Prudence probability premium

Economics Letters 109 (2010) 34–37

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Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t

Prudence probability premium Paan Jindapon ⁎ Department of Economics, University of Alabama, Box 870224, Tuscaloosa, AL 35487, United States

a r t i c l e

i n f o

Article history: Received 3 November 2009 Received in revised form 20 July 2010 Accepted 28 July 2010 Available online 6 August 2010

a b s t r a c t Prudence probability premium is defined in the risk apportionment model (Eeckhoudt and Schlesinger, 2006). For an increase in downside risk, we show sufficient conditions for comparing the probability premiums between two individuals when the apportioned risk is small and large. © 2010 Elsevier B.V. All rights reserved.

JEL classification: D81 Keywords: Probability premium Risk aversion Downside risk Prudence Risk apportionment

1. Introduction Arrow (1965) and Pratt (1964) define a local measure of absolute risk aversion and use it to compare the levels of risk aversion across individuals. It is well known that the Arrow–Pratt measure of risk aversion is equivalent to the risk premium of a gamble which is defined as π such that a decision maker is indifferent between the gamble and the expected value of the gamble minus π. Another equivalent concept is the probability premium, which is defined as the additional probability of winning the positive outcome of a zero-mean gamble yielding two outcomes with equal probabilities that makes the decision maker indifferent between the gamble and the status quo. Even though the risk premium concept has been widely used in the literature, it is rare to find a discussion on the probability premium. In this paper, we use Eeckhoudt and Schlesinger's (2006) framework to define a probability premium for downside risk and show sufficient conditions for comparing the probability premiums between two individuals. For random variables X and Y, let [X, Y] be a lottery that yields X or Y, each with probability 0.5. Consider two ˜ and B3 = ½˜ε; −k, where k is a positive lotteries, A3 = ½0; −k + ε constant and ε̃ is a zero-mean risk. Eeckhoudt and Schlesinger show that a prudent decision maker who always prefers B3 to A3 has a utility function of which the third derivative is positive. This condition is equivalent to downside risk aversion, defined by Menezes et al. (1980). Crainich and Eeckhoudt (2008) introduce a local measure of

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the intensity of downside risk aversion (u‴ / u′) and show that the measure is equivalent to a risk premium which is defined as the monetary value added to the payoff 0 in A3 such that an individual is indifferent between A3 and B3. While Crainich and Eeckhoudt look at the risk premium, we focus on the probability premium concept. We call the additional probability of winning 0 in A3 that makes an individual indifferent between A3 and B3 the prudence probability premium. A key difference between the Arrow–Pratt probability premium and the prudence probability premium is that the former is defined on preferences over a binary lottery and a risk-free state, while the latter is defined on preferences over two compound lotteries. For any small risk, we show that the Arrow–Pratt measure of risk aversion (− u″ / u′) and Kimball's (1990) measure of prudence (− u‴ / u″) of two individuals are sufficient for ranking the prudence probability premiums. Specifically, a decision maker who is more risk-averse and more prudent will have a higher prudence probability premium. The above conditions, however, are not sufficient when ε˜ is large. To compare the prudence probability premiums given large on, we define a measure called elasticity of pain which can be derived from the utility premium introduced by Friedman and Savage (1948) and generalized by Eeckhoudt and Schlesinger (2006). The utility premium given ε˜ at wealth level x is defined as w1 ðxÞ = Euðx + ε˜ Þ−uðxÞ, which is always negative if preferences are risk-averse. Its negative value can be interpreted as the pain associated with taking ε˜ . In this paper, we show that a decision maker who is more risk-averse and has more pain elasticity will always have a higher prudence probability premium. Throughout the paper we assume that a decision maker has a von Neumann–Morgenstern utility function which is 3 times differentiable.

P. Jindapon / Economics Letters 109 (2010) 34–37

We assume that u′(x) N 0,u″(x) b 0, and u‴(x) N 0 for all x. A constant k is strictly positive; a lottery ε˜ pays ε+ with probability α and ε− with probability 1 − α, with ε+ N 0, ε− b 0 and E ε˜ = 0.

35

Proof. Part 1 We can write x + εþ

w1 ðxÞ = Euðx + ε˜ Þ−uðxÞ = α∫

x

x

−ð1−αÞ∫

2. Small risk

x + ε−

Let A3 = ½0; −k + ε˜  and B3 = ½˜ε; −k and assume that ε˜ is small. Eeckhoudt and Schlesinger (2006) say that an individual is prudent if B3 c A3 for all x; k; ε˜ . We define the prudence probability premium p3 as the additional probability of winning 0 in A3 such that A3 ∼ B3. Then,

u′ ðyÞdy;

and hence, x−k + ε

x

Δuðx−kÞ−w1 ðx−kÞ = ∫

x−k

u′ ðyÞdy−α∫

x−k

We can write

Δw1 ðx−kÞ ; Δuðx−kÞ−w1 ðx−kÞ

= α∫

x−k + ε

ð2Þ

We define the utility premium of taking risk ε˜ as w1 ðxÞ = Euðx + ε˜ Þ−uðxÞ. Then, we have ð3Þ

where Δw1(x − k) = w1(x) − w1(x − k) and Δu(x − k) = u(x) − u(x − k). Eeckhoudt and Schlesinger (2006) prove that w1 ≤ 0 if and only if u″ ≤ 0, and w1′ ≥ 0 if and only if u‴ ≥0. Therefore, if a decision maker is riskaverse and prudent, i.e., u″ ≤ 0 and u‴ ≥ 0, then p3 ≥0. Dividing the numerator and the denominator in the right-hand side of Eq. (3) by u′(x − k) yields

=∫

ðx−kÞ = u′ ðx−kÞ

Δw1 : ½Δuðx−kÞ−w1 ðx−kÞ = u′ ðx−kÞ

ð5Þ

x Δuðx−kÞ−w1 ðx−kÞ u′ ðyÞ =∫ dy x−k + εþ u′ ðx−kÞ u′ ðx−kÞ

+ ð1−αÞ∫

rðx; uÞ =

and Kimball's measure of prudence, pðx; uÞ =

all y N x, and x

∫x−k + ε

x−k + εþ

∫x−k + ε

−

u′ ðyÞ dy: u′ ðx−kÞ

for all y b x. Therefore,

εþ

u′ ðyÞ u′ ðxÞ

ð11Þ v′ ðyÞ ≤ 0 for v′ ðxÞ

−

v′ ðyÞ dy ≤ 0: v′ ðx−kÞ

ð12Þ

v′ ðyÞ dy v′ ðx−kÞ

ð13Þ

εþ

x−k + u′ ðyÞ dy−∫ x−k + u′ ðx−kÞ

ε−

is ambiguous. If ε˜ is small, then Eq. (12) dominates Eq. (13), and hence

x

y + εþ

−ð1−αÞ∫

u‴ ðxÞ − ″ , u ðxÞ

ð7Þ

x

y

x−k

ð14Þ

u″ ðzÞdzdy

∫y + ε

−

u″ ðzÞdzdy:

Dividing Eq. (14) by u″(x − k) yields þ

x y + ε Δw1 ðx−kÞ u″ ðzÞ dzdy = α∫ ∫ x−k y u″ ðx−kÞ u″ ðx−kÞ x y u″ ðzÞ dzdy: −ð1−αÞ∫ ∫ x−k y + ε− u″ ðx−kÞ

u″ ðyÞ u″ ðxÞ x

ð8Þ

ð15Þ

Using Pratt's analogy, we find that p(x ; u) ≥ p(x ; v) implies v″ ðyÞ ≤0 v″ ðxÞ

−

þ

y + ε

∫x−k ∫y

for all y N x, and

u″ ðzÞ u″ ðx−kÞ

u″ ðyÞ u″ ðxÞ

x

dzdy−∫

x−k

−

v″ ðyÞ ≥0 v″ ðxÞ þ

y + ε

∫y

for all y b x. Therefore,

v″ ðzÞ dzdy≤0: v″ ðx−kÞ

ð16Þ

However, the sign of x

for all x, then p3 ≥ q3.

v′ ðyÞ − ′ ≥0 v ðxÞ

x u′ ðyÞ dy−∫ x−k + u′ ðx−kÞ

ð6Þ

for all x, and u is more prudent than v, i.e., u‴ ðxÞ v‴ ðxÞ ≥− u″ ðxÞ v″ ðxÞ

x−k + ε−

Eq. (6) is obtained. Part 2 Considering Eq. (9), we find that Δw1(x − k)=

Proposition 1. Given any small ε˜ , if u is more risk-averse than v, i.e.,

−

u′ ðyÞ u′ ðxÞ

x

to state the following proposition.

u″ ðxÞ v″ ðxÞ ≥− − u′ ðxÞ v′ ðxÞ

x−k + εþ

Pratt (1964) shows that r(x ; u) ≥ r(x ; v) implies

∫x−k w′1 ðyÞdy = α∫x−k ∫y

for all x. Our first result provides a sufficient condition for Eqs. (5) and (6). We adopt the Arrow–Pratt measure of absolute risk aversion, u″ ðxÞ − ′ , u ðxÞ

u′ ðyÞdy + ð1−αÞ∫

However, the sign of

and Δuðx−kÞ−w1 ðx−kÞ Δvðx−kÞ−z1 ðx−kÞ ≤ ; u′ ðx−kÞ v′ ðx−kÞ

x−k + εþ

u′ ðyÞdy x−k + ε− þ x−k + ε u′ ðyÞdy: x−k + ε−

ð4Þ

Consider another decision maker v with a utility function v(x). Let v's utility premium function be denoted by z1 ðxÞ = Evðx + ε˜ Þ−vðxÞ. Let p3 and q3 be the prudence probability premiums for agents u and v, respectively. Hence, for a given ε˜ and k, u's prudence probability premium is at least as great as v's if Δw1 ðx−kÞ Δz ðx−kÞ ≥ ′1 ; u′ ðx−kÞ v ðx−kÞ

x

u′ ðyÞdy + ð1−αÞ∫ þ

Dividing Eq. (10) by u′(x − k) yields

þ

2p3 =

x−k + ε x

x−k

x

½Euðx + ε˜ Þ−uðxÞ−½Euðx−k + ε˜ Þ−uðx−kÞ : uðxÞ−Euðx−k + ε˜ Þ

ð10Þ

u′ ðyÞdy

+ ð1−αÞ∫ u′ ðyÞdy x−k + ε−   þ x x−k + ε = α ∫ u′ ðyÞdy−∫ u′ ðyÞdy x−k x−k   x x−k ′ u′ ðyÞdy + ð1−αÞ ∫ u ðyÞdy + ∫ −

ð1Þ

2p3 =

þ

x−k

ð0:5 +p3 ÞuðxÞ + ð0:5−p3 ÞEuðx−k + ε˜ Þ = 0:5Euðx + ε˜ Þ + 0:5uðx−kÞ:

2p3 =

ð9Þ

u′ ðyÞdy

y

∫x−k ∫y + ε

−

x y u″ ðzÞ dzdy−∫ ∫ x−k y u″ ðx−kÞ

+ ε−

v″ ðzÞ dzdy v″ ðx−kÞ

ð17Þ

36

P. Jindapon / Economics Letters 109 (2010) 34–37

is ambiguous. If ε˜ is small, then Eq. (16) dominates Eq. (17), and hence Δw1 ðx−kÞ u″ ðx−kÞ

Δw ðx−kÞ − ″1 ≤0, u ðx−kÞ

which is equivalent to

Δw1 ðx−kÞ Δw1 ðx−kÞ ≥− : − u″ ðx−kÞ u″ ðx−kÞ

ð18Þ

Proposition 2. If u is more risk-averse and has more pain elasticity from ε˜ than v, then p3 ≥ q3. Proof. From Eq. (9), we can write

Since Eq. (7) also holds for x – k, then Eq. (18) implies Eq. (5). □ Proposition 1 states that, when ε˜ is small, if u is more risk-averse and more prudent than v, then u will have a higher prudence probability premium. It is possible for a more prudent utility function to be more risk-averse or less risk-averse (see examples in Eeckhoudt and Schlesinger, 1994). This condition is more restrictive than being more downside risk-averse as defined by Crainich and Eeckhoudt (2008).1 Crainich and Eeckhoudt show that their condition is equivalent to a monetary value added to the zero outcome of lottery A3, while we focus on the additional probability of winning zero in A3. 3. Large risk

x−k + εþ

w1 ðx−kÞ = Euðx−k + ε˜ Þ−uðx−kÞ = α∫

x−k

−ð1−αÞ∫

−Δw1 ðx−kÞ = w1 ðx−kÞ : 1−Δuðx−kÞ = w1 ðx−kÞ

ð19Þ

Δw1 ðx−kÞ Δz ðx−kÞ ≥− 1 ; w1 ðx−kÞ z1 ðx−kÞ

ð20Þ

y N x, and x−k + εþ

∫x−k

Δuðx−kÞ Δvðx−kÞ ≤− ; w1 ðx−kÞ z1 ðx−kÞ

ð21Þ

u′ ðyÞdy:

′

v ðyÞ v′ ðxÞ

≥

u′ ðyÞdy

for all y b x. Therefore, x−k + εþ

≤

∫x−k

v′ ðyÞdy

v′ ðx−kÞ

u′ ðyÞ u′ ðxÞ

≤

;

v′ ðyÞ v′ ðxÞ

for all

ð24Þ

x−k

x−k

∫x−k + ε− u′ ðyÞdy ∫x−k + ε− v′ ðyÞdy u′ ðx−kÞ

≥

v′ ðx−kÞ

:

ð25Þ

From Eqs. (23), (24), and (25), we have −

w1 ðx−kÞ z ðx−kÞ ≥− 1 : u′ ðx−kÞ v′ ðx−kÞ Pratt (1964) also shows that r(x;u)≥r(x;v) implies for yN x. Then,

v′ ðxÞ , vðyÞ−vðxÞ

u′ ðx−kÞ v′ ðx−kÞ ≥ : Δuðx−kÞ ΔvðxÞ

ð26Þ u′ ðxÞ uðyÞ−uðxÞ

≥

ð27Þ

Using Eqs. (26) and (27), we have −

w1 ðx−kÞ z ðx−kÞ ≥− 1 ; Δuðx−kÞ Δvðx−kÞ

ð28Þ

−½lnð−wðxÞÞ− lnð−wðyÞÞ ≥ −½lnð−zðxÞÞ− lnð−zðyÞÞ;

ð29Þ

which implies

Definition 1. Given risk ε˜ , u's measure of pain elasticity (with respect w′ ðxÞx to wealth) is given by ηðx; u; ε˜ Þ= − 1 , where w1 ðxÞ = Euðx + w1 ðxÞ

ε˜ Þ−uðxÞ. Let z1 ðxÞ = Evðx + ε˜ Þ−vðxÞ. We say that u has more pain elasticity from ε˜ than v if ηðx; u; ε˜ Þ ≥ ηðx; v; ε˜ Þ, or equivalently, w′1 ðxÞ z′ ðxÞ ≥− 1 w1 ðxÞ z1 ðxÞ

u ðyÞ u′ ðxÞ

u′ ðx−kÞ

for all x.

−

′

ð23Þ

and Eq. (21) is obtained. Integrating Eq. (22) from y = x − k to x yields

and −

x−k + ε−

Pratt (1964) shows that r(x ; u) ≥ r(x ; v) implies

We find that u's prudence probability premium is at least as great as v's if −

x−k

u′ ðyÞdy

and

The two conditions in Proposition 1, however, are not sufficient to conclude that p3 ≥ q3 when ε˜ is large. To compare the prudence ˜ we use a measure called pain probability premiums given large ε, elasticity which can be derived from the utility premium function introduced by Friedman and Savage (1948) and generalized by Eeckhoudt and Schlesinger (2006). Eeckhoudt and Schlesinger (2009) use the reaction of the utility premium to changes in wealth to explain the existence of a precautionary motive for saving. We can derive an individual's prudence probability premium from Eq. (3). Dividing the numerator and the denominator in the righthand side of Eq. (3) by − w1(x − k) yields 2p3 =

the pain elasticity together with the Arrow–Pratt measure of risk aversion to state the following result.

ð22Þ

for all x ≥ 0. If u has more pain elasticity from ε˜ than v, then the percentage change in pain from ε˜ due to a percentage change in wealth is higher for u than for v. In other words, we can say that u's pain from taking ε˜ increases at a higher rate than v's at any wealth level. Therefore, if the two individuals lose the same level of wealth, the one that has more pain elasticity will suffer a larger proportion of pain from ε˜ . We use 1 Crainich and Eeckhoudt (2008) propose a downside risk aversion meau‴ ðxÞ sure dðx; uÞ = ′ and it is obvious that r(x ; u) ≥ r(x ; v) and p(x ; u) ≥ p(x ; v) jointly u ðxÞ imply d(x ; u) ≥ d(x ; v).

−

w1 ðxÞ z ðxÞ ≥− 1 : w1 ðyÞ z1 ðyÞ

ð30Þ

Adding 1 to each side of Eq. (30) yields Eq. (20). □ In general, we need to know the utility function u and the zero-mean risk ε˜ to derive the pain elasticity measure. However, with a CARA utility + function u(x) = − e − ax, where a N 0, we have w1(x) = − e − ax[αe − aε + aε− (1− α)e − 1]. Therefore, ηðx; u; ε˜ Þ = ax for any ε˜ . Given another utility function v(x) = − e − bx, where a N b N 0, we can say that u is more risk-averse and has more pain elasticity than v. Hence u's prudence probability premium is larger than v's. 4. Summary We use Eeckhoudt and Schlesinger's (2006) model of risk apportionment to define the prudence probability premium. We show that, for any small zero-mean risk, if u is more risk-averse and more prudent than v, then u's prudence probability premium is higher than v's. However, when the apportioned risk is large, the two conditions are no longer sufficient. We develop a measure called pain

P. Jindapon / Economics Letters 109 (2010) 34–37

elasticity given ε˜ which is derived from the utility premium function introduced by Friedman and Savage (1948). We show that if u is more risk-averse and has more pain elasticity than v, then u's prudence probability premium is higher than v's even when ε˜ is large. The contributions of this paper include not only the probability premium concept that can be applied to downside risk, but also the applications of the pain elasticity measure. In the analysis of the prudence probability premium, we find that, for a given ε˜ , the pain elasticity can provide a more general result than Kimball's measure of prudence. Since greater pain elasticity means greater proportional change in pain from ε˜ when income decreases, the pain elasticity can be used as an alternative measure of downside risk aversion. In the literature, there are many ways to gauge the intensity of downside risk aversion. Chiu (2005) uses the measure of prudence to compare the optimal choice of self-protection expenditure against downside risk. Crainich and Eeckhoudt (2008) propose u‴ / u′ which is equivalent to the downside risk premium in risk apportionment. Keenan and Snow (2002, 2009) characterize the strength of downside risk aversion by proposing a measure that combines u‴ / u′ and the Arrow–Pratt measure of risk aversion, while Modica and Scarsini (2005) do so by strengthening u‴ / u′ the same way Ross (1981) does with the Arrow–Pratt measure of risk aversion. Jindapon and Neilson (2007) show that Modica and Scarsini's condition and Kimball's measure are equivalent to optimal shifts in the distribution function that reduce downside risk under different assumptions. Recently, Keenan and Snow (2010) establish restrictions on preferences under which greater prudence implies greater downside risk aversion. The relations between the pain elasticity measure and these downside risk aversion measures are left for future research. Acknowledgments The author thanks an anonymous referee, Henry Chiu, Louis Eeckhoudt, Harris Schlesinger, and seminar participants at the 2009 EGRIE conference for helpful comments.

37

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