In this section, we show that PSMs can be efficiently optimized over a convex set of random variables, a decision problem that arises in many contexts, including portfolio optimization. This is an advantage for PSMs relative to models of choice under risk that gain descriptive richness by means of, say, a weighting function applied on the cumulative distribution function, as the rank-dependent utility model (RDU) of Quiggin [33] and the cumulative prospect theory (CPT) of Tversky and Kahneman [38]. Indeed, in these models, probability weighting depends on the ranking of states of the world, and the ranking changes as different convex combinations of the random variable are chosen. Consequently, in the context of optimization problems, the weighting of the cumulative distribution function can be difficult to handle and in many applications of RDU or CPT it is ignored, losing the descriptive richness that was the very motivation for the model in the first place.
In contrast, these issues do not arise with PSMs, as we show in this section. We consider the problem of portfolio choice with our model. The use of PSM in a portfolio selection context may be natural in a setting when a manager has high incentives to outperform a pre-specified benchmark, particularly a very aggressive one.
Specifically, given a PSM ρ, we consider the problem
z∗= sup{ρ(X) : X ∈ X }, (8)
where X is the convex hull {Pni=1wiXi :
Pn
i=1wi = 1, wi ≥ 0, ∀i = 1, . . . , n} of n random variables
X1, . . . , Xn, which give the target premium over a given target τ . From a computational perspective,
finding a feasible solution in a convex set is relatively easy compared to finding a feasible solution in a non-convex one. Observe that for k > 0, the following set
S(k) = {X : ρ(X) ≥ k, X ∈ X },
which can be empty, is convex. If the secured set is nonempty, i.e., X++ 6= ∅, we can efficiently obtain the optimal solution to Problem (8) using the binary search procedure of Brown and Sim [5]. Otherwise, if the secured set is empty, we have z∗ ≤ 0. Moreover, each of the extreme points X
i must either be
in the neutral or vulnerable set. If one of them, say Xj, is in the neutral set, then if the secured set is
empty, ρ(Xj) = 0 attains the highest satisficing value over X . Suppose instead that Xi, i = 1, . . . , n,
are in the vulnerable set. Then, by quasi-convexity of ρ, there exists an extreme point that is optimal. Hence,
z∗= max
i=1...,n{ρ(Xi)},
and we can simply enumerate the PSM values for the n extreme points and choose the largest one in this case.
We now consider an asset allocation problem in which the underlying distributions of assets’ returns are not known exactly. Consider n assets with independently distributed returns Vi, i = 1, . . . , n. The
exact distribution of Vi is unknown but can be characterized by its support [vi, vi], i.e., the probability
that Vi belongs to [vi, vi] is one. Also, the mean of Vi is unknown and lies in [νi, νi] ⊆ [vi, vi]. We then
consider the following asset allocation problem: sup ρ Ã n X i=1 wiVi− τ ! s.t. n X i=1 wi = 1 wi≥ 0, i = 1, . . . , n.
where τ is a give target return.
Note that for any target τ (which, again, may be random) and portfolioPni=1wiVi, the portfolio’s
excess return over the target corresponds to (Pi=1wiVi) − τ = Pni=1wi(Vi− τ ) =Pni=1wiXi, where
Xi = Vi − τ is the excess return of asset i = 1, . . . , n. When τ is fixed, Xi possesses a unknown
distribution with support [vi− τ, vi− τ ] and unknown mean in [νi− τ, νi− τ ]. We denote these bounds
We restrict our analysis to the case of an entropic prospective satisficing measure in which the underlying risk measure is given by
µk(X) = 1
kln supP∈FEP[exp(−Xk)]
for k > 0 and k < 0, and F is the set of probability measures on (Ω, F) such that X possesses a feasible distribution (with the given support [x, x] and mean in the corresponding interval [µ, µ]).
Proposition 11. Let X be a random variable with support [x, x] and F be the set of all admissible distributions of X such that its mean lies in [µ, µ] ⊆ [x, x]. Then
sup
P∈FEP[exp(−aX)] = (
p exp(ax) + q exp(ax) if a ≥ 0 p exp(ax) + q exp(ax) otherwise, where p = (x − µ)/(x − x), q = 1 − p, p = (x − µ)/(x − x) and q = 1 − p.
Given a target τ , Proposition 11 enables us to compute the EPSM under the given distributional ambiguity. Suppose ρ(Vi− τ ) < 0 for all i, then all assets are in the vulnerable set and concentration is
favored. Hence, it is optimal to invest in the asset with the highest ρ(Vi− τ ). Otherwise, we solve the
following optimization problem sup k s.t. n X i=1 1 kln(piexp(−wik vi) + qi exp(−wik vi)) ≤ −τ n X i=1 wi = 1 k > 0, wi ≥ 0, i = 1, . . . , n,
where pi = (vi− νi)/(vi− vi), qi= 1 − pi. This is equivalent to the following problem: inf a
s.t.
n
X
i=1
a ln(pi, exp(−wivi/a) + qi exp(−wivi/a)) ≤ −τ n
X
i=1
wi = 1
a > 0, wi≥ 0, i = 1, . . . , n,
which is a convex optimization problem, and hence can be solved, in high dimension, efficiently using interior point methods.8
We now present a numerical example based on the information presented in Table 5. Note that the asset returns are defined in such a way that asset 1 is a risk-free asset that pays 2% in all states of the world. Assets 2 to 6 are risky assets, with asset 2 being the one with lowest downside and upside and asset 6 being the one the highest downside and upside. We solve the optimal asset allocation for various
8For this example, it is convenient to use a solver that can explicitly handle the “exponential cone”; here we use the software package ROME [17] to solve our example problem.
targets as presented in Table 6. The lowest target corresponds to the risk-free asset. In this case, the investor can reach the target for sure by investing in asset 1. As the target increases, the risk-free asset 1 becomes less attractive, because it fails to attain the target with certainty. The investor puts some of their wealth into the risky assets. If the target becomes very high, i.e., the investor is very ambitious, then they only hold asset 6, the asset with the highest upside potential and a positive probability to be above the target.
This example highlights the intuitive idea that if the investor possesses a high target return, then they will be willing to take more risk. This pattern is similar to that observed in mutual fund managers during the technology bubble of the 1990’s and discussed in Dass et al. [7]. Managers with high contractual incentives to rank at the top (i.e., those with a high target) adopted the risky and aggressive strategy to not invest in bubble stocks, as this was the only way to outperform the market. Such a strategy also carried with it a high probability of ranking at the bottom if the bubble continued. In contrast, mutual fund managers with a high incentive to follow the benchmark (i.e., those mainly concerned about not ranking at the bottom, thus with a low target) adopted the less risky strategy to follow the bubble (herding), which yields a small probability of ranking at the bottom. While the observation of Dass et al. [7] suggests that fund managers’ decisions may be target-based, we do not want to overemphasize here the ability of PSPs to explain real portfolio choices. Addressing portfolio selection and asset pricing implications of PSPs seems like an interesting subject for future research.
6
Conclusion
In this paper, we have developed prospective satisficing preferences, a target-based model of risky choice. Our model has several interesting features. First, we obtained a representation theorem that has practical relevance, since it links our framework to a more standard definition of a risk measure. In addition, under mild assumptions, the PSP model preserves first-order stochastic dominance over all premia, second-order stochastic dominance over secured premia, and risk-seeking second-order stochastic dominance over vulnerable premia. Moreover, PSPs can address several observed violations of expected utility theory, thus also displaying considerable descriptive power. Finally, the model is amenable to computationally efficient optimization.
There are a number of interesting directions for future research. One direction is to consider the use of the PSP model in various applications, such as portfolio choice or environments involving competition. It would be interesting to see what implications the model has for market portfolios and equilibrium behavior. Second, it would be useful to have more detailed experimental work that tests the implications of the model and examines elicitation of targets.
Finally, and related to this, is further investigation of the descriptive power of the PSP model. There is a large body of empirical evidence that suggests that performance relative to a target payoff is a critical factor driving the decision making of many individuals. We have found that the PSP model can also explain some recently observed “puzzles” in decision theory, in addition to the classical ones discussed earlier. For instance, Machina [27] recently pointed out that rank-dependent preferences that resolve the classical Ellsberg example can still struggle with other “Ellsberg-like” examples due to a required separability property. It turns out PSPs do not impose such separability and therefore
can address these instances. As another example, Wu and Markle [40] have recently pointed out that people demonstrate systematic violations of double matching, which is necessary for the representation of cumulative prospect theory and states that if a decision maker is indifferent between both the tail gains and tail losses of two gambles, then they must be indifferent between the composition of these two tails as well. One can verify that PSPs need not obey this stringent requirement of gain-loss separability either. What we find appealing is that the PSP model is fairly simple to motivate and not done with any particular descriptive intent in mind, yet it still can readily accommodate a fairly wide array of phenomena that seem to be present in the decision making of many individuals.
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Appendix
Proofs
Proof of Proposition 1
Let º be a prospective satisficing preference relation. Property 1 implies the existence of an upper semi-continuous function ˜ρ : X → R such that X º Y if and only if ˜ρ(X) ≥ ˜ρ(Y ) (Theorem 4, Bosi and
Mehta [4]).
Monotonicity for ˜ρ follows directly from Property 2 (that is, monotonicity of º).
Property 3 implies that ˜ρ is constant and maximal on {X ∈ X : X ≥ 0} and constant and minimal
on {X ∈ X : X < 0}. Assume that ˜ρ(X) = ˜ρufor all X ≥ 0 and ˜ρ(X) = ˜ρlfor all X < 0, then Property
3 also implies ˜ρ(Y ) ∈ [˜ρl, ˜ρu] for all Y ∈ X .
Property 4 implies that ˜ρ is constant on X0. Assume ˜ρ(X) = ρ0 for all X ∈ X0, then Property 4 also implies that ˜ρ(X) < ˜ρ0 for all X ∈ X−− and ˜ρ(X) > ˜ρ0 for all X ∈ X++. From this it follows that
˜
ρu > ˜ρ0 > ˜ρl.
Property 4(i) implies quasi-concavity9 for ˜ρ on X
++ and Property 4(ii) implies quasi-convexity for ˜
ρ on X−−.
Let ρ(X) = g(˜ρ(X)) − g(˜ρ0) where g : R → R ∪ {−∞, ∞} is defined as follows:
g(x) = −∞ x = ˜ρl 1 ˜ ρl−x x ∈ (˜ρl, ˜ρ0] 1 ˜ ρu−x+ 1 ˜ ρl−˜ρ0 − 1 ˜ ρu−˜ρ0 x ∈ (˜ρ0, ˜ρu) ∞ x = ˜ρu
Since g is strictly increasing, it preserves the ordering of ˜ρ and ρ satisfies properties (i), (ii) and (iii)(a).
Moreover, since g is continuous, ρ is also upper semi-continuous, quasi-concave on X++and quasi-convex on X−−. This proves the existence of ρ with properties (i)-(iv).
On the other hand, it is straightforward to show that a upper semi-continuous function ρ : X → R ∩ {−∞, ∞} with properties (i)-(iv) defines a prospective satisficing preference relation º with X º Y if and only if ρ(X) ≥ ρ(Y ). Note that when ρ is upper semi-continuous, then it follows from Bosi and Mehta [4] that º satisfies Properties (i) and (ii) in Property 1.
9Assume wlog that X º Y . Then λX + (1 − λ)Y º Y , so ρ(λX + (1 − λ)Y ) ≥ ρ(Y ) = min(ρ(X), ρ(Y )); an analogous argument follows for quasi-convexity on X−−.
Proof of Proposition 2
Clearly, all convex functions are also quasi-convex. It suffices to show that a quasi-convex function that satisfies translation invariance is always convex. We have:
µ(λX + (1 − λ)Y ) − (λµ(X) + (1 − λ)µ(Y ))
= µ(λ(X + µ(X)) + (1 − λ)(Y + µ(Y ))
≤ max{µ(X + µ(X)), µ(Y + µ(Y ))}
= max{0, 0} = 0. Hence,
µ(λX + (1 − λ)Y ) ≤ λµ(X) + (1 − λ)µ(Y ).
That a quasi-concave function that satisfies translation invariance is always concave follows by a an analogous argument.
Proof of Theorem 1
Suppose ρ takes the form (5) and µk is the family of risk measures described in Theorem 1; we will show that ρ is a prospective satisficing measure.
1. Upper semi-continuity:
Upper semi-continuity for ρ is equivalent to {X ∈ X : ρ(X) ≥ k} being closed for all k. Let
k ∈ (−∞, ∞)\{0} and take a sequence (Xn)n in {X ∈ X : ρ(X) ≥ k} such that Xn → X as n → ∞. Since ρ(Xn) ≥ k, then µk(Xn) ≤ 0. Therefore, the sequence (Xn)n belongs to the
acceptance set Aµk. Since Aµk is closed, than X ∈ Aµk, i.e., µk(X) ≤ 0. This implies ρ(X) ≥ k,
i.e., X ∈ {X ∈ X : ρ(X) ≥ k}. This proves that {X ∈ X : ρ(X) ≥ k} is closed for all k, and thus
ρ is upper semi-continuous.
2. Monotonicity:
Follows clearly from monotonicity of the underlying risk measures.
3. Satisficing behavior:
(a) Attainment content:
Note that if X ≥ 0, then monotonicity for µk implies µk(X) ≤ µk(0) = 0, for all k ∈ R\{0}. Hence, ρ(X) = ∞.
(b) Non-attainment apathy:
If X < 0, there exists ² < 0 such that X ≤ ². Hence, monotonicity for µk implies µk(X) ≥
µk(²) = −² > 0 for all k ∈ R\{0}; therefore, {k : µk(X) ≤ 0} = ∅, so ρ(X) = sup{} = −∞
(by our convention that sup ∅ = −∞).
4. Prospective behavior:
(a) Superiority of secured target premia and inferiority of vulnerable target premia:
(b) Quasi-concavity over secured target premia:
Let X, Y ∈ X++ and k∗ = min {ρ(X), ρ(Y )} > 0. Note that µ
k(X) ≤ 0 and µk(Y ) ≤ 0 for
all k ∈ (0, k∗). Then, using convexity of µ
k on k > 0, we have
ρ(λX + (1 − λ)Y ) = sup {k ∈ (−∞, ∞)\{0} : µk(λX + (1 − λ)Y ) ≤ 0}
≥ sup {k ∈ (0, ∞) : µk(λX + (1 − λ)Y ) ≤ 0}
≥ sup {k ∈ (0, ∞) : λµk(X) + (1 − λ)µk(Y ) ≤ 0}
≥ k∗
= min {ρ(X), ρ(Y )} .
(c) Quasi-convexity over vulnerable target premia:
Let X, Y ∈ X−−and k∗= max {ρ(X), ρ(Y )} < 0. Note that µ
k(X), µk(Y ) > 0 for all k > k∗.
Hence, for all k ∈ (k∗, 0),
µk(λX + (1 − λ)Y ) ≥ λµk(X) + (1 − λ)µk(Y ) > 0.
Since µk is nondecreasing in k, the above inequality also holds for k > k∗. Therefore, we
have
ρ(λX + (1 − λ)Y ) = sup {k ∈ (−∞, ∞)\{0} : µk(λX + (1 − λ)Y ) ≤ 0}
= sup {k ∈ (−∞, k∗] : µk(λX + (1 − λ)Y ) ≤ 0}
≤ k∗
= max {ρ(X), ρ(Y )} .
For the other direction, let µk take the form (6) in which ρ is a prospective satisficing measure. By
monotonicity of ρ, it is clear that µk is nondecreasing on k. To verify that µk is a risk measure with a closed acceptance set, we note the following:
1. Closed acceptance set:
We show that µk(X) ≤ 0 is equivalent to ρ(X) ≥ k. One direction is trivial, i.e., when ρ(X) ≥ k
then µk(X) ≤ 0. For the other direction, we note that upper semi-continuity for ρ implies upper semi-continuity for a → ρ(X + a), for all X ∈ X . Moreover, since a → ρ(a + X) is also increasing due to the monotonicity of ρ, then it is also right-continuous and the limit of Problem (6) is achievable. It follows that when µk(X) ≤ 0, then there exists an a ≤ 0 such that ρ(a + X) ≥ k.
Due to monotonicity of ρ we also have ρ(X) ≥ k. We have thus showed:
Aµk = {X ∈ X : µk(X) ≤ 0} = {X ∈ X : ρ(X) ≥ k}.
Since ρ is upper semi-continuous, {X ∈ X : ρ(X) ≥ k} is closed and thus so is Aµk.
2. Monotonicity:
3. Translation invariance: For all c ∈ R, µk(X + c) = inf{a : ρ(X + c + a) ≥ k} = inf{a − c : ρ(X + a) ≥ k} = µk(X) − c. 4. Convexity on k > 0:
Given X, Y ∈ X , notice that, by monotonicity of ρ and the definition of µk, we have for all ² > 0,
ρ(X + µk(X) + ²) ≥ k
and
ρ(Y + µk(Y ) + ²) ≥ k.
Since k > 0, we have X + µk(X) + ², Y + µk(Y ) + ² ∈ X++. For every λ ∈ [0, 1], define
aλ , λµk(X) + (1 − λ)µk(Y ). Then, for all ² > 0,
ρ(λX + (1 − λ)Y + aλ+ ²) = ρ(λ(X + µk(X) + ²) + (1 − λ)(Y + µk(Y ) + ²))
≥ min {ρ(X + µk(X) + ²), ρ(Y + µk(Y ) + ²)}
≥ k > 0.
Then
µk(λX + (1 − λ)Y ) = inf {a : ρ(λX + (1 − λ)Y + a) ≥ k}
≤ aλ
= λµk(X) + (1 − λ)µk(Y ).
5. Concavity on k < 0:
Since µk(X) = inf{a : ρ(X +a) ≥ k}, it follows that ρ(X +µk(X)+a) < k < 0 and ρ(Y +µk(Y )+
a) < k < 0 for all a < 0. Therefore, for all a < 0, X +µk(X)+a ∈ X−−, and Y +µk(Y )+a ∈ X−−;
hence,
ρ(λ(X + µk(X)) + (1 − λ)(Y + µk(Y )) + a) ≤ max{ρ(X + µk(X) + a), ρ(Y + µk(Y ) + a)} < k
for all λ ∈ [0, 1]. Therefore,
µk(λ(X + µk(X)) + (1 − λ)(Y + µk(Y )))
= inf{a : ρ(λ(X + µk(X)) + (1 − λ)(Y + µk(Y )) + a) ≥ k}
= inf{a : ρ(λ(X + µk(X)) + (1 − λ)(Y + µk(Y )) + a) ≥ k, a ≥ 0}
≥ 0.
Finally, we need to show that
ρ(X) = sup {k ∈ (−∞, ∞)\{0} : µk(X) ≤ 0} .
We have seen in (i) above that the limit of Problem (6) is achievable. Therefore,
sup {k ∈ (−∞, ∞)\{0} : µk(X) ≤ 0} = sup {k ∈ (−∞, ∞)\{0} : ∃a ≤ 0 s.t. ρ(X + a) ≥ k}
= sup {ρ(X + a) : a ≤ 0} = ρ(X),
which completes the proof. Proof of Proposition 3
It is straightforward to verify the concavity and nondecreasing properties of ¯µk. Moreover, for all k > 0, 0 = µk µ 1 2(X − X) ¶ ≤ 1 2µk(X) + 1 2 µ| {z }k(−X) =−¯µ−k(X) .
Hence, ¯µ−k(X) ≤ µk(X) for all k > 0. Therefore, for all s < 0, t > 0, ¯
µs(X) ≤ lim
k↑0µ¯k(X) ≤ limk↓0µk(X) ≤ µt(X).
Proof of Proposition 4
Let X ∈ X such that k 7→ µk(X) is strictly increasing. We have
ρ(−X) = sup{k ∈ (−∞, ∞) \ {0} : µk(−X) ≤ 0}
= sup{k ∈ (−∞, ∞) \ {0} : µ−k(X) ≥ 0}
= − inf{k ∈ (−∞, ∞) \ {0} : µk(X) ≥ 0}
= − sup{k ∈ (−∞, ∞) \ {0} : µk(X) ≤ 0}
= −ρ(X)
The first and the last equalities follow from Theorem 1, the second equality is given by the symmetric properties of the family of risk measures {µk : k ∈ R \ {0}}, and the fourth equality holds since
k 7→ µk(X) is strictly increasing.
If X is deterministic, then k 7→ µk(X) is also constant and the argument above does not hold.
In this case, however, if X > 0, then ρ(X) = ∞ and ρ(−X) = −∞. If X < 0, the opposite holds. Thus, if X is deterministic, X 6= 0, we also have ρ(−X) = −ρ(X). In contrast, if X = 0, then
Proof of Proposition 5
Suppose that X ≥(1) Y . Then E [u(X)] ≥ E [u(Y )] for all u nondecreasing and the inequality is strict
for at least one such u. Since u(x) is nondecreasing if and only if −u(−x) is also nondecreasing, we have −E [u(−X)] ≥ −E [u(−Y )], or, equivalently, E [u(−X)] ≤ E [u(−Y )] for u nondecreasing and the inequality is strict for at least one such u. This implies that −Y ≥(1)−X. Therefore,
¯
µ(X) = −µ(−X) ≤ −µ(−Y ) = ¯µ(Y ).
For SSD, we observe that a function u(x) is nondecreasing and concave if and only if −u(−x) is nondecreasing and convex. Hence, X ≥(2) Y if and only if −Y ≥(−2) −X. The result now follows
in similar fashion to above. Proof of Proposition 6
Note that if X ≥(1) Y , then µk(X) ≤ µk(Y ) for all k ∈ R\{0}, since µkpreserves FSD. By the definition
of ρ, it follows immediately that ρ(X) ≥ ρ(Y ), i.e., ρ also preserves FSD.
For the next claim, note that Y ∈ X++ implies that ρ(Y ) > 0. Since X ≥(2) Y and µρ(Y ) preserves
SSD, we have
µρ(Y )(X) ≤ µρ(Y )(Y ) ≤ 0. Therefore, ρ(X) ≥ ρ(Y ).
Likewise, Y ∈ X−− implies that ρ(Y ) < 0. Since X ≥(−2)Y and µρ(Y )preserves RSSD, we have
µρ(Y )(X) ≤ µρ(Y )(Y ) ≤ 0. Therefore, ρ(X) ≥ ρ(Y ).
Proof of Proposition 7
First, it is easy to see that law-invariance of the PSM implies law-invariance of the underlying family of risk measures (see Equation (6)). F¨ollmer and Schied [15] show that on atomless probability spaces any law-invariant risk measure preserves FSD, and any convex, law-invariant risk measure preserves SSD; the claim now follows by Propositions 5 and 6.
Proof of Theorem 2
The boundedness properties for the family {µk : k ∈ (−∞, ∞) \ {0}} imply µk(X) ≥ E [−X] for k > 0
and µk(X) ≤ E [−X] for k < 0. It follows that when E [X] < 0, then for all k > 0
µk(X) ≥ E [−X] > 0.
Hence, it follows from Theorem 1 that ρ(X) ≤ 0. Likewise, if E [X] ≥ 0, then for all k < 0,
µk(X) ≤ E [−X] ≤ 0.
Proof of Proposition 8
Since these are symmetric families of risk measures, it suffices to show that µk(X) ≥ E [−X] for k > 0,
which implies that for k < 0
µk(X) = −µ−k(−X) ≤ E [−X] .
Henceforth, we assume k > 0. For the entropic risk measure, we have, by Jensen’s inequality
µk(X) = 1
kln E [exp(−k X)] ≥
1
kln exp(E [−kX]) = −E [X] .
For the CVaR risk measure, we have
µk(X) = inf ν∈R © ν + ekE [(−X − ν)+]ª ≥ inf ν∈R{ν + E [(−X − ν) +]} ≥ inf ν∈R{ν + E [−X − ν]} = E [−X] .
Finally, for the homogenized entropic risk measure, we have
µk(X) = inf
a>0{a ln (E [exp(−X/a)]) + ak}
≥ inf
a>0{a ln (E [exp(−X/a)])}
≥ inf
a>0{a ln (exp(−E [X] /a))}
= E [−X] .
Proof of Proposition 9
First, it is clear that ρ(XA− τ ) = ∞ and ρ(XB− τ ) < ∞ for any τ ∈ (0, y]. We thus focus on comparing XC to XD over the range τ ∈ (0, y].
There is a one-to-one mapping between target levels and satisficing levels. In particular, for a particular satisficing level ρ, let τ (XC, ρ) and τ (XD, ρ) be the corresponding target levels that induce
the satisficing level ρ for gambles XC and XD, respectively. We have
τ (XC, ρ) = −1 ρlog £ 1 + p(e−xρ− 1)¤ τ (XD, ρ) = −1 ρlog £ 1 + q(e−yρ− 1)¤.
Note that τ (XC, ρ) and τ (XD, ρ) are both decreasing and continuous in ρ. We will compare these target
functions as ρ varies and will show that there exists a unique ρ? > 0 such that τ (X
C, ρ?) = τ (XD, ρ?),
and that τ (XD, ρ) > τ (XC, ρ) for all ρ > ρ?, and τ (X
D, ρ) < τ (XC, ρ) for all ρ < ρ?. This shows that
ρ(XC− τ ) > ρ(XD− τ ) if and only if τ > τ (XC, ρ?) = τ (XD, ρ?).
First, consider ρ < 0. Over this range, we have
τ (XC, ρ) > τ (XD, ρ) ⇔ −1ρlog £ 1 + p(e−xρ− 1)¤> −1 ρlog £ 1 + q(e−yρ− 1)¤ ⇔ p(e−xρ− 1) − q(e−yρ− 1) > 0.
Let g(ρ) = p (e−xρ− 1) − q(e−yρ− 1), the left hand side of the latter inequality. Over ρ < 0, we have
g0(ρ) = −pxe−xρ+ qye−yρ
< qy(e−yρ− e−xρ)
< 0,
where in the first line we use the fact that px > qy and in the second line we use ρ < 0 and y > x. In addition,
lim
ρ↑0g(ρ) = 0
lim
ρ→−∞g(ρ) = +∞.
In sum, g(ρ) is a strictly decreasing function from +∞ to 0 as ρ ↑ 0 and therefore must be strictly positive over the range, which implies that g(ρ) > 0 over ρ < 0, and thus τ (XC, ρ) > τ (XD, ρ) over this
range.
For ρ = 0, the target levels reduce to the expected values; thus, τ (XC, 0) = px > qy = τ (XD, 0).
Finally, consider ρ > 0. Similar to the first case, we have over this range
τ (XC, ρ) > τ (XD, ρ) ⇔ p (e−xρ− 1) − q(e−yρ− 1) < 0.
Let h(ρ) = p (e−xρ− 1) − q(e−yρ− 1), the left hand side of the latter inequality. We have lim
ρ↓0h(ρ) = 0
and limρ→∞h(ρ) = q − p > 0. Moreover, h0(ρ) = −pxe−xρ+ qye−yρ, so
h0(ρ) ≤ 0 ⇔ ρ ≤ µ 1 x − y ¶ log · px qy ¸ = ¯ρ > 0.
Thus, h(ρ) over ρ ≥ 0 has a left limit of zero, a right limit of the positive value q −p, and is nonincreasing