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 10-1: calculate DEAR for a position of $1 million of 7-year zero-coupon 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 five-day 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

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.

Historic simulation

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% worst-case (25^{th} lowest value of 500 days) outcomes

Advantages:

Simplicity

Does not require normal distribution of returns

Does not need correlations or standard deviations of individual asset returns