observations.
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