It is interesting that compared with theoretical results,firms in reality tend to hedge a constant amount between 0% and 100% (often 50%),if they do not have distinct expectations about the future course of the exchange rate[63, 159]. This behavior is called partial hedging and is justified by arguing that afull hedge (zero hedge) accompanied by a favorable (adverse) exchange rate development would lead to high losses, such that hedging 50% seems reasonable in theabsence of concrete expectations.
12.1.2 Plain Vanilla OptionIn contrast to forward contracts, options have an asymmetric distribution of rightsand obligations. A call option gives the holder the right (but not the obligation) tobuy the underlying asset by a certain date for a certain price. A put option gives theholder the right to sell the underlying asset by a certain date for a certain price.The price in the contract is known as the exercise price or strike price X. The date inthe contract is known as the maturity. American options can be exercised at any timeup to the expiration date. European options can be exercised only on the expirationdate itself.The fact that an option gives the holder the right to do something does not meanthat the right must be exercised. This fact distinguishes options from forwards,where the holder is obligated to buy or sell the underlying asset. Hence, there is afundamental difference between the use of forward contracts and options for hedging. Forward contracts are designed to neutralize transaction risk by fixing the pricethat the agent will pay or receive for the underlying asset. Options, in contrast, provide insurance by offering a way for agents to protect themselves against adverseprice movements in the future while still allowing them to benefit from favorableprice movements. The latter feature comes at an up-front fee which is referred to asthe option premium P.If we disregard the initial cost of the option, the payoff from a long position in aEuropean call option at maturity t = T isωT − X : ωT > X(12.6)CT = max(ωT − X, 0) =0 : ωT ≤ XEquation 12.6 reflects the fact that the option will be exercised if ωT > X and willnot be exercised if ωT ≤ X. The asymmetric nature of a long position in a Europeancall option is well captured by its payoff function (Fig. 12.2a). Short positions inEuropean call options are obtained conversely (Fig. 12.2b).The payoff to the holder of a long position in a European put option isX − ωT : ωT < X(12.7)PT = max(X − ωT , 0) =0 : ωT ≥ X
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Payoff
Payoff(a)
P0
0
ST
X–P
(a)
X
(b) Short Call
Long Call
ST
(b)
Fig. 12.2 Payoff functions of a European call option
If we take into account the option premium to be paid in t = 0, the payoff functionof the call is given by the difference between CT and the option premium multipliedby the risk free interest rate. The payoff functions for long and short positions in aEuropean put option are depicted in Fig. 12.3a and Fig. 12.3b.According to the principle of risk neutral valuation, the market value of European call and put option contracts at t < T are given by the Black and Scholesformulae [41]∗
CT = ωT × e−r f (T −t) N(d1 ) − X × e−r f (T −t) N(d2 )PT = X × e
−r f (T −t)
N(−d2 ) − ωt × e
−r∗f (T −t)
N(−d1 )
(12.8)(12.9)
withln ωXt + r f − r∗f + σ2√d1 =σ T −t
2
(T − t)
ln ωXt + r f − r∗f − σ2√d2 =σ T −t
(T − t)
2
√= d1 − σ T − t
and N denoting the cumulative standard normal distribution.Literature has recommended plain vanilla option strategies as an alternative tothe selective hedging strategy in order to avoid losses that may result from havingentered into a forward position when the exchange rate, ex post, took a positivedirection. First, the firm has a view on future exchange rate levels and wants toparticipate from favorable exchange rate movements while being protected againstadverse ones. In fact, it is this insurance property, i.e., the asymmetric profile, that isconsidered widely as the fundamental advantage of an option [40,299]. It guaranteesprotection against adverse exchange rate developments while allowing participationfrom movements in the agent’s favor. Thus, options can be motivated if a firm has
12.1 Currency Hedging Instruments
121
Payoff
Payoff
(b)P0
0X
–P
ST
X
ST
(a)
(a)
Long Put
(b) Short Put
Fig. 12.3 Payoff functions of a European put option
missing expectations about the future exchange rate, if its exchange rate expectations from the past have not been successful, or if foreign exchange markets arevolatile. Under homogeneous expectations about the underlying stochastic processof the exchange rate and therefore, also about its probability distribution, hedgingwith options cannot be optimal, since there is no deviating view from the risk-neutralvalue of the option. In addition, the future payoff of the option is uncertain. Hence,a full forward hedge resulting in a deterministic outcome would be optimal. Deviating expectations from those of market consensus imply a speculative position. Theuse of options for income enhancement has been analyzed by a streak of literaturethat examined whether firms incorporate exchange rate expectations in their hedgingstrategies [57, 237].
12.1.3 StraddleThe problem that arises when a combined position of forwards and plain vanillaoption instruments has to be found is that the two instruments are coupled withcomplementarity and substitution between them. First, the payoff profile from along forward can be replicated by a combined position in a long call and a shortput according to put-call parity ([200], pp. 174–175). Second, both market valuefunctions depend on the expected future exchange rate. The call option can onlybe distinguished from the forward by its functional dependency on the foreign exchange market’s expected standard deviation of the underlying. The first-momentoverlapping in the determinants of the price formation may lead to the problemthat the different characteristics of forwards and options are being disguised. Thisproblem can be avoided if straddles are used instead of plain vanilla options ([323],pp. 166–167).A long straddle involves buying a call and a put with the same strike price andmaturity date. Consequently, the payoff function of a long straddle is the result of
122Fig. 12.4 Payoff function of along straddle
12 Preferences over Probability DistributionsPayoff
0
X
the sum of the single payoff functions of a long call and a long put.ωT − X : ωT > XST = |ωT − X| =X − ωT : ωT ≤ X
wT
(12.10)
The payoff function of a long straddle is shown in Fig. 12.4. If the exchange rateis close to the strike price at maturity of the options, the straddle leads to a loss.However, for large movements in either direction, a profit can be realized. If theprice goes up (down) enough, the call (put) option comes into effect and the put(call) option is ignored.The advantage of the “buy straddle” investment tactic is that the firm still hasunlimited profit potential without knowing precisely what direction the exchangerate will move in, while total loss is limited to the price of purchasing the call andput options. As long as prices do not stagnate, the firm has limited risk and a bigpotential for gain. Hence, the main motivation of a long straddle is to take a positionon volatility play. If an upcoming news report is expected to cause an increase involatility or rate changes for a certain currency, a straddle strategy generates betterreturns if the realized volatility is as expected.
12.2 Formal Relationship Between Firm and Capital MarketExpectationsIt is argued that a way to understand corporate hedging behavior is in the contextof speculative motives that could arise from either overconfidence or informationalasymmetries. For this purpose, we assume the firm to have different future expectations than those implied by derivative prices.In the following, we assume that both the probabilistic perceptions of the firmand the risk-neutral idea of the capital market can be described by a normal distribution. Following [323], we describe deviating beliefs completely by the first two
12.3 Specification of Probability Distribution Function
123
moments of the normal probability distribution. In order to model these deviationsit is interesting to focus on the connection between exchange rate realizations withthe same probabilities of occurrence. Formally, we refer to the connection betweenthe exchange rate from a firm’s perspective ω u , and the risk-neutral exchange ratefrom a capital market’s view ω m as follows:
ωu = ωm + α
(12.11)
with α denoting the difference in exchange rate expectations. Differing ideas on thevariability of the exchange rate under homogenuous beliefs on the expected valuecan be modeled by introducing a mean-preserving spread [90]:
ω u = E[ω u ] + ε u = E[ω u ] + β × ε m
(12.12)
where ε u denotes the residual term from a firm’s perspective with E[ε u ] = 0, ε mdenotes the residual term from capital market’s perspective with E[ε m ] = 0, andβ > 0 represents the coefficient of deviation.From (12.11) and (12.12) we receive
ω u = E[ω m ] + α + β × ε m
(12.13)
The uncertainty regarding the exchange rate realization from a firm’s and capitalmarket’s view results from the error term ε m , respectively, ε u , which additively overlays the risk-neutral expected value E[ε m ] = 0, respectively E[ε u ] = 0. The deviatingprobabilistic perceptions between firm and capital market are captured by α = 0 andβ = 1, with α = 0 representing deviating perceptions regarding the first moment,and β = 1 representing deviating perceptions regarding the second moment.
12.3 Specification of Probability Distribution FunctionWe demonstrate how industrial corporations and investors can develop an approachto managing currency transaction risk, thereby adding economic value from currency fluctuations. Empirical studies on corporate hedging behavior suggest thatfirms are facing a conflict. On the one hand, they like to try to anticipate events. Onthe other hand, they cannot base risk management on second-guessing the market.For instance, if investors believe that the currency is going to move in an unfavorable direction, what should they use as a tool to hedge? If they expect the currencyto move favorably, but are not entirely sure, should they use a different risk management strategy? The following is useful for formalizing the dilemma that reductionof transaction exposure with hedging instruments directly creates economic risk.Definition 12.1. Given a finite set S of decision variables S = {xtF , xtO , xtS } ∈ [0, 1]where xtF , xtO , and xtS denote the forward, straddle option, and spot weights withxtF + xtO + xtS = 1, and t = 1, . . . , T . Let the discrete probability distribution functionof transaction exposure be
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12 Preferences over Probability DistributionsT
v(x, w) = ∑ zt × (xts × ωt + xtF × f0,t + xtO × X0,t )
(12.14)
t=1
and the discrete probability distribution function of economic exposure beT
w(x, ω ) = ∑ zt (xtF × (ωt − f0,t )
(12.15)
t=1+ xtO × (|ωt − X0,t | − p0,t ωt=0 − c × (xtF + xtO ))
where z denotes the transaction exposure and c the transaction cost for a singletransaction as imposed by the counterparty. The expressions in brackets, as givenin (12.15), denote the payoffs of long forward and straddle option contracts whichare used to reduce transaction exposure in (12.14). The random variable at t is givenby the spot exchange rate ωt , t = 1, . . . , T . For purposes of this analysis, it is assumed that an investor holds one unit of foreign exchange, i.e., zt = 1. The effectsof marking-to-market and margin requirements are disregarded since they are purelyaccounting related and do not impact economic performance.
12.4 Expected Utility Maximization and Three-MomentsRanking12.4.1 Preference Structures over LotteriesThe objective of finding a possibly optimal combination of linear and nonlinearfinancial instruments to hedge currency risk over a planning period requires formulating a goal function incorporating (12.14) and (12.15), as well as defining ameasure of preference in order to evaluate the quality of a feasible solution. In decision theory, utility is a measure of the desirability of consequences of courses ofaction in a decision made under uncertain conditions. Utility theory was formalizedmathematically by the classic work of [411]. Von Neumann and Morgenstern introduced a set of necessary and sufficient axioms for the rational decision-maker, andpromoted the development of methods to measure utilities on numerical scales, resulting in real numbers representing personal values – such that one alternative withprobabilistic consequences is preferred to another if and only if its expected utilityis greater than that of the alternative. The axiomatic system is based on preferencestructures and their numerical representations in form of utility functions which willbe described in the following.Definition 12.2. A preference structure can be formulated as a binary preferencerelation. Suppose Z is a set of outcomes resulting from an agent’s decisions. We assume that the agent’s preference structure over Z is captured by a preference relation on Z, defined as a binary relation with the following property: for all outcomesA, B ∈ Z
12.4 Expected Utility Maximization and Three-Moments Ranking
125
• A is preferred over B if and only if A  B• A is indifferent to B if and only if A ∼ B, that is neither A  B nor A ≺ B• A  B if and only if A  B or A ∼ BLemma 12.1. The preference relation  is a total order, that is, every pair of outcomes are comparable. Formally, the relation , as induced by  and thus ∼, is abinary relation with the following three properties:• Reflexivity: For all outcomes A ∈ Z, A ∼ A• Orderability: For all outcomes A, B ∈ Z, exactly one of the following holds A  B,B  A, or A ∼ B• Transitivity: For all outcomes A, B,C ∈ Z, if A  B, and A  C, then B  C.Preference structures can be extended to uncertain outcomes. Uncertain outcomesare represented using (discrete) probability distributions over Z, referred to aslotteries.Definition 12.3. A lottery L with a finite number of distinct outcomes C1 ,C2 , . . . ,Cn∈ Z and respective probabilities p1 , p2 , . . . , pn is written as L = {(p1 ,C1 ),(p2 ,C2 ), . . . , (pn ,Cn )} where ∑ni=1 pi = 1, pi ≥ 0, i = 1, . . . , n.Corollary 12.1. A lottery is degenerate if n = 1, in which case the probability in thenotation can be omitted. A lottery can also have a countably infinite number of distinct outcomes C1 ,C2 , . . . ,Cn , . . . ∈ Z with respective probabilities p1 , p2 , . . . , pn , . . .which can be written L = {(p1 ,C1 ), (p2 ,C2 ), . . . , (pn ,Cn ), . . .} where ∑∞i=1 pi = 1,pi ≥ 0, i = 1, . . . , n, . . . .We denote as L(Z) the set of all lotteries over the outcome set Z. The agent’s preference relation is correspondingly extended to lotteries, and we still use the notation for the preference relation on lotteries.Lemma 12.2. Preference structures over lotteries have additional properties thatare consistent with human intuition about preference structures involving uncertainty:• Continuity: For all lotteries A, B,C ∈ Z, if A  B  C, then there exists p ∈ [0, 1]such that ((p, A), (1 − p,C)) ∼ B• Substitutability: For all lotteries A, B ∈ L(Z), if A ∼ B then for all lotteries C ∈L(Z) and all p ∈ [0, 1], ((p, A), (1 − p,C)) ∼ ((p, B), (1 − p,C))• Monotonicity: For all lotteries A, B ∈ L(Z), if A  B then for all lotteries p, q ∈[0, 1], p ≥ q if and only if ((p, A), (1 − p, B)) ∼ ((q, B), (1 − q, B))• Decomposability: For all lotteries A, B,C ∈ Z and all p, q ∈ [0, 1], ((p, A),(1 − p, [(q, B), (1 − q,C)])) ∼ ((p, A), ((1 − p)q, B), (1 − p)(1 − q),C)).These properties imply that the extended preference relation is also a total order onL(Z). We will refer to the above set of seven properties as the utility axioms.
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12 Preferences over Probability Distributions
12.4.2 Preference Structures over Utility FunctionsFor practical decision problems under uncertainty, however, it is more convenient todefine preference structures based on real-valued functions on L(Z).Lemma 12.3. A real-valued function on L(Z) can define a preference structure satisfying the utility axioms if and only if it satisfies the functional form of the utilityaxioms, namely• Continuity: For all lotteries A, B,C ∈ L(Z), if U(A) > U(B) > U(C), then thereexists p ∈ [0, 1] such that U((p, A), (1 − p,C)) = B• Substitutability: For all lotteries A, B ∈ L(Z), if U(A) = U(B) then for all lotteriesC ∈ L(Z) and all p ∈ [0, 1], U((p, A), (1 − p,C)) = U((p, B), (1 − p,C))• Monotonicity: For all lotteries A, B ∈ L(Z), if U(A) = U(B) then for all lotteriesp, q ∈ [0, 1], p ≥ q if and only if U((p, A), (1 − p, B)) ∼ U((q, B), (1 − q, B))• Decomposability: For all lotteries A, B,C ∈ L(Z) and all p, q ∈ [0, 1], U((p, A),(1 − p, [(q, B), (1 − q,C)])) = U((p, A), ((1 − p)q, B), ((1 − p)(1 − q),C))Since the relation ≥ is a total order on R, the first three utility axioms: reflexivity,orderability, and transitivity do not need to be included.
12.4.3 Expected Utility MaximizationThe underlying assumption in utility theory is that the decision maker alwayschooses the alternative for which the expected utility is maximized. To determinethe expected utility, a utility value has to be assigned to each of the possible consequences of each alternative. A utility function maps utility to the range of outcomesof a decision, depending on the decision maker’s preferences and attitude towardrisk. Utility theory is therefore intrinsically related to the concepts of risk and uncertainty in decision making.Lemma 12.4. If the extended preference relation satisfies the utility axioms, thenthere exists a utility function U : L(Z) → R such that for all lotteries A, B ∈ L(Z),A  B if and only if U(A) ≥ U(B) where the utility of a lottery L is the expectationof the utilities of its components∞
U(L) = U((p1 ,C1 ), (p2 ,C2 ), . . . , (pn ,Cn ), . . .) = ∑ pi × U(Ci ) = E[U(z)] (12.16)i=1
where z is the random variable for the outcomes of lottery L.Therefore, the utility function only needs to be defined for elements in Z, and itsdefinition on L(z) can be obtained using (12.16).Definition 12.4. A is preferred to B if and only if terminal wealth satisfiesE[U(zA )] − E[U(zB )] ≥ 0 with at least one strict inequality U(zA ) − U(zB ) ≥ 0.
12.4 Expected Utility Maximization and Three-Moments Ranking
127
It must be emphasized, by convention, utility is purely an ordinal measure. In otherwords, utility can be used to establish the rank ordering of outcomes, but cannot beused to determine the degree to which one is preferred over the other. For example,consider two outcomes A and B with corresponding utilities of 100 and 25. We cansay that A is preferred over B, but we cannot say that A is four times more preferredthan B. As a consequence of this ordinality, utility functions unique up to a positivelinear transformation, that is, both U1 (·) and U2 (·) = a × U1 (·) + b (where a > 0),are utility functions corresponding to the same preference structure.1
12.4.4 Increasing Wealth PreferenceFirms can have different risk attitudes, that is, different ways of evaluating a (nondegenerate) lottery. Risk attitudes can be modeled using utility functions. Based onthe utility function, different types of risk attitudes can be identified [139].Definition 12.5. A utility function possesses increasing wealth preference if andonly if U (z) ≥ 0 for all z with at least one strict inequality.This feature captures the “more wealth is better” (non-satiability) philosophy offirm behavior and is generally considered a universal feature of utility functions. Forgreater wealth to be preferred, the utility function must be monotonically increasing.
12.4.5 Risk Aversion PreferenceThis feature captures the willingness of a firm to purchase insurance (i.e., to paymore than the expected loss to transfer an insurable loss). This is a subset of increasing wealth preference, a firm may have increasing wealth preference with orwithout exhibiting risk aversion, and is also generally considered a universal featureof utility functions. Mathematically this is expressed as:Definition 12.6. A utility function possesses risk aversion if and only if it satisfiesthe conditions for increasing wealth preference and U (z) ≤ 0 for all z with at leastone strict inequality.This mathematical definition of risk aversion is equivalent to the behavioral definition given above. Note, that Definition 12.6 defines a concave function. By applyingJensen’s inequality this yields E[U(z)] ≤ U(E[z]), i.e., under risk aversion the expected utility of a risky investment is less than the utility of the expected outcome.The reason for this phenomenon is that, by proposition, the firm has penalized theutility of the investment for the possibility of unfavorable outcomes. If we rewriteJensen’s inequality with a strict inequality we can show that E[U(z)] = U(E[z] − π ).1 For proof of this proposition, see [179].
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12 Preferences over Probability Distributions
This shows that the firm is indifferent between the return on a risky investment ora lower, risk-free wealth equal to E[z] − π where π is the premium that the firm iswilling to pay to eliminate risk.Corollary 12.2. If E[U(z)] = U(E[z]) for all z, then the utility function is riskneutral. Otherwise, if E[U(z)] > U(E[z]) for all z, then the utility function isrisk-seeking.Pratt [330] defined local properties of risk attitudes, as well as a measure of risksensitivity, which are convenient for risk attitudes involving both risk aversion andrisk seekingness, such as the risk attitude of an agent who buys insurance and lotterytickets at the same time. The local risk property is related to the local convexity ofthe utility function.Definition 12.7. If the utility function is twice differentiable and strictly monotonically increasing then the degree of absolute risk aversion is given by a local riskmeasureU (z)λ (z) = −(12.17)U (z)which relates to the global risk properties as follows: if λ (z) = 0 everywhere, thenthe utility function is linear, and thus the agent is risk-neutral. If λ (z) > 0 everywhere, then the utility function is strictly concave, and thus the agent is risk-averse.If λ (z) < 0 everywhere, then the utility function is strictly convex, and thus the agentis risk-seeking.
12.4.6 Ruin Aversion PreferenceInvestors typically distinguish between upside and downside risks implying thatmore than the mean and variance of returns is priced in equilibrium. Negativelyskewed probability distributions imply that the risk for a substantial loss is biggerthan the chance of a substantial gain, and firms are hence, averse to it. Positivelyskewed distributions imply that the chance of a substantial gain is bigger than therisk for a substantial loss which is typically a desirable state for an investor or a firm.Preferences for positive skewness have been shown to hold theoretically by [10,362], and empirically by, e.g., [258, 372, 373]. The preference for positive skewnessis classically presented as an individual’s willingness to play the lottery: to accepta small, almost certain loss in exchange for the remote possibility of huge returns.A firm’s concern, however, is with the opposite situation: unwillingness to acceptsmall, almost certain gain in exchange for the remote possibility of ruin [229]. Thelogic is therefore that a firm would be ready to trade off some average return for alower risk of high negative returns. Ruin aversion is a subset of risk aversion. A firmmay have risk aversion with or without exhibiting ruin aversion.Definition 12.8. A utility function possesses ruin aversion if and only if it satisfiesthe conditions for risk aversion and U (z) ≥ 0 for all z with at least one strict inequality.
12.5 Specification of Utility Function
129
As with risk aversion, it is not intuitively clear that the mathematical and behavioraldefinitions of ruin aversion are consistent. Formally, a positive skewness preferenceis related to the positivity of the third derivative of the utility function. This can beillustrated by taking a Taylor series expansion of the expected utilityU(z) = U(E[z]) + U (E[z]) × (z − E[z])U (E[z])+× (z − E[z])22!U (E[z])× (z − E[z])3+3!
(12.18)
which leads to the expressionE[U(z)] = U(E[z]) +
U (E[z])U (E[z])× Var[z] +× Sk[z]2!3!
(12.19)
From this expression it can be seen that U (E[z]) and U (E[z]) are, respectively,related to variance and skewness. While a negative second derivative U (E[z]) < 0of the utility function implies variance aversion, a positive third derivative of theutility function U (E[z]) > 0, entails a preference for positive skewness. Hence,any investment feature that decreases variance and increases positive skewness (orreduces negative skewness) acts to increase expected utility.Proof. From Definition 12.7, decreasing absolute risk aversion or constant absoluterisk aversion as an upper bound is present, if
∂ λ (z) −u × u + (u )2=≤0z(u )2
(12.20)
which implies u > (u )2 × (u )−1 > 0 in case of decreasing absolute risk aversionand u = (u )2 × (u )−1 > 0 in case of constant absolute risk aversion.Nondecreasing absolute risk aversion therefore implies a preference for positiveskewness (lower negative skewness) on the distribution of wealth.
12.5 Specification of Utility FunctionLike many other real world problems, our problem requires the simultaneous optimization of multiple competing criteria. Utility theory proposes to compute solutions to such problems by combining them into a single criterion to be optimized,according to some utility function. For half a century, Markowitz’s two-momentmean-variance model has been the default model in financial engineering and thebenchmark model for new theories of portfolio choice [278–281]. In this model, thechoice of asset allocations is asserted to depend solely on the expected return (mean)and risk (variances and covariances) of the admissible assets. From the outset,
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12 Preferences over Probability Distributions
portfolio selection was conceived as a two-step procedure: the determination of theefficient set of portfolios as the first step, preparing the selection of an optimal portfolio for a given preference structure in the second step. The idea of utility maximization as a methodology for portfolio optimization problems, based on the utilitytheory founded by [411], can be traced back at least to [391], and also appearedin the early assessments of the mean-variance approach (e.g., [257, 347, 348]).Risk-sensitive utility functions are defined in order to identify a (possibly) optimalportfolio. Levy and Markowitz [257] showed for various utility functions and empirical returns distributions, that the expected utility maximizing agent could typicallydo very well if he acted knowing only the mean and the variance of a probabilitydistribution. This makes the model simple to apply, but it is based on the assumptionthat either (1) the return distribution is normal or that (2) investors are indifferentto higher moments and equally averse to downside and upside risk.2 Regarding thefirst argument there has been extensive debate: in reality, financial returns rarely arenormal – more often than not, skewness and kurtosis deviate from normality, making the mean-variance approximation unlikely to select the true optimal portfolio[250, 273]. With regards to the second argument, it has become a stylized fact thatindividuals perceive risk in a nonlinear fashion. Apart from a firm’s natural desireto avoid large economic losses from derivative hedging, we motivate the skewnesscriterion for a more practical reason: empirical literature on the hedging behaviorof firms states that both currency forward and option vehicles are used to hedgeagainst currency risk. However, probability distributions cannot be formally distinguished by their first and second moments, if nonlinear instruments such as options[52, 53] and/or dynamic strategies are considered [189, 350]. As a result, we extendthe concept of mean-variance trade-off to include the skewness criterion in the currency hedging problem. A kurtosis criterion is not incorporated due to the abovementioned fact that a consideration of long horizon returns does usually not involvesignificant kurtosis.Definition 12.9. Let the standard measure of return be defined by mean, and thestandard measure of risk be defined by variance and skewness. With respect toDefinition 12.8, this results in a three-order polynomial utility functionU(v, w) = E[w] − {λ Var[v] − κ Sk[w]}
(12.21)
= E[w] − {λ E[(v − E[v]) ] − κ E[([w] − E[w]) ]}2
3
where λ > 0 denotes the coefficient of absolute risk aversion, and κ > 0 denotes thecoefficient of absolute prudence.Equation 12.21