CTIONOnce an investor has discovered a set of trading strategies that are expectedto have positive returns over time, the next step is to combine the trades intoa portfolio of diversified strategies. The goal is to combine strategies that areuncorrelated with each other to end up with a dampened-risk portfolio.
DiversificationFigure 15.4 illustrates a commodity futures portfolio from June 2000, whichcombined hedge-pressure trades with weather-fear-premium trades. The figure shows the effect of incrementally adding unrelated trades on portfoliovolatility.
14.013.0
Portfolio Volatility
12.011.010.09.08.07.06.01
2
3
4
5
6
7
Number of Strategies
FIGURE 15.4
Annualized Portfolio Volatility versus Number of CommodityInvestment Strategies, June 2000Source: Hilary Till, “Passive Strategies in the Commodity Futures Markets,”Derivatives Quarterly (2000), Exhibit 5.Copyright © Institutional Investor, Inc.
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Corn Futures Prices
2502452402352302252202152101.8
1.9
2
2.1
2.2
2.3
2.4
2.5
Natural Gas Futures Prices
FIGURE 15.5 September Corn Futures Prices versus September Natural GasFuture Prices, November 30, 1998, to June 28, 1999Source: Hilary Till, “Taking Full Advantage of the Statistical Properties ofCommodity Investments,” Journal of Alternative Investments (2001), Exhibit 3.Note: Using a sampling period of every three days, the correlation of the percentchange in corn prices versus the percent change in natural gas prices is 0.12.Copyright © Institutional Investor, Inc.
Inadvertent Concentration RiskA key concern for all types of leveraged investing is inadvertent concentration risk. In leveraged commodity futures investing, one must be careful withcommodity correlation properties. Seemingly unrelated commodity marketscan become temporarily highly correlated. This becomes problematic if acommodity manager is designing a portfolio so that only a certain amountof risk is allocated per strategy. The portfolio manager may be inadvertentlydoubling up on risk if two strategies are unexpectedly correlated.Figures 15.5 and 15.6 provide examples from the summer of 1999 thatshow how seemingly unrelated markets can temporarily become quiterelated.Normally natural gas and corn prices are unrelated, as shown in Figure15.5. But during July, they can become highly correlated. During a threeweek period in July 1999, the correlation between natural gas and cornprice changes was 0.85, as illustrated in Figure 15.6.Both the July corn and natural gas futures contracts are heavily dependent on the outcome of weather in the U.S. Midwest. And in July 1999, the
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PROGRAM EVALUATION, SELECTION, AND RETURNS
Corn Futures Prices
2152102052001951901852.16
2.21
2.26
2.31
2.36
2.41
Natural Gas Futures Prices
FIGURE 15.6 September Corn Futures Prices versus September NaturalGas Prices, June 29, 1999, to July 26, 1999Source: Hilary Till, “Taking Full Advantage of the Statistical Properties ofCommodity Investments,” Journal of Alternative Investments (2000),Exhibit 4.Using a sampling period of every three days, the correlation of thepercent change in corn prices versus the percent change in natural gasprices is 0.85.Copyright © Institutional Investor, Inc.
Midwest had blistering temperatures (which even led to some power outages). During that time, both corn and natural gas futures prices respondedin nearly identical fashions to weather forecasts and realizations.If a commodity portfolio manager had included both natural gas andcorn futures trades in a portfolio during this time frame, then that investorwould have inadvertently doubled up on risk.In order to avoid inadvertent correlations, it is not enough to measurehistorical correlations. Using the data in Figure 15.5, an investor wouldhave concluded that corn and natural gas price changes are only weaklyrelated. An investor needs, however, to have an economic understanding ofwhy a trade works in order to best be able to appreciate whether an additional trade will act as a portfolio diversifier. In that way, the investor willavoid doubling up on the risks that Figure 15.6 illustrates.
RISK MANAGEMENTThe fourth step in designing a commodity futures trading program is riskmanagement, because the portfolio manager needs to ensure that during
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both normal and eventful times, the program’s losses do not exceed a client’scomfort level.
Risk MeasuresOn a per-strategy basis, it is useful to examine each strategy’s:Value at risk based on recent volatilities and correlationsWorst-case loss during normal times■ Worst-case loss during well-defined eventful periods■ Incremental contribution to portfolio value at risk■ Incremental contribution to worst-case portfolio event risk■■
The last two measures give an indication if the strategy is a risk reduceror risk enhancer. On a portfolio-wide basis, it is useful to examine theportfolio’s:Value at risk based on recent volatilities and correlationsWorst-case loss during normal times■ Worst-case loss during well-defined eventful periods■■
Each measure should be compared to some limit, which has been determined based on the design of the futures product. So, for example, if clientsexpect the program to lose no more than, say, 7 percent from peak-totrough, then the three portfolio measures should be constrained to notexceed 7 percent. If the product should not perform too poorly during, say,financial shocks, then the worst-case loss during well-defined eventful periods should be constrained to a relatively small number. If that worst-caseloss exceeds the limit, then the manager can devise macro-portfolio hedgesaccordingly, as will be discussed later.For the purposes of extraordinary stress testing, we would recommendexamining how a portfolio would have performed during the four eventfulperiods listed in Table 15.2.
TABLE 15.2 Meaningful Eventful PeriodsOctober 1987 stock market crash1990 Gulf WarFall 1998 bond market debacleAftermath of 9/11/01 attacks
286TABLE 15.3
PROGRAM EVALUATION, SELECTION, AND RETURNS
Strategy-Level Risk Measures
Value at Risk
Worst-Case Lossduring NormalTimes
Worst-Case Lossduring EventfulPeriod
Deferred ReverseSoybean Crush Spread
2.78%
−1.09%
−1.42%
Long DeferredNatural Gas Outright
0.66%
−0.18%
−0.39%
Short DeferredWheat Spread
0.56%
−0.80%
−0.19%
Long DeferredGasoline Outright
2.16%
−0.94%
−0.95%
Long Deferred Gasolinevs. Heating Oil Spread
2.15%
−1.04%
−2.22%
Strategy
Long DeferredHog Spread
0.90%
−1.21%
−0.65%
Portfolio
3.01%
−2.05%
−2.90%
Source: Hilary Till, “Risk Management Lessons in Leveraged Commodity FuturesTrading,” Commodities Now (September 2002).
A commodity portfolio that would do poorly during these time framesmay be unacceptable to clients who are investing in a nontraditional investment for diversification benefits. Therefore, in addition to examining aportfolio’s risk based on recent fluctuations using value at risk measures, amanager also should examine how the portfolio would have performedduring the eventful times listed in Table 15.2.Tables 15.3 and 15.4 provide examples of the recommended risk measures for a particular commodity futures portfolio. Note, for example, theproperties of the soybean crush spread. It is a portfolio event-risk reducer,but it also adds to the volatility of the portfolio. An incremental contribution to risk measure based solely on recent volatilities and correlations doesnot give complete enough information about whether a trade is a riskreducer or risk enhancer.
Macro-Portfolio HedgingUnderstanding a portfolio’s exposure to certain financial or economicshocks can help in designing macro-portfolio hedges that would limit exposure to these events. For example, a commodity portfolio from the summer
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TABLE 15.4
Portfolio-Effect Risk Measures
StrategyDeferred ReverseSoybean Crush SpreadLong Deferred NaturalGas OutrightShort DeferredWheat SpreadLong DeferredGasoline OutrightLong Deferred Gasoline vs.Heating Oil SpreadLong Deferred Hog Spread
IncrementalContribution toaPortfolio Value at Risk
IncrementalContribution toWorst-Case PortfolioaEvent Risk
0.08%
−0.24%
0.17%
0.19%
0.04%
0.02%
0.33%
0.81%
0.93%0.07%
2.04%−0.19%
a
A positive contribution means that the strategy adds to risk while a negative contribution means the strategy reduces risk.Source: Hilary Till, “Risk Management Lessons in Leveraged Commodity FuturesTrading,” Commodities Now (September 2002).
of 2002 consisted of these positions: outright long wheat, a long gasolinecalendar spread, and short outright silver. When carrying out an event-riskanalysis on the portfolio, one finds that the worst-case scenario was a9/11/01 scenario. This is because the portfolio was long economicallysensitive commodities and short an instrument that does well during timeof flights to quality. Normally, though, these positions are unrelated to eachother. Given that the scenario that would most negatively impact the portfolio was a sharp shock to business confidence, one candidate for macroportfolio insurance was short-term gasoline puts to hedge against thisscenario.
LEVERAGE LEVELAnother consideration in designing a commodity futures program is howmuch leverage to use. Futures trading requires a relatively small amount ofmargin. Trade sizing is mainly a matter of how much risk one wants toassume. An investor is not very constrained by the amount of initial capitalcommitted to trading.
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What leverage level is chosen for a program is a product design issue.The manager needs to determine how the program will be marketed andwhat the client’s expectations will be.According to Barclay Managed Funds Report (2001), a number of topcommodity trading advisors (CTAs) have had losses in excess of −40 percent, which have been acceptable to their clients since these investment programs sometimes produce 100+ percent annual returns. Investors know upfront the sort of swings in profits and losses to expect from such managers.Choosing the leverage level for a futures program is a crucial issuebecause it appears that the edge that successful futures traders are able toexploit is small. Only with leverage do their returns become attractive.Table 15.5 shows how the returns to futures programs, here labeled “managed futures,” become competitive only after applying the most amount ofleverage of any hedge fund strategy.In Patel (2002), Bruce Cleland of Campbell and Company, a pioneer offutures investing, discusses how essential leverage is to his firm’s success:“Campbell’s long-term average rate of return compounded over 31 years is over17.6 percent net [of fees]. No market-place is going to be so inefficient asto allow any kind of systematic strategy to prevail over that period of time, to
TABLE 15.5
Levered and Delevered Returns by Hedge Fund Strategy,
1997 to 2001Style
a
Average Levered Return (%)
Short BiasedGlobal MacroEmerging MarketsEvent DrivenMerger AritrageLong/Short EquityFixed incomeConvertible ArbitrageManaged FuturesDistressed Securities
13.716.816.914.714.714.09.610.610.5n/a
a
Average Delevered Return (%)9.38.98.88.37.06.34.84.24.2n/a
a
Leverage analysis was done for funds with five-year historical leverage and performance data.Source: Altvest, CSFB/Tremont, EACM, HFR, Institutional Investor (June 2002),and CMRA.Leslie Rahl, “Hedge Fund Transparency: Unraveling the Complex andControversial Debate,” RiskInvest 2002, Boston, December 10, 2002, Slide 52.
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289
that extent. ‘Our true edge is actually only around 4 percent per year, butthrough leverage of between 4-1 and 5-1 you are able to get a much moreattractive return,’ Cleland says” (page 49). This quote from the president ofCampbell is very instructive for neophyte futures traders who must determine how much leverage to use in delivering their clients an attractive set ofreturns.
UNIQUE CONTRIBUTION TO THE INVESTOR’SOVERALL PORTFOLIOA final consideration in creating a futures trading program is to understandhow the program will fit into an investor’s overall portfolio. For investorsto be interested in a new investment, that investment must have a uniquereturn stream: one that is not already obtained through their other investments. More formally, the new investment must be a diversifier, either during normal times or eventful times.It is up to investors to determine how a new investment should fit intotheir portfolios. A futures trading program may be evaluated on how wellit diversifies an equity portfolio. Or it may be judged based on how well itdiversifies a basket of veteran CTAs. Finally, a new futures trading programmay be evaluated on how well it improves a fund of hedge fund’s riskadjusted returns. Examples of each kind of evaluation follow.
Equity Diversification ExampleOne potential commodity futures investment is based on the GoldmanSachs Commodity Index (GSCI). One way to evaluate its potential benefitsfor an international equity portfolio is to use a portfolio optimizer to create the portfolio’s efficient frontier both with and without an investment inthe GSCI. Figure 15.7 from Satyanarayan and Varangis (1994) illustratesthis approach. The efficient frontier with commodity assets lies everywherehigher than the portfolio without commodity assets, implying that for thesame levels of return (risk), the portfolio with commodity assets provideslesser (higher) risk (return). This would be regarded as attractive providedthat the historical returns, volatilities, and correlations used in the optimizer are expected to be representative of future results.
CTA Diversification ExampleA futures program that invests solely in commodities has a natural advantage in claiming diversification benefits for a portfolio of CTAs. As Table
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PROGRAM EVALUATION, SELECTION, AND RETURNS
0.12
30.21
Expected Monthly Return
2.5
0.30.38
2
0.40.42
1.5
With GSCIWithout GSCI
M
0.42
M
0.410.34
1
0.090.5
02.5
3
3.5
4
4.5
5
5.5
6
Monthly Standard Deviation
FIGURE 15.7 Optimal International Portfolios with and without Commodity AssetsSource: Sudhakar Satyanarayan and Panos Varangis, “An Efficient Frontier forInternational Portfolios with Commodity Assets,” Policy Research WorkingPaper 1266, The World Bank, March 1994, p. 19.Note: The numbers on the mean-standard deviation frontier refer to the percentage of the portfolio invested in commodity assets. M = minimum-risk portfolio.
15.6 illustrates, an index of managed futures returns is most stronglyrelated to investment strategies focused on currencies, interest rates, andstocks. Commodities are in fourth place.One way of demonstrating that a commodity investment strategy is ofbenefit to a diversified portfolio of CTAs is to calculate how the Sharperatio (excess return divided by standard deviation) would change once thenew investment is added to the portfolio. Table 15.7 shows how the addition of a particular commodity manager to three diversified portfolios increases the Sharpe ratio of each portfolio. The three diversified portfoliosare represented by CTA indices provided by Daniel B. Stark & Co.Figure 15.8 illustrates another way of confirming that a futures tradingprogram would be a diversifier for an existing investment in a basket offutures traders. Figure 15.8 shows that the Stark Diversified CTA indexalone has a Sharpe ratio of about 0.72. If 60 percent is allocated to theStark index and 40 percent to a specific advisor’s program, the Sharpe ratiorises to 1.0 even though the specific advisor’s program alone has a Sharperatio of below 1.0.
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How to Design a Commodity Futures Trading Program
TABLE 15.6
Regression of Managed Futures Returns on Passive Indices andEconomic Variables, 1996 to 2000Coefficient
Standard Error
T-Statistic
InterceptS&P 500Lehman USChange in Credit SpreadChange in Term SpreadMFSB/Interest RatesMFSB/CurrencyMFSB/Physical CommoditiesMFSB/Stock Indices
0.000.000.290.000.001.271.370.270.36
0.000.070.390.010.000.240.250.150.11
0.010.050.760.300.185.245.481.793.17
R-Squared
0.70
Source: Center for International Securities and Derivatives Markets (CISDM), 2ndAnnual Chicago Research Conference, May 22, 2002. Slide 48.Note: The Managed Futures Securities Based (MFSB) Indices are designed to mimicthe performance of CTAs who employ trend-following or countertrend strategies.
Fund of Hedge Fund Diversification ExampleSimilarly, if the futures program is expected to be a diversifier for a fund ofhedge funds portfolio, whether the Sharpe ratio of the enhanced portfolioimproves as well must be verified. This is illustrated in Table 15.8.
TABLE 15.7
Example of How the Sharpe Ratio of CTA Indices Changes with theAddition of a Particular Commodity Futures Program, September 1999 to March 2003b
Index Alone
With 10% GA Component
CARR
a
Vol %
SharpeRatio
CARR
Vol %
SharpeRatio
Stark Fund Index6.80%Stark 300 CTA Index 8.70%Stark Diversified CTA 9.50%
13.60%10.80%11.60%
0.500.800.82
7.80%9.40%10.10%
11.80%9.60%10.30%
0.660.980.98
Index
a
Compounded annualized rate of returnGlobal Advisors Discretionary Program, a futures trading programSource: “The Case for Commodities,” Global Advisors (June 2003).b
Copyright © Daniel B. Stark & Company.
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PROGRAM EVALUATION, SELECTION, AND RETURNS
1.151.101.05
Sharpe Ratio
1.000.950.900.850.800.750.700%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Stark 300 Index Allocation
Efficient Portfolio GALPa + Stark Diversified CTA Index, September1999 to March 2003Source: “The Case for Commodities,” Global Advisors (June 2003), Chart 1.aGlobal Advisors LP.
FIGURE 15.8
Note: The vertical axis is the Sharpe ratio. The horizontal axis is the amountallocated to the Stark Index; the balance is allocated to the GALP trading program.Copyright © Daniel B. Stark & Company.
TABLE 15.8 Example of How the Sharpe Ratio of a Fund of Hedge FundsChanges with the Addition of a Particular Commodity Futures Program,September 1999 to March 2003a
Index Alone
With 10% GA Component
Index
CARR
b
Vol%
SharpeRatio
CARR
Vol%
SharpeRatio
Model Fund ofFunds Portfolioc
7.80%
5.00%
1.56
8.50%
5.00%
1.7
a
Global Advisors Discretionary Program, a futures trading programCompounded annualized rate of returncThe model fund of funds portfolio comprises Edhec Business School indices in thefollowing weights: 40 percent Long/Short Equity, 10 percent Convertible Arbitrage,10 percent Global Macro, 10 percent Managed Futures, 5 percent Equity MarketNeutral, 5 percent Fixed Income Arbitrage, 5 percent Distressed Securities, 5 percentEmerging Markets, 5 percent Merger Arbitrage, and 5 percent Event Driven.Source: “The Case for Commodities,” Global Advisors (June 2003).b
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CONCLUSIONThis chapter has outlined the considerations involved in creating a commodity futures trading program. Commodity managers need to be awarethat trading strategies can exhibit periods of high correlation, which canlead to doubling risk. We showed that adding commodity futures to a portfolio can potentially reduce overall portfolio risk. We also showed thatfutures programs must employ leverage in order for their returns to becompetitive. To provide diversification benefits to investors, commoditymanagers must produce return streams that are sufficiently unrelated tothose of other manager strategies as well as to traditional investments.
CHAPTER
16
Choosing the Right CTA:A Contingent Claim ApproachZsolt Berenyi
anaged futures have enjoyed a significant increase as investments during the last one and a half decades, both on a stand-alone basis and aspart of a well-diversified portfolio. Managed futures accounts, indeed, seemto offer investors significant advantages not accessible elsewhere. Yet ranking such investment opportunities either on an ex-ante or an ex-post basisis still difficult because the risk and return structure of managed futuresaccounts often differs from that of (more or less) common benchmarks, andthe risk structure of such investments may be unstable since CTAs maychange the risk exposure of the funds individually.In this chapter we investigate the ex-post performance ranking of CTAsbased on a contingent claim performance approach. In this approach, theperformance of each managed futures fund is compared to individually created benchmark assets having the same risk profile in terms of particularhigher moments. Benchmark assets are constructed (“replicated”) using theS&P 500, options, and the risk-free asset. Using benchmark assets, we estimate the efficiency gain or loss each CTA produces and analyze the robustness of this kind of efficiency measurement with respect to the number ofmoments used.
M
INTRODUCTIONCommodity funds, which are managed by commodity trading advisors(CTAs), belong to the modern alternative investment class. Managed commodity funds (managed futures) are publicly offered investment vehiclesthat invest in futures and options of a wide range of financial assets as wellas commodities and may employ a variety of leverage-creating techniques.
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Choosing the Right CTA
295
Managed futures accounts offer investors significant advantages notaccessible elsewhere, due to their unconventional investment strategies.These forms of investment offer, in much the same way other modern alternative investment forms do, both diversification advantages and return profiles different from traditional investments.From this background, research on alternative investments, predominantly on CTAs and hedge funds, has mushroomed. A particularly interesting field continues to be the performance evaluation of those alternativeinvestments. Because they may offer highly nonnormal and optionlike returnprofiles, traditional performance measures used elsewhere suffer from serious disadvantages (i.e., they produce controversial results and, in particular,may be subject to gambling behavior). The performance of CTAs and hedgefunds remains, however, a particularly important issue because, in spite ofthe somewhat controversial theoretical results on persistence in CTA performance (see, e.g., Schneeweiss 1996), investors evaluate investments, at leastpartially, based on past performance.This chapter reviews the performance of a series of managed futuresfunds with a contingent claim–based efficiency measure, which is based ona moment-based performance evaluation methodology. First, we investigatethe efficiency of CTAs as stand-alone investments based on the comparison to option-based strategies. The basis for the comparison is the risk profile of the given CTA asset, where risk is defined as some set of statisticalmoments. Then we compare the moment-based efficiency measures to findout whether using a more complete replication pays off in terms of monetary advantages and accuracy.
MOMENT-BASED EFFICIENCY MEASUREDistributional Performance EvaluationAssessing performance in case of opaque or continuously changing portfolios such as managed funds remains difficult because finding or creating aproper benchmark is still not an easy task. Here we propose a methodologyin which the performance of CTA funds will be measured using synthetically created benchmarks. The main idea is to compare any investmentportfolio (especially those with nonnormal return distributions like managed funds) to artificial, so-called replicating benchmarks possessing riskcharacteristics similar to the primary investment.The idea that investors compare portfolios based on some statistical(or other) risk profile should not be very surprising. In the most fundamental consideration about investments, investors buy risky time, that is,
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a particular portfolio return profile provided by holding risky assets for apredefined period. Throughout this chapter, the distributional features ofreturn streams will be called risk characteristics (valid for the particularholding period).Return distributions certainly can be arbitrary, not just normal (aswould be the usual assumption in case of equity investments). Going onestep further, the particular portfolio payoff and return distribution are limited in their shape by the available investment opportunities. Performancemeasurement, consequently, denotes the evaluation of the particular riskcharacteristics of the individual payoff profiles.The payoff distribution pricing model of Dybvig (1988a), provides arelated perspective. Dybvig develops a pricing framework for assets witharbitrary return distributions. The basic idea of his work is that agents minimize the cost of any one-period return distribution, regardless of the factors that drive state probabilities. He calls the price of the minimum costportfolio for any return distribution the distributional price (to distinguishit from the normal asset price). That is, economic agents compare returndistributions resulting from any kind of investment opportunity directly.This approach neglects the underlying structure of portfolios, considering it as irrelevant for performance comparison. However, because investorsusually use cash returns from the noncash investments for consumption, weargue it is legitimate to do so. That is, it is of no relevance whether a portfolio contains common stocks or hedge funds, because only the distributionof the investment returns for the holding period is important for the performance assessment. This approach also may be justified by acknowledging factors like investment barriers and relative illiquidity.
Contingent Claim–Based Performance EvaluationThe possibility to create and transform arbitrary return distributions is animportant property of options that has been known and used by practitioners for a long time (cf. Reback 1975). Reback (1975) states that derivative assets are able to alter the pattern of any portfolio return to create anydesired shape of return distribution. Thus it is possible to create optionedportfolios mimicking other portfolios in risk characteristics by using options.Because the return distribution of optioned portfolios can be shapedarbitrarily, they can be used as a common benchmark asset. Thus, the use ofoptioned markets as the reference point suggests extending the performanceevaluation framework to multiple asset classes as well. Doing this facilitatesthe broadening of the classical one asset view to more asset classes competing with each other. In addition, also multimanager funds theoretically could
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297
be analyzed with optioned markets, if the underlying structure of the investments remains immaterial.Indeed, the use of optioned benchmark portfolios for performancemeasurement purposes itself is not a novel idea. The work of Dybvig(1988a, b) also signifies implicitly that optioned portfolios can be comparedto portfolios of other asset classes, regardless of the underlying asset.Glosten and Jagannathan (1994) propose the use of options to re-create contingent claims for mutual fund performance evaluation. Dynamic strategies,in much the same way as options, also can be used to create any particularpayoff profile. Recently Amin and Kat (2001) have proposed a similar methodology to evaluate hedge fund returns in using path-independent dynamicstrategies that have positive correlation with the underlying index. Thenovelty of this chapter in proposing optioned portfolios for benchmarkpurposes lies, however, in reducing the risk characteristics of the replicatingportfolio to a handful of higher statistical moments.
EFFICIENCY GAIN/LOSS MEASUREThis section proposes using so-called replicating portfolios for benchmarkpurposes. Replicating portfolios are optioned portfolios designed to reproduce the risk characteristics of a given asset by combining a benchmarkasset with options and the risk-free asset.The expected return on a replicating portfolio for a given risk shape (ofa particular asset) will be called the replicating return. The replicating returncan be interpreted as the alternative return an investor may achieve if, holding the risk exposure (defined here in terms of return variance, skewness, anda number of higher moments) constant, she chooses to invest in the optionedmarket instead of investing in a given portfolio.The efficiency gain/loss measure or excess replicating return is simplythe difference between the expected return of the asset under investigationand that of its synthetic benchmark asset. The expected return of this replicating benchmark asset will be termed as the replicating return.1This asset-specific replicating return embodies, at the same time, theminimum acceptable return on investments having the same risk structure,and serves thus as a natural benchmark. That is, investors always have the
1
Certainly the replicating benchmark asset will have to be computed to achievemaximum expected return within the set of possible replicating assets with the same(moment-based) risk characteristics.
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