Short-Term Investments and Indices of Risk
aa r X i v : . [ q -f i n . P M ] M a y Short-Term Investments and Indices of Risk
Yuval Heller ∗ and Amnon Schreiber †‡ May 15, 2020
Abstract
We study various decision problems regarding short-term investments in riskyassets whose returns evolve continuously in time. We show that in each problem,all risk-averse decision makers have the same (problem-dependent) ranking overshort-term risky assets. Moreover, in each problem, the ranking is represented bythe same risk index as in the case of CARA utility agents and normally distributedrisky assets.
JEL Classification:
D81, G32.
Keywords : Indices of riskiness, risk aversion, local risk, Wiener process.
We study various decision problems regarding investments in risky assets (henceforth,gambles), such as whether to accept a gamble, or how to choose the optimal capitalallocation. To rank the desirability of gambles with respect to the relevant decisionproblem, it is often helpful to use an objective riskiness index that is independent ofany specific subjective utility. For example, an objective riskiness index is needed whenpension funds are required not to exceed a stated level of riskiness (see, e.g., the discussionin Aumann & Serrano, 2008, p. 812). ∗ Department of Economics, Bar-Ilan University, Israel. [email protected]. † Corresponding author. Dept. of Economics, Bar-Ilan University, Israel. [email protected]. ‡ This paper replaces a working paper titled “Instantaneous Decisions.” We have benefited greatlyfrom discussions with Sergiu Hart, Ilan Kremer, Eyal Winter, and Pavel Chigansky, and from helpfulcomments from seminar audiences at the Hebrew University of Jerusalem and Bar-Ilan University. YuvalHeller is grateful to the European Research Council for its financial support (starting grant
1e analyze four decision problems that are important in economic settings. In general,different risk-averse agents rank the desirability of gambles differently. However, ourmain result shows that in each of these problems, all risk-averse agents have the same(problem-dependent) ranking over short-term investments in risky assets whose returnsevolve continuously. Moreover, in each problem, the ranking is represented by the samerisk index obtained in the commonly used mean-variance preferences (e.g., Markowitz,1952), which are induced by CARA utility agents and normally distributed gambles.
Brief Description of the Model
We consider an agent who has to make an invest-ment decision related to a gamble. We think of a gamble as the additive return on afinancial investment. We assume that the agent has (1) an initial wealth w , and (2) avon Neumann–Morgenstern utility u that is increasing and risk-averse (i.e, u ′ > u ′′ < decision function that assigns a number to each agent and eachgamble, where a higher number is interpreted as the agent finding the gamble to be moreattractive (i.e., less risky) for the relevant decision problem.We study four decision problems in the paper: (1) acceptance/rejection , in which theagent faces a binary choice between accepting and rejecting the gamble (e.g., Hart, 2011);(2) capital allocation , in which the agent has a continuous choice of how much to invest inthe gamble (e.g., Markowitz, 1952; Sharpe, 1964); (3) the optimal certainty equivalent , inwhich the agent evaluates how much an opportunity to invest in the gamble (accordingto the optimal investment level) is worth to the agent (e.g., Hellman & Schreiber, 2018);and (4) risk premium, in which the agent evaluates how much investing in the gamble isinferior to obtaining the gamble’s expected payoff (Arrow, 1970). A risk index is a function that assigns to each gamble a nonnegative number, whichis interpreted as the gamble’s riskiness. We say that a risk index is consistent with adecision function f over some set of agents and gambles, if each agent in the set ranks allgambles in the set according to that risk index; that is, f assigns for each agent a highervalue for gamble g than for gamble g ′ iff the risk index assigns a lower value to g . A risk-aversion index is a function that assigns to each agent a non-negative number, whichis interpreted as the agent’s risk aversion. We say that a risk-aversion index is consistent We use “risk premium” in its common acceptation in the economic literature since Arrow (1970). Inthe financial literature (and in practice), the “risk premium” of a security commonly has a somewhatdifferent meaning, namely, the security return less the risk-free interest rate (e.g., Cochrane, 2009).
Summary of Results
Agents, typically, have heterogeneous rankings of gambles, and,thus, no risk index (nor risk-aversion index) can be consistent with the rankings of allagents, unless one restricts the set of gambles. Our main result restricts the set of gam-bles to assets whose returns evolve continuously in time, where the local uncertainty isinduced by a Wiener process. Specifically, we focus on Ito processes which are continuous-time Markov processes. The class of Ito processes, is commonly used in economic andfinancial applications and includes, in particular, the geometric Brownian motion andmean-reverting processes (e.g., Merton, 1992). Our main result shows that in each of the four decision problems discussed above, allagents rank all gambles in the same (problem-dependent) way when they have to decideon short-term investments in gambles whose returns evolve continuously in time. More-over, the risk indices that are consistent with these decision functions are the same as inthe classic model of agents with CARA (exponential) utilities and normally distributedgambles. Specifically, we show that: (1) the variance-to-mean index Q V M ( g ) = σ [ g ] E [ g ] isconsistent with both the capital allocation function and the acceptance/rejection func-tion, (2) the inverse Sharpe index Q IS ( g ) = σ [ g ] E [ g ] is consistent with the optimal certaintyequivalent function, and (3) the standard deviation index Q SD ( g ) = σ [ g ] is consistentwith the risk premium function. Finally, we adapt the classic results of Pratt (1964) andArrow (1970) to the present setup, and show that the local Arrow–Pratt coefficient ofabsolute risk aversion ρ ( u, w ) = − u ′′ ( w ) u ′ ( w ) is consistent with all four decision functions. Related Literature and Contribution
Aumann & Serrano (2008) and Foster & Hart(2009) presented two “objective” indices of riskiness of gambles, which are independentof the subjective utility of the agent. These indices are either based on reasonable axiomsthat an index of risk should satisfy (e.g., Artzner et al. , 1999; Aumann & Serrano, 2008;Cherny & Madan, 2009; Foster & Hart, 2013; Schreiber, 2014; Hellman & Schreiber, 2018;see also the recent survey of Föllmer & Weber, 2015), or they are based on an “opera-tive” criterion such as an agent never going bankrupt when relying on an index of risk in Sec. 4.5 demonstrates that our results cannot be extended to continuous-time processes with jumps. We argue that risk is a multidimensional attribute that crucially depends on theinvestment problem. Different aspects of risk are relevant when an agent has to decidewhether to accept a gamble, compared with a situation in which an agent has to choosehow much to invest in a gamble, or has to evaluate the certainty equivalent of the optimalinvestment. Many existing papers focus on a single decision function. By contrast, wesuggest a framework for studying various decision problems, and associate each suchproblem with its relevant index of risk. We believe that this general framework may behelpful in future research on risk indices.In general, different agents make different investment decisions, based on the subjectiveutility of each agent. Thus, a single risk index cannot be consistent with the choices ofall agents, which, arguably, limits the index’s objectiveness (even when the index satisfiesappealing axioms or some operative criterion for avoiding bankruptcy). However, our mainresult shows that in various important decision problems, all agents rank all gambles inthe same way when deciding on short-term investments in gambles whose returns evolvecontinuously in time. This finding enables us to construct objective risk indices that areconsistent with the short-term investment decisions of risk-averse agents.There are pairs of gambles for which all risk-averse agents agree on which one ofthe gambles is more desirable. This happens if one gamble second-order stochasticallydominates the other gamble. However, the well-known order of stochastic dominance(Hadar & Russell, 1969; Hanoch & Levy, 1969; Rothschild & Stiglitz, 1970) is only a par-tial order and “most” pairs of gambles are incomparable. Interestingly, even if one gamblesecond-order stochastically dominates another gamble, it is not sufficient for a uniformranking among all risk-averse agents in every decision problem (see, e.g., the analysis ofcapital allocation decisions in Landsberger & Meilijson, 1993).A large body of literature uses the classic mean-variance capital asset pricing model(Markowitz, 1952; see Smetters & Zhang, 2013; Kadan et al. , 2016 for recent extensions).A well-known critique is that in a discrete-time setup the mean-variance preferencesare consistent with expected utility maximization only under severe restrictions, such asCARA utilities and normally distributed gambles (see, e.g., Borch, 1969; Feldstein, 1969; Aumann & Serrano’s (2008) and Foster & Hart’s (2009) indices of risk have been extended to gam-bles with an infinite support (Homm & Pigorsch, 2012; Schulze, 2014; Riedel & Hellmann, 2015) and togambles with unknown probabilities (Michaeli, 2014). These indices have been applied to study real-lifeinvestment strategies in Kadan & Liu (2014); Bali et al. (2015); Anand et al. (2016); Leiss & Nax (2018). t > In principle,one could apply a similar axiomatic method to our three other decision functions; weleave this interesting research direction for future research (for further discussion see Sec.5). Unlike the other papers mentioned above, Schreiber (2016) deals with returns in thecontinuous-time setup. Specifically, he analyzes acceptance and rejection of short-terminvestments. The key contributions of the present paper with respect to Schreiber (2016)consists in, first, extending the analysis to the other three decision functions (namely,capital allocation, optimal certainty equivalent, and risk premium) and, second, showingin all four cases an equivalence to the indices in the exponential-normal setup.
Structure
In Section 2 we present our model. In Section 3 we analyze the benchmarksetup of CARA utilities and normally distributed gambles. In Section 4 we adapt themodel to study risky assets whose returns evolve continuously in time, and present ourmain result. We conclude with a discussion in Section 5. Appendix A extends our modelto multiplicative gambles. The formal proofs are presented in Appendix B.
We consider an agent who has to make an investment decision related to a risky asset. Webegin by defining each of these components: agent, risky asset, and investment decision.A decision maker (or agent ) is modeled as a pair ( u, w ), where u : R → R is atwice continuously differentiable von Neumann–Morgenstern utility function over wealthsatisfying u ′ > u ′′ < w ∈ R is an initial wealth level. Let DM denote the set of all such decision makers.A gamble g is a real-valued random variable with a positive expectation and somenegative values (i.e., 0 < E [ g ], and P [ g < > x dollars and therandom payoff from the investment is y dollars, then the additive return g ≡ y − x is agamble. Let G denote the set of all such gambles.A decision function f : DM × G → R is a function that assigns to each agent and eachgamble a nonnegative number, where a higher value is interpreted as the agent finding Shorrer (2014) further applies analogous axioms in the related setup in which an agent has to ac-cept/reject an option to allocate a certain amount of money in a multiplicative gamble, and other inter-esting setups that deal with acceptance/rejection of cash flows and information transactions.
We study four decision functions in the paper:1.
Acceptance/rejection : We consider a situation in which an agent faces a binarychoice between accepting and rejecting the gamble. Specifically, the acceptancefunction f AR : DM × G → { , } is given by f AR (( u, w ) , g ) = E [ u ( w + g )] ≥ u ( w )0 E [ u ( w + g )] < u ( w ) . That is, f AR (( u, w ) , g ) is equal to one if accepting the gamble yields a weakly higherexpected payoff than rejecting it, and it is equal to zero otherwise. The acceptancefunction has been used to study risk indices in various papers (e.g., Foster & Hart,2009, 2013). In particular, our analysis of this decision function extends the analysisof Schreiber (2016), by showing the similarity between this function in the mean-variance setup and the corresponding decision function in the continuous-time setup.2. Capital allocation : Second, we study a situation in which an agent has a continuouschoice of how much to invest in the gamble. Specifically, the capital (or asset)allocation function f CA : DM × G → R + ∪ {∞} is given by f CA (( u, w ) , g ) = arg max α ∈ R + E h u (cid:16) w + αg (cid:17)i ; (1)if (1) does not admit of a solution (i.e., E h u (cid:16) w + αg (cid:17)i is increasing for all α -s), thenwe set f CA (( u, w ) , g ) = ∞ . That is, f CA (( u, w ) , g ) is the optimal level the agent( u, w ) chooses to invest in gamble g . An investment level of zero is interpreted as noinvestment in the gamble. An investment level in the interval (0 ,
1) is interpretedas a partial investment in the gamble. An investment level of one is interpreted asa total investment in the gamble (without leverage). Finally, an investment levelstrictly greater than one is interpreted as a more than total investment in the gamble(achieved, for example, through high leverage). The capital allocation function isprominent in classic analyses of riskiness of assets (e.g., Markowitz, 1952; Sharpe,7964), and, more recently, it has been used to derive an incomplete ranking over theriskiness of gambles (Landsberger & Meilijson, 1993).3.
The optimal certainty equivalent:
Third, we study a situation in which an agent hasto assess how much an opportunity to invest in the gamble g is worth to him (wherewe allow the agent to choose his optimal investment level). Specifically, the optimalcertainty equivalent function f CE : DM × G → R + ∪ {∞} is defined implicitly asthe unique solution to the equation u ( w + f CE ) = max α ∈ R + E h u (cid:16) w + α · g (cid:17)i ; (2)if (2) does not admit of a solution (which happens when E h u (cid:16) w + α · g (cid:17)i is increasingfor all α -s), then we set f CE = ∞ . That is, f CE (( u, w ) , g ) is interpreted as thecertain gain for which the decision maker is indifferent between obtaining this gainfor sure and having an option to invest in the gamble g , when the agent is allowedto optimally choose his investment level in g . Observe that one can express the RHSin (2) in terms of f CA and obtain the following equivalent definition of f CE as theunique solution to the equation u ( w + f CE ) = E h u (cid:16) w + f CA (( u, w ) , g ) · g (cid:17)i . Thefunction f CE has been studied axiomatically in Hellman & Schreiber (2018).4. Risk premium : Lastly, we study a situation in which the agent has to decide betweeninvesting in the gamble and obtaining a certain amount that is less than the gamble’sexpected payoff. Specifically, the risk premium function f RP : DM × G → R − ∪{−∞} is defined implicitly as the unique solution to the equation E [ u ( w + g )] = u ( w + E [ g ] + f RP ) ; (3)if such a solution does not exist then we set f RP (( u, w ) , g ) = −∞ . That is, f RP (( u, w ) , g ) is interpreted as the negative amount that has to be added to theexpected value of the gamble, to make the agent indifferent between investing inthe gamble, and obtaining the gamble’s expected payoff plus this negative amount.Here we use the common acceptation of risk premium in the economic literature(Arrow, 1970; see Kreps, 1990, Section 3.2, for a textbook definition), which has asomewhat different meaning in some of the finance literature (see Footnote 1).In the main text we study additive gambles, in which the gamble’s realized outcome is8dded to the initial wealth. In Appendix A we extend our model to multiplicative gambles,in which the realized outcome of the gamble is interpreted as the per-dollar return. We define a risk index as a function Q : G → R ++ that assigns to each gamble a positivenumber, which is interpreted as the gamble’s riskiness. We study three risk indices:1. The variance-to-mean index Q V M ( g ) is the ratio of the variance to the mean: Q V M ( g ) = σ [ g ] E [ g ] , where σ [ g ] ≡ E h ( g − E [ g ]) i .
2. The inverse Sharpe index Q IS ( g ) is the ratio of the standard deviation to the mean: Q IS ( g ) = σ [ g ] E [ g ] .
3. The standard deviation index Q SD ( g ) is equal to: Q SD ( g ) = σ [ g ] . We say that a risk index is consistent with a decision function over a domain of agents andgambles, if: each agent in the domain finds gamble g less attractive than g ′ with respectto the relevant decision function iff the risk index of g is higher than in g ′ . Formally: Definition 1.
Risk index Q is consistent with f over the domain DM × G ⊆ DM × G if Q ( g ) > Q ( g ′ ) ⇔ f (( u, w ) , g ) < f (( u, w ) , g ′ )for each agent ( u, w ) ∈ DM and each pair of gambles g, g ′ ∈ G .Our definition of consistency is restrictive and for a given domain of gambles andagents it may not apply at all. In particular observe that a domain DM × G ⊆ DM × G admits a consistent risk index iff all agents have the same ranking over gambles , i.e., if f (( u, w ) , g ) < f (( u, w ) , g ′ ) ⇔ f (( u ′ , w ′ ) , g ) < f (( u ′ , w ′ ) , g ′ )for each pair of agents ( u, w ) , ( u ′ , w ′ ) ∈ DM and each pair of gambles g, g ′ ∈ G .Note that consistency is an ordinal concept; i.e., a consistent risk index is unique up tomonotone transformations; if risk index Q is consistent with function f over the domain9 M × G , then risk index Q ′ is consistent with f over this domain iff there exists a strictlyincreasing mapping θ : Q ( G ) → Q ′ ( G ), s.t. Q ′ ( g ) = θ ( Q ( g )) for each gamble g ∈ G . We define a risk-aversion index as a function φ : DM → R ++ that assigns to each agent anon-negative number, which is interpreted as the agent’s risk aversion. We mainly studyone risk index in the paper, the Arrow–Pratt coefficient of absolute risk aversion, denotedby ρ : DM → R ++ , which is defined as follows: ρ ( u, w ) = − u ′′ ( w ) u ′ ( w ) . We say that a risk-aversion index is consistent with a decision function over a domainof agents and gambles, if, for each gamble and each pair of agents in the domain, the agentwith the higher index chooses a lower value for his investment decision in the gamble.
Definition 2.
Risk-aversion index φ is consistent with f over DM × G ⊆ DM × G if φ ( u, w ) > φ ( u ′ , w ′ ) ⇔ f (( u, w ) , g ) < f (( u ′ , w ′ ) , g )for each pair of agents ( u, w ) , ( u ′ , w ′ ) ∈ DM and each gamble g ∈ G .Here again, the definition of consistency is restrictive and for a given domain of gamblesand agents it may not apply at all. Specifically, a domain DM × G ⊆ DM × G admits aconsistent risk-aversion index iff all gambles induce the same ranking over agents , i.e., if f (( u, w ) , g ) < f (( u ′ , w ′ ) , g ) ⇔ f (( u, w ) , g ′ ) < f (( u ′ , w ′ ) , g ′ )for each pair of agents ( u, w ) , ( u ′ , w ′ ) ∈ DM and each pair of gambles g, g ′ ∈ G . Further,the consistency of a risk-aversion index is unique up to a strictly monotone transformation. We begin by presenting a claim, which summarizes known results for normal distributionsand CARA utilities. Specifically, we show that in each of the decision functions described10bove, all agents with CARA utilities have the same ranking over all normally distributedgambles, and that each of these rankings is consistent with one of the risk indices presentedabove. Moreover, all normally distributed gambles induce the same ranking over all agentswith CARA utilities, which is consistent with the Arrow–Pratt coefficient.Formally, let DM CARA ⊆ DM be the set of decision makers with CARA utilities: DM CARA = n ( u, w ) ∈ DM |∃ ρ > , s.t. u ( x ) = 1 − e − ρ · x o , and let G N ⊆ G be the set of normally distributed gambles with positive expectations: G N = { g ∈ G| g ∼ N orm ( µ, σ ) , for some µ, σ > } . Claim . Let u be a CARA utility with parameter ρ (i.e., u ( x ) = 1 − e − ρ · x ). Then:1. f RP (( u, w ) , g ) = − . · ρ · σ , which implies that the standard deviation index Q SD is consistent with the risk premium function f RP in the domain DM CARA × G N .2. (I) f AR (( u, w ) , g ) = 1 iff ρ · µσ ≥ f AR (( u, w ) , g ) = 0 otherwise), (II) f CA (( u, w ) , g ) = ρ · µσ , which imply that the variance-to-mean index Q V M is consistent with boththe acceptance/rejection function f AR and the capital allocation function f CA .3. f CE (( u, w ) , g ) = · ρ · (cid:16) µσ (cid:17) , which implies that the inverse Sharpe index Q IS isconsistent with the optimal certainty equivalent function f CE .4. The Arrow–Pratt coefficient of absolute risk aversion ρ is consistent with all fourdecision functions f CA , f AR , f CE , and f RP in the domain DM CARA × G N .For completeness, we present the proof of Claim 1 in Appendix B.1. Each agent with CARA utility is described by two parameters (initial wealth w , andArrow–Pratt coefficient ρ ). Similarly, each normal gamble is described by two parameters(expectation µ and standard deviation σ ). This implies that any decision function can beexpressed as a function g ( w, ρ, µ, σ ) of these four parameters. Clearly, one can extend the definition of DM CARA (without affecting any of the results) by allowingthe utilities to differ from 1 − e − ρ · x by adding a constant and multiplying by a positive scalar. ther Consistent Indices of Risk Aversion CARA utilities have the well-knownproperty that the initial wealth does not affect expected utility calculations with respectto investments in gambles. Thus, whenever the investment decision is made by choosingthe option that maximizes the agent’s expected utility (such as in all four of the decisionfunctions analyzed above), then the decision function is independent of w , which impliesthat the parameter ρ is a consistent risk-aversion index. By contrast, for investmentdecisions that are not determined by maximizing the agent’s expected utility, there mightbe different risk-aversion consistent indices. For instance, Foster & Hart (2009) analyze asituation in which an agent accepts or rejects gambles while his goal is to avoid bankruptcy.The index of risk aversion that is consistent with their decision function is the wealth level. Separability Condition for Having a Consistent Risk Index
The decision func-tions analyzed above have the additional separability property that each function f canbe represented as a product of two functions: one that depends only on the parametersdescribing the agent ( w and ρ ), and one that depends only on the parameters of the gam-ble, i.e., f (( u, w ) , g ) = ˜ f ( w, ρ, σ, µ ) = h ( w, ρ ) · ν ( µ, σ ). This separability implies that allagents with CARA utilities have the same ranking over normal gambles (as this rankingdepends only on ν ( µ, σ ), which does not depend on the agent’s parameters), which, inturn, implies that there exists a consistent risk index. Similarly, the separability impliesthat all normal gambles induce the same ranking over agents (as this ranking dependsonly on h ( w, ρ ), which does not depend on the parameters of the normal gamble).Other decision functions might not satisfy this separability property. One example ofsuch a non-separable decision function is the standard certainty equivalent of a continuousgamble f SCE (as opposed to the certainty equivalent of the optimal allocation of thegamble f CE (( u, w ) , g ) discussed above), which is implicitly defined by E [ u ( w + g )] = u ( w + f SCE ) . The definitions of f SCE and f RP imply that f SCE = E [ g ] + f RP . Substituting f RP = − . · ρ · σ (which is proven in Appx. B.1) yields: f SCE = E [ g ] + f RP = µ − . · ρ · σ , which is a non-separable function of ρ , µ, σ . The non-separability implies that agents withdifferent CARA utilities have different rankings for normal gambles and, therefore, norisk index can be consistent with these decisions.12 Short-Term Investments in Continuous Gambles
In what follows we adapt our model to short-term investment decisions regarding assetswhose value follows a continuous random process. Our description of the continuous-timesetup follows Shreve (2004).
Let (Ω , F , P ) be a probability space on which a Brownian motion W t is defined, withan associated filtration F ( t ). Let the process g be described by the following stochasticdifferential equation: dg t = µ t dt + σ t dW t , (4)where the drift µ and the diffusion σ are adapted stochastic processes (i.e., µ t and σ t are F ( t )-measurable for each t , see Shreve, 2004, Footnote 1 on page 97) and are bothcontinuous in t . We assume that µ > σ > σ t ≥ t >
0. Wealso assume that E R t σ s ds and E R t | µ s | ds are finite for every t >
0. This implies thatthe integrals E R t σ s dW s and E R t µds are well defined, and the Ito integral R t σ s dW s is amartingale; see Shreve (2004, Footnote 2 on page 143).The process g is interpreted as the additive return of some risky asset. Specifically,let P t be the price of some risky asset at time t and assume that P is known. Then, g t = P t − P , is the additive return of the asset at time t . In particular, observe that g = P t =0 − P ≡ g is bounded from below (i.e., there exists M g ∈ R , suchthat g t ≥ M g for each t > t , g t is a gamble (see Footnote 8 in Appendix B.2); i.e., it has positive expectation and takesnegative values with positive probability. Thus, we can apply our definitions of decisionfunctions and indices to g t for each specific value of t > P isa pure number and g t is a gamble, just as before. For ease of exposition we limit the Wiener process to one dimension. All the results remain the samewith a multidimensional Wiener process with the corresponding adjustments. ; Merton, 1992, Chapters 4 and 5) , and variants of arithmetic Brownian pro-cesses and of mean-reverting processes that are bounded from below (also known asOrnstein–Uhlenbeck processes; see, e.g., Merton, 1992, Chapter 5; Hull & White, 1987;Meddahi & Renault, 2004), such as Cox et al. ’s (1985) process for modeling interest rate. We define a local-risk index at time zero as a function Q l : Γ → R ++ that assigns to eachprocess g ∈ Γ a positive number, which is interpreted as the process’s short-term riskinessat t = 0. Given g ∈ Γ with initial parameters µ and σ , we define three specific local-riskindices (analogous to the corresponding definitions in Section 2.2):1. The variance-to-mean local index Q lV M ( g ) is equal to: Q lV M ( g ) = σ µ .
2. The inverse Sharpe local index Q lIS ( g ) is equal to: Q lIS ( g ) = σ µ .
3. The standard deviation local index Q lSD ( g ) is equal to: Q lSD ( g ) = σ . Given a continuous-time process g ∈ Γ, decision function f and agent ( u, w ) ∈ DM , let f ( u,w ) g ( t ) ≡ f (( u, w ) , g t ) be the value of the decision function of agent ( u, w ) with respectto an investment in g as a function of the duration of the investment t .The following definition, which deals with general real-valued functions, will be usefulfor defining the concept of consistency of indices in the continuous-time framework. When one models an asset’s price P by a geometric Brownian motion, then P t (the asset value attime t ) obtains only positive values. In this case, the additive return is defined as the difference betweenthe asset’s value at time t and its initial value, i.e., g t = P t − P . Obviously, the additive return can obtainboth positive and negative values (for any time t > g t be a gamble. Specifically, in the case of a geometric Brownian motion, dg t ≡ dp t = p t · µ · dt + p t · σ · dW ,which implies that µ t = p t · µ and σ t = p t · σ as in Equation (4). efinition 3. Let f, h : R + ⇒ R , and assume that lim t → f ( t ) h ( t ) is well defined. We saythat f is uniformly higher than h (around zero) and denote it by f >> h if (1) thereexists ¯ t >
0, such that f ( t ) > h ( t ) for each t ∈ (cid:16) , ¯ t (cid:17) and (2)lim t → f ( t ) h ( t ) = 1 . That is, f >> h if f ( t ) is strictly higher than h ( t ) for any sufficiently small t , and therelative difference between the two functions does not become negligible (as measured bythe ratio f ( t ) h ( t ) not converging to one) in the limit of t → g ′ is lower than the index of g iff all risk-averse agentsfind g t uniformly more attractive than g ′ t with respect to any short-term investment. Definition 4.
Local-risk index Q l : Γ → R + is consistent with decision function f overthe set of continuous returns Γ if, for each pair of continuous-time processes g, g ′ ∈ Γ andeach agent ( u, w ) ∈ DM , we have that Q l ( g ) > Q l ( g ′ ) ⇔ f ( u,w ) g >> f ( u,w ) g ′ . Note that a consistent risk index is unique up to strictly monotone transformations.We say that a risk-aversion index is consistent with a decision function over continuous-time returns, if the risk-aversion index of agent ( u, w ) is strictly higher than the index of( u ′ , w ′ ) iff agent ( u, w ) finds all gambles uniformly less attractive than agent ( u ′ , w ′ ). Definition 5.
Risk-aversion index φ : DM −→ R + is consistent with decision function f over the set of continuous returns Γ if, for each pair of agents ( u, w ) , ( u ′ , w ′ ) ∈ DM and for each gamble g ∈ Γ, we have that φ ( u, w ) > φ ( u ′ , w ′ ) ⇔ f ( u,w ) g << f ( u ′ ,w ′ ) g Note that a consistent risk-aversion index is unique up to monotone transformations.
Our main result shows that in each of the decision functions described above, all agentshave the same ranking over all short-term continuous returns. Moreover, the rankings15re consistent with the three risk indices presented above, and they are the instantaneousversions of the corresponding indices in the case of normally distributed gambles andCARA utilities analyzed in Claim 1. Finally, we adapt to the present setup the classicresult that all continuous short-term returns induce the same ranking over all agents,which is consistent with the Arrow–Pratt coefficient of absolute risk aversion (as in thecase of normally distributed gambles and CARA utilities). Formally:
Theorem 1.
The following conditions hold over the set of continuous returns Γ :1. The standard deviation index Q lSD is consistent with the risk premium function f RP .
2. The variance-to-mean index Q lV M is consistent with the capital allocation function f CA .3. The inverse Sharpe index Q lIS is consistent with the optimal certainty equivalentfunction f CE .4. The Arrow–Pratt coefficient of absolute risk aversion ρ is consistent with decisionfunctions f CA , f CE , and f RP .Sketch of proof; formal proof is presented in Appendix B.2. The value of an asset with acontinuous-time return g after a sufficiently small time t is represented by a gamble g t for which the magnitudes of all high moments are small relative to the magnitude of thesecond moment. Assuming random variables of this type allows us to use Taylor expansionto approximate the decision functions, and to obtain the consistent risk indices.Recall that lim t → σ [ g t ] t = σ and lim t → µ [ g t ] t = µ , which implies for sufficiently small t -s that σ [ g t ] ≈ t · σ and µ [ g t ] ≈ t · µ . We begin with a standard approximation of therisk premium function f RP (see, e.g., Eeckhoudt et al. , 2005, Chapter 1). Recall that therisk premium was defined implicitly as E [ u ( w + g t )] = u ( w + E [ g t ] + f RP ) . (5)A second-order Taylor expansion of the left-hand side around w + E [ g t ] yields E [ u ( w + g t )] ≈ E " u ( w + E [ g t ]) + u ′ ( w + E [ g t ])( g t − E [ g t ]) + 12 u ′′ ( w + E [ g t ]) ( g t − E [ g t ]) = u ( w + E [ g ]) + 12 u ′′ ( w + E [ g t ]) σ [ g t ] .
16 first-order Taylor expansion of the right-hand side around w + E [ g ] yields u ( w + E [ g t ] + f RP ) ≈ u ( w + E [ g t ]) + u ′ ( w + E [ g t ]) · f RP . Combining these equations and isolating f RP yields f RP ≈ u ′′ ( w ) u ′ ( w ) σ [ g t ] ≈ u ′′ ( w ) u ′ ( w ) t · σ , which implies that Q lSD = σ (resp., ρ = − u ′′ ( w ) u ′ ( w ) ) is a consistent risk index (resp., risk-aversion index) for decision function f RP .In order to analyze f CA , we define C ( α ) = ( f RP ( α · g t ) + E [ α · g t ]) to be the certaintyequivalent of investment α in g t . Substituting the value of f RP calculated above we get C ( α ) ≈ α u ′′ ( w ) u ′ ( w ) σ [ g t ] + α · E [ g t ] . In order to maximize C ( α ), we compare the derivative to zero, to get ∂C ( α ) ∂α = 0 ⇔ α ∗ · u ′′ ( w ) u ′ ( w ) σ [ g t ] + E [ g t ] = 0 ⇔ f CA = α ∗ ≈ − E [ g t ] u ′′ u ′ σ [ g t ] ≈ − µ u ′′ u ′ σ , which implies that the variance-to-mean index Q lV M (resp., the Arrow–Pratt coefficient ρ ) is a consistent risk index (resp., risk-aversion index) for decision function f CA .Finally, if we calculate f CE = C ( α ∗ ) = C ( f CA ) we get f CE ≈ E [ g ] u ′′ ( w ) u ′ ( w ) σ [ g t ] ! u ′′ ( w ) u ′ ( w ) σ [ g t ] + E [ g t ] − u ′′ ( w ) u ′ ( w ) σ [ g t ] E [ g t ]= 12 1 − u ′′ ( w ) u ′ ( w ) E [ g t ] σ [ g t ] ! ≈
12 1 − u ′′ ( w ) u ′ ( w ) t · µ σ ! , which implies that the inverse Sharp index Q lIS (the Arrow–Pratt coefficient ρ ) is a con-sistent risk (risk aversion) index for decision function f CE . Remark . The expressions that approximate the various functions inthe continuous-time setup consist of two elements: the coefficient of risk aversion withrespect to the initial wealth level, and a function of the first and second moment of the17small” gamble. In the CARA-normal setup of Section 3, the risk-aversion coefficient isconstant over all wealth levels, and, thus, it is relevant also to large gambles. In addition,the only moments that matter to an agent with CARA utility who invests in a normallydistributed gamble are the first two moments. To see that, recall that for CARA utility u (with coefficient of risk aversion ρ ) and normal gamble g , E [ u ( w + g )] = E [1 − e − ρ · ( w + g ) ] = 1 − e − ρ E [ w + g ]+0 . ρ σ [ g ] . Therefore, it seems plausible that the expressions that represent the decision functions inthe CARA-normal setup, depend only on the first two moments, and, thus, they coincidewith the approximated decision functions that are relevant for short-term investments inassets with continuous returns.
The case of the the acceptance/rejection function f AR has been analyzed in Schreiber(2016). As the function f AR has only two feasible values (0 or 1), it cannot admit ofconsistent risk indices, as in many cases in which one gamble is riskier than another, anagent may choose to reject both gambles (and his value of f AR of both gambles would bezero). Nevertheless, one can define the milder notion of weak consistency, and show thata corollary to Schreiber’s (2016) result is that the risk index Q lV M is weakly consistentwith the acceptance/rejection function f AR .A local-risk index is weakly consistent with a decision function over the set of con-tinuous returns, if each agent chooses a weakly lower value of his investment decision ingamble g t relative to g ′ t for a sufficiently small t if the local risk of g is strictly higher thanthe local risk of g ′ . Formally: Definition 6.
Local-risk index Q l : Γ → R + is weakly consistent with decision function f over the set Γ if for each agent ( u, w ) ∈ DM and each pair of continuous-time processes g, g ′ ∈ Γ, there exists time ¯ t , such that, for each time t < ¯ t , we have that Q l ( g ) > Q l ( g ′ ) ⇒ f (( u, w ) , g t ) ≤ f (( u, w ) , g ′ t ) . Note that weak consistency does not restrict the agents’ choices when both gambleshave the same local-risk index. As a result, a weakly consistent risk index is uniqueonly up to weakly monotone transformations; i.e., if Q is a weakly consistent local-risk18ndex with decision function f over the set of continuous returns Γ, then risk index Q ′ isconsistent with function f over this domain if there exists a weakly increasing mapping θ : Q ( G ) → Q ′ ( G ), such that Q ′ ( g ) = θ ( Q ( g )) for each g ∈ Γ. In particular, a constantindex is trivially a weakly consistent local-risk index of any decision function.We say that a risk-aversion index is weakly consistent with a decision function overcontinuous-time returns, if for each short-term return, an agent chooses a (weakly) highervalue for his investment decision in the asset relative to another agent’s decision if theformer agent’s risk aversion is smaller. Formally:
Definition 7.
Risk-aversion index φ : DM −→ R + is weakly consistent with decisionfunction f over the domain of short-term continuous gambles if for each continuous-timeprocess g ∈ Γ and each pair of agents ( u, w ) , ( u ′ , w ′ ) ∈ DM , there exists a time ¯ t , suchthat, for each time t < ¯ t , we have that φ ( u, w ) > φ ( u ′ , w ′ ) ⇒ f (( u, w ) , g t ) ≤ f (( u ′ , w ′ ) , g t ) . The following corollary, which is implied by Schreiber (2016, Theorems 2.2 & 3.3),shows that the standard deviation index Q lV M and the Arrow–Pratt coefficient of absoluterisk aversion ρ are weakly consistent with the acceptance/rejection function f AR . Corollary 1 (Implied by Schreiber (2016, Theorems 2.2 & 3.3)) . The following conditionshold over the domain of continuous short-term decisions:1. The variance-to-mean index Q lV M is weakly consistent with decision function f AR .2. The Arrow–Pratt coefficient ρ is weakly consistent with decision function f AR . The set of continuous-time gambles Γ analyzed in this paper does not allow for jumps. Inwhat follows we show that the absence of jumps is necessary for our main result. Specifi-cally, we demonstrate that risk-averse agents rank continuous-time processes with jumpsdifferently, even for short-term investments, which rules out the existence of consistentrisk indices. Consider, for example, the acceptance/rejection function f AR (similar conclu-sions can be drawn for the other decision functions). Hart (2011) observes that there aremany pairs of (discrete-time) gambles that are ranked differently by different risk-averseagents (see Hart 2011, Footnote 23). Let h, ˜ h be such a pair of gambles.19onsider the following compound Poisson processes g, ˜ g , where each has an initialvalue of zero. The value of each process changes only when there is a jump. The jumpsarrive randomly with a rate λ . In process g (resp., ˜ g ) the size of each jump is distributedaccording to h (resp., ˜ h ). Observe that for sufficiently short times the probability ofhaving two jumps is negligible, and the decision whether to accept or to reject a gambledepends only on what may happen after a single jump. This implies that agents whorank the gambles h, ˜ h differently, would also rank g t , ˜ g t differently, for any sufficientlyshort time t . This rules out the existence of a consistent risk index in this setup. Our main result is that in four central decision problems all risk-averse agents have thesame (problem-dependent) ranking over short-term investments in risky assets whosereturns evolve continuously, and these rankings are represented by simple well-knownindices of risk. The indices obtained are the same as in the classic model of CARAutilities and normally distributed gambles. Each problem relates to a different dimensionof risk, and, thus, its ranking is represented by a different risk index. Finally, adapting aclassic result to the present setup, we show in all of the decision functions analyzed above,the decisions of agents are consistent with their Arrow–Pratt coefficients of risk aversion.The proposed indices in our paper are all based on the first two moments. This is aresult of the known property of continuous stochastic processes for which higher momentsgo quickly to zero as the time parameter goes to zero. Hence, multiple indices of riskthat do use higher moments might coincide with our indices when they are applied tocontinuous-time processes and short-term investments. For instance, Schreiber (2016)shows that the index of Aumann & Serrano (2008) and that of Foster & Hart (2009)(which, in general, both depend on all moments of the gamble) coincide with the variance-to-mean index Q V M for continuous processes in the limit of t →
0, and Shorrer (2014)shows that there is a continuum of risk indices (which depend also on higher moments)that are consistent with acceptance/rejection decisions of agents with respect to smalldiscrete gambles. Indeed, under the assumption that returns evolve continuously in time,the only relevant parameters for measuring risk are the first two moments. Our resultscan be interpreted as characterizing a necessary condition for a plausible risk index,namely, that a plausible risk index (with respect to one of the four decision functionsanalyzed in the paper) should depend on the first two moments in the same way as20resented in our main result. We leave for future research the interesting question ofhow to choose among the various risk indices that satisfy this necessary condition. Onepossible direction for analyzing this question is the axiomatic approach applied in Shorrer(2014) to acceptance/rejection decisions.
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NBER Working Paper 19500 . A Multiplicative Gambles
In the main text we followed the recent literature of riskiness (initiated by Aumann & Serrano,2008 and Foster & Hart, 2009) and focused on decision problems with regard to additivegambles in units of dollars. However, in most financial applications, it is common to de-scribe the returns of an asset in relative terms, namely, percentages (see, e.g., Markowitz,1959 and Merton, 1992), as this is the way in which returns are described in practice inexchange markets. Hence, in this section we show that our results hold also with regardto multiplicative returns.In some sense, the difference between multiplicative and additive returns is only amatter of presentation: if one invests x dollars in a multiplicative gamble r , one’s payoff24ill be x (1 + r ) dollars, and this is just the same payoff as if one invests in an additivereturn of x · r dollars. Nevertheless, we think that presenting the results for multiplicativereturns is important for two reasons: first, as argued in Schreiber (2014), each investmentmight have two different aspects of riskiness, absolute and relative; given two assets, oneof them might be riskier in relative terms but less risky in absolute terms. Therefore itis worthwhile to study the difference between multiplicative and additive returns in oursetup. As it turns out, this potential difference vanishes when focusing on short-terminvestments and we derive in the multiplicative setup results analogous to those that wehave in the additive setup. Second, in many situations of decision making under risk,the risk-free interest rate should be taken into account. Since the risk-free interest rate iscalculated in terms of percentages, it is natural to combine it in decision problems withrelative return, as we do here. A.1 Adaptation to the Model
Let r f > multiplicative riskyasset (multiplicative gamble) r is a real-valued random variable with an expectation that isgreater than r f , and some negative values greater than − , i.e., E [ r ] > r f , P [ r < > r ≥ −
1. We interpret r as the per-dollar return of the asset. Let R denote the set ofall multiplicative risky assets.We adapt the definitions of our decision functions to the case of multiplicative gambles.1. The acceptance function f mAR : DM × R → { , } is given by f mAR (( u, w ) , r ) = E [ u ( w · (1 + r ))] ≥ u ( w · (1 + r f ))0 E [ u ( w · (1 + r ))] < u ( w · (1 + r f )) , where we consider a situation in which an agent faces a binary choice betweeninvesting his entire wealth in a multiplicative gamble r and investing it in the risklessasset with return r f .2. The capital allocation function f mCA : DM × R → R + ∪ {∞} is given by f mCA (( u, w ) , r ) = arg max α ∈ R + E h u (cid:16) w · (1 + r f ) + α · w · ( r − r f ) (cid:17)i ; (6)if Equation (6) does not admit of a solution (i.e., E h u (cid:16) w · (1 + r f ) + α · w · ( r − r f ) (cid:17)i
25s increasing for all α -s), then we set f mCA (( u, w ) , r ) = ∞ . This function deals witha situation in which an agent decides on the optimal share α ≥ w to invest in the multiplicative gamble r (where α > f mCE : DM × R → R + ∪ {∞} is definedimplicitly as the unique solution to the equation u ( w · (1 + f mCE )) = max α ∈ R + E [ u ( w · (1 + r f ) + α · w · ( r − r f ))] ≡ E h u (cid:16) w (1 + r f ) + f mCA (( u, w ) , r ) · w · ( r − r f ) (cid:17)i ; (7)if Equation (7) does not admit of a solution (i.e., E h u (cid:16) w · (1 + r f ) + α · w · ( r − r f ) (cid:17)i is increasing for all α -s), then we set f mCE (( u, w ) , r ) = ∞ . This function describesthe rate of a constant return that is equivalent to investing optimally in a multi-plicative gamble r , where the remaining wealth is invested in the riskless asset.4. The risk-premium function f mRP : DM × R → R − is defined implicitly as the uniquesolution to the equation E [ u ( w · (1 + r ))] = u ( w · (1 + E [ r ] + f mRP )) , where f mRP represents the constant (negative) return that makes the agent indifferentbetween investing all his wealth in the multiplicative gamble r and investing in anasset with a constant return that is equal to the expectation of r plus f mRP .Let R N ⊆ R be the set of normally distributed multiplicative gambles (defined analogouslyto the definition of G N ). The Arrow–Pratt coefficient of relative risk aversion, denoted by ̺ : DM → R ++ , is defined as follows: ̺ ( u, w ) = − w · u ′′ ( w ) u ′ ( w ) . We adapt the three indices of risk in the main text to the multiplicative setup and theexistence of a risk-free interest rate. Specifically:26. The variance-to-mean index Q mV M ( r ) is equal to Q mV M ( r ) = σ [ r ] E [ r ] − r f , where σ [ r ] ≡ E h ( r − E [ r ]) i .
2. The inverse Sharpe index Q mIS ( r ) is equal to Q mIS ( r ) = σ [ r ] E [ r ] − r f .
3. The standard deviation index Q mSD ( r ) is equal to Q mSD ( r ) = σ [ r ] . A.2 Adapted Results
The adaptation of Claim 1 and Theorem 1 to multiplicative gambles is as follows. Observethat all the results remain the same, except that the Arrow–Pratt coefficient of relativerisk aversion replaces the coefficient of absolute risk aversion.
Claim . The following conditions hold over the domain DM CARA × R N :1. The standard deviation index Q mSD is consistent with decision function f mRP .
2. The variance-to-mean index Q mV M is consistent with both the capital allocationfunction f mCA and the acceptance/rejection function f mAR .3. The inverse Sharpe index Q mIS is consistent with the decision function f mCE .4. The Arrow–Pratt coefficient of relative risk aversion ̺ is consistent with all fourdecision functions: f mAR , f mCA , f mCE , and f mRP .The proof of Claim 2 is made analogous to the corresponding proof in the additivecase by using the following identities (details are omitted for brevity):1. f mAR (( u, w ) , r ) ≡ f AR (( u, w (1 + r f )) , w ( r − r f )),2. f mCA (( u, w ) , r ) ≡ f CA (( u, w (1 + r f )) , w ( r − r f )),3. f mCE (( u, w ) , r ) ≡ f CE (( u, w (1 + r f )) , w ( r − r f )) /w , and4. f mRP (( u, w ) , r ) ≡ f RP (( u, w (1 + r f )) , w ( r − r f )) /w .27ecall that in the continuous-time setup, the decision problems are parameterized by t ,which is the investment horizon. Previously, we assumed that a continuous-time randomprocess g represents the additive return of a financial investment. Now the continuous-time random process r represents the excess multiplicative return: r t = ( P t − P ) /P .We assume that the compound risk-free interest rate is r f and hence the riskless returnover period t is r f ( t ) = e µ f · t −
1. The adapted definitions of the risk indexes in themultiplicative setup for the local risk indices are as follows:1. The variance-to-mean local index Q l,mV M ( g ) is equal to Q l,mV M ( g ) = σ µ − µ f .
2. The inverse Sharpe local index Q l,mIS ( g ) is equal to Q l,mIS ( g ) = σ µ − µ f .
3. The standard deviation local index Q l,mSD ( g ) is equal to Q l,mSD ( g ) = σ . The analogous result to Theorem 1 is as follows.
Theorem 2.
The following conditions hold over the domain of continuous short-termdecisions with respect to multiplicative gambles:1. The standard deviation index Q l,mSD is consistent with the risk premium function f mRP .
2. The variance-to-mean index Q l,mV M is consistent with the capital allocation function f mCA , and it is weakly consistent with the acceptance/rejection function f mAR .3. The inverse Sharpe index Q l,mIS is consistent with the decision function f mCE .4. The Arrow–Pratt coefficient of relative risk aversion ̺ is consistent with decisionfunctions: f mCA , f mCE , and f mRP , and it is weakly consistent with f mAR . The proof of Theorem 2 is made analogous to the corresponding proof in the additivecase by using the following identities (details are omitted for brevity):1. f mAR (( u, w ) , r t ) ≡ f AR (( u, w (1 + r f ( t ))) , w ( r t − r f ( t ))),28. f mCA (( u, w ) , r t ) ≡ f CA (( u, w (1 + r f ( t ))) , w ( r t − r f ( t ))),3. f mCE (( u, w ) , r t ) ≡ f CE (( u, w (1 + r f ( t ))) , w ( r t − r f ( t ))) /w , and4. f mRP (( u, w ) , r t ) ≡ f RP (( u, w (1 + r f ( t ))) , w ( r t − r f ( t ))) /w . B Proofs
B.1 Proof of Claim 1
The following well-known fact, which describes the expectation of a log-normal distribu-tion, will be useful in our proofs (the standard proof, which relies on the Laplace transformof the normal distribution, is omitted for brevity; see, e.g., Forbes et al. , 2011, page 132).
Fact 1. If y is normally distributed with expectation µ and standard deviation σ , then E [ e y ] = e µ +0 . σ . Next, we prove Claim 1. Let g be a normally distributed random variable with ex-pectation µ and standard deviation σ . Let u be a CARA utility with parameter ρ , i.e., u ( x ) = 1 − e − ρ · x . Let w be the arbitrary initial wealth.1. Q SD and ρ are consistent with f RP . The risk premium x is defined implicitly by E [ u ( w + g )] = E h − e − ρ ( w + g ) i = u ( w + E [ g ] + x ) = 1 − e − ρ ( w + µ + x ) ⇔ − E h e − ρ ( w + g ) i = 1 − e − ρ ( w + µ + x ) ⇔ E h e − ρ ( w + g ) i = e − ρ ( w + µ + x ) . By Fact 1 E h e − ρ ( w + g ) i = e − ρ ( w + µ )+0 . ρ σ , which implies e − ρ ( w + µ )+0 . ρ σ = e − ρ ( w + µ + x ) ⇔ − ρ ( w + µ )+0 . ρ σ = − ρ ( w + µ + x ) ⇔ x = 0 . ρσ . Thus, f RP (( u, w ) , g ) = 0 . ρσ , which implies that Q SD = σ is a consistent riskindex (and that ρ is a consistent risk-aversion index with respect to f RP . Q V M and ρ are consistent with f AR . The agent accepts the gamble iff E (cid:16) − e − ρ ( w + g ) (cid:17) > E (cid:16) − e − ρw (cid:17) e − ρ ( w + µ )+0 . ρ σ < e − ρw ⇔ . ρ < µσ . Thus, f AR (( u, w ) , g ) = { . ρ< µσ } , which implies that Q V M = σ µ is a consistentrisk index (and that ρ is a consistent risk-aversion index) with respect to f AR . Q V M and ρ are consistent with f CA . The capital allocation function is given by f CA (( u, w ) , g ) = arg max α ∈ R + E h u (cid:16) w + αg (cid:17)i = arg max α ∈ R + E " − e − ρ (cid:16) w + αg (cid:17) .It follows from Fact 1 that the r.h.s. of the above equation is equivalent toarg max α ∈ R + E " − e − ρ (cid:16) w + αg (cid:17) = arg max α ∈ R + (cid:16) − e − ρw − ραµ +0 . ρ α σ (cid:17) = arg min α ∈ R + (cid:16) − ραµ + 0 . ρ α σ (cid:17) . The first-order condition is − µ + ρα ∗ σ = 0 ⇔ α ∗ = 1 ρ µσ . Thus, f CA (( u, w ) , g ) = ρ µσ , which implies that Q V M = σ µ is a consistent risk index(and that ρ is a consistent risk-aversion index) with respect to f CA . Q IS and ρ are consistent with f CE . The optimal certainty equivalent function isgiven by 1 − e − ρ ( w + f CE ) = u ( w + f CE ) = E h u (cid:16) w + f CA (( u, w ) , g ) · g (cid:17)i = E " u (cid:16) w + 1 ρ µσ · g (cid:17) = E " − e − ρ · (cid:16) w + ρ µσ · g (cid:17) = 1 − e − ρw − µ σ +0 . µ σ , where the last equality uses Fact 1. This implies that1 − e − ρ ( w + f CE ) = 1 − e − ρw − µ σ +0 . µ σ ⇔ − ρ ( w + f CE ) = − ρw − µ σ +0 . µ σ ⇔ f CE = 12 ρ µ σ . Thus, f CE (( u, w ) , g ) = ρ µ σ , which implies that Q IS = σµ is a consistent risk index30resp., ρ is a consistent risk-aversion index) with respect to f CE . B.2 Proof of Theorem 1
The following three lemmas will be useful in our proofs. The first lemma is a simpleversion of Ito’s well-known lemma (see, e.g., Shreve, 2004, Equation 4.4.24).
Lemma 1 (Ito’s lemma) . Let s ( t ) be a random process described by ds t = µ t dt + σ t dW .Let f ( t, s ) be a twice-differentiable function; then df = µ t ∂f∂s + 0 . σ t ∂ f∂s + ∂f∂t ! dt + ∂f∂s σ t dW. The next two lemmas are standard calculus results.
Lemma 2.
Let F t ( y ) be a set of real-valued, continuous, and weakly increasing functions,with < t ≤ T and y ∈ R . Assume that there exists a continuous and strictly increasingfunction F ( y ) such that (1) ∀ y, F ( y ) = lim t → F t ( y ) , and (2) ∃ y ∗ , s.t. F ( y ∗ ) = 0 . Then,there exists ¯ t > s.t. ∀ t < ¯ t ∃ y t s.t. F t ( y t ) = 0 , and lim t → y t = y ∗ . Proof.
Let δ >
0. We have to show that there exists ¯ t s.t. ∀ t < ¯ t there is a value y t satisfying | y t − y ∗ | < δ and F t ( y t ) = 0. Since F ( y ) is strictly increasing there exists apositive number C such that F ( y ∗ − δ ) < − C and F ( y ∗ + δ ) > C . Condition (1) impliesthat there exists ¯ t s.t. ∀ t < ¯ t , | F t ( y ∗ + δ ) − F ( y ∗ + δ ) | < C, and | F t ( y ∗ − δ ) − F ( y ∗ − δ ) | < C. Hence, F t ( y ∗ − δ ) < F t ( y ∗ + δ ) >
0. Since F t is continuous, ∃ y t ∈ ( y ∗ − δ, y ∗ + δ )s.t. F t ( y t ) = 0. Lemma 3.
Let F t ( α ) be a set of twice-differentiable strictly concave functions where < t ≤ T and α ∈ R , and let F be a twice-differentiable strictly concave function suchthat (1) ∀ α, F ( α ) = lim t → F t ( α ) , and (2) ∃ α ∗ ∈ R such that α ∗ = arg max α ∈ R F ( α ) .Then, there exists ¯ t > such that ∀ t < ¯ t, ∃ α t ∈ R s.t. α t = arg max α F t ( α ) , and lim t → α t = α ∗ . roof. We have to show that, given δ >
0, there exists ¯ t > ∀ t < ¯ t , ∃ α t , whichmaximizes F t ( α ), and that | α t − α ∗ | < δ . Let δ = min { F ( α ∗ ) − F ( α ∗ − δ ) , F ( α ∗ ) − F ( α ∗ + δ ) } . There exists ¯ t s.t. ∀ t < ¯ t , | F t ( α ∗ ) − F ( α ∗ ) | < δ / , | F t ( α ∗ + δ ) − F ( α ∗ + δ ) | < δ / , and | F t ( α ∗ − δ ) − F ( α ∗ − δ ) | < δ / . Hence, ∀ t < ¯ t , F t ( α ∗ ) > F t ( α ∗ − δ ) and F t ( α ∗ ) > F t ( α ∗ + δ ) . Since for all t , F t is weakly concave, there exists α t ∈ ( α ∗ − δ, α ∗ − δ ), which is the argmaxof F t .Next, we prove the main theorem. Let g ∈ Γ be a continuous-time random process,and let ( u, w ) ∈ DM be a decision maker.1. Q lSD and ρ are consistent with f RP . For every t >
0, let F t be defined as follows: F t ( x ) = u (cid:16) w + E [ g t ] + x · t (cid:17) − E h u (cid:16) w + g t (cid:17)i t . By definition, if for some value of x , F t ( x ) = 0, then x · t = f RP (( u, w ) , g t ). Tocalculate the limit of F t as t goes to zero, it is simpler to look at F t as the differencebetween two functions k t and h t , defined by k t ( x ) = u ( w + E [ g t ] + x · t ) − u ( w ) t , and h t = E " u w + g t ! − u ( w ) t . (8)Clearly, F t ( x ) = k t ( x ) − h t for every value of x. The limit of k t ( x ) as t goes to zero is simply the derivativewith respect to t at w : lim t → k t ( x ) = u ′ ( w ) · ( µ + x ) . (9)32y applying Ito’s lemma h t = E hR t (cid:16) µ q u ′ q + σ q u ′′ q (cid:17) dq i t + E hR u ′ q σ q dW i t , where u ′ q ≡ du ( w q ) /d ( w q ), u ′′ q ≡ du ( w q ) /d ( w q ), and w q = w + g q . Since we assumedthat g is bounded from below, the concavity and monotonicity of u implies that u ′ q is bounded. In addition, we assumed that the σ t satisfies the square-integrabilitycondition and, therefore, that E " R t σ q dq is finite. These two assumptions implythat E " R t ( u ′ q σ q ) dq is finite and, therefore, that R t u ′ q σ t dW is a martingale; seeShreve (2004, Theorem 4.3.1. on page 134). Hence, h t can be rewritten as follows: h t = E hR t (cid:16) µ q u ′ q + σ q u ′′ q (cid:17) dq i t . Since µ q , σ q , u ′ q and u ′′ q are all continuous, according to the mean-value theorem forintegration, for each realization of g there exists some x ∈ (0 , t ) for which R t (cid:16) µ q u ′ q + σ q u ′′ q (cid:17) dqt = µ x u ′ x + 12 σ x u ′′ x . As t goes to zero this expression converges to µ u ′ ( w ) + σ u ′′ ( w ). Since for everyrealization of g it converges to the exact same number, the expectation of thisexpression also converges to this number. Therefore,lim t → h t = µ u ′ ( w ) + 12 σ u ′′ ( w ) . (10)It follows from Equations (9) and (10) that F ( x ) ≡ lim t → F t ( x ) = u ′ ( w ) x − σ u ′′ ( w ) . Let x ∗ be the real number s.t. F ( x ∗ ) = 0, i.e., x ∗ = 12 u ′′ ( w ) u ′ ( w ) σ . It is easy to see that the two conditions of Lemma 2 are satisfied: for all t , first F t
33s continuous as it is the sum of continuous functions, and second, F t is a strictlyincreasing function since u is an increasing function. It follows from the lemma thatthere exists ¯ t and x t such that F t ( x t ) = 0 for each t < ¯ t , andlim t → x t = x ∗ , where, by definition, f RP (( u, w ) , g t ) = x t t . Note that since u ′′ ( w ) u ′ ( w ) is negative, x ∗ isnegative as well, and therefore x ∗ (and x ∗ · t for all t >
0) is strictly decreasing with ρ = − u ′′ ( w ) u ′ ( w ) and with Q lSD = σ .Next, we would like to show that Q lSD and ρ are consistent with f RP . We beginby showing that Q lSD ( g ) > Q lSD ( g ′ ) implies that ( f RP ) ( u,w ) g << ( f RP ) ( u,w ) g ′ for any( u, w ) ∈ DM . Fix a decision maker ( u, w ), and let x ∗ ( g ) ≡ x ∗ (( u, w ) , g ) (and usea similar notation for x t ( g )). Let g, g ′ ∈ Γ be two processes satisfying Q lSD ( g ) >Q lSD ( g ′ ). Then x ∗ ( g ) < x ∗ ( g ′ ), and from the fact that x t → x ∗ it follows that thereexists ¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , x t ( g ) · t < x t ( g ′ ) · t , which implies that f RP (( u, w ) , g t ) < f RP (( u, w ) , g ′ t ). In addition, lim t → f RP (( u, w ) , g t ) f RP (( u, w ) , g ′ t ) = lim t → x t ( g t ) tx t ( g ′ t ) t = x ∗ ( g ) x ∗ ( g ′ ) = (cid:16) Q lSD ( g ) (cid:17) (cid:16) Q lSD ( g ′ ) (cid:17) = 1 , which proves that ( f RP ) ( u,w ) g << ( f RP ) ( u,w ) g ′ . Similarly, we show that ρ ( u ′ , w ′ ) >ρ ( u ′′ , w ′′ ) implies that ( f RP ) ( u ′ ,w ′ ) g << ( f RP ) ( u ′′ ,w ′′ ) g for any g ∈ Γ. Fix a process g ∈ Γ, and let x ∗ ( u, w ) ≡ x ∗ (( u, w ) , g ) (and use a similar notation for x t ( u, w )).Let ( u ′ , w ′ ) , ( u ′′ , w ′′ ) ∈ DM be two agents satisfying ρ ( u ′ , w ′ ) > ρ ( u ′′ , w ′′ ). Then x ∗ ( u ′ , w ′ ) < x ∗ ( u ′′ , w ′′ ), and from the fact that x t → x ∗ it follows that there exists¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , x t ( u ′′ , w ′′ ) · t < x t ( u ′ , w ′ ) · t , implying that f RP (( u ′′ , w ′′ ) , g t ) < f RP (( u ′ , w ′ ) , g t ). In addition, lim t → f RP (( u ′′ , w ′′ ) , g t ) f RP (( u ′ , w ′ ) , g t ) = lim t → x t ( u ′′ , w ′′ ) tx t ( u ′ , w ′ ) t = x ∗ ( u ′′ , w ′′ ) x ∗ ( u ′ , w ′ ) = ρ ( u ′′ , w ′′ ) ρ ( u ′ , w ′ ) = 1 , which proves that ( f RP ) ( u ′ ,w ′ ) g << ( f RP ) ( u ′′ ,w ′′ ) g .For the other direction, given some agent ( u, w ), if for two processes g and g ′ thereis some ¯ t s.t. f RP (( u, w ) , g t ) < f RP (( u, w ) , g ′ t ) for every 0 < t < ¯ t , and theratio f RP (( u, w ) , g t ) /f RP (( u, w ) , g ′ t ) does not go to 1 when t goes to zero, then34 t < x ′ t for all t < ¯ t , implying that the limits also satisfy x ∗ < x ′∗ and, therefore, σ > σ ′ . Similarly, given some process g , if for two agents ( u ′ , w ′ ) and ( u ′′ , w ′′ ) thereis some ¯ t s.t. f RP (( u ′ , w ′ ) , g t ) < f RP (( u ′′ , w ′′ ) , g t ) for every 0 < t < ¯ t , and theratio f RP (( u, w ) , g t ) /f RP (( u ′ , w ′ ) , g t ) does not go to 1 when t goes to zero, then x ∗ < x ′∗ , implying that ( u ′ , w ′ ) is locally more averse to risk than ( u ′′ , w ′′ ).2. Q lV M and ρ are consistent with f CA . The capital allocation function is defined by f CA (( u, w ) , g t ) = arg max α ∈ R + E h u (cid:16) w + α · g t (cid:17)i , where f CA (( u, w ) , g t ) equals infinity if there is no internal solution. For every t > F t be the function defined as follows: F t ( α ) = E " u w + αg t ! − u ( w ) t . (11)By Ito’s lemma, F t ( α ) = E hR t αµ q u ′ q + α σ q u ′′ q dq i t + E " R t αu ′ q σ q dW t , where u ′ q ≡ du ( w q ) /d ( w q ), u ′′ q ≡ du ( w q ) /d ( w q ), and w q = w + αg q . For the samereason as in the case of f RP , the expression on the right-hand side E " R t αu ′ q σ q dW is zero and therefore it can be omitted.We define F ( α ) to be the limit of F t ( α ) as t goes to zero. For the same reason asin the case of f RP it equals to: F ( α ) ≡ lim t → F t ( α ) = αµ u ′ ( w ) + 12 α σ u ′′ ( w ) . (12)We denote by α ∗ the value of α that maximizes F ( α ): α ∗ = arg max α F ( α ) = − u ′ ( w ) u ′′ ( w ) µ σ . (13)The two conditions of Lemma 3 are satisfied: first, by definition, the limit of F t is35 . Second, we represent F t as the sum of two expressions F t ( α ) = α · E o hR t µ q u ′ q dq i t + α · E o h R t σ q u ′′ q dq i t . Since we assume that u ′′ is negative, F t is strictly concave with α and the secondcondition of the lemma is satisfied.By the lemma, there exists ¯ t > α t maximizes F t for all t < ¯ t , and lim t → α t = α ∗ . Note that the limit α ∗ is strictly decreasing with ρ = − u ′′ ( w ) /u ′ ( w ), and with Q lV M = σ /µ .Next we would like to show that Q lV M and ρ are consistent with f CA , where bydefinition f CA (( u, w ) , g t ) = ( f CA ) ( u,w ) g ( t ) = α t (( u, w ) , g t ) . For the first direction,we have to show that if for two processes g, g ′ ∈ Γ , Q lV M ( g ) > Q lV M ( g ′ ), then( f CA ) ( u,w ) g << ( f CA ) ( u,w ) g ′ . Indeed, Q lV M ( g ) > Q lV M ( g ′ ) implies that α ∗ ( g ) < α ∗ ( g ′ ),and from the convergence of α t it follows that there exists ¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , α t ( g ) < α t ( g ′ ). Since α ∗ ( g ) < α ∗ ( g ′ ), it follows that lim t → α t ( g ) /α t ( g ′ ) =1 and therefore that ( f CA ) ( u,w ) g << ( f CA ) ( u,w ) g ′ . Similarly, if for two agents ( u ′ , w ′ )and ( u ′′ , w ′′ ), ρ ( u ′ , w ′ ) > ρ ( u ′′ , w ′′ ), then α ∗ ( u ′ , w ′ ) < α ∗ ( u ′′ , w ′′ ), and from theconvergence of α t it follows that there exists some ¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , α t ( u ′ , w ′ ) < α t ( u ′′ , w ′′ ). Since α ∗ (the limit of α t ) is positive, it followsthat lim t → α t ( u ′ , w ′ ) /α ( u ′′ , w ′′ ) = 1 and therefore that ( f CA ) ( u ′ ,w ′ ) g << ( f CA ) ( u ′′ ,w ′′ ) g .For the other direction, let g, g ′ ∈ Γ be two processes for which, for any deci-sion maker ( u, w ), ( f CA ) ( u,w ) g << ( f CA ) ( u,w ) g ′ . Indeed, the limits of f CA (( u, w ) , g t )and f CA (( u, w ) , g ′ t ) when t goes to zero satisfy α ∗ ( g ′ ) > α ∗ ( g ) and, therefore, Q lV M ( g ) > Q lV M ( g ′ ). Similarly, let ( u, w ) and ( u ′ , w ′ ) be two decision makersfor which ( f CA ) ( u ′ ,w ′ ) g << ( f CA ) ( u ′′ ,w ′′ ) g . Indeed, the limits of f CA (( u, w ) , g t ) and The analysis implies that g t is a “gamble” for each t < ¯ t . To see that, note that we have shownthat for every process g , and for every strictly concave utility function, there exists ¯ t such that for every t < ¯ t , the solution of the maximization problem is internal. This implies that for every such t , E [ g t ] > α t = 0, contradicting our result here that α t > t . Similarly, the analysis implies that P ( g t < > t . Otherwise, for every α t and ǫ >
0, ( α t. + ǫ ) g t would first-order stochastically dominate α t g t and therefore any agent would be better off enlarging any given α t. , which implies that the solutionis not internal, contradicting our result that some finite α t > F t . These two properties of g t imply that g t is a gamble. CA (( u, w ) , g ′ t ) when t goes to zero satisfy α ∗ ( u ′ , w ′ ) < α ∗ ( u ′′ , w ′′ ) and, therefore, ρ ( u ′ , w ′ ) > ρ ( u ′′ , w ′′ ).3. Q lIS and ρ are consistent with f CE . For every t >
0, let F t be defined as follows: F t ( z ) = u ( w + z · t ) − E " u w + αg t ! t . It is easy to see that if α is the optimal allocation and F t ( z ) = 0 then z · t = f CA (( u, w ) , g t ). To calculate the limit of F t as t goes to zero, it is simpler to lookat F t as the difference between two functions k t and h t , defined by k t ( z ) = u ( w + zt ) − u ( w ) t , and h t = E " u w + αg t ! − u ( w ) t . (14)Clearly, F t ( z ) = k t ( z ) − h t for every value of z . The limit of k t ( z ) as t goes to zero is simply the derivative:lim t → k t ( z ) = u ′ ( w ) · z. Using Ito’s lemma, and taking the limit (as we did in Equations 11 and 12), we getlim t → h t = αµ u ′ ( w ) + 12 α σ u ′′ ( w ) . Recall that according to Equation (13), α ∗ = − u ′ ( w ) u ′′ ( w ) µ σ . Plugging α = α ∗ into h t , we getlim t → h t = − ( u ′ ( w )) µ u ′′ ( w ) σ + 12 ( u ′ ( w )) µ u ′′ ( w ) σ = −
12 ( u ′ ( w )) µ u ′′ ( w ) σ .
37e define F ( z ) to be the limit of F t ( z ), where t goes to zero: F ( z ) ≡ lim t → F t ( z ) = lim t → k t ( z ) − lim t → h t ( z ) = u ′ ( w ) z + 12 ( u ′ ) u ′′ (cid:16) µ σ (cid:17) . We define z ∗ to be the value that results in F ( z ∗ ) = 0: z ∗ = − u ′ ( w ) u ′′ ( w ) (cid:16) µ σ (cid:17) . For every t , F t ( z ) is continuous and strictly increasing satisfying the conditions ofLemma 2, therefore by the lemma there is ¯ t such that F t ( z t ) = 0 for every t ∈ (0 , ¯ t ),implying that z t · t is the certainty equivalent of the optimal investment in the gamblewith horizon t , and that lim t → z t = z ∗ . It is easy to see that z ∗ (and therefore z ∗ t for all t ) is strictly decreasing with ρ = − u ′′ ( w ) u ′ ( w ) and with Q lIS = σµ .Next, we would like to show that Q lIS and ρ are consistent with f CE . We beginby showing that Q lIS ( g ) > Q lIS ( g ′ ) implies that ( f CE ) ( u,w ) g << ( f CE ) ( u,w ) g ′ for any( u, w ) ∈ DM . Fix a decision maker ( u, w ), and let z ∗ ( g ) ≡ z ∗ (( u, w ) , g ) (and usea similar notation for z t ( g )). Let g, g ′ ∈ Γ be two processes satisfying Q lIS ( g ) >Q lIS ( g ′ ). Then z ∗ ( g ) < z ∗ ( g ′ ), and from the fact that z t → z ∗ it follows that thereexists ¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , z t ( g ) · t < z t ( g ′ ) · t , which implies that f CE (( u, w ) , g t ) < f CE (( u, w ) , g ′ t ). In addition, lim t → f CE (( u, w ) , g t ) f CE (( u, w ) , g ′ t ) = lim t → z t ( g t ) tz t ( g ′ t ) t = z ∗ ( g ) z ∗ ( g ′ ) = (cid:16) Q lIS ( g ) (cid:17) (cid:16) Q lIS ( g ′ ) (cid:17) = 1 , which proves that ( f CE ) ( u,w ) g << ( f CE ) ( u,w ) g ′ . Similarly, we show that ρ ( u ′ , w ′ ) >ρ ( u ′′ , w ′′ ) implies that ( f CE ) ( u ′ ,w ′ ) g << ( f CE ) ( u ′′ ,w ′′ ) g for any g ∈ Γ. Fix a process g ∈ Γ, and let z ∗ ( u, w ) ≡ z ∗ (( u, w ) , g ) (and use a similar notation for z t ( u, w )).Let ( u ′ , w ′ ) , ( u ′′ , w ′′ ) ∈ DM be two agents satisfying ρ ( u ′ , w ′ ) > ρ ( u ′′ , w ′′ ). Then z ∗ ( u ′ , w ′ ) < z ∗ ( u ′′ , w ′′ ), and from the fact that z t → z ∗ it follows that there exists¯ t >
0, such that for each t ∈ (cid:16) , ¯ t (cid:17) , z t ( u ′′ , w ′′ ) · t < z t ( u ′ , w ′ ) · t , implying that38 CE (( u ′′ , w ′′ ) , g t ) < f CE (( u ′ , w ′ ) , g t ). In addition, lim t → f CE (( u ′′ , w ′′ ) , g t ) f CE (( u ′ , w ′ ) , g t ) = lim t → z t ( u ′′ , w ′′ ) tz t ( u ′ , w ′ ) t = z ∗ ( u ′′ , w ′′ ) z ∗ ( u ′ , w ′ ) = ρ ( u ′′ , w ′′ ) ρ ( u ′ , w ′ ) = 1 , which proves that ( f CE ) ( u ′ ,w ′ ) g << ( f CE ) ( u ′′ ,w ′′ ) g .For the other direction, given some agent ( u, w ), if for two processes g and g ′ thereis some ¯ t s.t. f CE (( u, w ) , g t ) < f CE (( u, w ) , g ′ t ) for every 0 < t < ¯ t , and theratio f CE (( u, w ) , g t ) /f CE (( u, w ) , g ′ t ) does not go to 1 when t goes to zero, then x t < x ′ t for all t < ¯ t , implying that the limits also satisfy z ∗ < z ′∗ and, therefore, σ > σ ′ . Similarly, given some process g , if for two agents ( u ′ , w ′ ) and ( u ′′ , w ′′ ) thereis some ¯ t such that f CE (( u ′ , w ′ ) , g t ) < f CE (( u ′′ , w ′′ ) , g t ) for every t ∈ (cid:16) , ¯ t (cid:17) , andthe ratio f CE (( u ′′ , w ′′ ) , g t ) /f CE (( u ′ , w ′ ) , g t ) does not go to 1 when t goes to zero,then z ∗ < z ′∗ , implying that ( u ′ , w ′ ) is locally more averse to risk than ( u ′′ , w ′′′′