Kevin Dowd, Measuring Market Risk, 2nd Edition

Transcription

1 P1.T4. Valuation & Risk Models Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM

2 Dowd, Chapter 2: Measures of Financial Risk DESCRIBE THE MEAN-VARIANCE FRAMEWORK AND THE EFFICIENT FRONTIER EXPLAIN THE LIMITATIONS OF THE MEAN- VARIANCE FRAMEWORK WITH RESPECT TO ASSUMPTIONS ABOUT THE RETURN DISTRIBUTIONS

3 Dowd, Chapter 2: Measures of Financial Risk Describe the mean-variance framework and the efficient frontier. Explain the limitations of the mean- variance framework with respect to assumptions about the return distributions. Define the Value-at-risk (VaR) measure of risk, discuss assumptions about return distributions and holding period, and explain the limitations of VaR. Define the properties of a coherent risk measure and explain the meaning of each property. Explain why VaR is not a coherent risk measure. Explain and calculate expected shortfall (ES), and compare and contrast VaR and ES. Describe spectral risk measures and explain how VaR and ES are special cases of spectral risk measures. Describe how the results of scenario analysis can be interpreted as coherent risk measures. Describe the mean-variance framework and the efficient frontier. The mean-variance framework In the mean-variance framework, we model financial risk in terms of the mean and variance (or standard deviation, as the square root of the variance) of Profit/Loss (or returns). As a related convenience, we assume the daily profit and loss (P/L) or returns obey a normal distribution. Note that, by specifying only the first two moments (mean and variance) we implicitly suggest a normal distribution; e.g., a normal does not require a third (skew) or fourth (kurtosis) moment specification. A random variable X is normally distributed with mean, μ and variance σ 2 (or standard deviation, σ) if the probability that X takes the value x, f(x), obeys the following probability density function (pdf): ( ) = 1 2 ( ) 2 Where, the range of x is defined as <x<. A normal pdf with mean 0 and standard deviation 1, known as a standard normal, is illustrated in Figure 1 below. 3

4 Figure 1 The normal probability density function As the mean is zero and the figure shows that outcomes (or x-values) are more likely to occur close to the mean (μ=0); it also tells us that the spread of the probability mass around the mean depends on the standard deviation σ, i.e., the greater the standard deviation, the more dispersed the probability mass. Properties of probability density function (pdf): The pdf is also symmetric around the mean: X is as likely to take a particular value μ+x as to take the corresponding negative value μ x. The pdf falls as we move further away from the mean, and outcomes well away from the mean are very unlikely, because the tail probabilities diminish exponentially as we go further out into the tail. The pdf is a useful way of depicting outcomes in the left-hand tail, which corresponds to high negative returns or big losses, typically an area of concern in risk management. 4

5 As Dowd explains: A related attraction of particular importance is that the normal distribution requires only two parameters the mean and the standard deviation (or variance), and these parameters have ready financial interpretations: the mean is the expected return on a position, and the standard deviation can be interpreted as the risk associated with that position. This latter point is perhaps the key characteristic of the mean variance framework: it tells us that we can use the standard deviation (or some function of it, such as the variance) as our measure of risk. And conversely, the use of the standard deviation as our risk measure indicates that we are buying into the assumptions normality or, more generally, elliptically on which that framework is built. The Efficient Frontier Before we introduce the efficient frontier, let us first understand how the mean variance approach works; suppose we wish to construct a portfolio from a particular universe of financial assets. We are concerned about the expected return on the portfolio, and about the variance or standard deviation of its returns. The expected return and standard deviation of return depend on the composition of the portfolio, and assuming that there is no risk-free asset for the moment, the various possibilities are shown by the curve in Figure 2: any point inside this region (i.e., below or on the curve) is attainable by a suitable asset combination. Points outside this region are not attainable. Figure 2 The mean variance efficient frontier without a risk-free asset 5

6 Since the investor regards a higher expected return as good and a higher standard deviation of returns (i.e., in this context, higher risk) as bad, the investor wants to achieve the highest possible expected return for any given level of risk; or equivalently, wishes to minimize the level of risk associated with any given expected return. This implies that the investor will choose some point along the upper edge of the feasible region, known as the efficient frontier. The point chosen will depend on their risk-expected return preferences (or utility or preference function): an investor who is more risk-averse will choose a point on the efficient frontier with a low risk and a low expected return, and an investor who is less risk-averse will choose a point on the efficient frontier with a higher risk and a higher expected return. The efficient frontier refers either to the universe without the risk-free asset or with the risk-free asset. Before the introduction of the risk-free asset, the efficient frontier refers to the combination (allocation) of risky assets which include the market portfolio that are superior. Thus, the efficient frontier is the set of points for which we cannot find an obvious improvement: a point is efficient if any increase in portfolio return implies an increase in risk (i.e., a trade-off). The points on the lower (red-ish) segment below are inefficient because they are vertically inferior to points with equivalent risk and higher return: we can improve the risk without sacrificing returns, and vice-versa. Expected Return 19.0% Risky Portfolio 17.0% CML 15.0% Market Portfolio 13.0% 11.0% 9.0% 7.0% 5.0% 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% Standard Deviation Figure 3 6

7 Then, if we add the risk-free asset, we can draw a line segment from the risk-free rate (on the y- axis) that is tangent to the curved efficient segment and contacts the formerly efficient frontier segment exactly at the market portfolio (and, because it is a tangency line, only overlaps at the market portfolio). The new, straight capital market line (CML) becomes the efficient frontier in the presence of the risk-free rate. Explain the limitations of the mean- variance framework with respect to assumptions about the return distributions. Although, the mean variance framework addresses the twin problems of measuring risks and choosing between risky alternatives, yet measuring risk by the standard deviation of returns can be a very unsatisfactory risk measure when we are dealing with seriously nonnormal distributions. Any risk measure at its most basic level involves an attempt to capture or summarise the shape of an underlying density function, and although the standard deviation does that very well for a normal (and up to a point, more general elliptical) distribution, it does not do so for others. Since, any statistical distribution can be described in terms of its moments or momentbased parameters such as mean, standard deviation, skewness and kurtosis. In the case of the normal distribution, the mean and standard deviation can be anything (subject only to the constraint that the standard deviation can never be negative), and the skewness and kurtosis are 0 and 3. However, other distributions can have quite different skewness and/or kurtosis, and therefore have quite different shapes than the normal distribution, and this is true even if they have the same mean and standard deviation. Thus, the normality assumption (implied by mean-variance framework) is only appropriate if we are dealing with a symmetric (i.e., zero-skew) distribution that also has normal tails (i.e., kurtosis = 3). If our distribution is skewed or has heavier tails as is typically the case with financial returns then the normality assumption is inappropriate and the mean variance framework can produce misleading estimates of risk. 7