## Example 188 FRM Exam 1997Question 11Credit Risk

A commercial loan department lends to two different BB-rated obligors for one year. Assume the one-year probability of default for a BB-rated obligor is 10 and there is zero correlation (independence) between the obligor's probability of defaulting. What is the probability that both obligors will default in the same year FIGURE 1S-3 Distribution of Credit Losses 100 90 80 70 60 50 40 30 20 10 1 credit of 100 million N 1, E(Loss) 1 million, V(Loss) 99 million - 100 - 90 - 80 - 70 - 60 - 50...

## Cross Rate Volatility

Exchange rates are expressed relative to a base currency, usually the dollar. The cross rate is the exchange rate between two currencies other than the reference currency. For instance, say that S1 represents the dollar pound rate and that S2 represents the dollar euro EUR rate. Then the euro pound rate is given by the ratio s3 eur bp sS 5 131 ln S3 ln Si - ln S2 13.2 of of of - 2p12a1 lt r2 13.3 Thus we could infer the correlation from the triplet of variances. Note that this assumes both the...

## Combination of Options

Options can be combined in different ways, either with each other or with the underlying asset. Consider first combinations of the underlying asset and an option. A long position in the stock can be accompanied by a short sale of a call to collect the option premium. This operation, called a covered call, is described in Figure 6-3. Likewise, a long position in the stock can be accompanied by a purchase of a put to protect the downside. This operation is called a protective put. We can also...

## Application The Basel Rules

The Basel market risk charge requires VAR to be computed with the following parameters a. A horizon of 10 trading days, or two calendar weeks b. A 99 percent confidence interval c. An observation period based on at least a year of historical data and updated at least once a quarter The Market Risk Charge MRC is measured as follows which involves the average of the market VAR over the last 60 days, times a supervisor-determined multiplier k with a minimum value of 3 , as well as yesterday's VAR,...

## Example IBM Preferred Stock

IBM issued 11.25 million preferred shares in June 1993. These are traded as 45 million depositary shares, each representing one-fourth of the preferred, under the ticker IBM-A on the NYSE. Dividends accrue at the rate of 7.50 per annum, or 1.875 per depositary share. As of April 2001, the depositary shares were trading at 25.4, within a narrow 52-week trading range of 25.00, 26.25 . Using the valuation formula for a consol, the shares trade at an implied yield of 7.38 . The total market...

## Other Option Contracts

The options described so far are standard, plain-vanilla options. Since the 1970s, however, markets have developed more complex option types. Binary options, also called digital options pay a fixed amount, say Q, if the asset price ends up above the strike price where I x is an indicator variable that takes the value of 1 if x gt 0 and 0 otherwise. Because the probability of ending in the money in a risk-neutral world is N d2 , the initial value of this option is simply These options involve a...

## Lognormal Distribution

The normal distribution is a good approximation for many financial variables, such as the rate of return on a stock, r P1 - P0 Po, where P0 and P1 are the stock prices at time 0 and 1. Strictly speaking, this is inconsistent with reality since a normal variable has infinite tails on both sides. Due to the limited liability of corporations, stock prices cannot turn negative. This rules out returns lower than minus unity and distributions with infinite left tails, such as the normal distribution....

## Simulating Yields

The GBM process is widely used for stock prices and currencies. Fixed-income products are another matter. Bond prices display long-term reversion to the face value unless there is default . Such process is inconsistent with the GBM process, which displays no such mean reversion. The volatility of bond prices also changes in a predictable fashion, as duration shrinks to zero. Similarly, commodities often display mean reversion. These features can be taken into account by modelling bond yields...

## Coupon Curve Duration

FIGURE 1-3 Effective Duration and Convexity to the true value of 862.48. In general, however, effective duration is a by-product of the pricing model. Inaccuracies in the model will distort the duration estimate. Finally, this numerical approach can be applied to get an estimate of the duration of a bond by considering bonds with the same maturity but different coupons. If interest rates decrease by 100 basis points bp , the market price of a 6 30-year bond should go up, close to that of a 7...