^{1}

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This paper proposes a decomposition of the cost of risk (as measured by a risk premium) across intervals/quantiles of the payoff distribution. The analysis is based on general smooth risk preferences. While this includes the expected utility model as a special case, the investigation is done under a broad class of non-expected utility models. We decompose the risk premium into additive components across quantiles. Defining downside risk as the risk associated with a lower quantile, this provides a basis to evaluate the cost of exposure to downside risk. We derive a local measure of the cost of risk associated with each quantile. It establishes linkages between the cost of risk, risk preferences and the distribution of risky prospects across quantiles (as measured by quantile variance and skewness). The analysis gives new and useful information on how risk aversion, exposure to downside risk and departures from the expected utility model interact as they affect the risk premium.

For risk-averse decision makers, the cost of risk can be measured by the risk premium reflecting the willingness-to-pay to replace a risky outcome by its mean. In general, the cost of risk depends on the nature of risk exposure. Special attention has also focused on the role played by downside risk, i.e. the risk associated with unfavorable events. Previous research has examined safety first models (e.g., [^{1}. But the strong linkages established by Pratt between risk aversion “in the small” and “in the large” have proved more difficult to extend to downside risk. For example, Keenan and Snow ([

The complexities associated with a global analysis of aversion to downside risk suggest a need to explore a different approach. Like Pratt [^{2}. This paper examines the cost of risk (as measured by the risk premium) using a quantile approach. This is done by dividing the range of stochastic payoff into intervals, each interval characterizing a quantile of the underlying distribution. This allows us to examine the nature and welfare effect of risk exposure in each interval/quantile. We first show that the risk premium can be decomposed into additive components across quantiles. This result is global and applies “in the large”. It provides a direct measure of the cost of exposure to downside risk. Indeed, defining downside risk as the risk located in the lower quantile(s) of the payoff distribution, the cost of downside risk is just the component of the risk premium associated with such quantile(s). We also show our global decomposition can generate local measures that apply “in the small”. In turn, these local measures provide some useful information about linkages between risk preferences and moment-based measures of risk exposure.

We present our arguments under general risk preferences represented by a general smooth preference functional over the probability distribution function of payoff. While most previous research on downside risk aversion has focused on the expected utility model (e.g., [_{p}-Fréchet differentiable preference functional over the probability distribution function of payoff. As showed by Wang [

This paper makes four contributions. The first two contributions were noted above. First, our analysis of the cost of risk and downside risk is presented under a non-expected utility model. This extends previous analyses that have focused on the expected utility model. Second, we propose an additive decomposition of the cost of risk across quantiles, each component identifying the role of risk associated with each quantile. Besides being global and applying “in the large”, this result is useful in the sense that the component of the risk premium associated with lower quantile(s) gives anexplicit measure of the cost of exposure to downside risk. This provides a basis to assess the relative importance of downside risk in the evaluation of the cost of risk^{3}.

Our third contribution is to derive a local measure of the cost of risk associated with each quantile. This measure is approximate and applies “in the small.” It relies on quantile moments to evaluate risk exposure in each quantile^{4}. A quantile moment is a moment defined over a specific interval/quantile of the payoff distribution. Our local measure establishes linkages between the cost of risk and quantile moments (including quantile variance and quantile skewness) in each and every quantile. This is of practical value as moments have been commonly used in empirical investigations of risk and downside risk (e.g., [

Our fourth contribution is to use our local quantile-based measures to examine how departures from the expected utility model affect the risk premium. This is of particular interest when such departures occur for low probability events located in the lower tail of distribution. Our analysis identifies interaction effects between the degree of risk aversion and non-linearity in the probabilities of facing unfavorable events. It shows how departures from expected utility and exposure to downside risk can interact to increase the cost of risk.

Consider a decision maker facing an uncertain payoff

Assume that the decision maker’s preferences are represented by the real-valued utility functional^{5}. As showed by Wang [

model when ^{6}. When

in the probabilities. As noted by Kaheman and Tversky [

this reduces to the expected utility model with

Denote the overall mean of

Let

where R is the decision maker’s willingness-to-pay to replace π by its overall mean

In this paper, we explore the economics of exposure to downside risk using a quantile approach. For that purpose, let

_{k}’s are chosen such that

Define

Define the mean of π in the interval

_{k}_{1} the k-th quantile mean. Note that the overall mean _{k}_{1}’s:

Consider the willingness-to-pay to eliminate the risk in the first quantile, moving it to the mean payoff

Next, consider the incremental willingness-to-pay to eliminate the risk of the k-th quantile, moving it the mean payoff

for

using (4). Equation (5c) makes it clear that eliminating the risk in the k-th quantile (and moving it to

Definition: The k-th incremental risk premium is defined as the sure amount

By adjusting the willingness-to-pay

Proposition 1: Under the risk preferences

where

Equation (6b) is our first main result. Importantly, it applies under general risk preferences^{7}. In this case, while the decomposition of the risk premium R given in (6b) remains globally valid, evaluating the cost of risk associated with each quantile becomes more complex^{8}. Alternatively, measuring the incremental risk premium would become simpler in the absence of income effects^{9}. In this context, the

Equation (6b) shows that the overall risk premium R is equal to the sum of the incremental risk premium

This section explores local measures of the risk premium, with a focus on the decomposition identified in Proposition 1. The analysis proceeds in several steps. In a first step, following Machina [

Following Machina [

Lemma 1: Given two distribution functions

where

Lemma 1 shows that the welfare effect of a change from

(8). This result is global in the sense that it does not restrict the distribution functions

strictly increasing in π and satisfies

cal in the sense that the distribution function

We have defined the overall mean of π by

Note that quantile moments in (4) and (10) are related to partial moments. First, when

(non-central) quantile moment in the interval

being in the k-th quantile. Noting that ^{10}, this establishes the relationship existing between partial moments and quantile moments^{11}.

The central moments

Our derivation of a local quantile-based measure of the cost of risk is long and tedious. It starts with an alternative representation of the risk premium (presented in lemma 2 below), following a two-step approach. From Equation (2), recall that the risk premium R is the willingness-to-pay to replace the random payoff π by its mean

mean

Consider the first step. Letting

where the quantile mean

For a given

where

For given

where

Next, consider the second step. Letting

Note that

For a given in (13), let

where

For a given s, Lemma 1 implies that there exists a distribution function

where

Noting that

Lemma 2: The risk premium R given in (2) can be decomposed as follows

where

Equation (15) states that the risk premium R can be decomposed into two additive parts:

We now proceed with deriving expressions that provide a local approximation of the decomposition of the risk premium given in Proposition 1. Note that, in contrast with the Arrow-Pratt analysis of risk aversion, the linkages between local and global characterizations of downside risk aversion are complex. As noted in the introduction, Keenan and Snow [

Our analysis proceeds first with the local measurement of ^{12}.

Under differentiability, we now present an approximate measure of

Proposition 2: A local measure of the risk premium component

where

Equation (16) gives an approximate measure of

An approximate measure of

Proposition 3:A local measure of the risk premium component

where

Combining Lemma 2 with Propositions 2 and 3, we obtain the following key result.

Proposition 4: The overall risk premium R can be approximated as

Equation (18) provides an approximate decomposition of the overall risk premium R in terms of the contribution made by each interval

Equation (18) also shows how risk exposure across intervals affects the (approximate) cost of risk. It meas-

ures risk exposure by the probability of being in the k-th interval

Finally, Equation (18) shows that the overall cost of risk R is (approximately) equal to the sum of the interval-specific cost of risk across all intervals,

This section discusses the implications of the quantile-based measures of the risk premium and its decomposition given in Proposition 4. As noted above, our analysis applies under non-expected utility preferences. This section proceeds in three steps. First, we study the implications of Proposition 4 under general risk preferences. Second, we examine the special case where risk preferences satisfy the expected utility model. Third, we evaluate how departures from expected utility affect the cost of risk. The analysis provides new information on the role of downside risk exposure and its effects on the risk premium.

Combining Propositions 1, 2, 3 and 4, we obtain the following result. (See the proof in the Appendix).

Proposition 5: Let V be the class of functions

where

Equations (19a)-(19c) are consistent with the decomposition of the risk premium R given in (6b) (defining

And they are consistent with the approximations given in (16), (17) and (18). As such, Equation (19b) defining

Each of these terms is weighted by the probability of being in the k-th interval,

rences with respect to variance. Under risk aversion (where

weighted by the term

skewness. Under downside risk aversion (where

Similarly, Equation (19c) defining

cludes two additive terms: a variance component (including the squared deviation from the mean,

sion (where

These results provide useful information on the economics of downside risk. To the extent that most decision makers are averse to downside risk, they provide a way to assess the cost of downside risk exposure. First, since our analysis applies to all intervals

Note that our analysis includes the expected utility model as a special case. Indeed, as noted above, the expected utility model holds when

Under (20), we have

^{ 13}. Substituting these expressions into (19b)-(19c) shows how our analysis can

be used to provide simple measurements of the component of the risk premium associated with the k-th quantile of the payoff distribution. When focusing on the lowest quantile (

More importantly, our analysis applies to non-expected utility models. This corresponds to situations where the local utility function

where

where

where

where

derivatives of the local utility function

Of special interest here is the case of risk preferences that tend to “overweight” the probability of rare events located in the lower tail of the distribution (e.g., [

function

_{12} on ΔR_{a}_{1}. The quantile variance _{1} can contribute to increasing the risk premium.

Finally, our investigation also provides useful information on how quantile skewness affects the risk premium. To see that, rewrite Equation (23c) as

Comparing the rank dependent utility model (21) with the expected utility model (obtained when^{14}. For_{1}) and on the (non-linear) way in which probabilities enter risk preferences.

This paper has developed a quantile-based analysis of the cost of risk (as measured by the risk premium) reflecting risk exposure across different intervals of the risk distribution. Our analysis applies under general smooth risk preferences. While this includes the expected utility model as a special case, it covers the case of non-expected utility models. Using a quantile-based analysis of the cost of risk, we show how the risk premium can be decomposed into additive components across the range of stochastic outcomes. We identify the components of the cost of risk associated with specific quantiles of the payoff distribution. In this context, the lower quantile(s) correspond to exposure to downside risk, i.e. exposure to unfavorable risky events. This decomposition applies “in the large”. It means that the component(s) of the risk premium associated with the lower quantile(s) provide an assessment of the relative role of downside risk in the evaluation of the cost of risk. Using a quantile approach, we also derive risk premium measures associated with each interval “in the small”. They generalize local measures of risk aversion and downside risk aversion presented in previous research by extending them across multiple intervals of the distribution. We show how quantile variance and skewness across intervals affect the cost of risk.

Finally, we show how departures from the expected utility model affect the risk premium. Our local quantile-based measures identify interaction effects between the degree of risk aversion and non-linearity in the probabilities. They show how risk aversion, departure from the expected utility model and exposure to downside risk interact as they affect the cost of risk.

Note that our paper has focused on the case of risk represented by a distribution function. Extending the analysis to cover situations of uncertainty or ambiguity (where probability assessments may be deemed inappropriate) seems to be a good topic for further research.

This work was partly supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014S1A3A2044459).

Jean-PaulChavas,KwansooKim, (2015) Aversion to Risk and Downside Risk in the Large and in the Small under Non-Expected Utility: A Quantile Approach. Theoretical Economics Letters,05,784-804. doi: 10.4236/tel.2015.56090

Proof of Proposition 1: Equation (5a) defines DV_{1} as the decision maker’s sure willingness-to-pay to eliminate the risk in the first quantile, moving it to the mean payoff

payoff. Noting that

Proof of Lemma 1: Following Wang [

Using integration by parts and (8), we obtain

Q.E.D

To prove Propositions 2 and 3, first consider a function

Lemma 3: Let

Then,

Proof: Evaluated at

Differentiating (A1) with respect to s gives

Evaluating (A6) at

Under the strong monotonicity of f in t (with

Differentiating (A6) with respect to s, we obtain

Evaluating (A8) at

Note that, if

Differentiating (A8) with respect to s, we obtain

Evaluating (A10) at

Note that, if

Under the strong monotonicity of f in t (with

Note that lemma 3 has one important implication. If

Proof of Proposition 2: From Equation (11), we have

where

since

Note that (A14) implies that

and

And from (A5) in lemma 3, letting

Given

A third-order Taylor series expansion of

where

Proof of Proposition 3: From (13), we have

Note that Equation (14b) corresponds to (A1) in lemma 3. It follows from (A2) that

where

for all F. In lemma 3, this corresponds to

and

And from (A5) in lemma 3, letting

Since

A third-order Taylor series expansion of

Using

Proof of Proposition 5: Under the strong monotonicity of

Similarly, using (8), note that the right-hand side of (19a), (19b) or (19c) is equal to 0 for all

Given_{k}_{1}.

illustrates that a change from