THE LONG VIEW OF FINANCIAL RISKLisa R. Goldberg a,∗ and Michael Y. Hayes a
We discuss a practical and effective extension of portfolio risk management and construction bestpractices to account for extreme events. The central element of the extension is (expected) shortfall,which is the expected loss given that a value-at-risk limit is breached. Shortfall is the most basicmeasure of extreme risk, and unlike volatility and value at risk, it probes the tails of portfolio returnand profit/loss distributions. Consequently, shortfall is (in principle) a guide to allocating reservecapital. Since it is a convex measure, shortfall can (again, in principle) be used as an optimizationconstraint either alone or in combination with volatility. “In principle” becomes “in practice” onlyif shortfall can be forecast accurately. A recent body of research uses factor models to generate robust,empirically accurate shortfall forecasts that can be analyzed with standard risk management toolssuch as betas, risk budgets and factor correlations. An important insight is that a long history ofreturns to risk factors can inform short-horizon shortfall forecasts in a meaningful way.
[Leading up to the credit crisis, financial models took a] view ofthe world that was far more benign than it was reasonable to take,emphasizing recent inputs over more historic numbers.—MyronScholes, quoted in “Efficiency and Beyond,” The Economist, 16July 2009
Even the tamest financial markets can produceunpredictably wild return dynamics. Low volatilityregimes are punctuated with extreme events, andinterspersed with bursts of turbulence that differ inmagnitude, duration, and other details. Figure 1shows a relatively long history of daily returns tothe MSCI USA Index. The highlighted region
marks the four years preceding the credit crisis (June2003–May 2007).1 This limited window exhibitstractable dynamics that may be compatible withsimple statistics (such as Gaussian). But the longerview shows less regularity and a wider range ofbehavior. This irregularity is also seen in how pricesmove together. Figure 2 shows contemporaneousdaily returns to Coca-Cola and Exxon-Mobil overdifferent four-year periods. These plots illustrate thefamiliar observation that assets may appear uncor-related, even for a long time, and then suddenlyappear highly correlated. Calm periods can persistfor years, so a deep historical perspective is requiredto appreciate the market’s potential for surprise.
Nonetheless, the practice of quantitative risk man-agement is typically not informed by a long view of
FIRST QUARTER 2010 39
40 LISA R. GOLDBERG AND MICHAEL Y. HAYES
-25%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
1972 1978 1984 1990 1996 2002 2008
Figure 1 Daily returns to the MSCI USA Index; highlighted time-span is four years prior to the credit crisis (June2003–May 2007). Even a comparatively tame index can produce wild return dynamics. This may not be evidentover time intervals that are typically used to forecast volatility at horizons up to three months.
1981-1983 1984-1987 1988-1991 1992-1995
1996-1999 2000-2003 2004-2007 2008-July 2009
Figure 2 Contemporaneous daily returns to Coca-Cola (horizontal axis) and Exxon-Mobil (vertical axis) in four-year snapshots (January 1981–December 2007) and a shorter final snapshot (January 2008–July 2009). Stocks thatappear to be uncorrelated in one time period can appear to be correlated in another.
history. One reason is that the cornerstone of riskmanagement, volatility (or equivalently, its square,variance), shows uncommon predictability acrossconsecutive periods of up to three months. Figure 3shows realized volatility on adjacent periods of oneweek, one month, and three months over the historyof the MSCI USA Index, which begins in 1972. Thestraight-line relationship implies that tomorrow’svolatility is largely an echo of today’s.
Figure 3 illustrates the well-known point thatrecent history may be the best guide to forecast-ing volatility, and indeed volatility over horizonsup to three months is most commonly measuredusing a relatively short history. For daily volatility, acommon choice is to exponentially weight the his-tory with a half-life of approximately 23 days. Thismeans that data from the past six months almostcompletely determine tomorrow’s forecast, so daily
JOURNAL OF INVESTMENT MANAGEMENT FIRST QUARTER 2010
THE LONG VIEW OF FINANCIAL RISK 41
0%
5%
10%
15%
20%
25%
0% 5% 10% 15% 20% 25%
Vol
atili
ty (
perio
d t+
1)
Volatility (period t)
week month quarter
Figure 3 Realized volatility of the MSCI USA Index inadjacent one-week, one-month, and three-month peri-ods (January 1972–July 2009). Volatility estimates arebased on equally weighted daily returns and neglect auto-correlation. In the time-scales shown, recent volatility isa good predictor of near-term volatility.
volatility is not ideally suited to take full advantageof a deep history.
This paper focuses on shortfall, which is an extremerisk measure that harnesses market history whileretaining volatility’s intuitive, predictable, and use-ful properties. The concept of shortfall is not new.However, practical advances that enable investors totake account of shortfall in portfolio constructionand risk analysis are of recent vintage. The longview of history that is required to accurately esti-mate shortfall implies that it has the potential tocarry information that complements volatility. Asa result, shortfall can be a meaningful adjunct tovolatility in the investment process.
However, shortfall is not the only possible volatil-ity supplement. Different aspects of a portfolioreturn distribution can be probed with convex risk
measures, which can be analyzed with standard riskmanagement tools such as beta and correlation.In Section 1, we discuss convexity, which is themathematical embodiment of diversification. Bothvolatility and shortfall are convex risk measures,and in Section 2 we argue that convexity is essen-tial to sound risk management. This argument iscontroversial, since value at risk is not a convexmeasure. Subsequently, in Section 3, we describean empirically based methodology that uses a longview of history to forecast shortfall over horizons ofone to ten days. We conclude with three empir-ical questions concerning leverage, the premiumassociated with extreme risk, and the fair valueof insurance. Answers to these questions wouldimprove our understanding of financial risk and ourability to manage it.
1 What makes a good risk measure?
In a landmark paper, Markowitz (1952) mathemati-cally formulated portfolio construction as a tradeoffbetween expected return (desirable) and risk (to beavoided). Markowitz proposed variance to measurerisk, but repeatedly emphasized the broader goal ofseeking and measuring diversification:
Diversification is both observed and sensible; a rule of behaviorwhich does not imply the superiority of diversification must berejected.
Since Markowitz, many alternatives to variancehave been suggested. Examples include skewness,kurtosis, and value at risk. A drawback to theseexamples is that they do not consistently rewarddiversification. A related consequence is that theserisk measures are difficult or impossible to optimizeagainst.
More recently a series of academic papers haveaddressed the question of what makes a good riskmeasure; see for example Artzner et al. (1999),Föllmer and Schied (2002, 2004, Chapter 4), Rock-afellar et al. (2006), and references therein. Whilethere is not a consensus on all details, every response
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42 LISA R. GOLDBERG AND MICHAEL Y. HAYES
echoes Markowitz by demanding that a “good” riskmeasure promotes diversification. This condition isguaranteed by the mathematical property of con-vexity. The set of convex risk measures includesshortfall, acceptability indices, MinMaxVaR, andMaxMinVaR. Further details are in Eberlein andMadan (2007) and Cherny and Madan (2006).
Convexity can be visualized in terms of a curvethat represents portfolio risk as a function of assetweight. If the risk measure is convex, a straightline between any two points on the curve will lieon or above the curve. Convexity is guaranteed(in one dimension) by a non-decreasing slope (orfirst derivative), when it exists. This means that asthe weight in a risky position increases (and theweight in cash decreases), the slope of portfoliorisk also increases or stays flat, but cannot decrease.Some convex and non-convex curves are sketchedin Figure 4. The top left-hand curve in Figure 4depicts the shape of portfolio variance as a functionof weight in a single asset. The only parameter isthe ratio of height to width of the curve.
Figure 4 illustrates the connection between convex-ity and diversification: the risk of any combinationof two assets (intermediate points) can never begreater than their average stand-alone risks (straight
line). Artzner et al. (1999) make the incisive point2
that
[Diversification] does not create extra risk.
If a diversified portfolio had greater risk than theaverage stand-alone risks, this would imply thatdiversification was intrinsically risky. Convexityallows the diversified risk to equal the average stand-alone risks, but not exceed it. Non-convexity admitsthe possibility that the act of putting two assetstogether in a portfolio entails its own risk, so non-convex risk measures may encourage concentrationover diversification.
More generally, consider a decomposition of a port-folio return into weighted return of n positions,which can be assets, factors, or sectors3:
P =n∑
i=1
xiAi . (1)
Here xi denotes the weight of position i, and Aidenotes its return. A measure µ of the risk of P isconvex if
µ
(n∑
i=1
xiAi
)≤
n∑i=1
xiµ(Ai), (2)
whenever∑
i xi = 1 and xi ≥ 0.4 Paraphrasing thesuccinct statement of Föllmer and Schied (2004,Page 154):
If one diversifies, spending only the fraction xi on component Ai ,one obtains
∑ni=i xiAi . Thus convexity [Ineq. (2)] gives a precise
meaning to the idea that diversification should not increase risk.
2 Managing convex risk
Quantitative risk management relies on a constella-tion of analysis tools that can uncover unintendedbets on common risk factors, illuminate underly-ing sources of risk, and reveal subtle opportunitiesfor diversification. The standard toolkit includesbetas, risk budgets, and correlations, and these toolsare usually applied to analyze portfolio volatility σ.
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THE LONG VIEW OF FINANCIAL RISK 43
However, a broader analysis stems from the obser-vation that these tools can be expressed in terms ofthe slope of a position’s risk profile (Figure 4), alsoknown as the marginal contribution to risk (MCR):
MCRµ(P , Ai) = ∂µ(P)
∂xi. (3)
We illustrate the connection between the MCR andthe standard toolkit with the beta of a positionwith respect to a portfolio. A few lines of alge-bra transform the familiar definition of beta (interms of covariance and variance) into a portfoliorisk-rescaled marginal contribution:
βσ(P , Ai) = Cov(P , Ai)
Var(P)
= MCRσ(P , Ai)
σ(P). (4)
Similarly, the familiar definition of linear correla-tion between a portfolio and a position (in terms ofcovariance and standard deviation) can be recast asa marginal contribution rescaled by position risk5:
Corrσ(P , Ai) = Cov(P , Ai)
σ(P)σ(Ai)
= MCRσ(P , Ai)
σ(Ai). (5)
Equation (5) indicates that the marginal contribu-tion of a position to volatility is the product of theposition risk and the correlation between the posi-tion and the portfolio. This leads to the concept ofrisk-implied correlation:
MCRµ(P , Ai) = ρµ(P , Ai)µ(Ai), (6)
which is explored in Cherny and Madan (2007) andGoldberg et al. (2009). Convex risk-implied corre-lation is non-decreasing in position weight. Note,however, that this need not hold for a non-convexrisk measure. Generalizations of Eqs. (4) and (5)to a convex measure of extreme risk are exploredby Goldberg et al. (2009), who show that a par-allel decomposition of two risk measures provides
insights that cannot be attained by analyzing a singlemeasure of risk.
Since it is the slope of a risk profile (Eq. (3)),the MCR describes the impact of a sufficientlysmall trade on portfolio risk. Therefore, by exam-ining marginal contributions across a collection ofpositions, an investor can determine the set oftrades that have the greatest impact on portfoliorisk.6
The gradient (or multidimensional slope) of a riskmeasure is a vector of MCRs. If the risk mea-sure is convex, the gradient has special utility toinvestors seeking to optimize against risk since italways leads to a global minimum risk portfolio. Incontrast, a non-convex measure may have multipleminima, so a particular minimum risk portfo-lio is not guaranteed to be the global minimum.Consequently, following the gradient of a non-convex risk measure may lead away from the globalminimum.
Consider an unconstrained portfolio that is mean-risk optimal, so that the position weights x maxi-mize an objective function:
f (x) = E[x · A] − �µ(x · A), (7)
which is expressed in terms of a risk aversion param-eter �. Since the derivative of the objective fevaluated at the optimal weights is zero:
∂f∂x
= E[A] − �MCRµ(P , A) = 0, (8)
the expected return of any position is proportionalto its MCR:
E[Ai] = �MCRµ(P , Ai). (9)
Running the logic of optimization backwards leadsto reverse optimization (see, for example, Sharpe,2001 or Pearson, 2002). Assuming a portfolio isoptimal, Eq. (9) determines the ratios of expectedexcess returns to positions in terms of MCR. A man-ager who is concerned about particular measure ofrisk (e.g., volatility or extreme risk)7 can compare
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44 LISA R. GOLDBERG AND MICHAEL Y. HAYES
risk-implied expected excess returns with his ownfundamental views. If there is disagreement, he canrebalance accordingly. Since the slope of a convexrisk profile is non-decreasing in the weight of anasset, selling the asset decreases its MCR and buy-ing the asset increases its MCR. For a non-convexmeasure, this basic investment intuition may be vio-lated: selling an asset can increase its MCR, andhence its implied excess expected return.
The marginal contribution of a position to a con-vex risk measure is bounded by the stand-alone riskof the position. It may be useful to explicitly con-sider these bounds, as they define the bounds onexpected return in any portfolio through reverseoptimization.
3 The special case of shortfall
Shortfall (s) is the average loss beyond a given (high)percentile of possible losses, so it is an estimate of theexpected value of an especially severe loss. The levelof severity is specified by the percentile. Shortfall isalso known as expected shortfall, conditional value atrisk, average value at risk, and expected tail loss. Asan average over worst outcomes, shortfall is a guideto capital reserve allocation and risk monitoring.Shortfall provides a forecast of those extreme lossesthat capital reserves are designed to cover.
Since it is a convex measure of risk, shortfall, likevolatility, promotes diversification and is a sensibleoptimization constraint. Unlike volatility, which ismost accurately estimated using recent history, highpercentiles of shortfall demand abundant data toobtain reasonable estimation error (see Yamai andYoshiba, 2002). Generating adequate relevant datais the most serious challenge to accurately estimat-ing shortfall. Goldberg et al. (2008) develop a modelthat uses returns to risk factor and time series anal-ysis to generate rich data sets that reflect the riskcharacteristics of a wide class of portfolios.8 Forexample, consider a portfolio P whose exposures toa set of risk factors is given by the vector X. Then a
history of common factor return P f to the portfoliois given by:
P ft = X · ft ,
where ft is the vector of time t returns to the factors.The model developed in Goldberg et al. (2008) isimplemented in Barra Extreme Risk (BxR), whichforecasts and analyzes shortfall at horizons of one toten days.
Goldberg et al. (2008) develop test statistics to assessthe accuracy of shortfall forecasts.9 Based on out-of-sample tests over the 11-year period 1996–2007,Goldberg et al. (2008) show that BxR is more accu-rate than a conditionally normal model, which takesshortfall to be a fixed multiple of volatility. Thissuggests that a relatively long, consistently appliedhistory provides a more accurate forecast of extremerisk over a one- to ten-day horizon than a shorthistory.
Forecasts of shortfall in BxR are called ExtremeShortfall or xShortfall, and they can be written asa time-dependent volatility term multiplied by atime-independent constant (st = σt C ). Estimatesof time-independent constant C change as newinformation becomes available, but are stable whenestimated with a long history. A measure NN ofnon-normality is the ratio of C to the correspondingconstant in a Gaussian model (C G ) minus one:
NN = CC G − 1.
The quantity NN is the fractional differencebetween xShortfall and a Gaussian shortfall esti-mate. Figure 5 shows values of NN for style factorsin two Barra factor models: Global Equity Model(GEM2) and Europe Equity Model (EUE3). Thevalues were computed on 30 December 2008 using12 years (GEM2) and 14 years (EUE3) of dailyfactor history. Protective factors such as Size inboth models, and Value in GEM2 appear less riskythan normal, while aggressive factors such as Lever-age, Growth, and Momentum appear riskier than
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THE LONG VIEW OF FINANCIAL RISK 45
-0.1 -0.05 0 0.05 0.1 0.15 0.2
Dividend YieldSize
Earnings YieldMomentum
ValueLeverageLiquidityGrowth
Volatility
ValueSize
Size NonlinearityMomentum
GrowthLiquidityVolatility
Leverage
GEM2
EUE3
Figure 5 Non-normality (NN) of 95% one-day shortfall of style factors in two Barra factor models: Global EquityModel 2 (GEM2) and European Equity Model 3 (EUE3). The non-normality measure displayed is the percentdifference between normal and Barra Extreme Risk estimates of 95% one-day shortfall. A positive value of NNmeans that the empirically observed ratio of shortfall to volatility exceeds the normal estimate. A negative ratiomeans that the normal estimate exceeds the empirically observed ratio.
normal. This result may be familiar to a veteranof financial markets, as it reflects empirical factorbehavior across history.
Marginal contributions to shortfall admit to a sim-ple and testable formulation. If the distribution ofportfolio return is a smooth function of weights,
MCR s(P , Ai) = ∂s(P)
∂xi
= E [Ai |P > VaR]. (10)
In other words, the marginal contribution to short-fall is equal to the expected return of asset i when theportfolio value at risk is exceeded. MCR s describesthe behavior of assets when the portfolio sufferslarge losses, revealing which assets can be expectedto mitigate a large loss.
Test statistics that evaluate the accuracy of shortfallare developed in Goldberg et al. (2008), Watewai(2007), and Barbieri et al. (2008). These statis-tics compare forecast shortfall to realized loss (−P)
when a value-at-risk limit is exceeded. They are aver-ages of ratios (realized to forecast) or differences(realized minus forecast). Using Eq. (10), we canconstruct statistics to test marginal contributionsof assets, sectors, and factors to shortfall. Furtherout-of-sample evaluation of shortfall and marginalcontributions to shortfall for diverse classes ofportfolios over different market climates and timehorizons is required to assess the value of shortfallforecasts to investors.
The long view of history in BxR provides insightinto extreme market dynamics, accounting forvolatility uncertainty, sudden spikes in correla-tion, frequent outliers, and asymmetry betweengains and losses. Out-of-sample results raise thepossibility that financial extremes are, to someextent, predictable. Importantly, they are intrinsi-cally less predictable than volatility. Nevertheless, theinclusion of shortfall in the portfolio constructionprocess may lead to better performance, as shortfalladdresses an aspect of risk and a source of data thatare outside the purview of volatility.
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46 LISA R. GOLDBERG AND MICHAEL Y. HAYES
4 Suggestions for empirical studies
Accurate shortfall forecasts and analysis advance ourability to manage financial risk, but this is onlyone aspect of a larger program. New sources ofdata will reveal new opportunities for empirical riskmanagement, allowing refinements of old statisticsor the estimation of new ones. Creative applicationof the theory of convex risk may lead to even morerelevant and useful measures. On the theme of thisspecial edition on the future of risk management,we provide a list of empirical questions for futureresearch.
What is the impact of leverage on risk? In theconcluding remarks to his analysis of the Long-Term Capital Management (LTCM) investmentstrategies, Jorion (2000) comments:
…such [undiversified and highly leveraged] strategies are fun-damentally dangerous and much riskier than measured bytraditional risk management systems.
Leverage scales the range of portfolio outcomeswithout changing the initial investment. The scalefactor or leverage ratio (λ) in a portfolio is theamount invested divided by initial capital. In aninfluential article, Artzner et al. (1999) propose thatrisk depends linearly on leverage:
µ(λP) = λµ(P), λ > 0. (11)
While there are theoretical benefits to linear scal-ing, such as the Euler decomposition of risk intosources (described in Goldberg et al. (2009) andelsewhere), it is not clear that Eq. (11) is flexibleenough to measure the real impact of leverage onrisk. Linear scaling implies that investing $100 ofcapital in a stock is only half as risky as borrowingan additional $100 to invest a total of $200 in thesame stock. In the first investment, the most you canlose is your entire stake of $100. In the second, youstand to lose double your initial investment, andyou are at the mercy of the market. It may be dis-advantageous, or even impossible, to borrow whatyou owe.
Volatility and shortfall satisfy the linear scalingproperty in Eq. (11). Therefore, even in combina-tion, they may not fully describe the risk of leveragein a portfolio. All convex risk measures satisfy aless restrictive leverage rule (as long as the risk of aconstant-valued portfolio is zero):
µ(λP) ≥ λµ(P), λ ≥ 1
µ(λP) ≤ λµ(P), 0 ≤ λ ≤ 1. (12)
Föllmer and Schied (2004, Chapter 4) analyze con-vex measures that do not scale linearly, such as theminimum entropy risk measure, given by log E[eP ].LTCM reported their risk as comparable to that ofthe S&P 500; a convex risk measure that scales non-linearly could alert investors to the hidden impactof leverage.
Is there a shortfall risk premium? The capital assetpricing model (CAPM) assumes that the price ofan asset is determined by its covariance with themarket and market volatility. According to Famaand French (2004), most explanations for depar-ture from the CAPM are of two types. The firstis that equity indices and other proxies for themarket portfolio used in empirical studies are notsufficiently representative. In other words, studiesthat purport to test the CAPM are, in fact, test-ing something else. The second type of explanationfocuses on the simplifying assumptions underlyingthe CAPM, such as a consensus view of asset meansand variances, market equilibrium, and absence ofconstraints (see, for example, Markowitz, 2005).Fama and French point out that the empirical depar-ture from the CAPM calls into question many of ittextbook applications.
It is conceivable that markets demand a shortfallrisk premium in certain climates. This raises thepossibility that the market portfolio may sometimesbe closer to mean-shortfall efficient than to mean-variance efficient. Or perhaps the market is efficientwith respect to a convex risk measure that is sensi-tive to leverage. Tests described in Fama and French
JOURNAL OF INVESTMENT MANAGEMENT FIRST QUARTER 2010
THE LONG VIEW OF FINANCIAL RISK 47
(2004) can be applied to analyze these questions. Ifthere is a shortfall risk premium, accounting for itcould lead to a better asset pricing model.
Is portfolio insurance worth the price? Lo (2008)posits a fictitious hedge fund in which the chiefrisk officer (CRO) implements a strategy to hedgedownside risk exposure to collateralized debt obli-gations (CDOs) in a bull market.
From 2004 to 2006, such a hedging strategy would likely haveyielded significant losses, and the reduction in earnings due to thishedge, coupled with the strong performance of the CDO business,would be sufficient grounds for dismissing the CRO.
To be of any use, out of the money puts and otherforms of downside protection must be purchasedbefore a crisis occurs. However, incentive structuresthroughout the financial services industry tend tobe aligned with short-term goals. The institution ofless myopic incentives can be supported by empir-ical cost–benefit analysis of the value of downsideprotection. Since the results of any particular testdepend on the details of the scenario used, it is desir-able to consider the widest possible range of marketconditions, asset classes, time horizons, investmentstrategies, and insurance types.
5 Taking the long view
Bender and Nielsen (2009) elaborate on three tenetsof investing:
• Risk management is not limited to the risk manager.Anyone involved in the investment process ... shouldbe thinking about risk.
• If you can’t assess the risk of an asset, maybe youshouldn’t invest in it.
• Proactive risk management is better than reactiverisk management.
These common-sense principles are typical of therhetoric that has accompanied the global marketturmoil that began in 2007, but they are oftenignored.
The empirically and scientifically motivated ideasreviewed in this paper have the potential to supportnew financial markets that work better than the oldones. But they are no substitute for judgment, pru-dence, or an incentive structure that takes the longview.
Acknowledgements
The authors thank Angelo Barbieri, VladislavDubikovsky, Alexei Gladkevich, Jose Menchero,and Indrajit Mitra for their significant contribu-tions to the ideas in this paper. The authors alsothank Robert Anderson, David Brierwood, PatrickBurke-d’Orey, Jason Draut, Steve Evans, MiriamKaminsky, Robert Korajczyk, Julie Lefler, FrankNielsen, Baer Pettit, Jo Robbins, Oleg Ruban, andJeremy Staum for their valuable editorial input, andJessica Guan, Andrea Pasqua, and Jackson Wang fortheir technical support. The authors are grateful toan anonymous referee for a thoughtful review thathas led to improvements in the exposition.
Notes
1 Time intervals begin on the first business day of theindicated month or year, and end on the last businessday.
2 Artzner et al. (1999) were referring to the property of subad-ditivity, which is closely related to convexity. For example,a convex risk measure that scales linearly is subadditive; fur-ther details are in Föllmer and Schied (2004, Chapter 4).The insight from Artzner et al. (1999) about diversificationand subadditivity applies equally well when “subadditivity”is replaced with “convexity.”
3 Change in portfolio value over an investment period canalso be expressed in units of profit and loss. This is more gen-eral, since it allows for any starting value. However, units ofreturn are generally preferred for long-only portfolios andthose with relatively low leverage.
4 Even though the definition of convexity is expressed interms of positive weights, convex measures are appropri-ate for measuring the risk of long-short portfolios. Furtherdetails are in Föllmer and Schied (2004, Chapter 4).
5 Further details are in Menchero and Poduri (2008).
FIRST QUARTER 2010 JOURNAL OF INVESTMENT MANAGEMENT
48 LISA R. GOLDBERG AND MICHAEL Y. HAYES
6 The impacts of a single trade on different risk measuressometimes differ. Further details are in Goldberg et al.(2009).
7 Or more generally, a risk aversion-weighted combinationof different measures of risk.
8 Extensions of this model also use bootstrapping techniquesto generate synthetic data.
9 These statistics are based on ratios of realized to forecastshortfall as well as differences between realized and forecastshortfall. The test statistics are asymptotically normal underassumptions strong enough for the Central Limit Theoremto hold.
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