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Short-Term Investments and Indices of Risk

We study various decision problems regarding short-term investments in risky assets whose returns evolve continuously in time. We show that in each problem, all risk-averse decision makers have the same (problem-dependent) ranking over short-term risky assets. Moreover, in each problem, the ranking is represented by the same risk index as in the case of CARA utility agents and normally distributed risky assets.


Introduction
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 capital allocation. To rank the desirability of gambles with respect to the relevant decision problem, it is often helpful to use an objective riskiness index that is independent of any specific subjective utility. For example, an objective riskiness index is needed when pension funds are required not to exceed a stated level of riskiness (see, e.g., the discussion in Aumann & Serrano, 2008, p. 812). with a decision function over some set of agents and gambles, if for each gamble and each pair of agents in the set, the agent with the higher index of risk-aversion invests less in the gamble than the other agent. Observe that different decision functions may correspond to different concepts of risks, and, may induce different indices of risk and of risk aversion.

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 all agents, unless one restricts the set of gambles. Our main result restricts the set of gambles to assets whose returns evolve continuously in time, where the local uncertainty is induced by a Wiener process. Specifically, we focus on Ito processes which are continuoustime Markov processes. The class of Ito processes, is commonly used in economic and financial applications and includes, in particular, the geometric Brownian motion and mean-reverting processes (e.g., Merton, 1992). 2 Our main result shows that in each of the four decision problems discussed above, all agents rank all gambles in the same (problem-dependent) way when they have to decide on short-term investments in gambles whose returns evolve continuously in time. Moreover, the risk indices that are consistent with these decision functions are the same as in the classic model of agents with CARA (exponential) utilities and normally distributed gambles. Specifically, we show that: (1) the variance-to-mean index Q V M (g) = σ 2 [g] E[g] is consistent with both the capital allocation function and the acceptance/rejection function, (2) the inverse Sharpe index Q IS (g) = σ [g] E [g] is consistent with the optimal certainty equivalent function, and (3) the standard deviation index Q SD (g) = σ [g] is consistent with the risk premium function. Finally, we adapt the classic results of Pratt (1964) and Arrow (1970) to the present setup, and show that the local Arrow-Pratt coefficient of absolute risk aversion ρ (u, w) = −u ′′ (w) u ′ (w) is consistent with all four decision functions. Aumann & Serrano (2008) and Foster & Hart (2009) presented two "objective" indices of riskiness of gambles, which are independent of the subjective utility of the agent. These indices are either based on reasonable axioms that 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 "operative" criterion such as an agent never going bankrupt when relying on an index of risk in deciding whether to accept a gamble (Foster & Hart, 2009; and see also Meilijson, 2009, for a discussion of operative implication of Aumann & Serrano's index of risk). 3 We argue that risk is a multidimensional attribute that crucially depends on the investment problem. Different aspects of risk are relevant when an agent has to decide whether to accept a gamble, compared with a situation in which an agent has to choose how much to invest in a gamble, or has to evaluate the certainty equivalent of the optimal investment. Many existing papers focus on a single decision function. By contrast, we suggest a framework for studying various decision problems, and associate each such problem with its relevant index of risk. We believe that this general framework may be helpful in future research on risk indices. In general, different agents make different investment decisions, based on the subjective utility of each agent. Thus, a single risk index cannot be consistent with the choices of all agents, which, arguably, limits the index's objectiveness (even when the index satisfies appealing axioms or some operative criterion for avoiding bankruptcy). However, our main result shows that in various important decision problems, all agents rank all gambles in the same way when deciding on short-term investments in gambles whose returns evolve continuously in time. This finding enables us to construct objective risk indices that are consistent with the short-term investment decisions of risk-averse agents.

Related Literature and Contribution
There are pairs of gambles for which all risk-averse agents agree on which one of the gambles is more desirable. This happens if one gamble second-order stochastically dominates the other gamble. However, the well-known order of stochastic dominance (Hadar & Russell, 1969;Hanoch & Levy, 1969;Rothschild & Stiglitz, 1970) is only a partial order and "most" pairs of gambles are incomparable. Interestingly, even if one gamble second-order stochastically dominates another gamble, it is not sufficient for a uniform ranking among all risk-averse agents in every decision problem (see, e.g., the analysis of capital 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 preferences are consistent with expected utility maximization only under severe restrictions, such as CARA utilities and normally distributed gambles (see, e.g., Borch, 1969;Feldstein, 1969;Hakansson, 1971). By contrast, the seminal results of Robert Merton (as summarized and discussed in Merton, 1975Merton, , 1992 show that in a continuous-time model with log-normally distributed asset prices mean-variance preferences are consistent with the optimal portfolio allocation of all risk-averse agents. Merton's results present an important theoretical foundation for the classic model. The present paper extends this idea by showing that the equivalence between the decisions of agents with CARA utilities with respect to normally distributed gambles and the decisions of risk-averse agents with respect to short-term investments holds more broadly: (1) it holds with respect to various decision functions beyond optimal portfolio allocation, and (2) it holds with respect to a broad class of continuous-time processes beyond log-normally distributed asset prices. On the other hand, our analysis is less general than Merton's in that we analyze situations in which the agent acts only at the beginning, at time zero, and cares about his wealth at a single future nearby point t > 0, rather than allowing the agent to act continuously in time.
Our paper is also related to the literature on local risks. This literature focuses on discrete-time gambles, rather than continuous-time returns (which are the focus of the present paper). Pratt (1964) shows that if the distribution of the returns is sufficiently concentrated, i.e., the third absolute central moment is sufficiently small relative to variance, then for any agent, the magnitude of the risk premium depends on the local level of the agent's risk aversion. Samuelson (1970) shows that classic mean-variance analysis (Markowitz, 1952), applies approximately to all utility functions in situations that involve what he calls "compact" distributions. More recently, Schreiber (2015) shows that if one gamble is riskier than another gamble according to the Aumann & Serrano's index of risk, then every decision maker who is willing to accept a small proportion of the riskier gamble is also willing to accept the same proportion of the less risky gamble.
In this context, two papers are close to the present paper: Shorrer (2014)  5). Unlike the other papers mentioned above, Schreiber (2016) deals with returns in the continuous-time setup. Specifically, he analyzes acceptance and rejection of short-term investments. 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, showing in 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 benchmark setup of CARA utilities and normally distributed gambles. In Section 4 we adapt the model to study risky assets whose returns evolve continuously in time, and present our main result. We conclude with a discussion in Section 5. Appendix A extends our model to multiplicative gambles. The formal proofs are presented in Appendix B.

Model
We consider an agent who has to make an investment decision related to a risky asset. We begin 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 a twice continuously differentiable von Neumann-Morgenstern utility function over wealth satisfying u ′ > 0 (i.e., utility is increasing in wealth) and u ′′ < 0 (i.e., risk aversion), and 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 some negative values (i.e., 0 < E [g], and P [g < 0] > 0). We think of a gamble as the additive return on a risky investment; for example, if the initial investment is x dollars and the random payoff from the investment is y dollars, then the additive return g ≡ y − x is a gamble. 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 each gamble a nonnegative number, where a higher value is interpreted as the agent finding the gamble to be more attractive (i.e., less risky) for the relevant investment decision.

Decision Functions
We study four decision functions in the paper: 1. Acceptance/rejection: We consider a situation in which an agent faces a binary choice between accepting and rejecting the gamble. Specifically, the acceptance function f AR : DM × G → {0, 1} is given by That is, f AR ((u, w) , g) is equal to one if accepting the gamble yields a weakly higher expected payoff than rejecting it, and it is equal to zero otherwise. The acceptance function 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 analysis of Schreiber (2016), by showing the similarity between this function in the meanvariance 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 continuous choice of how much to invest in the gamble. Specifically, the capital (or asset) allocation function f CA : DM × G → R + ∪ {∞} is given by if (1) does not admit of a solution (i.e., E u w + αg is increasing for all α-s), then 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 no investment in the gamble. An investment level in the interval (0, 1) is interpreted as a partial investment in the gamble. An investment level of one is interpreted as a total investment in the gamble (without leverage). Finally, an investment level strictly greater than one is interpreted as a more than total investment in the gamble (achieved, for example, through high leverage). The capital allocation function is prominent in classic analyses of riskiness of assets (e.g., Markowitz, 1952;Sharpe, 1964), and, more recently, it has been used to derive an incomplete ranking over the riskiness of gambles (Landsberger & Meilijson, 1993).
3. The optimal certainty equivalent: Third, we study a situation in which an agent has to assess how much an opportunity to invest in the gamble g is worth to him (where we allow the agent to choose his optimal investment level). Specifically, the optimal certainty equivalent function f CE : DM × G → R + ∪ {∞} is defined implicitly as the unique solution to the equation if (2) does not admit of a solution (which happens when E u w + α·g is increasing for all α-s), then we set f CE = ∞. That is, f CE ((u, w) , g) is interpreted as the certain gain for which the decision maker is indifferent between obtaining this gain for sure and having an option to invest in the gamble g, when the agent is allowed to optimally choose his investment level in g. Observe that one can express the RHS in (2) in terms of f CA and obtain the following equivalent definition of f CE as the unique solution to the equation . The function f CE has been studied axiomatically in Hellman & Schreiber (2018).

Risk premium:
Lastly, we study a situation in which the agent has to decide between investing in the gamble and obtaining a certain amount that is less than the gamble's expected payoff. Specifically, the risk premium function f RP : DM × G → R − ∪ {−∞} is defined implicitly as the unique solution to the equation if such a solution does not exist then we set f RP ((u, w) , g) = −∞ . That is, is interpreted as the negative amount that has to be added to the expected value of the gamble, to make the agent indifferent between investing in the 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 a somewhat 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 is added 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.

Risk Indices
We define a risk index as a function Q : G → R ++ that assigns to each gamble a positive number, 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: 2. The inverse Sharpe index Q IS (g) is the ratio of the standard deviation to the mean: We say that a risk index is consistent with a decision function over a domain of agents and gambles, if: each agent in the domain finds gamble g less attractive than g ′ with respect to the relevant decision function iff the risk index of g is higher than in g ′ . Formally: 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 and agents 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 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 to monotone transformations; if risk index Q is consistent with function f over the domain DM × G, then risk index Q ′ is consistent with f over this domain iff there exists a strictly increasing mapping θ : Q (G) → Q ′ (G), s.t. Q ′ (g) = θ (Q (g)) for each gamble g ∈ G.

Risk-Aversion Indices
We define a risk-aversion index as a function φ : DM → R ++ that assigns to each agent a non-negative number, which is interpreted as the agent's risk aversion. We mainly study one risk index in the paper, the Arrow-Pratt coefficient of absolute risk aversion, denoted by ρ : DM → R ++ , which is defined as follows: We say that a risk-aversion index is consistent with a decision function over a domain of agents and gambles, if, for each gamble and each pair of agents in the domain, the agent with the higher index chooses a lower value for his investment decision in the gamble.
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 gambles and agents it may not apply at all. Specifically, a domain DM × G ⊆ DM × G admits a consistent risk-aversion index iff all gambles induce the same ranking over agents, i.e., if 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.

Result
We begin by presenting a claim, which summarizes known results for normal distributions and CARA utilities. Specifically, we show that in each of the decision functions described above, all agents with CARA utilities have the same ranking over all normally distributed gambles, and that each of these rankings is consistent with one of the risk indices presented above. Moreover, all normally distributed gambles induce the same ranking over all agents with CARA utilities, which is consistent with the Arrow-Pratt coefficient.
Formally, let DM CARA ⊆ DM be the set of decision makers with CARA utilities: 5 and let G N ⊆ G be the set of normally distributed gambles with positive expectations: Claim 1. Let u be a CARA utility with parameter ρ (i.e., u (x) = 1 − e −ρ·x ). Then: ρ · µ σ 2 , which imply that the variance-to-mean index Q V M is consistent with both the acceptance/rejection function f AR and the capital allocation function f CA .
2·ρ · µ σ 2 , which implies that the inverse Sharpe index Q IS is consistent with the optimal certainty equivalent function f CE .
4. The Arrow-Pratt coefficient of absolute risk aversion ρ is consistent with all four For completeness, we present the proof of Claim 1 in Appendix B.1.

Discussion
Each agent with CARA utility is described by two parameters (initial wealth w, and Arrow-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 be expressed as a function g (w, ρ, µ, σ) of these four parameters.
Other Consistent Indices of Risk Aversion CARA utilities have the well-known property that the initial wealth does not affect expected utility calculations with respect to investments in gambles. Thus, whenever the investment decision is made by choosing the option that maximizes the agent's expected utility (such as in all four of the decision functions analyzed above), then the decision function is independent of w, which implies that the parameter ρ is a consistent risk-aversion index. By contrast, for investment decisions that are not determined by maximizing the agent's expected utility, there might be different risk-aversion consistent indices. For instance, Foster & Hart (2009) analyze a situation 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 functions analyzed above have the additional separability property that each function f can be represented as a product of two functions: one that depends only on the parameters describing the agent (w and ρ), and one that depends only on the parameters of the gam- . This separability implies that all agents with CARA utilities have the same ranking over normal gambles (as this ranking depends only on ν (µ, σ), which does not depend on the agent's parameters), which, in turn, implies that there exists a consistent risk index. Similarly, the separability implies that all normal gambles induce the same ranking over agents (as this ranking depends only 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 of such a non-separable decision function is the standard certainty equivalent of a continuous gamble f SCE (as opposed to the certainty equivalent of the optimal allocation of the gamble f CE ((u, w) , g) discussed above), which is implicitly defined by which is a non-separable function of ρ,µ, σ. The non-separability implies that agents with different CARA utilities have different rankings for normal gambles and, therefore, no risk index can be consistent with these decisions.

Short-Term Investments in Continuous Gambles
In what follows we adapt our model to short-term investment decisions regarding assets whose value follows a continuous random process. Our description of the continuous-time setup follows Shreve (2004).

Continuous-Time Random Processes
Let (Ω, F , P) be a probability space on which a Brownian motion W t is defined, 6 with an associated filtration F (t). Let the process g be described by the following stochastic differential equation: 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 both continuous in t. We assume that µ 0 > 0 and σ 0 > 0 and that σ t ≥ 0 for all t > 0. We also assume that E t 0 σ 2 s ds and E t 0 |µ s |ds are finite for every t > 0. This implies that the integrals E t 0 σ s dW s and E t 0 µds are well defined, and the Ito integral t 0 σ s dW s is a martingale; 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 0 is known. Then, is the additive return of the asset at time t. In particular, observe that g 0 = P t=0 −P 0 ≡ 0. For simplicity we assume that g is bounded from below (i.e., there exists M g ∈ R, such that g t ≥ M g for each t > 0). Under these assumptions, for a sufficiently small time t, g t is a gamble (see Footnote 8 in Appendix B.2); i.e., it has positive expectation and takes negative values with positive probability. Thus, we can apply our definitions of decision functions and indices to g t for each specific value of t > 0. In our setup, a decision maker has to make a decision at time zero, where he cares only about his wealth at some time t. From the perspective of this decision maker, P 0 is a pure number and g t is a gamble, just as before.
Let Γ denote the set of all continuous-time random processes satisfying these assumptions. Observe that the set Γ is quite general. In particular, it includes returns on assets whose prices are described by geometric Brownian processes (Black & Scholes, 1973;Merton, 1992, Chapters 4 and 5), 7 and variants of arithmetic Brownian processes and of mean-reverting processes that are bounded from below (also known as Ornstein-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.

Adapted Definitions
We define a local-risk index at time zero as a function Q l : Γ → R ++ that assigns to each process g ∈ Γ a positive number, which is interpreted as the process's short-term riskiness at t = 0. Given g ∈ Γ with initial parameters µ 0 and σ 0 , we define three specific local-risk indices (analogous to the corresponding definitions in Section 2.2): 1. The variance-to-mean local index Q l V M (g) is equal to: 2. The inverse Sharpe local index Q l IS (g) is equal to: 3. The standard deviation local index Q l SD (g) is equal to: Q l SD (g) = σ 0 .
Given a continuous-time process g ∈ Γ, decision function f and agent (u, w) ∈ DM, let f (u,w) w) , g t ) be the value of the decision function of agent (u, w) with respect to 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 useful for defining the concept of consistency of indices in the continuous-time framework.
Definition 3. Let f, h : R + ⇒ R, and assume that lim t→0 We say that f is uniformly higher than h (around zero) and denote it by f >> h if (1) there existst > 0, such that f (t) > h (t) for each t ∈ 0,t and (2) That is, f >> h if f (t) is strictly higher than h (t) for any sufficiently small t, and the relative difference between the two functions does not become negligible (as measured by the ratio f (t) h(t) not converging to one) in the limit of t → 0.
We say that a local-risk index is consistent with a decision function over continuous returns, if the local-risk index of g ′ is lower than the index of g iff all risk-averse agents find 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 over the set of continuous returns Γ if, for each pair of continuous-time processes g, g ′ ∈ Γ and each agent (u, w) ∈ DM, we have that 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 continuoustime 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 Note that a consistent risk-aversion index is unique up to monotone transformations.

Main Result
Our main result shows that in each of the decision functions described above, all agents have the same ranking over all short-term continuous returns. Moreover, the rankings are consistent with the three risk indices presented above, and they are the instantaneous versions of the corresponding indices in the case of normally distributed gambles and CARA utilities analyzed in Claim 1. Finally, we adapt to the present setup the classic result 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 the case 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 l SD is consistent with the risk premium function f RP . t = µ 0 , which implies for sufficiently small t-s that σ 2 [g t ] ≈ t · σ 2 0 and µ [g t ] ≈ t · µ 0 . We begin with a standard approximation of the risk premium function f RP (see, e.g., Eeckhoudt et al., 2005, Chapter 1). Recall that the risk premium was defined implicitly as

The variance-to-mean index
A second-order Taylor expansion of the left-hand side around w + E[g t ] yields A first-order Taylor expansion of the right-hand side around w + E[g] yields Combining these equations and isolating f RP yields ) is a consistent risk index (resp., riskaversion 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 certainty equivalent of investment α in g t . Substituting the value of f RP calculated above we get In order to maximize C (α), we compare the derivative to zero, to get which implies that the variance-to-mean index Q l V M (resp., the Arrow-Pratt coefficient ρ) is a consistent risk index (resp., risk-aversion index) for decision function f CA . Finally which implies that the inverse Sharp index Q l IS (the Arrow-Pratt coefficient ρ) is a consistent risk (risk aversion) index for decision function f CE .
Remark 1 (on why the indices in the continuous-time setup coincide with the indices in the CARA-normal setup). The expressions that approximate the various functions in the continuous-time setup consist of two elements: the coefficient of risk aversion with respect to the initial wealth level, and a function of the first and second moment of the "small" gamble. In the CARA-normal setup of Section 3, the risk-aversion coefficient is constant 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 normally distributed 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, Therefore, it seems plausible that the expressions that represent the decision functions in the CARA-normal setup, depend only on the first two moments, and, thus, they coincide with the approximated decision functions that are relevant for short-term investments in assets with continuous returns.

Weak Consistency for Acceptance/Rejection
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 of consistent risk indices, as in many cases in which one gamble is riskier than another, an agent may choose to reject both gambles (and his value of f AR of both gambles would be zero). Nevertheless, one can define the milder notion of weak consistency, and show that a corollary to Schreiber's (2016) result is that the risk index Q l V M is weakly consistent with the acceptance/rejection function f AR .
A local-risk index is weakly consistent with a decision function over the set of continuous returns, if each agent chooses a weakly lower value of his investment decision in gamble g t relative to g ′ t for a sufficiently small t if the local risk of g is strictly higher than the 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 timet, such that, for each time t <t, we have that Note that weak consistency does not restrict the agents' choices when both gambles have the same local-risk index. As a result, a weakly consistent risk index is unique only up to weakly monotone transformations; i.e., if Q is a weakly consistent local-risk index with decision function f over the set of continuous returns Γ, then risk index Q ′ is consistent 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 constant index 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 over continuous-time returns, if for each short-term return, an agent chooses a (weakly) higher value for his investment decision in the asset relative to another agent's decision if the former agent's risk aversion is smaller. Formally: Definition 7. Risk-aversion index φ : DM −→ R + is weakly consistent with decision function f over the domain of short-term continuous gambles if for each continuous-time process g ∈ Γ and each pair of agents (u, w) , (u ′ , w ′ ) ∈ DM, there exists a timet, such that, for each time t <t, we have that The following corollary, which is implied by Schreiber (2016, Theorems 2.2 & 3.3), shows that the standard deviation index Q l V M and the Arrow-Pratt coefficient of absolute risk aversion ρ are weakly consistent with the acceptance/rejection function f AR .

Continuous-Time Processes with Jumps
The set of continuous-time gambles Γ analyzed in this paper does not allow for jumps. In what follows we show that the absence of jumps is necessary for our main result. Specifically, we demonstrate that risk-averse agents rank continuous-time processes with jumps differently, even for short-term investments, which rules out the existence of consistent risk indices. Consider, for example, the acceptance/rejection function f AR (similar conclusions can be drawn for the other decision functions). Hart (2011) observes that there are many pairs of (discrete-time) gambles that are ranked differently by different risk-averse agents (see Hart 2011, Footnote 23). Let h,h be such a pair of gambles.
Consider the following compound Poisson processes g,g, where each has an initial value of zero. The value of each process changes only when there is a jump. The jumps arrive randomly with a rate λ. In process g (resp.,g) the size of each jump is distributed according to h (resp.,h). Observe that for sufficiently short times the probability of having two jumps is negligible, and the decision whether to accept or to reject a gamble depends only on what may happen after a single jump. This implies that agents who rank the gambles h,h differently, would also rank g t ,g t differently, for any sufficiently short time t. This rules out the existence of a consistent risk index in this setup.

Conclusion
Our main result is that in four central decision problems all risk-averse agents have the same (problem-dependent) ranking over short-term investments in risky assets whose returns evolve continuously, and these rankings are represented by simple well-known indices of risk. The indices obtained are the same as in the classic model of CARA utilities and normally distributed gambles. Each problem relates to a different dimension of risk, and, thus, its ranking is represented by a different risk index. Finally, adapting a classic 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 a result of the known property of continuous stochastic processes for which higher moments go quickly to zero as the time parameter goes to zero. Hence, multiple indices of risk that do use higher moments might coincide with our indices when they are applied to continuous-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 varianceto-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 small discrete gambles. Indeed, under the assumption that returns evolve continuously in time, the only relevant parameters for measuring risk are the first two moments. Our results can 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 functions analyzed in the paper) should depend on the first two moments in the same way as presented in our main result. We leave for future research the interesting question of how to choose among the various risk indices that satisfy this necessary condition. One possible direction for analyzing this question is the axiomatic approach applied in Shorrer (2014)

A Multiplicative Gambles
In the main text we followed the recent literature of riskiness (initiated by Aumann &Serrano, 2008 andHart, 2009) and focused on decision problems with regard to additive gambles in units of dollars. However, in most financial applications, it is common to describe the returns of an asset in relative terms, namely, percentages (see, e.g., Markowitz, 1959 andMerton, 1992), as this is the way in which returns are described in practice in exchange markets. Hence, in this section we show that our results hold also with regard to multiplicative returns.
In some sense, the difference between multiplicative and additive returns is only a matter of presentation: if one invests x dollars in a multiplicative gamble r, one's payoff will be x(1 + r) dollars, and this is just the same payoff as if one invests in an additive return of x · r dollars. Nevertheless, we think that presenting the results for multiplicative returns is important for two reasons: first, as argued in Schreiber (2014), each investment might have two different aspects of riskiness, absolute and relative; given two assets, one of them might be riskier in relative terms but less risky in absolute terms. Therefore it is worthwhile to study the difference between multiplicative and additive returns in our setup. As it turns out, this potential difference vanishes when focusing on short-term investments and we derive in the multiplicative setup results analogous to those that we have 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 is calculated in terms of percentages, it is natural to combine it in decision problems with relative return, as we do here.

A.1 Adaptation to the Model
Let r f > 0 be the risk-free interest rate available for all investors. A multiplicative risky asset (multiplicative gamble) r is a real-valued random variable with an expectation that is greater than r f , and some negative values greater than −1, i.e., E [r] > r f , P [r < 0] > 0, and r ≥ −1. We interpret r as the per-dollar return of the asset. Let R denote the set of all multiplicative risky assets.
We adapt the definitions of our decision functions to the case of multiplicative gambles.
1. The acceptance function f m AR : DM × R → {0, 1} is given by where we consider a situation in which an agent faces a binary choice between investing his entire wealth in a multiplicative gamble r and investing it in the riskless asset with return r f .

The capital allocation function
if Equation (6) does not admit of a solution (i.e., E u w · (1 + r f ) + α · w · (r − r f ) is increasing for all α-s), then we set f m CA ((u, w) , r) = ∞. This function deals with a situation in which an agent decides on the optimal share α ≥ 0 of his wealth w to invest in the multiplicative gamble r (where α > 1 can be induced by leverage), where his remaining wealth is invested in the riskless asset.
3. The optimal certainty equivalent function f m CE : DM × R → R + ∪ {∞} is defined implicitly as the unique solution to the equation if Equation (7) does not admit of a solution (i.e., E u w · (1 + r is increasing for all α-s), then we set f m CE ((u, w) , r) = ∞. This function describes the rate of a constant return that is equivalent to investing optimally in a multiplicative gamble r, where the remaining wealth is invested in the riskless asset.
4. The risk-premium function f m RP : DM × R → R − is defined implicitly as the unique solution to the equation where f m RP represents the constant (negative) return that makes the agent indifferent between investing all his wealth in the multiplicative gamble r and investing in an asset with a constant return that is equal to the expectation of r plus f m RP .
Let R N ⊆ R be the set of normally distributed multiplicative gambles (defined analogously to the definition of G N ). The Arrow-Pratt coefficient of relative risk aversion, denoted by ̺ : DM → R ++ , is defined as follows: We adapt the three indices of risk in the main text to the multiplicative setup and the existence of a risk-free interest rate. Specifically: Recall 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 random process g represents the additive return of a financial investment. Now the continuoustime random process r represents the excess multiplicative return: r t = (P t − P 0 )/P 0 . We assume that the compound risk-free interest rate is r f and hence the riskless return over period t is r f (t) = e µ f ·t − 1. The adapted definitions of the risk indexes in the multiplicative setup for the local risk indices are as follows: ((u, w(1 + r f (t))), w(r t − r f (t)))/w, and 4. f m RP ((u, w), r t ) ≡ f RP ((u, w(1 + r f (t))), w(r t − r f (t)))/w.

B.1 Proof of Claim 1
The following well-known fact, which describes the expectation of a log-normal distribution, will be useful in our proofs (the standard proof, which relies on the Laplace transform of 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+0.5σ 2 .
Next, we prove Claim 1. Let g be a normally distributed random variable with expectation 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 Thus, f RP ((u, w) , g) = 0.5ρσ 2 , which implies that Q SD = σ is a consistent risk index (and that ρ is a consistent risk-aversion index with respect to f RP . 2. Q V M and ρ are consistent with f AR . The agent accepts the gamble iff E 1 − e −ρ(w+g) > E 1 − e −ρw which, by Fact 1 is equivalent to e −ρ(w+µ)+0.5ρ 2 σ 2 < e −ρw ⇔ 0.5ρ < µ σ 2 .
Thus, f AR ((u, w) , g) = 1 {0.5ρ< µ σ 2 } , which implies that Q V M = σ 2 µ is a consistent risk index (and that ρ is a consistent risk-aversion index) with respect to f AR .
3. Q V M and ρ are consistent with f CA . The capital allocation function is given by It follows from Fact 1 that the r.h.s. of the above equation is equivalent to The first-order condition is Thus, f CA ((u, w) , g) = 1 ρ µ σ 2 , which implies that Q V M = σ 2 µ is a consistent risk index (and that ρ is a consistent risk-aversion index) with respect to f CA . 4. Q IS and ρ are consistent with f CE . The optimal certainty equivalent function is given by where the last equality uses Fact 1. This implies that Thus, f CE ((u, w) , g) = 1 2ρ µ 2 σ 2 , which implies that Q IS = σ µ is a consistent risk index (resp., 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 simple version 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 .
1. Q l SD and ρ are consistent with f RP . For every t > 0, let F t be defined as follows: By definition, if for some value of x, F t (x) = 0, then x · t = f RP ((u, w) , g t ).
To calculate the limit of F t as t goes to zero, it is simpler to look at F t as the difference between two functions k t and h t , defined by Clearly, for every value of x. The limit of k t (x) as t goes to zero is simply the derivative with respect to t at w: lim By applying Ito's lemma where u ′ q ≡ du(w q )/d(w q ), u ′′ q ≡ du 2 (w q )/d 2 (w q ), and w q = w+g q . Since we assumed that 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-integrability condition and, therefore, that E t 0 σ 2 q dq is finite. These two assumptions imply that E t 0 (u ′ q σ q ) 2 dq is finite and, therefore, that t 0 u ′ q σ t dW is a martingale; see Shreve (2004, Theorem 4.3.1. on page 134). Hence, h t can be rewritten as follows: Since µ q , σ q , u ′ q and u ′′ q are all continuous, according to the mean-value theorem for integration, for each realization of g there exists some x ∈ (0, t) for which As t goes to zero this expression converges to µ 0 u ′ (w) + 1 2 σ 2 0 u ′′ (w). Since for every realization of g it converges to the exact same number, the expectation of this expression also converges to this number. Therefore, It follows from Equations (9) and (10) that Let x * be the real number s.t. F (x * ) = 0, i.e., It is easy to see that the two conditions of Lemma 2 are satisfied: for all t, first F t x t < x ′ t for all t <t, implying that the limits also satisfy x * < x ′ * and, therefore, σ 0 > σ ′ 0 . Similarly, given some process g, if for two agents (u ′ , w ′ ) and (u ′′ , w ′′ ) there is somet s.t. f RP ((u ′ , w ′ ) , g t ) < f RP ((u ′′ , w ′′ ) , g t ) for every 0 < t <t, and the ratio 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 l V M and ρ are consistent with f CA . The capital allocation function is defined by where f CA ((u, w) , g t ) equals infinity if there is no internal solution. For every t > 0, let F t be the function defined as follows: By Ito's lemma, where u ′ q ≡ du(w q )/d(w q ), u ′′ q ≡ du 2 (w q )/d 2 (w q ), and w q = w + αg q . For the same reason as in the case of f RP , the expression on the right-hand side E t 0 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 as in the case of f RP it equals to: We denote by α * the value of α that maximizes F (α): The two conditions of Lemma 3 are satisfied: first, by definition, the limit of F t is F . Second, we represent F t as the sum of two expressions Since we assume that u ′′ is negative, F t is strictly concave with α and the second condition of the lemma is satisfied.
By the lemma, there existst > 0 such that α t maximizes F t for all t <t, and 8 lim t→0 α t = α * .
Note that the limit α * is strictly decreasing with ρ = −u ′′ (w)/u ′ (w), and with Q l V M = σ 2 0 /µ 0 . Next we would like to show that Q l V M and ρ are consistent with f CA , where by definition 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, implies that α * (g) < α * (g ′ ), and from the convergence of α t it follows that there existst > 0, such that for each t ∈ 0,t , α t (g) < α t (g ′ ). Since α * (g) < α * (g ′ ), it follows that lim t→0 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 the convergence of α t it follows that there exists somet > 0, such that for each t ∈ 0,t , α t (u ′ , w ′ ) < α t (u ′′ , w ′′ ). Since α * (the limit of α t ) is positive, it follows that lim t→0 α t (u ′ , w ′ )/α(u ′′ , w ′′ ) = 1 and therefore that (f CA ) (u ′ ,w ′ ) For the other direction, let g, g ′ ∈ Γ be two processes for which, for any decision 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, Similarly, let (u, w) and (u ′ , w ′ ) be two decision makers for which (f CA ) (u ′ ,w ′ ) g << (f CA ) (u ′′ ,w ′′ ) g . Indeed, the limits of f CA ((u, w) , g t ) and 8 The analysis implies that g t is a "gamble" for each t <t. To see that, note that we have shown that for every process g, and for every strictly concave utility function, there existst such that for every t <t, the solution of the maximization problem is internal. This implies that for every such t, E[g t ] > 0. Otherwise, a risk-averse agent would be better off by choosing α t = 0, contradicting our result here that α t > 0 for a sufficiently short time t. Similarly, the analysis implies that P (g t < 0) > 0 for a sufficiently short time 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 solution is not internal, contradicting our result that some finite α t > 0 maximizes F t . These two properties of g t imply that g t is a gamble.
3. Q l IS and ρ are consistent with f CE . For every t > 0, let F t be defined as follows: 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 look at F t as the difference between two functions k t and h t , defined by 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→0 k t (z) = u ′ (w) · z.