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Value at risk is an adaptation of classical statistics to risk measurement
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All these techniques have problems as risk management tools for hedge funds. Hedge funds often have exposure to stocks, bonds, and currencies, and a risk measurement tool should accommodate mixed positions. Hedge fund portfolios may not be well-diversified so it may be inappropriate to ignore nonsystematic risk. Funds with currency exposure may want more precise hedging methods than proxy hedging.
The risk management tools described next incorporate uncertainty in the risk calculations. Value at risk, the most popular credit risk models, and option risk management all rely on statistics and dont necessarily require an a priori understanding of the relationships between different assets in the portfolio. As a result, these methods allow the hedge fund to monitor the risk of portfolios containing different types of assets.
Value at Risk (VaR )
Value at risk is an adaptation of classical statistics to risk measurement. The first step is to accumulate the risks of the individual positions. The returns on all the assets held in the portfolio are assumed to be normally distributed. The expected return and risks of the portfolio are reduced to the mean and standard deviation of a familiar bell curve. VaR transforms these values into a loss amount.
This shows the normal distribution of two assets similar to the bell curves used in Chapter 6. Asset A has a volatility or annualized standard deviation of 20 percent, and asset B has a volatility of 40 percent. Notice that the distribution is actually a probability function. The area under each line corresponds with the probability of that outcome. Half of the outcomes occur to the left of the midpoint of each curve (because it represents half of the area).
The taller bell curve for asset A is actually the less volatile of the two assets. Returns close to the midpoint are likely and either extremely high or extremely low returns are less likely than the asset with the flatter bell curve for asset B. At the same time, both distributions share one characteristic: About 5 percent of the time, return for the year is lower than 1.65 standard deviations below the mean (the one-tailed test in statistics). In other words, 95 percent of the time, asset A with the 20 percent standard deviation will have a return of -39 percent or better (0 percent mean return less 1.65 20 percent) and 95 percent of the time, asset B with the 40 percent standard deviation will have a return of -78 percent or better (0 percent mean return less 1.65 40 percent).
The 95 percent confidence level is the basis of the VaR calculation. If a hedge fund portfolio holds $2.5 million of asset A, 95 percent of the time the asset will lose less than $975,000 ($2.5 million 39 percent) over a year. If the hedge fund portfolio holds $5.25 million of asset B, 95 percent of the time, the asset will lose less then $4,095,000 ($5.25 million 78 percent) over a year.
Adapting the VaR Format into a Risk Model The one-year horizon is not typical for portfolio managers. Typically, the VaR calculations are based on a one-day holding period. The volatilities of these assets must be adjusted for the shorter time horizon. Equation (11.12) shows the adjustment from annual to daily:
Calculating Portfolio VaR The VaR on a portfolio is not simply the sum of the VaR on each asset held in the portfolio. The portfolio VaR is based on the sum of the squared VaR values of individual assets. However, the portfolio VaR must also account for the effects of diversification.
CREDIT RISK MEASUREMENT
Several alternative ways to measure credit risk are sponsored by software vendors. The most successful commercial product to evaluate and manage credit risk is the KMV Credit Monitor family of products, now a division of Moodys Investors Service. The KMV suite of programs attaches probably to default and measures the expected liquidation value upon default. The software quantifies the default risk on a wide range of securities and loans and allows traders to compare the expected returns of different types of fixed income assets.
OPTION RISK MEASUREMENT
Many hedge funds trade derivative securities, whose price depends on the price of some underlying asset. This option risk measurement section will not describe the specific options because most of the concepts apply to all derivative instruments.
There are many option models, but most models base the value on the current price of the underlying asset, the volatility of that asset, a shortterm borrowing rate, the time to expiration, and the dividend or coupon (if any). Because these factors determine the fair value of an option, changes in any one of these inputs would affect the fair value of the option.
Introduction to the Greeks
The sensitivity of the derivative value to the inputs is described by a series of Greek letters. The formulas of these Greeks differ from model to model but are usually derived along with each valuation model. These sensitivities are a measure of risk in these derivative positions.
Delta Sensitivity to Changes in the Underlying Asset When the price of an asset rises, calls gain value and puts lose value. The sensitivity of the option price change to changes in the underlying asset is called delta and is frequently labeled with the Greek letter ?. For example, a delta of .60
means that a call option rises in value only 60 percent as much as the underlying asset. A hedge fund could buy a call on $10 million in securities or buy $6 million of the security outright. At least with respect to small changes in the price of the asset, the two positions would behave similarly. The risk modeler can treat this $10 million derivative as equivalent to the outright ownership of $6 million of the security for the purposes of determining whether the hedge fund is long or short overall.
Gamma Sensitivity of Delta to Changes in the Underlying Asset The delta of an option changes based on changes in the option inputs. The delta of an option is most influenced by the value of the underlying asset. The sensitivity of the hedge ratio delta to changes in the underlying asset is called gamma and is frequently labeled with the Greek letter ?. Changes in the delta are important to risk measurement because they mean that the risk characteristics of the portfolio can change in rising or falling markets. Often, these changes warrant changes in the composition of the portfolio or of hedges.
Theta Sensitivity to Time to Expiration The value of the underlying security moves around randomly. In contrast, the time to expiration trends steadily down over time. Theta measures the impact of passing time on the value of a derivative. Long positions nearly always lose value under the scenario of no change in price or volatility.
QUESTIONS AND PROBLEMS
11.1What are some of the advantages of using risk control models thatdo not rely on probability?11.2What advantages do probability-based risk control models haveover other techniques that do not rely on probability?11.3Why is risk management not equivalent to risk elimination or riskminimization?
11.4 What is distinctive about bonds that allows the risk manager to use tools such as duration and convexity that may be of limited value for other types of assets?
11.5 Why is the full price or dirty price as high as or higher than the price conventionally used in securities trading?
.6 Why is duration a better measure of bond risk than average life? 11.7 For many years, the Argentine peso was pegged to the U.S. dollar.
Describe some of the advantages and disadvantages of using the U.S. dollar as a hedge, even if your real exposure is in pesos.
11.8 Why might beta provide an improper hedge for long and short equity positions?
11.9 You are long a call on a futures contract that is correctly delta hedged versus the underlying future. You dont plan to adjust the hedge before expiration. Explain how this position resembles an option straddle.
11.11 You calculated the trade weights for a two-year note versus a fiveyear note. Your calculations suggest that you should sell $2 million face amount of five-year notes for each $5 million face amount of two-year notes. However, your weightings were based on modified duration and the value of the positions. You are concerned because the yield on the two-year note seems to move 25 percent more than the yield on the five-year note (that is, a 4-basispoint shift in the five-year yield is associated with a 5-basis-point shift in the two-year yield). How should you adjust your hedge ratio, if at all?
11.12 What would happen if you ignored accrued interest in the trade weighting in question 11.11?
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