Risk Measurement From Theory to Practice: Is Your Risk Metric Coherent and Empirically Justified?
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Risk Measurement From Theory to Practice: Is Your Risk Metric Coherent and Empirically Justified?

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I present desirable features for a risk metric, incorporating the coherent risk framework and empirical features of markets. I argue that a desirable risk metric is one that is coherent and focused ...

I present desirable features for a risk metric, incorporating the coherent risk framework and empirical features of markets. I argue that a desirable risk metric is one that is coherent and focused on measuring tail losses, which significantly affect investment performance. I evaluate 5 risk metrics: volatility, semi-standard deviation, downside deviation, Value at Risk (VaR) and Conditional Value at Risk (CVaR). I demonstrate that CVaR is the only coherent risk metric explicitly focused on measuring tail losses, which are an important, empirical feature of markets. CVaR is the most practically useful risk metric for an investor interested in minimizing declines in the value of a portfolio at stress points while maximizing returns. Through several examples, I demonstrate that the choice of a risk metric may lead to very different portfolios and investment performance due to differences in investment selection, portfolio construction and risk management. I also demonstrate that the focus on tail losses as opposed to volatility results in superior performance - much smaller declines in value at stress points with improvements in average and cumulative returns; similar results can be achieved with other risk metrics, which are not designed to measure tail losses like CVaR Based on empirical data, practical recommendations for investment analysis, portfolio construction and risk management are included throughout the article.

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Risk Measurement From Theory to Practice: Is Your Risk Metric Coherent and Empirically Justified? Risk Measurement From Theory to Practice: Is Your Risk Metric Coherent and Empirically Justified? Document Transcript

  • Risk Measurement from Theory to Practice: Is Your Risk Metric Coherent and Empirically Justified? The Abstract I present desirable features for a risk metric, incorporating the coherent risk framework and empirical features of markets. I argue that a desirable risk metric is one that is coherent and focused on measuring tail losses, which significantly affect investment performance. I evaluate 5 risk metrics: volatility, semi-standard deviation, downside deviation, Value at Risk (VaR) and Conditional Value at Risk (CVaR). I demonstrate that CVaR is the only coherent risk metric explicitly focused on measuring tail losses, which are an important, empirical feature of markets. CVaR is the most practically useful risk metric for an investor interested in minimizing declines in the value of a portfolio at stress points while maximizing returns. Through several examples, I demonstrate that the choice of a risk metric may lead to very different portfolios and investment performance due to differences in investment selection, portfolio construction and risk management. I also demonstrate that the focus on tail losses as opposed to volatility results in superior performance - much smaller declines in value at stress points with improvements in average and cumulative returns; similar results can be achieved with other risk metrics, which are not designed to measure tail losses like CVaR Based on empirical data, practical recommendations for investment analysis, portfolio construction and risk management are included throughout the article. Mikhail Munenzon, CFA, CAIA mikhailmunenzon@gmail.com
  • The devil is in the tails…. Catherine Donnelly and Paul Embrechts (2010) Introduction Starting with the work of Markowitz (1959), volatility has played a central role as risk metric in classical finance and industry practice, despite its reliance on assumptions (e.g., returns are best characterized by normal or Gaussian distribution) that find very limited support. Since then a number of new risk metrics were proposed – semi-standard deviation, downside deviation, Value at Risk (VaR), Conditional Value at Risk (CVaR) – which are supposed to improve on volatility’s deficiencies as a risk measurement tool. The choice of a risk metric plays a key role in determining the potential attractiveness of an investment, the construction of a portfolio and the development of a risk management process. Which risk metric or metrics should an investor rely on? More importantly, which features should a risk metric have to be useful to a practitioner in investment analysis, portfolio construction, risk measurement and management in light of how markets actually behave? Based on the framework of coherent risk proposed by Artzner et al (1999), I evaluate the above metrics. I also critically compare assumptions, goals of various metrics and their practical value by considering their ability to capture empirical features of markets important for portfolio performance. Finally, I illustrate their usefulness with several practical examples. The article is structured as follows. After introducing the coherent risk framework and empirical features of markets a risk metric should be able to handle, I evaluate the above mentioned risk metrics. Then, I discuss several practical applications to further illustrate concepts; concluding remarks follow. 2
  • Coherent Risk Framework Artzner et al (1999) proposed that a desirable risk metric, r, must satisfy four properties. Such a metric is then coherent (see Table 1). First, it must be monotonic (if asset X ≥ 0, r(X) ≤ 0), or if there are only positive returns, then risk is non-positive. In other words, risk cannot increase because of positive returns. Second, it must be sub - additive (r(X+Y) ≤ r(X) + r(Y)), or the risk of a portfolio of 2 assets should be less than or equal to the sum of the risks of individual assets. While this property may not be important for all investors, it is generally important for institutional investors: as one adds more assets to a portfolio, one want to be sure that the chosen risk metric for the portfolio cannot be larger than the sum of such risk metric for individual assets. Otherwise, there is no incentive to own portfolios of securities, resulting in highly concentrated investments. Third, a risk metric must exhibit positive homogeneity (for any positive real number c, r(cX) = cr(X)), or if the portfolio is increased c times, the risk becomes c times larger. Consequently, risk preferences are separate from a risk metric and risk measurement. If an investor is risk averse (risk loving), doubling his investments may more (less) than double his risk in his view. However, that reflects the investor’s perception of risk, rather than its measurement. Finally, a risk metric must exhibit translation invariance (for any real number c, r(X+c) ≤ r(X) – c), or cash or another risk free asset does not contribute to portfolio risk. Consequently, risk metrics should be measured in value terms (e.g., in dollars), rather than relative terms (e.g., volatility). Then, capital invested in a risk asset can be offset by capital invested in a risk free asset or cash. Risk metrics stated in percentage terms can be easily converted to value terms by multiplying it with the value of a portfolio. 3
  • Empirical Features of Markets and Risk Measurement A useful risk metric should also be able to handle empirical features of markets important for risk measurement, which I briefly summarize (for more detail, see Munenzon (2010a,2010b) and Cont and Tankov (2004)). First, returns do not follow a Gaussian (normal1) distribution of a symmetrical, bell-shaped curve, which is a key tool in classical finance. This is particularly true at stress points in markets. Return time series have significant skewness and kurtosis. More importantly, many investment choices available to investors (e.g., equity securities) are typically characterized by negative skewness (returns below the mean are more likely than returns above the mean) and large kurtosis (extreme events are more likely than for the Gaussian distribution)2. With such features, a return time series has regular, large, negative extreme events. Additionally, such features are the opposite of what a typical investor may prefer – positive skewness and small kurtosis, resulting in consistent, positive returns. Secondly, losses (and gains) are concentrated. Finally, there is gain/loss asymmetry: large declines are generally larger in magnitude than large price increases. Consequently, it is important that a risk metric is focused on measuring tail losses, as they will have a very significant impact on portfolio performance. As a result, throughout the remainder of the article, I assume that an investor is interested in minimizing tail losses while maximizing returns. Of course, an investor may be focused on minimizing volatility. However, as I demonstrate below, such a focus is likely to lead to portfolios that are likely to experience significant declines at stress points, from which it will be hard or impossible to recover. Semi-standard deviation and downside deviation 1 In fact, the term normal distribution is misleading since it wrongly implies that the empirically observed behavior of markets (e.g., negative skewness and large kurtosis) is the exception rather than the rule. 2 For Gaussian distribution, skewness is 0 (symmetrical) and kurtosis is 3. 4
  • will produce a similar result as, like volatility or VaR, they are not focused on measuring tail losses (see Table 1). Risk Metrics Volatility Volatility (or standard deviation) is simply the square root of the average squared deviations from the mean of a time series. Mathematically, volatility or σ is defined as N σsemi = √ ( N-1 ∑ (Rn – μ)2) n=1 Where R is a series of returns, and μ is the mean return and N is the number of observations in a time series. In cases of limited or incomplete data for a time series which is common in practice, it is more appropriate to divide by (N – 1), rather than N. Therefore, as the related name of standard deviation suggests, the primary purpose of volatility is to measure the average range of deviations from the average of a time series. Its purpose is not to measure the magnitude of potential losses or their probability. It can be a good approximation of potential losses only if returns follow the Gaussian distribution, which is a very special case3. It is also not designed to measure tail losses. Additionally, it penalizes deviations above the mean (above average returns) as much as deviations below the mean (below average returns). Consequently, an investment which never has a negative return period and produces only positive returns 3 For the Gaussian distribution, all events can be completely defined by the mean and volatility. However, for non-Gaussian distributions, one also needs to know skewness and kurtosis, as the reliance on the mean and volatility will present an incomplete picture. 5
  • of varying magnitude will have positive volatility (positive risk) though the probability of a loss of capital with such investment is zero. Because of this, one can conclude that volatility cannot be a coherent risk metric as it fails to satisfy the property of monotonicity of a coherent risk metric discussed above. Moreover, if the range of potential losses is very broad, the average may leave the portfolio unprepared for unfavorable environments. For example, if the average volatility is 2% but can be as high as 15% for some periods, is such a portfolio ready for so high a level of stress? Finally, as with all historical based metrics, it is important to make sure that the data one uses to calculate results is large enough to be representative4 and still relevant for the future. In summary, volatility is not a coherent metric as it penalizes both positive and negative deviations from the mean, resulting in positive risk for an investment that never loses money. Moreover, it is not designed to measure tail losses, which is an important, practical deficiency. As the average measure of deviations, it may not allow one to fully appreciate the potential range of outcomes. It can be a good approximation of average losses only in the special case when a time series follows the Gaussian distribution. In practice, significant deviations from normality are observed, which further limit the usefulness of volatility as a risk metric. In fact, some non-normal distributions with fat tails will not even have a well – defined standard deviations. Semi-Standard Deviation (SSD) 4 For example, one may measure the sample error of a time series as 1 / square root of the number of observations. Therefore, with 100 observations, there is 10% chance that one’s data is not complete or representative. 6
  • Markowitz (1959) noted the practical usefulness of focusing only on downside deviations from the mean, as opposed to penalizing all deviations equally. SSD is just the square root of the average squared deviations below the mean. Mathematically, SSD is defined as N σsemi = √ ( N ∑ min (Rn – μ, 0)2) , -1 n=1 Where R represents a series of returns, μ is the mean return and N is the number of observations. As with volatility, the primary purpose is to measure the average range of deviations below the mean, not the magnitude of potential losses or their probability. It is also not focused on measuring tail losses. Similar to volatility, negative skewness and fat tails may produce an understated SSD as large, negative events receive a low weigh in the formula despite their potentially large impact on the portfolio. Moreover, semi- deviation type measures fail the sub-addivity property and thus are not coherent risk measures (for more detail, see Artzner et al (1999)). Downside Deviation (DV) DV measures the average deviation relative to some minimum acceptable return (MAR), rather than the mean return of a time series. Typically, MAR is set to 0. Mathematically, DV is defined as σdv = √ ( N-1 ∑ min (Rn – MAR, 0)2) 7
  • Where R represents a series of returns, MAR is the minimum acceptable return and N is the number of observations5. With DV, one can measure risk with a broad range of return thresholds. DV also satisfies all properties of a coherent risk metric. However, like the above metrics, DV measures risk as the average range of deviations from some number. Therefore, all practical limitations of such metrics discussed above apply: it is not focused on measuring tail losses, it provides no information on the likelihood of a potential event, and the average may significantly understate the range of potential losses with significant skewness and kurtosis. Value at Risk (VaR) Typically, VaR is defined as the level of losses that will not be exceeded at some confidence level. For example, one can be 99% confident that losses will not exceed 5% on a daily basis. Mathematically, VaR(1-α) = - RVaR6 so that P(R ≥ RVaR) = 1-α Where R is an observed return and α is confidence level (e.g., 1% or 5%). Therefore, VaR(99%, daily) = 5% means that in 99% of days, losses will not exceed 5%. A related and more practically useful definition is as the minimum level of losses that will be exceeded with some probability, leading to P(R ≤ RVaR) = α 5 More generally, DV is a lower partial moment function (LPM) of order 2 described as LPM2,MAR. LPM can be defined as LPMp,,MAR = E(│min(R – MAR, 0) │p ) 1/p. 6 VaR will therefore be a positive number, consistent with the first condition of coherent risk. Of course, VaR can also be reported as a negative number, which further emphasizes that it is a measure of losses. 8
  • Where R is an observed return and α is confidence level (e.g., 1% or 5%). Thus, VaR(99%, daily) = 5% means that in 1% of cases, losses will be at least 5%. Using a historical time series, one can easily find VaR as an appropriate percentile. By using historical data, one avoids making any assumptions about the distribution of a return series, letting the data speak. However, one must be careful in drawing conclusions from data that may not be large enough to be representative or, due to structural breaks, relevant for the future. One can also easily use analytical formulas assuming Gaussian7 or other distributional shapes without relying on quantile analysis of historical data. At first glance, VaR appears as a much more practically useful risk metrics than the ones discussed above. It is explicitly concerned with the measurement of potential loss levels, rather than the average deviations from the mean or some other threshold. Moreover, one can also obtain the likelihood associated with a particular loss level. Such ease of use and convenience make this method widespread for business and regulatory purposes8. However, VaR has a number of very serious practical deficiencies. First of all, except in the special case of the Gaussian distribution analytic formula, VaR is not sub-additive. Therefore, it is not a coherent risk metric. If one combines securities or portfolios, one cannot simply add their VaR at some confidence level and be sure that the combined VaR is at most no higher than the sum of the individual elements. Moreover, VaR is a very incomplete risk metric since it cannot provide any information about the magnitude of losses once the VaR limit is exceeded. As noted above, most investment options tend to have concentrated losses, which deviate very significantly from the 7 Despite its mathematical elegance and convenience, such an assumption has very limited empirical support as noted above. 8 For example, Basel II requires that banks measure VaR(99%). 9
  • ‘normal’ market environment. Knowing that in 99% of cases one is not likely to lose more than 1% is not helpful if in 1% of cases when such losses are exceeded, they may be exceeded by a very large margin and may average at a level much higher than losses in a more typical environment. That is also the reason why it is useful to remember that VaR is the minimum level of losses. Conditional Value at Riks (CVaR) (Expected Tail Loss or Expected Shortfall) CVaR measures the expected or average losses in the tail. It is related to VaR in that CVaR measures losses once VaR is exceeded – the area which will drive most of potential losses. In other words, while VaR(99%) measures the maximum loss in 99% of cases (or the minimum loss in 1% of cases), CVaR(99%) measures the average loss in the 1% of the worst cases. Mathematically, CVaR can be defined as CVaR = -E(R│R < -VaR) CVaR satisfies all 4 properties of a coherent risk metric. Like VaR9, it is an easy and convenient metric to report and explain, measuring not only the magnitude but also the likelihood of losses. However, whereas VaR stops at the start of the tail leaving one unprepared for market stress, CVaR calculates losses one may experience in the worst cases10. This feature gives CVaR a very important practical advantage over VaR and other risk metrics: as noted above, empirically, losses are concentrated and reside in the tails of a distribution of a time series, which CVaR is designed to measure so that a 9 Like VaR, CVaR can also be calculated not just from historical data but via a formula with the assumption of some distributional shape for the tail such as those from the Extreme Value Theory (for example, see Meucci (2007)). 10 In a very special case of limited tails, CVaR can be as large as VAR. However, the larger is the tail as descibed by skewness and kurtosis, the larger is the difference between CVaR and VaR. 10
  • portfolio might survive a stress period. As one calculates the average of the tail, one must be mindful of the fact that data limitations may at times be significant. For example, with 100 observations, CVaR(99%) will have just 1 observation (1% of 100). One must also be aware of the limitations of the average discussed above. Therefore, as a coherent risk metric which is focused on measuring the tails, CVaR appears to be the most practically useful risk metric for an investor interested in minimizing declines in portfolio values at stress points while maximizing returns. I now turn to several examples of practical applications of the above metrics to illustrate and compare them further. Examples A Hypothetical Hedge Fund Manager A hypothetical manager has 200 months long track record. In 100 months, returns are 0.8%; in 90 months, losses are -0.1%; in 8 months, losses are -5%; in 2 months, losses are -7% and -10%. I do not make any assumptions about the path of returns, though typically, such a manager may have many months of relatively consistent positive performance followed by concentrated losses, which eliminate multiple months of positive performance. This example is not as contrived as might seem at first glance as the manager’s statistical features may parallel those for sub-par funds involved in several alternative investments strategies such convertible arbitrage (see next example). Risk metrics for such manager are presented in Table 2. The manager’s average return is only slightly above 0. Volatility is quite controlled at 1.4% and so are semi-standard and downside deviations. Volatility is higher than semi-standard and downside deviations 11
  • because it also penalizes large positive deviations from the mean. If one excludes the 10 worst months (or 5% of data), the manager’s average return jumps to 0.37%. VaR(95%) is much lower than deviation-based metrics (0.34% vs 1.4% for volatility) as it finds the percentile in a time series where most losses are only 0.1% . VaR(99%) at 5% is much higher than the prior metrics as it get closer to the tail and then stops. The manager’s large negative skewness and kurtosis should serve as signals that the above analysis for potential losses is far from complete, as deviation-based averages and VaR cannot capture fat, negative tails, where losses are concentrated. CVaRs, which measures average losses in the tail, are larger than VaR, especially at 99% confidence level. It correctly alerts an investor to the true size of potential declines. Therefore, CVaR provides a more practically relevant estimate of downside risk for a manager as compared to the estimate of other metrics. Traditional Asset Classes and Alternative Investment Strategies I now apply the above risk metrics to traditional asset classes and alternative investment strategies (for more detail, see Munenzon (2010a,2010b)). I used data for the following traditional asset classes: equities – SP 500 Total Return Index (SPX); bonds - JPM Morgan Aggregate Bond Total Return Index (JPMAGG); commodities – SP GSCI Commodities Index (GSCI); real estate – FTSE EPRA/NAREIT US Total Return Index (NAREIT)11. Performance data for alternative investment strategies are Center for International Securities and Derivatives Markets (CISDM) indices. I use 9 common alternative investment strategies: convertible arbitrage (CA), 11 Some investors consider commodities and real estate alternative asset classes, as compared to stocks and bonds. However, for the purposes of this analysis, I consider all such asset classes to be traditional ingredients in an investment program. 12
  • distressed (DS), merger arbitrage (MA), commodity trading advisor (CTA), macro, equity long/short (LS), equity market neutral (EMN), emerging markets (EM), event driven (ED) (for more detail on strategies, the reader is referred to Anson (2006)). The monthly data for the indices was downloaded via Bloomberg. The full historical time horizon for this analysis is 12/31/1991 (the first month available for all CISDM indices via Bloomberg) to 1/29/2010 to allow for all asset classes and strategies to have the same historical time period. One can notice that all investment choices are strongly non-Gaussian and that the worst month is generally much larger in magnitude than the best month. Also, most have negative skewness and large kurtosis, signaling that deviation-based metrics and VaR will provide an incomplete picture of potential losses. Only CTA and Macro have positive skewness12. Such departures from the Gaussian distribution are on average much larger for alternative investment strategies than traditional asset classes, particularly convertible arbitrage, distressed, event driven and emerging markets. Similar to the prior example, different risk metrics may lead to very different asset allocation and risk management decisions. For example, the gap between CVaR, VaR and deviation – based metrics is particularly large for large deviations from the Gaussian assumptions, such as convertible arbitrage and distressed. Deviation-based metrics and VaR might lead an investor to favor bonds, CA and EMN. However, once the focus shifts to the magnitude of the negative tails as measured by CVaR (particularly, at 99%), CA is no longer attractive in relative terms as its low volatility with respectable returns comes at the cost of large, negative tails, as signaled by its large negative skewness and 12 Macro also has large kurtosis, which, combined with positive skewness, may be attractive to some investors as it suggests that when extreme events do occur, such events are likely to be positive for returns, not negative when negative skewness is combined with large kurtosis. 13
  • large kurtosis. An investor may also make different decisions when comparing investments not just based on their risk metric but also when comparing their risk/return relationship. For example, based on the classical Sharpe ratio, which compares the excess return above the risk free rate of an investment to its volatility, one might prefer EMN, MA, ED, CA and DS. However, once investments are penalized for the size of their tails with the adjusted Sharpe ratio (excess return divided by CVaR), ED, CA and DS become less attractive while Macro, CTA, LS and bonds rise significantly in their individual ranking13. Portfolio Optimization I now illustrate the application of volatility and CVaR to portfolio optimization through mean variance and mean – CVaR optimizations14 for 4 strategies: CA, DS, CTA and LS. I find the global minimum variance (GMV) and global minimum CVaR (min – CvaR(95%)) portfolios with the constraints that portfolio weights are non-negative and sum up to 1 (see Table 4). The GMV approach has a relatively low estimation error as it does not require any return inputs, which have the largest estimation error. For the CVaR optimization, historical data was used as return scenarios. Unsurprisingly, portfolios weights are quite different for 2 optimization methods. For volatility-based optimization, CA and DS are heavily weighed as their volatilities are lower than those for CTA and LS15. However, CVaR - based optimization is not ‘fooled’ by low volatility, which 13 It must be noted that ‘smooth’ returns, as described by significant positive autocorrelation further understate potential losses. One should unsmooth such a time series to get a better understanding the potential range of outcomes. This is beyond the scope of this article, but the reader referred to Davies et al (2005) for an example. 14 For mathematical details regarding implementation, the reader is referred to Fabozzi et al (2007). 15 For the same reason, similar result can be reached with semi-standard deviation and downside deviation. VaR will generally produce results similar to those for deviation – based metrics since, though it is 14
  • typically hides large tails. Instead, such method is strongly attracted to small, negative tails. Therefore, in the global minimum CVaR portfolio, weights shift to CTA / LS away from CA / DS as tails are much smaller for CTA / LS as compared to those for CA / DS. More importantly, these differences in weights result in meaningful differences in economic performance experienced by an investor in this historical sample. Average and cumulative returns are higher for min - CVaR portfolio as compared to GMV portfolio as large declines in the value of a portfolio are avoided. For example, the worst month is - 2.6% for the min – CVaR portfolio as compared to the decline of -6.6% for GMV portfolio, an improvement of over 60%. CVaR metrics also experience an improvement of 18 – 50% for the min – CVaR relative to the GMV portfolio. It is also interesting to note skewness is positive and kurtosis is small for the min – CvaR, creating a portfolio with small, negative tails and returns likely to be higher than the mean. Such features are typically what investors prefer. By contrast, the GMV portfolio has large, negative skewness and large kurtosis, producing fat, negative tails. Also, gain/loss asymmetry was eliminated in the CVaR portfolio: its best month is much larger in magnitude than the worst month, whereas the opposite is true for the GMV portfolio. Additionally, the gap between volatility and CVaR(95% and 99%) is very large (over 4x for volatility vs CVaR(99%) and 1.9x for volatility and CVaR(95%)) for the GMV portfolio, shrinking meaningfully for the min – CVaR portfolio (1.6x for volatility and CVaR(99%) and 1.2x for volatility and CVaR(95%). This fact further emphasizes that the reliance on volatility may leave a portfolio unprepared for stress, as volatility is likely to significantly understate potential declines, especially in the presence of typical departures from concerned with measuring loss levels and their probabilities, it is not focused on measuring tails, where such losses are concentrated. 15
  • Gaussian assumptions. Finally, the min – CVaR portfolio performs much worse than the GMV portfolio if the focus, as is typically the case in the industry, is on volatility or VaR(95%). Thus, paradoxically, an organization holding the min – CVaR portfolio may be required to hold more capital in good times or may appear unattractive to potential clients than an organization with the GMV portfolio, though the GMV portfolio may result in substantial declines from which it may be hard or impossible to recover. Concluding remarks I presented desirable features a risk metric, incorporating the coherent risk framework and empirical features of markets. I argue that a desirable risk metric is one that is coherent and focused on measuring tail losses, which significantly affect investment performance. I evaluated 5 potential risk metrics: volatility, semi-standard deviation, downside deviation, VaR and CVaR. Volatility is not a coherent metric as it penalizes positive deviations from the mean as much as negative deviations. Moreover, it provides no information about the likelihood of a particular loss level, focusing on the average. Also, it does not focus on the tails of a time series, where potential losses will be concentrated. Semi-standard deviation and VaR are not coherent risk metrics because they are not sub-additive. Like volatility, semi-standard deviation measures the average deviation from some threshold and does not focus on the tails. VaR provides an estimate of a potential loss and its likelihood. However, like deviation – based metrics, it is not focused on measuring the tail. Downside deviation is a coherent risk metric. However, it is concerned with the average downside deviation, rather than the tail loss. CVaR is a coherent risk metric. Moreover, it provides an estimate of a loss and its associated 16
  • likelihood. More importantly, CVaR is explicitly designed to measure the tail loss. Therefore, CVaR is the most practically useful risk metric for an investor interested in minimizing declines in portfolio values at stress points while maximizing returns. Through several examples, I demonstrated that the choice of a risk metric may lead to very different portfolios and investment performance due to differences in investment selection, portfolio construction and risk management. I also demonstrate that the focus on tail losses as opposed to volatility results in superior performance - much smaller declines in value at stress points with improvements in average and cumulative returns. References Alexander, C. 2008. Value at Risk Models. John Wiley and Sons. Anson, M. 2006. Handbook of Alternative Assets. John Wiley and Sons. Artzner, P., Delbaen F., Eber J. and Heath D. 1999. Coherent measures of risk. Mathematical Finance 9, 203-228. Cont, R. and Tankov P. 2004 Financial modeling with jump processes. CRC Press. Markowitz, H. 1959. Portfolio Selection. John Wiley & Sons. Davies, R., Kat, H., and Lu, S. 2005. Fund of Hedge Funds Portfolio Selection: a Multiple Objective Approach. Working Paper. Donnelly, C. and Embrechts, P. 2010. The Devil is in the Tails: Actuarial Mathematics and the Subprime Mortgage Crisis. Working Paper. Fabozzi F., Kolm P, Pachmanova D. and Focardi S. 2007. Robust Portfolio Optimization and Management. John Wiley and Sons. 17
  • Meucci, A. 2007. Risk and Asset Allocation. Springer. Munenzon, M. 2010a. “20 Years of VIX: Fear, Greed and Implications for Traditional Asset Classes.” Working Paper. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1583504 Munenzon, M. 2010b. “20 Years of VIX: Fear, Greed and Implications for Alternative Investment Strategies.” Working Paper. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1597904 18
  • Table 1 Coherent Risk Framework and Empirical Features of Markets Risk Metric Semi-standard Downside Volatilty deviation Deviation VaR CVaR Does the metric satisfy a coherent risk property? Monotonicity No Yes Yes Yes Yes Sub-additivity Yes No Yes No Yes Positive homogeneity Yes Yes Yes Yes Yes Translation invariance Yes Yes Yes Yes Yes Does it focus on the tails? No No No No Yes
  • Table 2 A hypothetical hedge fund manager 200 months of observations 100 months - returns are 0.8% 90 months - losses are -0.1% 8 months - losses are -5% 2 months - losses are -7% and -10% Mean return (%) 0.07 Mean return with the worst 10 months excluded (%) 0.37 Skewness -4.04 Kurtosis 18.72 Volatility or standard deviation 1.44 Semi-standard deviation 1.34 Downside deviation 1.32 VaR(95%) 0.34 VaR(99%) 5.02 CVaR(95%) 5.70 CVaR(99%) 8.50
  • Table 3 12/31/1991 - 1/31/2010 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return 0.7% 0.7% 1.1% 0.5% 1.5% 0.8% 0.9% 0.8% 0.7% 0.8% 1.0% 0.7% 0.9% 1.0% Compounded avg return 0.6% 0.5% 0.9% 0.5% 0.1% 0.8% 0.9% 0.8% 0.7% 0.8% 0.9% 0.7% 0.8% 0.9% max 9.8% 21.1% 31.7% 4.6% 90.8% 4.7% 5.3% 4.7% 7.9% 8.6% 9.4% 2.8% 12.1% 4.8% min -16.8% -27.8% -32.2% -3.5% -32.7% -11.5% -10.6% -5.6% -5.4% -5.4% -9.4% -2.1% -26.3% -7.3% Normality at 95% confidence level? No No No No No No No No No No No No No No pval 0.1% 0.1% 0.1% 2.3% 0.1% 0.1% 0.1% 0.1% 4.5% 0.1% 0.1% 0.1% 0.1% 0.1% No serial correlation at 95% confidence level? Yes No No Yes No No No No Yes No Yes No No No pval 69% 0% 0% 51% 35% 0% 0% 0% 20% 4% 7% 0% 0% 0% Volatility 4.3% 6.1% 6.0% 1.2% 17.9% 1.4% 1.8% 1.1% 2.5% 1.6% 2.2% 0.6% 3.8% 1.7% Semi-standard deviation 3.3% 4.4% 4.6% 0.8% 10.5% 1.2% 1.5% 0.8% 1.6% 1.0% 1.6% 0.4% 3.1% 1.3% Downside deviation 3.0% 4.1% 4.2% 0.6% 9.5% 1.0% 1.2% 0.6% 1.3% 0.6% 1.2% 0.2% 2.7% 1.0% VaR (95%) 7.6% 9.4% 7.8% 1.4% 21.1% 1.0% 1.5% 1.0% 3.3% 1.2% 2.4% 0.1% 4.4% 1.5% VaR (99%) 12.1% 14.3% 22.3% 2.7% 29.9% 4.4% 6.0% 2.4% 4.4% 2.5% 4.5% 1.1% 12.6% 6.9% CVaR(95%) 10.1% 12.9% 15.0% 2.1% 26.4% 3.1% 3.9% 2.0% 4.0% 2.2% 3.9% 0.7% 9.2% 3.7% CVaR(99%) 14.0% 19.2% 25.9% 2.9% 31.1% 7.4% 8.1% 3.5% 4.8% 3.6% 6.3% 1.5% 17.6% 7.1% Skewness -0.8 -0.3 -0.9 -0.2 1.4 -3.9 -1.9 -0.8 0.4 1.2 -0.2 -0.4 -2.1 -1.6 Kurtosis 4.4 5.2 11.6 3.9 6.8 33.6 13.4 8.7 3.0 7.6 5.7 6.3 16.0 9.4 cumulative return for full sample 269.32% 174.79% 604.87% 214.90% 27.50% 435.40% 596.83% 450.51% 306.00% 413.66% 637.00% 322.57% 516.20% 651.63% % of months with positive returns 64.1% 56.2% 65.0% 68.2% 46.5% 85.7% 79.7% 86.2% 55.3% 71.0% 70.0% 92.2% 71.4% 80.2% Sharpe Ratio (0% risk free rate) 0.141 0.077 0.150 0.457 0.006 0.542 0.492 0.712 0.260 0.469 0.418 1.155 0.222 0.557 Rank 12 13 11 7 14 4 5 2 9 6 8 1 10 3 CVaR(99%) Adjusted Sharpe Ratio (0% risk free rate) 0.043 0.025 0.035 0.181 0.004 0.105 0.112 0.225 0.137 0.213 0.146 0.450 0.048 0.131 Rank 11 13 12 4 14 9 8 2 6 3 5 1 10 7 Notes: Jarque-Bera test was used to evaluate normality of a time series; null hypothesis is stated in the question. Ljung-Box test with 20 lags was used to evaluate serial correlation of a time series; null hypothesis is stated in the question. SPX - SP500 Total Return GSCI - SP GSCI NAREIT - FTSE EPRA/NAREIT US Total Return JPMAGG - JPM Morgan Aggregate Bond Total Return VIX - VIX Index CA - convertible arbitrage DS - distressed LS - equity long/short MA - merger arbitrage EM - emerging markets EMN - equity market neutral ED - event driven
  • Table 4 CISDM Alternative Strategy Indices 12/31/1991 - 12/31/2009 monthly data Convertible Arb Distressed CTA Long Short Minimum variance portfolio weights 55.4% 16.0% 26.5% 2.1% Minimum CVaR(95%) portfolio weights 0.0% 32.1% 47.8% 20.1% Constraints: Non-negative weights. Weights sum up to 1. % change from Min Portfolio statistics Min Variance Min CVAR(95%) Variance Arithmetic avg return 0.78% 0.81% 3.85% max 4.16% 4.84% 16.35% min -6.60% -2.62% -60.30% volatility 1.18% 1.46% 23.74% Semi-standard deviation 0.93% 0.99% 5.97% Downside deviation 0.67% 0.56% -16.44% VaR(95%) 0.80% 1.41% 76.81% VaR(99%) 3.28% 2.15% -34.45% CVaR(95%) 2.22% 1.81% -18.46% CVaR(99%) 4.74% 2.35% -50.42% Skewness -1.74 0.28 -115.94% Kurtosis 12.26 2.77 -77.43% Cumulative return 434.30% 461.70% 6.31%