Multiplicative Risk Prudence *

Transcription

1 Multiplicative Risk Prudence * Xin Chang a ; Bruce Grundy a ; George Wong b,# a Department o Finance, Faculty o Economics and Commerce, University o Melbourne, Australia. b Department o Accounting and Finance, Faculty o Business and Economics, Monash University, Australia. First Drat: November 004 This Drat: December 006 Abstract We examine the optimal saving decision o individuals ho ace a multiplicative risk. An individual is deined to be multiplicative risk prudent i multiplying a pure risk to her uture ealth raises her optimal savings. We sho that convex marginal utility is not suicient to induce multiplicative risk prudent. Instead, an individual is multiplicative risk prudent i and only i her relative prudence o uture consumption uniormly exceeds to. We then study jointly the impact o correlated additive and multiplicative risks on optimal savings decision and demonstrate that the concept o multiplicative risk prudence is stronger than additivemultiplicative risk prudence. Our results suggest one should take the condition o multiplicative risk prudence as a natural restriction on preerence. In addition, our indings provide an explanation to the risk-ree rate puzzle. JEL Classiication: E1, D81 Keyords: Multiplicative risk, Risk prudence, Background risk, Precautionary savings.. * An important part o this paper as done hen Wong as pursuing his Ph.D. study. Wong thanks his supervisors, Christine Bron, Paul Koman and Richard Stapleton or guidance. We thank Ning Gong and seminar participants at the University o Melbourne or helpul comments. # Corresponding author. Department o Accounting and Finance, Building 11E, Monash University, VIC, 3800, Australia. George.Wong@BusEco.monash.edu.au. Tel: Fax:

2 1. Introduction It is a natural belie that an individual ill save more i she is less certain about the level o her uture ealth. The individual ho behaves in this ay is said to be risk prudent and the uncertainty induced additional savings is called precautionary savings. The uncertainty could be either additive or multiplicative to uture ealth. 1 Leland (1968) shos that an individual ho aces an additive risk is risk prudent, i and only i, her marginal utility o uture ealth is convex. Hoever, one question that has not been addressed so ar is: Under hat condition ill the individual subject to multiplicative risk behave in a similar risk prudent ashion? There has been a air amount o literature ocusing on the eect o additive background risk. Undiversiiable labour income risk (hich is an additive background risk) have been studied in the context o optimal portolio choice (e.g., see Heaton and Lucas (1997, 000), Vicera (001), and Franke, Peterson, and Stapleton (005)) and equilibrium asset prices (e.g., see Weil (199)). Moreover, various preerence conditions have been established under hich the presence o additive background risk increases risk aversion toards other independent risks (e.g., see Pratt and Zeckhauser (1987), Kimball (1993) and Gollier and Pratt (1996)). Surprisingly, there has been little attention given to the case here the background risk is multiplicative. Franke, Schlesinger, and Stapleton (005) point out that examples ith multiplicative risk are at least as prevalent as those ith additive background risk and extend some results on optimal bond-stock investment. They establish a set o conditions on preerences under hich the presence o multiplicative background risk makes the investor behave in a more risk-averse manner. 1 Speciically, or an investor ith certain level o ealth, the impact o an additive risk x% on her uture ealth can be ritten as + x%, and the eect o a multiplicative risk y% is given by the product y%. Note that Leland (1968) examines the impact o uture endoment uncertainty on savings here the uture endoment is an additive component o uture ealth. Under risk aversion, the marginal utility is convex i and only i the degree o absolute prudence is uniormly greater than zero. Absolute prudence is deined as the negative o the ratio o the third to the second derivative o the utility unction, The notion o absolute and relative prudence ere irst introduced by Kimball (1990).

3 We believe that multiplicative risk matters to both corporate and individual savings decisions. For instance, a multinational company may be exposed to exchange rate risk hich is multiplicative to its overseas proit, and this additional risk may aect the optimal cash holding decisions o the company. Similarly, an individual savings decision may be inluenced by the inlation uncertainty hich is multiplicative to her uture real ealth. This paper investigates the impact o multiplicative risk on optimal savings decision. We introduce a concept called multiplicative risk prudence that is analogous to the deinition o additive risk prudence. Multiplicative risk prudence is deined as ollos. 3 Deinition 1 An individual is multiplicative risk prudent, i multiplying a pure risk to her uture ealth raises her optimal savings. Since the seminal paper by Leland (1968), the eect o absolute prudence on savings has became obuscated by generality. A prudent individual (hose marginal utility is convex) is expected to save more i she is less certain about the level o her uture ealth. Our analytical and numerical results indicate that this ell-accepted generality does not hold hen the uncertainty attached to the uture ealth is multiplicative. Using a simple numerical example, e irst sho that the ell-accepted condition o risk prudence (i.e. convex marginal utility) ails to induce multiplicative risk prudence. Assuming additively-seperable (AS) utility, e then derive the necessary and suicient condition o multiplicative risk prudence, hich is stronger than the one suggested by Leland (1968). We then analyze the optimal savings in response to an increase in multiplicative uncertainty olloing Rothschild 3 Note that a multiplicative risk is deined to be a pure risk i the expected value o the uture ealth is unaected by the multiplicative risk (i.e. the multiplicative risk has an expected value o one). 3

4 and Stiglitz (1971), and ind that a multiplicative risk prudent individual ill alays raise her optimal savings in response to an increase in multiplicative uncertainty. 4 We urther generalize our analysis o multiplicative risk prudence and the sensitivity o optimal responses to the case o non-additively-separable (NAS) utility. Utility unction is oten assumed additively-seperable in the literature or its mathematical convenience in deriving analytical results. Yet, this condition can be restrictive. For example, it does not allo or habit ormations, a term or the idea that the marginal utility o uture consumption is increasing ith the level o past consumption. 5 Our results are unaected by the introduction o this more general orm o utility unction. Finally, e study jointly the impact o correlated additive and multiplicative risks on optimal savings decision. An individual is deined to be additive-multiplicative risk prudent, i the joint presence o additive and multiplicative pure risks raise her optimal savings. Many prior studies on background risk assume that background risks are independent. Tsetlin and Winkler (005) point out that the independence assumption is oten unrealistic, ignoring the correlation o background risk may lead to a poor decision. Their results suggest that the optimal risky choices in the correlated setting can be very dierent rom those that ould appear optimal i the correlation ere ignored. In practice, additive and multiplicative risks and their correlation can be very important to corporations and individuals in making their cash holding and saving decisions. Additive risk can come in the orm o overseas proit uncertainty or a multinational company or uture endoment uncertainty or an individual. One may expect overseas proit or uture endoment to be correlated ith the exchange rate or inlation and jointly aect optimal savings. Our analysis suggests that hen additive and multiplicative risks are independent, the concept o multiplicative risk prudence and additivemultiplicative risk prudence are equivalent. Hoever, hen additive and multiplicative risks 4 Kimball (1990) also exploits the observation o Rothschild and Stiglitz (1971) to analyze the sensitivity o an optimal responses, though he ocuses on the case o additive risks. 5 See Constantinides (1990) and Campbell and Cochrane (1999). 4

5 are positively correlated, the concept o multiplicative risk prudence is stronger than additivemultiplicative risk prudence. Our results on the impact o multiplicative risk on aggregate savings also provide an explanation to the risk-ree rate puzzle. As stated by Mehra and Prescott (1985), the model o the representative consumer o Lucas (1978) predicts a risk-ree rate that is too high given the observed lo variability o consumption groth. Intuitively, an increase in inlation uncertainty in an economy should induce aggregate precautionary motive to save. This must be compensated by a reduction o risk-ree rate. Fail to take inlation uncertainty into the account may lead to an overestimation o risk-ree rate. The remainder o the paper is organized as ollos. We set up our basic model and present a motivating example in Section to demonstrate the insuiciency o convex marginal utility to induce multiplicative risk prudence. Section 3 derives the necessary and suicient condition or multiplicative risk prudence and studies the eects o an increase in multiplicative uncertainty on optimal savings. We urther generalize the concept o multiplicative risk prudence and the sensitivity o optimal responses to case o NAS utility. Section 4 studies the impact o correlated additive and multiplicative risks on optimal savings decision and Section 5 concludes.. Is a prudent individual really risk prudent? An individual is called prudent i her marginal utility is convex. As suggested by Leland (1968), a risk prudent individual is expected to save more i she is less certain about the level o her uture ealth. Is a prudent individual really risk prudent? We sho that, indeed, a prudent individual need not behave in a risk prudent ashion..1 Model 5

6 Assume a to-dated model here the dates are indexed 0 and. Consider the saving (s) decision o an individual ho is endoed ith initial ealth 0. In addition to the initial ealth, the individual is also endoed ith ealth at time. The individual chooses optimal savings at time 0 in order to maximize the expected utility unction deined over the consumption plan (c 0, c ). The utility unction U(c 0, c ) is assumed to be additively-separable, so that U(c 0, c ) = u 0 (c 0 ) + u (c ), here, u 0 and u represent the utility unctions deined over the consumption at time 0 and, respectively. 6 Let R represent the gross deposit rate o savings. To simpliy our analysis, e ollo Leland (1968) by assuming a ixed rate o return on savings. 7 The consumptions c 0 and c can be ritten as c 0 = 0 - s and c = + R s. The maximization problem o the individual ith certain uture ealth is the olloing max u ( s) + u ( + R s) (1) s 0 0 The F.O.C. o problem (1) is u s = R u + R s, () * * 0( 0 ) ( ) here s * represents the optimal savings and u 0 and u denote the marginal utilities at time 0 and, respectively. Suppose that the individual is exposed to a pure multiplicative risk y% at time such that E y% =1. Since our ocus is on the impact o multiplicative risk on savings decision, the uture endoment, hich is additive to uture ealth, is assumed to be constant or the 6 The case o nonseparable utility is addressed in Section 3. 7 Leland (1968) argues that most savings, including bank deposits and government bonds, oer constant rates o return. 6

7 moment. 8 The maximization problem and its corresponding F.O.C. are ritten, respectively, as max ( s) = u ( s) + Eu (( + R s) y% ), (3) s 0 0 and u ( s$ ) = R Eyu % (( + R s$ ) y% ), (4) 0 0 here ŝ represents the optimal savings or an individual acing a multiplicative risk. An individual is said to be multiplicative risk prudent i the existence o multiplicative risk y% results in precautionary savings (i.e. ŝ s * ). It is orth pointing out that in contrast to problem (3), Leland (1968) examines the optimal savings problem o an individual hose uture endoment is stochastic, max ( s) = u ( s) + Eu ( % + R s), (5) s 0 0 here % is the stochastic uture endoment. The F.O.C. o problem (5) can be ritten as u ( s) = R Eu ( % + R s), (6) 0 0 here s represents the optimal savings or an individual ith an additive risk.. A numerical example Assume the additively-seperable constant relative risk aversion (CRRA) utility unction, U(c 0, c ) = u 0 (c 0 ) + u (c ), here, u ( c ) = βu ( c ), (7) 0 8 It is noteorthy that this assumption is merely or simplicity. The case o the stochastic and correlated uture endoment is discussed in Section 4. 7

8 γ c0 u0( c0) = βu0( c ), (8) γ and γ < 1 under risk aversion. 0 β 1 is the discount actor or the utility on consumption at time. For simplicity, e isolate the ealth eect rom the consumption smoothing eect in our analysis by assuming R = 1, β = 1 and 0 = = 1. Given these assumptions, there is no incentive or any individual to save under certainty (i.e. s * = 0). 9 Deine a tri-variate random variable, y% 0 = (1.3, 1, 0.8), ith probability distribution, P( y% 0 ) = (0.3, 0., 0.48). y% 0 is a pure risk since E y% 0 =1 and it is a multiplicative (additive) risk i it is multiplicative (additive) to the uture ealth. 10 It is idely believed that a prudent individual ill save more i she is less certain about the level o her uture ealth. This is the notion o precautionary savings. While the motive o precautionary savings is natural, the convex marginal utility is not suicient to induce precautionary savings. This can be illustrated by considering the optimal savings o a prudent individual ith γ = Table 1 shos the optimal savings o the individual ith γ = 0.5. Optimal savings are computed under three situations - here her uture ealth is exposed to no uncertainty, additive uncertainty and multiplicative uncertainty, respectively. 1 The individual ill save % (-0.585%) o her ealth hen y% 0 is additive (multiplicative). Contrary to the common belie, the prudent individual saves less hen her uture ealth is exposed to multiplicative uncertainty. 9 One can check that s * = 0 by solving F.O.C. () or any value o γ. 10 Speciically, y% 0 is a multiplicative (or additive) risk i the uture ealth is given by ( + R s) y% 0 (or y% 0 + R s). 11 Given the CRRA utility in equation (8), the degree o absolute prudence is given by ( - γ) /. It is positive or all γ < (i.e. all individuals ith γ < are prudent). 1 The optimal savings under multiplicative (additive) uncertainty is computed by solving equation (4) (equation (6)), here u and u 0 are given by equation (7) and (8). 8

9 Figure 1 plots the degree o relative prudence on the horizontal axis ith the optimal savings on the vertical axis. 13 The solid line (dashed line) represents the optimal savings or the case here the uncertainty is multiplicative (additive). Note that the optimal savings are increasing ith the degree o relative prudence (i.e. an individual ill save more hen she is more prudent). More importantly, Figure 1 shos that the convex marginal utility is insuicient to induce precautionary savings or the case here uncertainty is multiplicative (i.e. optimal savings are negative hen the relative prudence is belo ). This implies that a prudent individual need not behaves in a risk prudent ashion. Like risk aversion, risk prudence is generally vieed as a natural human behaviour. Given that the uncertainty o uture ealth can take dierent orms, i one believes that additive risk prudence is a natural assumption, then there is no reason hy e should ignore the assumption o its multiplicative counterpart. Our numerical results suggest that a stronger condition than the traditional convex marginal utility is required or an individual to be risk prudent. 3. The condition o multiplicative risk prudence In this section, e derive the condition on preerences or an individual to be multiplicative risk prudent. We irst look at the case here the utility unction is additivelyseparable (AS), olloed by the derivation or the non-additively-separable (NAS) utility. 3.1 Additively-separable (AS) utility Recall that an individual is said to be multiplicative risk prudent, i ŝ s *. This implies that ( s ) = u ( s ) + R Eyu % (( + R s ) y% ) 0. (9) * * * 0 0 Rearranging inequality (9) and using equation () yields 13 The relative prudence is deined as ealth multiplied by the absolute prudence (i.e. u ( )/ u ( ) ). Given the CRRA utility in equation (8), the degree o the relative prudence is given by - γ. 9

10 Eyu % R s y% u R s (10) * * (( + ) ) ( + ). Thus, the equivalent condition to multiplicative risk prudence is y %, : Ey% = 1 Eg ( y% ) g (1), (11) here g y yu y R s * ( ) = ( ), and = +. We ind that condition (11) holds, i and only i, p, here u ( ) p ( ) = u ( ) represents the degree o relative prudence at time. The suiciency is immediate rom Jensen's inequality and simple algebra. That is, Eg ( y% ) g (1) i g ( y) is convex in y (i.e. p ). To sho the necessity, consider the risk y% ε ith the probability distribution, P( % ε = 1 + ε) = P( % ε = 1 ε) = 1/. Using a Taylor series expansion, one can sho that or suiciently small ε, 1 Eg( y% ε ) g(1) = g (1) E % ε, (1) here ε% is a risk ith the probability distribution, P( % ε = ε) = P( % ε = ε) = 1/. Equation (1) is positive, i and only i, g(1) 0, or equivalently, p. Thus, i p < or some, there ill exist some risk y% ε ith small ε such that the individual is not multiplicative risk prudence. This leads directly to the olloing proposition. Proposition 1 An individual is multiplicative risk prudent, i and only i, or equivalently, g( y) = yu ( y) is convex in y, p ( ), here u ( ) p ( ) = represents the degree o relative prudence at time. u ( ) 10

11 Leland (1968) demonstrates that or an individual acing an additive pure risk to raise savings, the degree o the absolute risk prudence, u ( ), must be greater than zero. In u ( ) contrast, our Proposition 1 states that the uncertainty o uture ealth, caused by a pure multiplicative risk, raises optimal savings i and only i the relative prudence uniormly exceeds to. Given that an individual cannot be a net borroer hen she retires (i.e. 0 at time ), the concept o the multiplicative risk prudence is stronger than that o the additive risk prudence. Thus our Proposition 1 suggests a ne necessary and suicient condition that needs to be imposed on preerences or individuals to be risk prudent. Our result o the multiplicative risk prudence may have implications or the risk-ree rate puzzle. Speciically, the risk-ree rate puzzle means that the representative consumer model o Lucas (1978) predicts a risk-ree rate that is too high given the observed lo variability o consumption groth. Consider the representative consumer hose maximization problem is deined in equation (3). Using equation (4) and assuming that the market clears at ŝ = 0, one can have R, y% = u 0( 0), Eyu % ( y% ) (13) here R, y % represents the risk-ree rate in the presence o inlation uncertainty y%. I the representative consumer is risk prudent, condition (11) holds and the risk-ree rate is loer in the presence o inlation uncertainty. Intuitively, the presence o inlation uncertainty gives rise to an aggregate precautionary motive to save. This must be accompanied by a reduction in risk-ree rate. The ailure to take inlation uncertainty into account may lead to an overestimation o risk-ree rate Note that the model in equation (13) is used only to illustrate that the implication o multiplicative risk prudence or the risk-ree rate puzzle. It assumes a deterministic consumption groth and the only ay o shiting consumption through time is via a nominal bond. The deterministic groth assumption is merely or 11

12 A natural extension o Proposition 1 is to examine the eect o an increase in multiplicative risk on optimal savings. Consider a pair o multiplicative risks, y% 1 and y%, such that y% 1 is an increase in risk o y% in the sense o Rothschild and Stiglitz (1970) (i.e. y% second order stochastically dominates y% 1 ). Intuition suggests that the individual ill save more under y% 1. We sho that, hoever, the individual ill save more under y% 1, i and only i, the individual is multiplicative risk prudent. Consider an individual ith utility as a unction o some control parameter α and a random variable θ %. The individual chooses α in order to maximize her expected utility max E U ( % θ, α). α Rothschild and Stiglitz (1971) sho that i U ( θ, α) is a convex unction o θ and α decreasing in α, an increase in risk o θ % ill raise the optimal α. 15 Let U( y%, s) = u ( s) + Eu (( + R s) y% ), this gives rise to an immediate generalization o 0 0 Proposition 1. Proposition For any pair o multiplicative risks y% 1 and y%, such that y% second order stochastically dominates y% 1, y% 1 induces greater optimal savings, i and only i, p ( ). Taken together, our Proposition suggests that an increase in inlation uncertainty ill encourage optimal savings i the individual is risk prudent. By the same token, i the simplicity and ill be relaxed in Section 4. The equilibrium model hich incorporates inlation uncertainty and allos individuals to shit consumption through time via nominal bond or stock can be ounded in Chang, Grundy, and Wong (006). 15 Kimball (1990) also exploits the observation o Rothschild and Stiglitz (1971) to analyze the sensitivity o an optimal response, though he ocuses on the case o the additive risk. 1

13 representative consumer is risk prudent, then an increase in inlation uncertainty should lead to a reduction in risk-ree rate. 3. Non-additively-separable (NAS) utility We extend our analysis o multiplicative risk prudence to the case o non-additivelyseparable (NAS) utility. Since AS utility is only a special case o NAS utility, the conditions o multiplicative risk prudence in the case o NAS utility ill be more restrictive than (or, at least, as restrictive as) p. Without the loss o generality, e assume = 0 and R = 1. The individual chooses optimal savings at time 0 in order to maximize the expected utility unction deined over the consumption plan (c 0, c ), max E U( s, sy% ). (14) s 0 The optimal solution o problem (14) exists i U is concave in s, i.e., du U 11 U 1 U 1 U = + 0. (15) ds For the case o AS utility, U 1 = 0, condition (15) holds naturally. For the case o NAS utility, condition (15) holds i U 1 0, i.e., a condition hich allos or internal habit ormation - a term or the situation here the marginal utility o uture consumption is increasing ith the level o past consumption (e.g., see Constantinides (1990) and Campbell and Cochrane (1999)). Let * s s U 0 s s = arg max (, ), the individual is multiplicative risk prudent, i and only i, EyU % ( s, s y% ) EU ( s, s y% ). (16) * * * * implies that * * * * Note that, U ( s, s ) EU ( s, s y% ), i and only i, U 1 is concave in y. This

14 U ( s, s ) = U ( s, s ) EU ( s, s y% ), (17) * * * * * * Thus, condition (16) holds i the unction g y = yu s s y is convex in y (i.e. * * ( ) ( 0, ) U c U ( c, c ) 0 by algebra). Given that AS utility is only a special case o NAS utility, e ( c 0, c ) have the olloing proposition hich is an extension o Proposition 1 to the case o NAS utility. Proposition 3 A suicient condition o multiplicative risk prudence or NAS utility is U c U ( c, c ) 0 and 1 0. ( c 0, c ) U (18) A necessary condition is U ( c, c ) 0 c U ( c 0, c ). U( c0, c ) As in the case o AS utility, the term c is a measure o relative prudence U ( c, c ) o the time consumption. In particular, i the utility is additively-separable, U 1 = 0, and 0 = U c0 c u c 0 U ( c, c ) u ( ) c c c, Proposition 3 is reduced to Proposition 1. (, ) ( ) Campbell and Cochrane (1999) sho that habit ormation provides an explanation to the risk-ree rate puzzle, i.e., R ith habit ormation R ithout habit ormation. In contrast, our Proposition 3 suggests that the presence o multiplicative risk and habit ormation together may provide an even stronger explanation to the risk-ree rate puzzle, i.e., R ith habit ormation and y% R ithout habit ormation. According to Proposition 3, equation (19) holds i the relative prudence o time consumption uniormly exceeds to (the necessary condition o multiplicative risk prudence), and the marginal utility o past consumption is increasing (the necessary condition o internal 14

15 habit ormation) and concave in the level o uture consumption. Folloing the observation o Rothschild and Stiglitz (1971), one can urther generalize Proposition 3 as ollos. Proposition 4 For any pair o multiplicative risk y% 1 and y%, such that y% second order stochastically dominates y% 1, y% 1 induces greater amount o optimal savings, i U ( c, c ) 0 c and U1 0. U ( c 0, c ) U( 0 s, sy) The proo is immediate given that s is convex in y, i and only i, su + U + syu 0. (0) 1 U( c0, c ) Equation (0) holds i c and U1 0. U ( c, c ) 0 Note that Proposition 4 is simply a generalization o Proposition to case o NAS utility. Similar to Proposition, Proposition 4 also predicts that an increase in inlation uncertainty ill encourage optimal savings i the individual ith NAS utility is risk prudent. 4. Additive-multiplicative risk prudence So ar, the uture endoment has assumed to be constant. To ensure that our results are robust to the stochastic uture endoment, e assume that in addition to multiplicative risk y%, the individual is exposed to uture endoment uncertainty, % = (1 + % ), here E % = 0 and % > 1. We study the concept o additive-multiplicative risk prudence. An individual is said to be additive-multiplicative risk prudent, i the joint presence o additive and multiplicative pure risks raises her optimal savings. We sho that the concept o multiplicative risk prudence is stronger than additive-multiplicative risk prudence. This implies that the 15

16 condition o multiplicative risk prudence is alays suicient to induce risk prudent behaviour. Most research on background risk assumes that background risks are independent o each other. As pointed out by Tsetlin and Winkler (005), the independence assumption is oten unrealistic and ignoring the correlation o background risks may lead to a poor decision. Accordingly, e allo % and y% to be correlated. We only consider the case here the correlation beteen % and y% is positive, simply because a positive correlation beteen % and y% ill make uture ealth more uncertain and should urther induce precautionary motive to save. Whereas a negative correlation beteen % and y% gives rise to a hedging eect hich can be an ally and need not induce precautionary savings. In Appendix e ill demonstrate ho a negative correlation beteen % and y% gives rise to a hedging eect. To allo or the correlation beteen % and y% hile isolating the change in endoment risk eect, e let % ( a) = 1 ax% + az%, here 0 a 1, z% = y% 1, and x% and z% are i.i.d. a is the correlation coeicient beteen % and y%. % and y% are independent hen a = 0 and perectly positively correlated hen a =1. 16 Note that the variance o a %( ) is constant in a, i.e., the change in the correlation does not aect the endoment risk. have The utility is assumed to be additively-seperable. By the convex marginal utility, e Eyu % ax% az% s R y% Eyu % az% s R y% * * (( ( ) + ) ) (( (1 + ) + ) ). 16 I y% is the rate o inlation hich is random, the independence o % and y% implies the neutrality o money. While the positive correlation beteen % and y% leads to Tobin's portolio shit eect (namely, the inlation increases capital per capita hich in turn increases output per capita along the groth path). 16

17 The presence o the correlated multiplicative and additive risks induces precautionary savings, i and only i, Eyu % az% s R y% u s R (1) * * (( (1 + ) + ) ) ( + ). By applying the technique similar to the one employed or proving Proposition 1, one can sho that condition (1) holds i, y, : g ( y) 0, and only i, g (1) 0, here g y = yu + a y + s R y By simple calculus and algebra, e have * ( ) (( (1 ( 1)) ) ). y, : g ( y) 0, i and only i, 17 p y,, ( + (1 a + s R / )/ ay) * () and g (1) 0, i and only i, p. ( + (1 a + s R / )/ a) * (3) Given that * (1 a + s R / ) / a 0, 3 p (a condition stronger than convex marginal utility) is necessary to induce precautionary savings. When % and y% are independent (i.e. a = 0), conditions () and (3) become p (the equivalent condition o multiplicative risk prudence). For all 0 a 1, the right hand side o condition () alays lies beteen 3 and. This leads to the olloing proposition. Proposition 5 I % and y% are independent, the concepts o additive-multiplicative risk prudence and multiplicative risk prudence are equivalent; I % and y% are positively correlated, the concept o multiplicative risk prudence is stronger than additive-multiplicative risk prudence. The suicient and the necessary conditions o additive-multiplicative risk prudence are given by conditions () and (3), respectively. 17 The detailed proo can be ound in Appendix. 17

18 In other ords, Proposition 5 suggests that hen % and y% are independent, the equivalent condition o additive-multiplicative risk prudence is p. Whereas, hen % and y% are positively correlated, the relative prudence must be at least greater than 3/ in order to induce precautionary savings. To see hy the condition o multiplicative risk prudence is stronger than that o additive-multiplicative risk prudence, note that in comparing to the presence o an independent additive risk, the presence o a positively correlated additive risk makes the uture ealth more uncertain. Thereore, a loer relative risk prudence is required to induce precautionary savings. 5. Conclusion Leland (1968) shos that an individual ho aces an additive risk is risk prudent, i and only i, her marginal utility o uture ealth is convex. Our analysis suggests that convex marginal utility is not suicient to induce risk prudent behaviour and a stronger condition on preerence is required or an individual to be risk prudent. The stronger concept called multiplicative risk prudence is introduced and its corresponding condition on preerence is derived. An individual is multiplicative risk prudent i and only i her relative prudence o uture consumption uniormly exceeds to. A multiplicative risk prudent individual ill alays raise her optimal savings hen acing a multiplicative uncertainty or in response to an increase in multiplicative uncertainty. We generalize the concept o multiplicative risk prudence to the case o nonadditively-separable utility. An individual ith NAS utility is multiplicative risk prudent i her relative prudence o uture consumption uniormly exceeds to and her marginal utility o current consumption is concave in uture consumption. 18

19 The concept o multiplicative risk prudence provides an explanation to the risk-ree rate puzzle. Intuitively, i the representative consumer is multiplicative risk prudent, the presence o inlation uncertainty ill induce aggregate precautionary motive to save. This must be compensated by a reduction o risk-ree rate. Fail to take inlation uncertainty into the account may lead to an overestimation o risk-ree rate. Finally, e study jointly the impact o correlated additive and multiplicative risks on optimal savings decision and ind that our result is robust and is unaected by the presence o correlated additive background risk. An individual is said to be additive-multiplicative risk prudent, i the presence o positively correlated additive and multiplicative pure risks raise her optimal savings. We sho that the concept o multiplicative risk prudence is stronger than additive-multiplicative risk prudence. Like risk aversion, risk prudence has been regarded as a natural human behaviour. As uncertainty o uture ealth can take dierent orms, i one believes that additive risk prudence is a natural assumption, then there is no reason hy e should ignore the assumption o its multiplicative counterpart. Our results suggest that the condition o multiplicative risk prudence is alays suicient to induce risk prudent behaviour in dierent situations. Accordingly, one should take the condition o multiplicative risk prudence as a natural assumption. 19

20 Reerences Campbell, J., and J. Cochrane, 1999, By orce o habit: A consumption-based explanation o aggregate stock market behaviour, Journal o Political Economy 107: Constantinides, G. M., 1990, Habit ormation: A resolution o the equity premium puzzle, Journal o Political Economy 98: Chang, X., and B. D. Grundy and G. Wong, 005, Inlation uncertainty and asset pricing, Working paper, Monash University. Franke, G., S. Peterson, and R. Stapleton, 005, Long-term portolio choice given uncertain personal savings, Working Paper, The University o Manchester. Franke, G., H. Schlesinger, and R. Stapleton, 006, Multiplicative background risk, orthcoming in Management Science. Gollier, C., and J. W. Pratt., 1996, Risk vulnerability and the tempering eect o background risk, Econometrica 64: Heaton, J., and D. J. Lucas, 1997, Market rictions, saving behaviour and portolio choice. Macroeconomic Dynamics 1: Heaton, J., and D. J. Lucas, 000, Portolio choice and asset prices: The importance o entrepreneurial risk, Journal o Finance 55: Kimball, M. S., 1990, Precautionary savings in the small and in the large, Econometrica 58: Kimball, M. S., 1993, Standard risk aversion, Econometrica 61: Leland, E. H., 1968, Savings and uncertainty: the precautionary demand or saving, Quarterly Journal o Economics 8: Lucas, R., 1978, Asset prices in an exchange economy, Econometrica 46: Mehra, R., and E. Prescott, 1985, The equity premium: A puzzle, Journal o Monetary Economics 10: Pratt, J. W., and R. Zeckhauser, 1987, Proper risk aversion, Econometrica 55: Rothschild, M., and J. Stiglitz, 1970, Increasing risk: I. A deinition, Journal o Economic Theory : Rothschild, M., and J. Stiglitz, 1971, Increasing risk: II. Its economic consequences, Journal o Economic Theory 3: Tsetlin, I., R. Winkler, 005, Risky choices and correlated background risk, Management Science 51:

21 Vicera, L. M Optimal portolio choice or long-horizon investors ith nontradable labour income. Journal o Finance 55: Weil, P., 199, Equilibrium asset prices ith undiversiiable labor income risk, Journal o Economic Dynamics and Control 16:

22 Table 1: The optimal savings o a prudent individual ith γ = 0.5. The optimal savings is expressed in percentage ealth. NU, MU and AU represent the case here the individual is exposed to no uncertainty, multiplicative uncertainty and additive uncertainty, respectively. Optimal savings (%) NU MU AU Figure 1: The optimal savings under uncertainty vs relative prudence. The optimal savings (in percentage) is plotted on vertical axis and the degree o relative prudence is plotted on the horizontal axis. The solid line (dashed line) represents the optimal savings or the case here the uncertainty is multiplicative (additive) Optimal Savings (%) Relative Prudence

23 Appendix The case here % and y% are negatively correlated: Since y% 0 and an individual cannot be a net borroer hen she retires, % and y% cannot be negatively correlated. I % and y% are negatively correlated, i.e., % ( a) = 1 ax% az%, then can be negative or some y > 0. Thereore, to examine the hedging eect o negatively related background risks, e let a %( ) = 1 ax% az %, (A1) here % = % and 1 z y μ μ 1 = Ey%. x% and z % are i.i.d. a measures the degree o negative relation beteen % and y%. % and y% are independent hen a = 0 and perectly negatively related hen a = 1. Given equation (A1), it is obvious that 0 or all y 0. By convex marginal utility e have Eyu % ax% az% R s y% Eyu % az% R s y% * * (( ( ') + ) ) (( (1 + ') + ) ). Let g y = yu + a y + R s y then 1 * ( ) (( (1 ( μ) ) ), Eyu % az% Rs y% u a Rs (A) * * (( (1 + ') + ) ) ( (1 + (1 μ)) + ), i y, : g ( y) 0, and only i, g (1) 0. Since μ > 1 and by risk aversion, the olloing condition holds naturally, u a R s u R s * * ( (1 + (1 μ)) + ) ( + ). and only i, By simple calculus and algebra, condition (A) holds i 1 ay p +, y (A3) * (1 aμ) + R s a p +. (A4) * (1 aμ) + R s Again, hen % and y% are independent (i.e. a = 0), conditions (A3) and (A4) reduce to p (the equivalent condition o multiplicative risk prudence). Note that i condition (A4) ails to hold, then it is possible to exist some y% such that 3

24 u a R s u R s Eyu % a y% s R y% * * 1 * ( (1 + (1 μ)) + ) ( + ) (( (1 + ( μ)) + ) ). Given that y 0 and an individual cannot be a net borroer hen she retires, the right hand side o conditions (A3) and (A4) are greater than to (i.e. condition hich is stronger than multiplicative risk prudence). This demonstrates ho a negative relation beteen % and y% gives rise to a hedging eect. The hedging eect is stronger hen % and y% are more negatively related. As a increases, the second term in conditions (A3) and (A4) increases. This is because a higher a value gives rise to a stronger hedging eects, thus a higher degree o prudence is required in order to induce precautionary savings. Proo o condition (): g y ay a s R u y a s R ay u * * ( ) = (6 + ((1 ) + / )) + ((1 ) + / + ) 0 By algebra, the equivalent condition o g ( y) 0 is p Substituting into the above inequality * * (6 ay+ ((1 a) + s R / ))(1 a + s R / + ay) *. (1 a + s R / + ay) (6 ay + ((1 a) + s R / )) u + y((1 a) + s R / + ay) u = + + * ((1 a) s R / ay) ay. and rearranging terms yield conditions () and (3). * * 4