Asset Management with Endogenous Withdraws under a Drawdown Constraint

Asset Management with Endogenous ithdraws under a Drawdown Constraint Hervé Roche Finance Department, Universidad Adolfo Iáñez, Avenida Diagonal las Torres 2640, Peñalolén, Santiago, Chile March 18, 2014 Astract Individuals driven y capital accumulation may e reluctant to experience large wealth downfalls. Implications for optimal consumption and investment policies are explored in a dynamic setting where wealth is restrained from falling elow a fraction of its all-time high. Risky investment regulates wealth growth and mitigates the ratchet effect of the constraint, and may decrease as wealth approaches its maximum. The correspondence found etween hait formation over consumption and wealth ratcheting provides a rational explanation for the extensive use of such a practice in investment management. An extension emeds the spirit of capitalism using wealth as an index for social status. Journal of Economic Literature Classification Numers: D81, E21, G11 Keywords: Optimum Portfolio Rules, High ater Mark, Ratchet Effect, Endogenous Hait Formation. Tel: +56 2 2331 1642, Fax: +56 2 2331 1906; E-mail address: herve.roche@uai.cl Hervé Roche) 1

1 Introduction The desire for wealth accumulation is well estalished in the literature. According to Max eer 1958), man is dominated y the making of money, y acquisition as the ultimate purpose of his life. Economic acquisition is no longer suordinated to man as the means for the satisfaction of his material needs. The recent explosion and success of capital guarantee funds reveal that investors are looking for downside protection ut at the same time upside potential, in particular for long term investments such as 401k) s and especially during financial trouled times. as a university or a foundation may also seek asset preservation. Fund trusts and institutions such hen a non-profit organization receives an endowment and other long-term funding, it has to manage these resources prudently y estalishing a spending policy that accommodates the need for asset protection and portfolio growth. Usually donors require endowment assets to e kept permanently and prohiit grantees from using or orrowing against principals. The aim of such spending rules is to preserve financial independence and to avoid the purchasing power erosion over time 1. This suggests that investors often use all-time record levels as a enchmarks to measure fund performances and high-water marks 2 are common in the investment management industry. In everyday life, households perform similar mental accounting: many homeowners still ear in mind the value of their property once reached at the dawn of the financial crisis and may feel today disappointed when comparing it with current market value, even though their equity may have significantly increased since they moved in. This psychological feature has prompted some financial services firms to offer their customers the following portfolio insurance strategy: an investor who stays invested until the fund matures is guaranteed to receive a value equal to the highest value of the fund ever achieved, even if the fund s daily value has fallen since its highest point. In this paper, we analyze the intertemporal investment-consumption rules for an infinite lived individual maximizing her expected discounted utility under wealth ratcheting. Namely, the agent 1 In the US, trustees and charity professionals who run foundations are only oliged to spend as little as 5% a year of the capital. In many foundations, capricious and poorly thought out projects or programs were undertaken to fulfill the interests of trustee managers not the wishes of the founder 2005). 2 The high-water mark is a target value that can depend on the current asset value of the fund. It is periodically adjusted due to withdraws, allocated expenses and a contractual growth rate. In the simplest case, the high-water mark is the highest level the asset has reached in the past. 2

does not tolerate losing more than a fixed percentage of her all-time high level of wealth 3. Our motivation for such specification of preferences is twofold. The first one deals with the investor s ego that dictates an extreme form of loss aversion over wealth since utility can e defined to e minus infinity if the drawdown constraint is violated. The concept of loss aversion postulates that the impact of a loss is greater than that of an equally sized gain. Investors may like to exhiit their superior forecasts aout the market movements, ut are less keen on acknowledging some wrong predictions. After a significant loss in the stock market, an investor may lament or lame herself aout her decision to invest in stocks. At a personal level, she may interpret her loss as a sign that after all, she is not the first rate investor she thought she was, thus inflicting a painful low to her ego. At a social level, she may feel humiliation as the news spread among colleagues, friends and family memers. The second motivation captures the fact that when an investor s wealth goes up, she may psychologically commit part of the profits made in order to finance future expenditures, including sometimes important lifetime decisions such as early retirement. Being forced to revise downward her plans after a sudden drop in wealth may turn out to e quite painful. The key intuition ehind most of the results is driven y two effects. First, as in any portfolio selection prolem with restrictions on asset positions, the agent is concerned with hedging motives that in the future the constraint may e inding. As a enchmark, hedging concerns are addressed in a simpler framework when the investor is required to maintain her wealth aove a fixed floor university charter requirement). Essentially, risk aversion is enhanced, which leads to smaller stock holdings and lower consumption plans with respect to the unconstrained case. Both optimal allocations are found increasing in wealth. Second, the drawdown constraint displays a ratcheting feature since each time financial wealth reaches a new record high, the minimum floor rises and the restriction ecomes more stringent. The agent has two margins of adjustment at her disposal to regulate the growth of her wealth: consumption and risky investment. The latter is the most sensitive of the two as it governs the diffusion component of the wealth process. The optimal solution of the model reflects the tradeoff etween consuming today and deferring consumption to take advantage of investing in the stock market, which may e thwarted y the presence of the ratchet. e derive conditions under which, as 3 This constraint was first introduced y Grossman and Zhou 1993) to examine the prolem of maximizing the long term growth rate of expected utility of final wealth. Their analysis is quite insightful ut they do not allow for endogenous withdraws from the fund to finance intermediate consumption. Cvitanic and Karatzas 1995) extend their work to a more general class of stochastic processes. More recently, Sekine 2013) and Cherny and O lój 2013) generalize the analysis to the case of a non-linear drawdown constraint when the riskfree rate is equal to zero. 3

wealth approaches its all-time high, the fraction of wealth invested in stocks decreases and possily approaches zero. This last case occurs when the investor is fairly impatient, namely when the pure time discount rate exceeds the interest rate, and the drawdown coefficient exceeds a cut-off value. Impatient agents wish to consume a large fraction of their current wealth so they are very reluctant to ratchet up the minimum floor; as a consequence, the optimal wealth process never achieves a new height. Tracking wealth movements, the optimal consumption policy exhiits a ratcheting ehavior and large drawdowns from its all-time consumption level never occur. e highlight the correspondence etween wealth ratcheting and hait formation in the spirit of Duesenerry 1949). This twin ratcheting is an important result that rationalizes the loss aversion over wealth for an investor who aims to maintain her standard of living. An extension of the asic model emeds the spirit of capitalism y including wealth, an index of social status, inside the utility function 4. Persistent enefits derived from uilding up status lead to a more aggressive risky investment policy whereas consumption ecomes less appealing. This paper uilds on the dynamic portfolio choice literature. Early works on optimal consumptioninvestment allocations in a frictionless market and no orrowing restrictions include Samuelson 1969) and Merton 1971). Further, attention has een paid on more real world situations where investors face constraints in their portfolio investments 5. In general, the optimal strategy differs from the unconstrained one as the agent aims at hedging against the constraint at some cost) since even though the constraint may not e inding, there is a possiility that it does in the future. Recent papers focus on portfolio allocations under wealth performance relative to an exogenous enchmark such as in Browne 1999) and Tepla 2001) or suject to growth ojectives required y the decision maker as in Hellwig 2004). In Carpenter 2000), the fund manager is compensated with a call option on the wealth she manages with a enchmark index as strike price. The author shows that the option compensation does not necessarily lead to more risk seeking. Goetzmann, Ingersoll and Ross 2003) and Panageas and esterfield 2009) study hedge fund compensation schemes when managers 4 For instance see Bakshi and Chen 1996) and Smith 2001). 5 Cvitanic and Karatzas 1992) and Cuoco 1997) develop a general martingale approach to cope with convex contemporaneous constraints on trading strategies which includes the case of incomplete markets and prohiited short sales. Cuoco and Liu 2000) analyze the optimal consumption portfolio choice prolem under margin requirements and evaluate the cost of the constraint. He and Pages 1993) and El Karaoui and JeanBlanc-Picqué 1998) treat the case of non-negative wealth in presence of laor income. Grossman and Villa 1993) followed y Villa and Zariphopoulou 1997) study the consumption-portfolio prolem for a CRRA investor facing a leverage constraint. 4

perceive a regular fee proportional to the portfolio asset value and an incentive fee ased on the fund return each year in excess of the high-water mark. Consistent with empirical evidence, they otain that a significant proportion of managers compensation can e attriuted to the incentive fee, in particular for high volatility asset funds for which high manager skills are required. Janeček and Sîru 2011) examine a similar prolem while allowing for endogenous withdraws from the fund. Guasoni and Olój 2013) analyze the optimization prolem faced y an isolastic utility fund manager whose compensation includes oth maintenance and incentive fees that maximizes her long term expected utility of terminal wealth while eing restricted to invest the collected fees into the risky asset. They otain that the optimal investment strategy consists in investing a constant fraction of the wealth into the risky asset that only depends on the fund manager s CRRA coefficient and incentive fee. Finally, in a closely related paper, Elie and Touzi 2008) analyze the consumption-investment prolem under a drawdown constraint for a general class of utility functions and stock returns ut restrict their attention to a zero risk free rate setting. As shown in Grossman and Zhou 1993), when the riskfree rate is equal to zero, the optimal allocations have a simpler structural form as the drawdown constraint never inds. More importantly the optimal policy has very different properties from the ones derived in this paper that assumes a positive interest rate; in particular, when the riskfree rate is set to zero, the optimal consumption pattern does not exhiit hait formation. The paper is also related to the trend of research that strives to provide some alternative to the usual time separale von Neumann-Morgenstein preferences whose performance has een poor from an empirical point of view. Such preferences have failed to explain important facts aout stock returns; for instance the equity premium see Mehra and Prescott 1985)). A popular attempt to provide a more sophisticated specification for utility is hait formation 6 that postulates that agents not only derive utility from current consumption ut also from consumption history, typically captured y a standard of living index. For tractaility reasons, many models assume that the agent derives utility from the excess etween current consumption and the hait level. If the marginal utility at zero is infinite, the standard of living index acts as a floor level elow which current consumption does not fall. This addictive feature - optimal consumption levels can only increase across time regardless of the state of the economy- is not supported y empirical evidence. Detemple and Karatzas 2003) address this issue and investigate the case of finite marginal utility of consumption at zero when imposing a non-negativity constraint on consumption plans. hen the shadow price of consumption is high, the 6 See for instance Sundaresan 1989), Constantidines 1990), Detemple and Zapatero 1991), Campell and Cochrane 1999). 5

agent optimally reduces her consumption along with her standard of living and the associated cost of haits as well. An alternative approach proposed y Dyvig 1995) is to ratchet current consumption. Originally, Duesenerry 1949) argued that consumption may not e entirely reversile over time ut instead may increase along with income and decline less than proportionally with it. Dyvig 1995) formalizes this idea y looking at an extreme form of hait formation where consumption is prevented from falling over time. It is straightforward to extend Dyvig s analysis to the case of an agent who is intolerant to any decline in consumption that exceeds a fixed proportion of her all-time consumption. e show that ratcheting wealth induces a ratcheting ehavior on consumption with a strong parallel with Dyvig 1949). Essentially, the framework developed in this paper is a mirror image of the one proposed in Dyvig 1995). Finally, our model orrows concepts from the psychology literature. The idea that investors are more sensitive to wealth reductions than wealth increases loss aversion) is the central feature of the prospect theory developed y Kahneman and Tversky 1979) and Tversky and Kahneman 1991). Bareris, Huang and Santos 2001) explore the implications on asset prices of loss aversion y considering an investor who derives utility not only from consumption ut also from changes in the value of her financial wealth. Here we take a different approach. The enchmark for loss aversion experienced y the investor is the all-high level of wealth. The paper is organized as follows. Section 2 descries the economic setting and contains an heuristic derivation of the optimal consumption and portfolio allocations for a CRRA utility agent. In section 3, we discuss an extension of the asic model that emeds the spirit of capitalism using wealth as a proxy for social status. Section 4 displays a verification theorem that formally proves that the heuristic solution is indeed valid. Section 5 concludes. Proofs of all results are collected in the appendix. 2 The economic setting Time is continuous. An infinitely lived investor, who is reluctant to let her wealth fall more than a fraction of its historical maximum, has to optimally allocate her wealth etween a risk-free ond, a risky asset and consumption. Individual preferences. There is a single perishale good availale for consumption, the numéraire. 6

Preferences are represented y a time additive utility function Uc) = E 0 [ 0 ] uc s )e θ s ds, where the instantaneous utility function u is twice continuously differentiale, increasing and strictly concave and θ denotes the sujective time discount rate. In addition, u satisfies the following Inada conditions: lim c) = and lim u c) = 0. In the sequel, we focus our analysis on an individual c 0 +u c with constant relative risk aversion preferences c 1 1 uc) =, 1 ln c, = 1. Following Cuoco 1997), in order to ensure that the expected utility U is well defined, we require consumption plans ĉ to satisfy 7 min {E 0 [ 0 ] uĉ s ) + e θs ds, E 0 [ 0 ]} uĉ s ) e θs ds <, 1) where x + respectively x ) denotes the positive respectively negative) part of x. Financial market. There are two securities availale in the financial market: - a risk-free ond whose price B evolves according to db s = r B s ds, where r is the constant interest rate, and, - a stock index whose price S follows a geometric Brownian motion ds s = S s µ ds + σdw s ), suject to some given initial stock price S 0, where dw s is the increment of a standard iener process w, µ is the mean return of the stock index S and σ 2 is its instantaneous variance. All the stochastic processes considered in the paper are assumed to e adapted on a common filtered proaility space whose filtration is the one induced y the oservations of w. 7 This assumption is only needed for the logarithmic preferences case. hen uc) has a constant sign, Leesgue Monotone Convergence Theorem can e applied directly to prove step 4 of the Verification Theorem provided in section 4 of the paper. 7

Let x and ẑ e respectively the amount of dollars invested in the riskless ond B and risky security S, so that the wealth process Ŵ is equal to x+ẑ. A consumption plan ĉ is feasile if there is a trading strategy ẑ such that dŵs = r Ŵ s ĉ s + ẑ s µ r ))ds + σẑ s dw s, Ŵ s > K, 2) with K > 0. The condition Ŵs > K rules out aritrage opportunities, such as douling strategies presented in Harrison and Kreps 1979). Drawdown constraint. For λ > 0, define M t = sup { M 0 e λt, Ŵse λt s) ; 0 s t}. 0 s t λ is the minimum) growth rate of the floor required y the investor. As introduced in Grossman and Zhou 1993), the drawdown constraint is Ŵ s α M s, 3) for some α in 0, 1]. This constraint indicates that the investor is reluctant to let her wealth fall elow a fraction of its maximum to date adjusted for some minimum growth rate λ. Grossman and Zhou 1993) argue that a large drawdown typically aove 25 percent) is often a reason for firing fund managers. In the investment management industry, a realistic estimate of α ranges from 75 to 88 percent. In practice, different values of α may apply to different types of traders. For instance, for proprietary traders internal hedge fund traders) who invest money elonging to their company, α can depend on the target amount of money a trader is required to generate during the year and could e as high as 94 percent. Stop loss strategies are often used y traders for downside protection. These strategies allow an investor to automatically unwind a position; they can e very effective at limiting losses if the market moves quickly against her. e first review the main results for the unconstrained prolem studied y Merton 1971). 2.1 Merton prolem ithin our financial market framework, the Merton prolem 1971) for a CRRA investor is [ F Ŵt) = max E ĉ 1 ] s t s t) ĉ,ẑ) 1 e θ ds, suject to the udget constraint 2) and Ŵt > 0 given. t 8

Merton 1971) shows that the optimal strategy c f, z f ) is linear in wealth with ẑf s and σ ) 2 ĉ f s = 1 Ŵ s A, provided that 1 A = θ + 1 r + µ r ) 2 > 0. The optimal wealth process 2σ Ŵ f is a geometric 2 Brownian motion whose dynamics are dŵ f t = Ŵ f t r 1A + µ r ) 2 σ 2 )dt + µ r ) σ dw t. Ŵ s = µ r In order to gain insights aout the effects of the drawdown constraint 3), we examine a simpler consumption-portfolio choice prolem where wealth is required to e kept aove a fixed minimum floor adjusted for some required growth rate. quantify hedging motives. This natural enchmark allows us to isolate and 2.2 Benchmark case: fixed minimum floor prolem Consider a foundation whose charter stipulates that the endowment αm > 0 adjusted for some required growth rate λ > 0 cannot e used for expenditures only the returns are eligile). No other constraint is assumed regarding the growth ojectives of the trust fund of a charitale foundation or a university. At any time t, wealth Ŵt must e maintained aove a minimum level αme λt. Let us define t Ŵte λt, c t ĉ t e λt and z t ẑ t e λt. Due to the linearity of the wealth dynamics and the homogeneity of the utility function, the investor s prolem can e written [ F t ) = max E c 1 ] s t c,z) 1 e θs t) ds t s.t. d s = r s c s + z s µ r)) ds + σz s dw s s αm, t > 0 given, where the parameters adjusted for growth are r = r λ, µ = µ λ, and the adjusted time discount rate θ = θ + 1)λ is assumed to e strictly positive. e still require A > 0 and in addition we make the following assumptions: A1. The interest rate adjusted for inflation is positive: r > 0. A2. The excess mean return of the risky asset is positive: µ r > 0. Assumption A1. is required for feasiility; when r = 0, the prolem also admits a solution that is analyzed separately. Assumption A2. is made for convenience and without loss of generality. First of all, note that the value function F is increasing and concave 8 in. Then, for αm, the 8 The strict concavity of F comes from the fact that the utility function is strictly concave and the constraint is linear so that if and are admissile wealth processes, then for all λ in [0, 1], λ + 1 λ) is also admissile. 9

Hamilton Jacoi Bellman HJB) equation of this prolem is θf = max c,z) The optimality conditions are c 1 1 + r c + zµ r)) F + σ2 2 z2 F. and F satisfies the following non-linear ODE c = F ) 1 z µ r)f = σ 2 F, θ F = F ) 1 1 + r F 1 µ r) 2 F ) 2 2 σ 2 F. 4) Lemma 1 The general solution of ODE 4) is such that = AF )) 1 + L1 F )) β 1 1 + L 2 F )) β 2 1, 5) where β 1 and β 2 are respectively the positive and negative roots of the quadratic 1 µ r) 2 1 2 2 σ 2 x 2 + A r 1 µ r) 2 ) 2 2 σ 2 x = 1 A, 6) and L 1 and L 2 are two constants to e determined. Proof. See the Appendix. Useful results β 1 > 1 and 1 β 2 > 0 are proved in the Appendix. Boundary condition at the minimum floor. At = αm, we have αm = AF αm)) 1 + L1 F αm)) β 1 1 + L 2 F αm)) β 2 1, 7) and in order not violate the constraint with some positive proaility in a near future, stock holdings must e zero, which implies A = β 1 1)L 1 F αm)) β 1 + β 2 1)L 2 F αm)) β 2. 8) hen is large, the constraint is equivalent to 0, so the solution is equivalent to the one for the unconstrained case, i.e. F ) A ). Since β 2 1 relationships 7) and 8) yields αm L 1 = β 1 ) β1 ) A 1 β1 > 0. β 1 1 10 < 1, we must have L 2 = 0. Comining

) Note that at = αm, the wealth dynamics are deterministic as d t = r β 1 1 β 1 A t dt. Since r β 1 1 β 1 A = 1 µ r ) 2 2 σ β1 1) > 0, the wealth process ounces ack upward after hitting the minimum floor. Finally, oserve that as r goes to zero, we have β 1 = 1. In this case, we find that F ) = A αm) 1 1 with c = αm A and z = µ σ 2 αm), i.e., the optimal investment strategy elows to the constant proportion portfolio insurance class CPPI) as developed y Black and Perold 1992). The optimal wealth process is given y t = αm + 0 αm)e θ + 1+ µ 2 2 2 σ 2 σ2 2 )t+σwt and clearly the minimum floor is never reached within a finite time. 2.2.1 Properties of the optimal allocations The consumption-wealth ratio c is given y c = 1. A + L 1 F )) β 1 It is increasing in wealth and always smaller than in the unconstrained case. The fraction of wealth invested in the stock is given y z = µ r σ 2 1 β 1 + β 1 A A + L 1 F )) β 1 This ratio is monotonic increasing) in wealth and always smaller than in the unconstrained case. The reason is the rise of the relative risk aversion of the lifetime utility in wealth since ) F β 1 L 1 F )) β 1 F = 1 + >. A + 1 β 1 )L 1 F )) β 1 At the floor = αm, the relative risk aversion is infinite and consequently holdings in stock are zero. Note that, in general, the risky investment strategy is not of the CPPI type. As wealth increases, lifetime utility relative risk aversion decreases and as wealth ecomes very large, the effects of the constraint vanish: optimal allocations converge to the optimal unconstrained ones. Our analysis has shown that in presence of a fixed minimum floor, hedging motives induce a reduction in consumption and risky investment and enhance risk aversion. In the next section, the investor s aility to control the minimum floor comined with a ratchet effect lead to quite different properties of the optimal strategy as stock holdings and consumption plans serve as wealth growth regulators. ). 11

2.3 Consumption-portfolio choice prolem under a drawdown constraint As efore define c t ĉ t e λt, z t ẑ t e λt, t Ŵte λt and M t M t e λt so that the drawdown constraint 3) is equivalent to M t = sup {M 0, s ; 0 s t}. 0 s t Choosing adapted consumption plan c and adapted risky investment strategy z, the agent aims at maximizing her lifetime utility [ F t, M t ) = max E c 1 ] s t c,z) t 1 e θs t) ds s.t. d s = r s c s + z s µ r)) ds + σz s dw s s αm s, t > 0, M t > 0 given. 9) It is assumed that α 0, 1] and parameters θ, r, µ r and A are all positive 9. This preference specification intends to capture two aspects that we think are important for understanding investor ehavior. The first aspect is loss aversion over wealth driven y ego considerations. The second feature we wish to take into account is the fact that as an investor ecomes richer, she may psychologically commit part of the profits made to finance future expenditures such as a new house or early retirement and savor the thought of extra future consumption. A sudden significant loss in wealth may really hurt. e start the analysis y reviewing some useful properties of the maximum process M and the value function F. 2.3.1 Properties of the maximum process PM1. As mentioned in Grossman and Zhou 1993), M is a continuous increasing process and thus a finite variation process. PM2. Denoting y [X, Y ] the quadratic covariation etween processes X and Y, we have d [M, ] t = 0 and d [M, M] t = 0. 2.3.2 Properties of the value function P1. F is strictly increasing in, decreasing in M and strictly concave in, M). P2. F is homogeneous of degree 1 in, M). 9 The case r = 0 has een extensively studied y Elie and Touzi 2008) and leads to optimal allocations that exhiit distinct properties than the ones derived in this paper. 12

Proof. See the Appendix. Property P2 implies that F, M) = M 1 fu), with u = M and some smooth, concave and strictly increasing function f. 2.3.3 Derivation of the value function Given the properties of the maximum process M, for αm, M), the HJB associated to the investor s program is θf = max c,z) The optimality conditions can e written c 1 1 + r c + zµ r)) F 1 + σ2 2 z2 F 11. 10) c = Mf u)) 1 z = µ r)f u) σ 2 uf u), and for u α, 1) the reduced) value function f satisfies the non-linear ODE θfu) = f u)) 1 1 + ruf u) 1 µ r) 2 f u)) 2 2 σ 2 f u). 11) As shown in lemma 1, the general solution f of the ODE 11) is such that u = Af u)) 1 + K1 f u)) β 1 1 + K 2 f u)) β 2 1, 12) where K 1 and K 2 are two constants to e determined. In the sequel, we find that K 1 > 0 and K 2 < 0. Interpretation of the solution The optimal wealth process is the sum of three terms: = AMf u)) 1 + K1 Mf u)) β 1 1 + K 2 Mf u)) β 2 1. The first term is the one found for the Merton prolem and indicates that wealth is used to finance consumption plans. The second term is positive and incorporates hedging motives as in the fixed minimum floor prolem. This term can e related to portfolio insurance strategies involving simple options such as in Black and Perold 1992) and Tepla 2001). Finally, the third term is also a hedging component: it is negative and regulates the growth rate of the wealth to mitigate the ratchet effect of the stochastic floor. The next step is to derive the oundary conditions at u = α and u = 1. 13

2.3.4 Boundary conditions The oundary conditions are derived in the Appendix. In summary, at u = α, holdings in the risky asset must e zero. At u = 1, the condition must ensure that the Hamilton Jacoi Bellman equation still holds. There are two possiilities depending on the parameters: either F 2 M, M) = 0 or holdings in the risky asset is set to zero, z 1) = 0. henever condition F 2 M, M) = 0 leads to a gloally concave 10 value function F, this is the optimality condition. Otherwise, z 1) = 0 must e imposed. Denoting Y = f 1)) 1 and X = f α)) 1, the oundary conditions are αx = A + K 1 X β 1 + K 2 X β 2 A = β 1 1)K 1 X β 1 + β 2 1)K 2 X β 2 Y = A + K 1 Y β 1 + K 2 Y β 2 { A β 1 1)K 1 Y β 1 β 2 1)K 2 Y β 2 = max 0, 1 β } 1β 2 A 0 Y ) 1 1, where A 1 0 = θ+ 1)r and when = 1, Y = A = A 0 = 1 θ. The existence and uniqueness of the quadruple K 1, K 2, X, Y ) are shown in the Appendix and we find that K 1 > 0 and K 2 < 0. The first three oundary conditions are the same whether F 2 M, M) = 0 or z 1) = 0 is imposed. For 1, when F 2 M, M) = 0 is imposed, the fourth condition is A β 1 1)K 1 Y β 1 β 2 1)K 2 Y β 2 = 1 β 1β 2 A 0 Y ), 13) 1 and for = 1, Y = A = A 0, whereas when z 1) = 0 is imposed, the fourth condition is A = β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2. Let S denote the 4 y 4 non-linear system defined y the first three oundary conditions and equation 13). Proposition 1 If the solution of system S is such that, whenever 1 1), Y A 0 Y A 0 ), then F 2 M, M) = 0 is the optimality condition, otherwise, z 1) = 0 is optimal. If θ r, in fact F 2 M, M) = 0 is always optimal. Conversely, if θ > r, then for 1, there exists a critical value α 0, 1) such that for all α 0, α ), F 2 M, M) = 0 is optimal. At α = α, we have F 2 M, M) = 0 and z 1) = 0. Then, for all α α, 1], z 1) = 0 is optimal and wealth can never reach a new height. Proof. See the Appendix. 10 In the Appendix, we show that the value function F is gloally concave iff β 1 1)K 1Y β 1 + β 2 1)K 2Y β 2 A. 14

As developed in more details in the sequel, the intuition ehind the results of proposition 1. is the agent s willingness of mitigating the ratchet impact and the irreversile associated cost) of the drawdown constraint. An individual who highly discounts the future θ large) would like to consume most of her current wealth. Increasing the minimum floor αm implies putting a cap on the amount of current wealth that can e used to finance current consumption, which may turns out e too costly for such an agent, especially when the drawdown coefficient α is large. In the sequel, we show that maintaining wealth within a fixed and [αm 0, M 0 ] has interesting implications for the optimal consumption process. A more precise characterization of the cut off value α is availale. At α = α, we have Y = A 0. Let X e the unique) root of the equation with A = β 1 1)K 1 X β 1 + β 2 1)K 2 X β 2, K 1 = β 2A β 2 1)A 0 β 1 β 2 A β 1 0 and K 2 = β 1A + β 1 1)A 0 β 1 β 2 A β 2 0. Then the cut off value α is given y α = AX 1 + K 1 X β 1 1 + K 2 X β 2 1. For the logarithmic preferences case, = 1, Proposition 1 implies that F 2 M, M) = 0 is always optimal since in this case solution Y of system S is equal to A 0. hen z 1) = 0 is optimal, we show in the Appendix that we always have Y A 0. Finally, note that when the drawdown constraint inds = αm, the wealth dynamics ecome deterministic In the Appendix, we estalish that r > 1 αx d t = r 1 αx ) tdt. is always satisfied, so that after hitting the minimum floor the wealth process remains aove the minimum floor αm in the next instant. Having solved the HJB equation and determined the oundary conditions at u = α and u = 1, we now analyze the properties of the optimal allocations. Special case r = 0. Closed form solutions are availale in this limit case. hen r = 0, we have β 1 = 1 and consumption must e zero at = αm. It follows that we must have X = and K 1 = α. Solving for Y and K 2, we find that Y = β 2 A β 2 1)1 α)+[1 α α ]+ and K 2 = A β 2 1 Y β 2, with α = These results are consistent with the findings y Elie and Touzi 2008). 15 1 β 2.

2.4 Properties of the optimal allocations 2.4.1 Consumption Optimal consumption c is implicitly defined y the relationship ) c M = G, 14) M where Gx) = Ax + K 1 x 1 β 1 + K 2 x 1 β 2. Since G > 0, the optimal consumption c is increasing in current wealth. Then, recall that the consumption wealth ratio is given y c = 1 u f u)) 1, so ) c = 1 f u u 2 u) ) 1 uf ) u) f u) > 0, since due to hedging motives, we estalish in the sequel that z < µ r σ 2, which implies that the lifetime utility relative risk aversion uf u) f u) is aove its unconstrained level. The consumption-wealth ratio c is increasing in the ratio current wealth over its peak M, so in particular increasing in current wealth and decreasing in the historical maximum level of wealth M. At the ceiling = M, we have c M = 1 Y. If > 1, we have Y A 0 A so we can conclude that for all u in [α, 1], c < 1 A. Recall that the intertemporal elasticity of sustitution IES) s is equal to 1. An investor who is reluctant s < 1) to alter her consumption plans overtime chooses to consume a lower fraction of her wealth than she does in the unconstrained case. If < 1 and F 2 M, M) = 0 is optimal, we have Y < A. An investor willing to alter her consumption plans s > 1) has a consumption-wealth ratio that is larger than in the unconstrained case for large values of the ratio M. For α close to 1, this property is gloal 11 i.e. for all u in [α, 1], c > 1 A. If < 1 and z 1) = 0, oth cases 1 Y 1 Y < 1 A are possile. > 1 A and Next, we show that optimal consumption inherits a ratcheting ehavior from wealth and hait formation endogenously arises. All-time high consumption and hait formation. Denoting c M t = sup {c s} the maximum to 0 s t date level of consumption, for 0 s t, we have 1 X c s M s 1 Y. Since M s M t, it follows that for all date t, Y X c t c M t. The current consumption level c t over its peak c M t remains within the fixed and [α c, 1], with α c = Y X < 1. The maximum drawdown in consumption from its previous all-time high is 1 α c and 11 In the limit case α = 1, when F 2M, M) = 0 is optimal, we have X = Y = A 0. Since for < 1, continuity in α, for α large enough, it follows that αx < A, which implies that c > 1, for all u in [α, 1]. A 16 1 A 0 > 1 A, y

it decreases as α goes up see the Appendix). Imposing ratcheting on the wealth process induces a ratcheting ehavior of the optimal consumption plan as posited y Duesenerry 1949) and analytically derived y Dyvig 1995). hen the investor does not tolerate any decline in consumption, Dyvig estalishes [ that ] for all times t, the wealth process t must optimally e maintained within the and c Mt r, β 2 1)c M t β 2 r. This implies that current wealth t must e kept aove the proportion β 2 β 2 1 of its peak M t. Grossman and Zhou 1993) claim that the reason for such a restriction on the manager s investment policy is that the owner of the fund psychologically and often physically) commits to use part of the profit when reaching the peak. Dyvig argues that imposing a drawdown constraint on wealth seems ad hoc from an economic point of view, and his motivation was to offer an alternative to the work y Grossman and Zhou 1993). Although the prolem studied here and Dyvig s model are not equivalent, our analysis provides a ridge etween the two approaches as well as an economic justification in terms of preferences hait formation) over consumption for downside protection on wealth. The drawdown constraint 3) is a practical and effective way to ensure that standard of living will not have to e lowered y too much in the case of an adverse shock. e now investigate the impact of the magnitude of the drawdown proportion α on the consumptionwealth ratio. Proposition 2 If z 1) = 0 is optimal, the more stringent the drawdown constraint higher α), the smaller the consumption-wealth ratio for all u in [α, 1]. hen F 2 M, M) = 0 is optimal, if 1, the previous result remains valid. However, if < 1, there is a critical value u α in α, 1), such that the consumption-wealth ratio decreases in α on [α, u α] and increases on [u α, 1]. Proof. See the Appendix. Proposition 2 suggests that for an investor with a high IES s > 1), when wealth is aout to reach its peak, for large values of α, the investor relies on the consumption margin to regulate the growth of her wealth and dampen the ratchet effect. 17

Figure 1 : Consumption-wealth ratio c as a function of M µ = 0.12, r = 0.04, σ = 0.2, θ = 0.06, = 2.5 Figure 2 : Consumption-wealth ratio c as a function of M µ = 0.12, r = 0.04, σ = 0.2, θ = 0.06, = 0.8 The consumption-wealth ratio c constraint parameter α. In Figure 1, for > 1, as α goes up, c is displayed in Figures 1 and 2 for several values of the drawdown 18 uniformly shrinks and remains elow

the unconstrained ratio 1 A = 0.0672. The reduction in consumption is large when wealth is close the minimum floor. For α = 0.6, 0.8, 0.9 and 0.95, the endogenous ratchet coefficient for consumption α c is 0.29, 0.43, 0.53 and 0.61 respectively. In figure 2, for < 1, curves cross with one another and as asserted in proposition 2 when u is high enough, an increase in α leads to a higher consumption-wealth ratio that significantly exceeds the unconstrained ratio 1 A = 0.04. For α = 0.6, 0.8, 0.9 and 0.95, the values otained for α c are 0.14, 0.22, 0.28 and 0.34 respectively. Oserve that when < 1, larger drawdowns 1 α c from all-time high consumption level are allowed than in the case > 1, reflecting the fact that individuals with a higher IES tolerate larger changes in their consumption plans across time. 2.4.2 Asset allocations The fraction of wealth invested in the risky asset is given y z = µ r σ 2 1 β 1K 1 f u)) β1 + β 2 K 2 f u)) β 2 A + K 1 f u)) β 1 + K 2 f u)) β 2 ). 15) The fraction of wealth invested in the risky asset is lower than in the unconstrained case, i.e. This is due in part to the hedging motives as descried in the section 2.2. In addition, numerical simulations displayed in the sequel) indicate that the investor s desire to dampen the ratchet effect plays a significant role in explaining the reduction in risky investment. Finally, note that the optimal risky investment strategy is not of the CPPI type as in Dyvig 1995) where the demand for stocks is proportional to the excess of wealth over the perpetuity value of current consumption or as in Grossman and Zhou 1993) for the case λ = 0, Sekine 2013) and Cherny and O lój 2013) where risky investment is linear in the excess of wealth over the minimum stochastic floor. µ r σ 2. Proposition 3 hen F 2 M, M) = 0 is optimal, if > 1 < 1) and θ < 1 Y θ > 1 Y ), the fraction of wealth invested in the risky asset is non-decreasing in the ratio M ; otherwise it is hump-shaped. hen z 1) = 0 is optimal, the fraction of wealth invested in the stock and the ratio M an inverted U-relationship. are linked y Proof. See the Appendix. Conditions for the logarithmic investor are more cumersome and are presented in the Appendix. 19

Proposition 3 deserves several oservations. First, raising the fraction of wealth invested in stocks when wealth goes up is optimal only if the cost associated with the ratchet effect is not too large. Oserve that θ is the consumption-wealth ratio at u = 1 for the myopic investor = 1). hen > 1, the investor s IES is low s < 1) and she is mainly concerned with the current consumption-wealth ratio and is reluctant to defer consumption. Proposition 3 suggests that the agent optimally chooses an increasing risky investment policy provided that at u = 1, c M = 1 Y is aove the corresponding value for the myopic investor. Conversely, if < 1, the individual is eager to defer consumption and accepts a low level of her current consumption-wealth ratio elow that of the myopic investor) at u = 1; consequently the fraction of wealth invested in stocks is increasing. Second, decreasing stock holdings as a percentage of wealth when is close to M depart from the results otained in Grossman and Zhou 1993) where the fraction of wealth invested in stock always increases in the ratio M. Recall that in Grossman and Zhou 1993) there is no intermediate consumption so intertemporal consumption sustitution plays no role. Nevertheless, the hump-shaped relationship corroorates the intuition pointed out y these authors, i.e. αm is expected to grow at a faster rate than and therefore investment in the risky asset is expected to fall. The lifetime utility relative risk aversion uf u) f u) is no longer decreasing as current) wealth rises ut instead is U-shaped. The condition for the ratio z to e non-decreasing in M depends on all the parameters of the model. hen F 2 M, M) = 0 is optimal, we estalish in the Appendix that a raise in the drawdown coefficient α increases decreases) Y whenever > 1 < 1) and lim Y = A 0. From Proposition 3, we α 1 deduce that a sufficient condition for z to e non-decreasing in M is simply θ < r, i.e. the investor must e sufficiently patient. Proposition 4 The more stringent the drawdown constraint higher α), the smaller the fraction of wealth invested in the risky asset. Proof. See the Appendix. Proposition 4 formally states that when the drawdown constraint ecomes more stringent the fraction of wealth invested in the stocks z is the favored channel to achieve wealth growth regulation. is uniformly reduced, suggesting that indeed risky investment 20

Figure 3 : Fraction of wealth invested in stocks z as a function of M µ = 0.12, r = 0.04, σ = 0.2, θ = 0.06, = 2.5 Figure 3 depicts the fraction of wealth invested in the risky asset z for several values of the drawdown constraint parameter α. As α goes up, risky investment is reduced and when α is sufficiently large, the curve z fraction µ r σ 2 is hump shaped. As a enchmark, when α = 0, the unconstrained allocation the = 0.8. Indeed, oserve that even when the current wealth t is far from the minimum floor αm t, the reduction in stock holdings can e sustantial. Oviously, the analysis performed comined oth hedging and ratchet effects. In order to disentangle the two effects, consider a fixed minimum floor equal to αm and compute the fraction of wealth invested in the stock z z when wealth varies from αm up to M. Note that the ratio is 21

independent of the choice of M. Tale I: Disentangling hedging and ratchet effects α u 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.925 0.95 0.975 1 0.6 z 0 0.283 0.383 0.450 0.499 0.537 0.568 0.581 0.592 0.603 0.613 z 0 0.279 0.375 0.436 0.478 0.506 0.526 0.532 0.537 0.540 0.541 0.8 z - - - - 0 0.248 0.339 0.372 0.402 0.428 0.450 z - - - - 0 0.230 0.299 0.318 0.331 0.336 0.334 0.9 z - - - - - - 0 0.168 0.234 0.283 0.322 z - - - - - - 0 0.143 0.184 0.199 0.190 Tale I reports stock holdings z and those corresponding to the drawdown prolem z for several values of α. Recall that for the Merton prolem, this ratio is constant and equal to 0.8. Oserve that hedging motives explain a significant share of the reduction in risky investment. Nevertheless, the ratchet effect ecomes significant when the ratio M constraint ecomes more stringent higher α). approaches 1 and is enhanced as the drawdown Taking the unconstrained portfolio allocation as a enchmark, at u = 1, the ratchet effect accounts for 9%, 14.5% and 16% for α = 0.6, 0.8 and 0.9 respectively of the total reduction in stock holdings. 2.4.3 Representation of the optimal wealth process The optimal policy c, z ) has een expressed in terms of state variales, M) using dynamic programming. Alternatively, it is possile to provide a representation in terms of simple regulated stochastic processes and gain some insights aout the dynamics across time. Details of the derivation are presented in the Appendix. c M Recall that M = G c M ), so that c M = H M ), with H = G 1. First, we estalish that the process is a two sided regulated geometric Brownian motion12 with lower arrier 1 X and upper arrier 1 Y and for u in α, 1), the dynamics are given y ) c d t = c t r M t M t 1A + µ r)2 σ 2 12 For the definition of a regulated Brownian motion, see Harrison [23], p14. )dt + µ r ) σ dw t. 22

Oserve that this law of motion is the same as the one that governs the optimal consumption process in the Merton prolem section 2.1). Second, the representation of current wealth, all time maximum wealth M and consumption c as stochastic processes depends on the oundary condition at u = 1. ) hen F 2 M, M) = 0 is optimal, we show that the process log H t M t is a one sided regulated arithmetic Brownian motion with lower arrier log Y, and for u in α, 1), ) t d log H = r θ µ r)2 + 2σ 2 )dt + µ r σ dw t. M t If z 1) = 0 is optimal, we estalish in proposition 1 that wealth can never exceed its initial) all time high level M 0. The ceiling = M is an upper reflecting arrier; the optimal consumption process c is a two sided regulated geometric Brownian motion. Next, we estimate the cost induced y the constraint. 2.5 Cost of the drawdown constraint There are several ways of estimating the cost of the drawdown constraint. e can assess the loss in terms of forgone lifetime utility; alternatively, we can measure it in terms of the numéraire. e start with the first measure and to keep things simple, we derive the maximum cost when α = 1. 2.5.1 Cost in terms of forgone lifetime utility ealth must always e maintained at its maximum, so in order not to violate the drawdown constraint, holdings in the stock must e zero z 0. It is easy to verify that if r θ, the optimal solution is c = A 0 and wealth dynamics are deterministic with d t = r θ tdt 0. Conversely, if r θ, we have a corner solution with c = r and d t = 0: wealth remains at its initial level 0. The corresponding value function is F 0 ) = K 1 1, with A 0, if r θ K = r 1 θ 1, if r θ. The maximum) cost of the drawdown constraint in terms of loss of the lifetime utility is the relative difference etween the constrained and unconstrained value functions. hen 1, it is simply given y ) K ε 1, A where ε = 1ε = 1) if > 1 < 1). For µ = 0.12, σ = 0.2, θ = 0.05, = 2.5, the loss is approximately 55.4% when r = 0.06. It decreases with the instantaneous variance σ 2 ut increases with the mean 23

return µ. hen r = 0.04, the loss is approximately 151%, which illustrates that imposing a drawdown constraint severely penalizes impatient individuals. 2.5.2 Cost in terms of the numéraire e calculate the percentage k increase in wealth necessary to ring the level of the lifetime utility to the level of those of an unconstrained investor, i.e. we want to determine k such that F 1 + k) ) = A 1 1. e otain k = ) K 1 1, A and for the parameters chosen previously, we find that the percentage increase k is approximately 34.2% when r = 0.06. It also decreases with the instantaneous variance σ 2 and increases with the mean return µ. hen r = 0.04, the necessary percent increase k is 84.7%, again reflecting the high cost of the constraint for impatient individuals. For oth measures used, the cost induced y the constraint is economically significant. 3 Extension of the asic model In this section, we consider the case of an agent who derives utility from current consumption and also from her status. There are two rival theories of social status: ascription versus achievement. Individual position can e ascried y virtue of their age, sex, race, and family memership or connection. Alternatively, individuals can achieve their own position y their own performance and merits. Here, we interpret a society in which higher wealth confers a higher status. Status can confer power, privileges, access to political circles or social events, and at a more personal level, enhance self esteem. As argued in Cole, Mailath, and Postlewaite 1992), social status can determine the degree of success one group memer may have with non-market decisions such as finding a good mate for instance. eer 1968) refers to a status group as a collection of individuals who happen to have a common lifestyle and share the same economic interest. Maintaining one s memership of a status group is certainly desirale and amition may dictate social climing; however, individuals may e reluctant to lower their position in society. People who experience a downward social shift may experience depression or poor psychological well eing. 13 13 The University of Newcastle upon Tyne study y Parker, Pearce and Tiffin 2005) indicates that women are twice as likely to e downwardly moile. The study involved men and women orn in 1947 in Newcastle and followed them 24

Following Bakshi and Chen 1996) and Smith 2001), to keep things simple, we retain current wealth level as an index of status and status seeking is modeled as direct preference for financial wealth. More specifically, preferences 14 are given y c+a ) 1 1, αm uc,, M) =, otherwise, where parameter a > 0 governs how much the agent cares aout her social status. e first examine the optimal consumption-portfolio choices for an unconstrained investor. 3.1 Benchmark case In the asence of status downfall fear, for t > 0 given, the agent aims at maximizing her lifetime utility [ F a t ) = max E t c,z) t c s + a s ) 1 1 ] e θs t) ds s.t. d s = r s c s + z s µ r)) ds + σz s dw s. This optimization prolem can e rewritten F a t ) = max c,z) E t [ t c s ) 1 ] 1 e θs t) ds s.t. d s = r + a) s c s + z s µ + a r + a)))ds + σz s dw s, with c s = c s + a s. hen the optimal consumption allocation c is non-negative, the prolem can e nested in a standard Merton prolem as in section 2.1 with a riskfree rate r + a and a mean return of the stock µ + a. Optimal allocations are given y provided that A 1 a = θ + 1 r + a + 1 2 c = 1 A a a z = µ r σ 2, ) ) 2 µ r σ > 0 and 1 A > a, i.e., a sufficiently small. For our choice of the functional form of the utility function, status seeking only affects the optimal consumption plans. Recall that the agent derives utility through two channels: current consumption c and current wealth. These two channels compete with each other: the higher the consumption, the from childhood to age 50. Researchers noted the findings might e explained y the fact that men orn during that era gained much of their self-esteem from their careers, whereas women found fulfillment from social pursuits outside of work, such as children and friendships. 14 The choice of the functional form of the utility function is motivated y tractaility reasons. 25

lower wealth accumulation and therefore the lower the future status. An increase in status enjoyment higher a) leads to a decrease in the consumption-wealth ratio: the agent chooses to foster wealth accumulation, which is reflected y a higher mean growth of the wealth process. e now study the case when the agent is reluctant to accept large status downfalls. 3.2 Maintaining social status It is easy to realize that the analysis performed in section 2.3 for the case a = 0 still applies if we sustitute 1 A with 1 A a and replace µ, r) with µ + a, r + a) in the definition of roots β 1 and β 2 and provided that the optimal consumption allocation c is non-negative. This last condition is equivalent to f au)) 1 au for all u [α, 1]. As previously derived in section 2.4, function u 1 u f au)) 1 increasing, so it is enough to have the condition met at u = α, i.e. 1 X sufficiently small. e now investigate the quantitative impact of status on the optimal allocations. is αa, which is satisfied for a 0.8 0 0.6 0.4 a 0.045 a 0.03 a 0.015 a 0 0.2 0.80 0.85 0.90 0.95 1.00 Figure 4 : Fraction of wealth invested in stocks z as a function of M µ = 0.12, r = 0.04, σ = 0.2, θ = 0.06, = 2.5, α = 0.8 26

0.07 0 0.06 0.05 a 0 a 0.015 a 0.03 0.04 a 0.045 0.03 0.02 0.01 0.85 0.90 0.95 1.00 Figure 5 : Consumption-wealth ratio c as a function of M µ = 0.12, r = 0.04, σ = 0.2, θ = 0.06, = 2.5, α = 0.8 Figures 4 and 5 represent the fraction of wealth invested in the risky asset z and the consumptionwealth ratio c for several values of the parameter a. As a goes up, the investor increasingly values social status, which leads to heavier stock holdings in an attempt to achieve a higher growth rate of her wealth Figure 4). In parallel, oserve in Figure 5 that the consumption-wealth ratio uniformly shrinks. Priority shifts towards uilding up status that has a persistent impact whereas consumption is less appealing since its effect is only instantaneous. hen a ratcheting ehavior is imposed on wealth, introducing wealth as a proxy for social status in the utility function fosters risky investment in a sustantial manner. 4 The verification theorem The goal of this section is to formally estalish that the heuristic proposed optimal strategy c, z ) is indeed valid. e closely follow Dyvig 1995). The proof consists of four steps. For any feasile strategy c, z), let define the process M t = t 0 uc s )e θs ds + e θt V t, M t ), where V denotes the value function that corresponds the feasile strategy c, z). 27

Step 1: M is a local supermartingale for all admissile strategies c, z) and a local martingale for the proposed) optimal strategy c, z ). There are three regions to examine. First, at t = αm t, any admissile strategy should e such z t = 0 otherwise the drawdown constraint would e violated with a positive proaility in the next instant t + h, h > 0) and it follows that dm t e θt = uc t) θv + 1 2 σ2 z 2 t V 11 )dt + V 1 d t + V 2 dm t. Since z t = 0 and dm t = 0 as t < M t ), we have [ ] E t dm t = r t V 1 θv )dt + uc t ) c t V 1 )dt. Bellman s principle states that e θt 0 = sup uc t ) θv + r t V 1 + µ r) z t V 1 + 1 c t,z t) 2 σ2 zt 2 V 11. Furthermore, since V 1 > 0 and u is strictly concave, for the proposed optimal strategy c, we have u c t ) = V 1, and it follows that r V 1 θv + uc t ) c t V 1 = 0. Since it is a maximum, for all other admissile strategies c, we must have r V 1 θv + uc t ) c t V 1 0, which implies that M is a local martingale for the proposed) optimal strategy c, z ) and a local supermartingale for all feasile strategies c, z). Second, for αm t < t < M t, we have [ ] E t dm t = r t V 1 θv + uc t ) c t V 1 + µ r) z t V 1 + 1 2 σ2 zt 2 V 11 )dt. e θt Since for the proposed) optimal strategy c, z ) V 11 < 0, risky investment z t is such that z t = µ r)v 1 σ 2 V 11 and as efore u c t ) = V 1. Since it is the maximum, all other possile strategies must have a non-positive drift, meaning that M is a local supermartingale. The drift is exactly zero for the optimal proposed strategy, so in this case M is a local martingale. Third, at t = M t, using Ito lemma for semi-martingales, we have E t [ dm t ] e θt = r t V 1 θv + uc t ) c t V 1 + µ r) z t V 1 + 1 2 σ2 z 2 t V 11 )dt + E t [V 2 dm t ], since M is an asolutely continuous process. For h > 0, over the interval of time [t, t + h], the Bellman equation is [ ] V t, M t ) = max E t uc t )h + e θh V t+h, M t+h ), c t,z t) 28

again applying Ito lemma for semi-martingales, we have 0 = max c t,z t) uc t)h + E t +E t [ t+h t [ t+h t V 2 dm s ]. Since E t [M t+h M t t = M t ] = θv + r s c s + z s µ r)) V 1 + 12 σ2 z 2sV ) ] 11 ds 2 π σ z t h + Oh), when h is small, h dominates h so in order for the Bellman equation to hold at t = M t, for the proposed) optimal strategy c, z ) we must have F 2 M t, M t ) = 0 or z t achieves a new maximum so dm t = 0. = 0. In the latter case, we estalish in the Appendix that wealth never Thus, for strategy c, z ), at t = M t, we always have V 2 dm t = 0. Furthermore, y construction, as within the second region, in this case, the drift of dm t is exactly zero, which shows that M is a local martingale for the proposed) optimal strategy c, z ). Since it is a maximum, for all other feasile strategies the drift must e non-positive r V 1 θv + uc t ) c t V 1 + µ r) z t V 1 + 1 2 σ2 z 2 t V 11 + E t [V 2 dm t ] 0. e can conclude that Mt is a local supermartingale for all admissile strategies. Step 2: M is supermartingale for all feasile strategies c, z) and a martingale for the optimal proposed) strategy c, z ). Given some initial condition 0, M 0 ), for any admissile strategy, the corresponding value function cannot e smaller than the one otained with the same admissile strategy ut with initial condition αm 0, M 0 ). At t = αm t, the wealth dynamics are deterministic Choosing c t t r d t = r t c t )dt. induces that d t 0 and for t 0, we have t αm 0. Such a strategy is the worst admissile one and leads to the value function V αm 0, M 0 ) = u αm 0 ) r θ. It follows that M t t 0 = u αm 0 θ u αm 0 r r ). )e θs ds + e θt u αm 0 r ) θ This implies that M is a supermartingale ecause a local supermartingale ounded from elow is a supermartingale. Then, to show that for the optimal proposed strategy, M is a martingale, we follow the proof provided in Dyvig 1995). Lemma 4. in Dyvig 1995) p307 remains valid for our proposed optimal strategy since the wealth process is ounded away from 0 as all t 0, t αm 0 > 0 29

and for the proposed optimal strategy the fraction of wealth invested in the risky asset zs s is in the compact set [ 0, µ r ] σ. Furthermore, notice that Lemma 5 in Dyvig 1995) p307 also remains valid. 2 Comining oth Lemmas, we can conclude that M is a martingale for the proposed optimal strategy. Step 3: Asymptotic ehavior of the residual term e θt V t, M t ). For all feasile strategies V t, M t ) u αm 0 r ) θ, which implies that for all > 0 αm0 lim inf V t, M t )e θt θt u r ) lim e 0. t t θ For > 1, for all feasile strategies, the value function V is always negative, so the previous inequality implies that lim V t, M t )e θt = 0. t Then, for < 1, differentiating relationship 12) with respect to recall u yields f u) f u) = A f u)) 1 β 1 1 + K 1 f u)) β 1 1 + β 2 1 K 2 f u)) β 2 1, 16) Since f < 0, comining relationships 12) and 16) leads to β 1 Af u) 1 β 2 β 1 )K 2 f u)) β 2 1 + β 2 1)u β 2 Af u) 1 β 1 β 2 )K 1 f u)) β 1 1 + β 1 1)u, for u [α, 1]. This implies that A β2 β 2 1 ) ) u f u) A β1 u. 17) β 1 1 From the reduced) HJB equation 11), using relationship 16) we find that θfu) f u)) 1 1 + ruf u) + 1 2 µ r σ ) 2 Af u)) 1. Using relationship 17), we otain that for u [α, 1], fu) Ku 1, where 1 µ r) 2 ) ) 1 K = 1 + 1 θ 2 2 σ 2 A A 1 β2 + r ) β1 β 2 1 θ A > 0 β 1 1 is a constant, and we have F, M) K 1. For the case = 1, the HJB equation is θfu) = ln f u) 1 + ruf u) 1 µ r) 2 f u)) 2 2 σ 2 f u), and again using relationships 16) and 17), we otain that for all u [α, 1], fu) ln u θ L = ln θ 1 + ln β 2 1 β 2 30 + r θ θ β 1 1 β 1 + 1 2 µ r) 2 A σ 2 + L, where

is a constant, and we have F, M) ln θ u αm 0 r ) θ 1 + L. In addition, recall that for any feasile strategy V, M). Following the same steps as Dyvig 1995) in lemma 6 p308, it follows that, for lim F t, M t )e θt = 0. t Step 4: Optimality of the proposed solution c, z ). Assumption 1) implies that a consumption plan c that satisfies E 0 [ 0 uc s )e θs ds ] < must e such that 0 uc s ) + e θs ds < and 0 uc s ) e θs ds < almost surely). Then, for all t 0, we have t 0 uc s) + e θs ds < 0 uc s ) + e θs ds, and y assumption E 0 [ 0 uc s ) + e θs ds ] <. Using Leesgue Monotone Convergence Theorem, we otain that [ t ] [ ] lim E 0 uc s ) + e θs ds = E 0 uc s ) + e θs ds. t 0 0 The same property holds for the negative part u of the utility function u. Finally since uc) = uc) + uc), it follows that [ t ] [ ] lim E 0 uc s )e θs ds = E 0 uc s )e θs ds. t 0 0 For the proposed optimal strategy c, z ), M is a martingale and lim t E 0 [ e θt F t, M t ) ] = 0. Therefore F 0, M 0 ) = M0 ] = E 0 [Mt for all times t 0 ] = lim E 0 [Mt t [ t ] = lim E 0 uc s )e θs ds + e θt F t, M t ) t 0 [ t ] [ ] = lim E 0 uc s )e θs ds + lim E 0 e θt F t, M t ) t 0 t [ ] = E 0 uc s )e θs ds, 0 where exchanging the expectation and the limit is justified y uc s ) u Ms X ) u M 0 X ). For any feasile 31

strategy c, z), M is a supermartingale and lim t inf E 0 [ e θt V t, M t ) ] 0. Therefore This completes the proof. V 0, M 0 ) = M0 ] E 0 [Mt for all times t 0 ] lim E 0 [Mt t [ t ] = lim E 0 uc s )e θs ds + e θt V t, M t ) t 0 [ t ] [ ] = lim E 0 uc s )e θs ds + lim E 0 e θt V t, M t ) t 0 t [ ] E 0 uc s )e θs ds. 0 5 Conclusion e have examined the implications of the intolerance of a large decline in wealth on optimal consumption and portfolio policies for an investor with constant relative risk aversion preferences. Our specification of preferences is an attempt to capture two features we think are important for understanding investor ehavior. The first one is related to the concept of loss aversion over wealth. Essentially, a heavy loss in the stock market may inflict a severe low to the ego of an investor who comes to realize that she is not the gifted investor she once thought to e. She may experience some regret aout her investment and even a feeling of humiliation eing a loser) as the news spread among friends and family memers. The second motivation reflects the fact that when an investor s wealth is increasing, she may psychologically commit part of the profits made for future expenditures, including sometimes important lifetime decisions such as early retirement for instance. Being forced to revise downward her plans after a drop in wealth may e painful. e find that wealth ratcheting induces a ratcheting ehavior of consumption as current optimal consumption is always maintained aove a fixed percentage of its all-time maximum. Hedging motives and mitigating the cost associated with the ratchet feature of the constraint govern the agent s intertemporal choices. e have isolated the impact of hedging y analyzing asset management for a university that is required to preserve its endowment. Essentially, the lifetime utility relative risk aversion rises, which leads to smaller stock holdings. Looking at the wedge etween optimal allocations for the fixed floor and a ratchet floor allows us to quantify the ratchet impact and uncover its significance. In particular, impatient investors may cur investment in stocks as wealth approaches its peak to 32

limit its growth and the risk of raising the minimum floor. In fact, for large values of the drawdown coefficient, investors whose pure time discount rate exceeds the interest rate find it optimal not to let the wealth process achieve a new maximum. An extension of the asic model incorporates the spirit of capitalism and interprets wealth as an index for social status. Lasting enefits from current and future status levels provide incentives for a higher growth of the wealth and induce a more aggressive risky investment strategy at the expense of consumption. Possile extension of this paper would e to include laor income. If the correlation etween laor income and the stock market is small or negative, the investor naturally would like to orrow against her future income to increase risky investment, which could drive down her financial wealth and exacerate oth hedging motives and the ratchet effect. A detailed analysis is left for future research. Acknowledgments. This is revised version of the paper previously circulated under the title Optimal Consumption and Investment Strategies under ealth Ratcheting. I wish to thank Sanjiv Jaggia, Bruce Mcilliams, Christopher Pink, Gariel Sod Hoffs, Stathis Tompaidis, Fernando Zapatero and seminar participants at the 2008 European Economic Association Meeting, 2013 French Finance Association Meeting, 2013 Latin American Meeting of the Econometric Society, Cal Poly, Universidad Adolfo Iáñez and UTexas at Austin for helpful comments and suggestions. I am especially grateful to Philip Dyvig whose insightful comments resulted in significant improvements to the paper. All errors remain mine. 33

6 Appendix Appendix A Proof of lemma 1. Consider the following changes of variales: X = F ), = J X) and F ) = JX) XJ X). Using relationship 11), we find that the function J must e solution of the following linear ODE θjx) = 1 X 1 + θ r)xj X) + 1 2 µ r) 2 X 2 J X). σ 2 The general solution of this ODE is JX) = 1 AX 1 L 1 β 1 1 + X β1 1+ L 2 β 2 1 + X β2 1+, 18) where L 1 and L 2 are constants. X = F ) and = J X) provides the desired result. Proof of properties β 1 > 1 and 1 β 2 > 0. Differentiating 18) with respect to X and using the fact that Qx) = 1 µ r) 2 1 2 2 σ 2 x 2 + A r 1 2 Recall that β 1 is the positive root of the quadratic µ r) 2 ) 2 σ 2 x 1 A. Since Q1) = r < 0, we must have β 1 > 1. Then, using the fact that 1 2 β 1 +β 2 µ r) 2 2 = 1 2 σ 2 A + r + 1 µ r) 2 θ 2, we find that β 2 σ 2 1 + 1)1 β 2 ) = 1 we have 1 β 2 > 0. Appendix B Proof of properties P1. µ r) 2 2 2 σ 2 µ r) 2 Aβ 2 σ 2 2 β 1 = 1 and. Since β 1 > 1, indeed F is strictly increasing in and decreasing in M since given, the higher M, the more stringent the drawdown constraint. Let λ 0, 1), 0, M 0 ) and 0, M 0 ) e two initial states and c, z) and c, z ) the associated optimal strategies. Then, for initial wealth level λ 0 + 1 λ) 0 and initial all-time high wealth level λm 0 + 1 λ)m 0, λc + 1 λ)c, λz + 1 λ)z ) is also a feasile strategy as the wealth dynamics are linear in variales c, z) and λ t + 1 λ) t λαm t + 1 λ)αm t α max{λm 0 + 1 λ)m 0, λ s + 1 λ) s, s t}. Finally, y strict concavity of the utility function u [ ] [ E 0 uλc s + 1 λ)c s)e θs ds > E 0 0 34 0 ] λucs ) + 1 λ)uc ) s )e θs ds,

which implies that F λ 0 + 1 λ) 0, λm 0 + 1 λ)m 0 ) > λf 0, M 0 ) + 1 λ)f 0, M 0 ). Proof of properties P2. Let c, z) e a feasile strategy for an initial state 0, M 0 ). Then for λ > 0, λc, λz) is feasile for the initial state λ 0, λm 0 ) since the dynamics of the corresponding wealth process λ are d λs = λr s λc s + λz s µ r))ds + λz s σdw s = λd s, so λs = λ s and therefore λs = λ s αλm s = αm λs. It follows that F λ, λm) λ 1 F, M) y homogeneity of degree 1 the utility function. Then, oserve that F, M) = F λ 1 λ, λ 1 λm) λ 1 F λ, λm), so in fact we have F λ, λm) = λ 1 F, M). Appendix C Derivation of oundary conditions Condition for a well defined value function. The oundary conditions must e such that f is well defined and the drawdown constraint is satisfied. Differentiating relationship 12) with respect to u yields Since f u) f u) f u) f u) = Af u)) 1 β1 1)K 1 f u)) β1 1 β 2 1)K 2 f u)) β 2 1. 19) is non-negative, for all u in [α, 1], we must have β 1 1)K 1 f u)) β 1 + β 2 1)K 2 f u)) β 2 A. 20) Then, set Y = f 1)) 1 and X = f α)) 1 and note that Y X. Given relationship 12), it must e the case that for all y in [Y, X], the function G : y Ay 1 + K 1 y β1 1 + K 2 y β2 1 is invertile so we can write f u) = G 1 u) ), for all u in [α, 1]. Since f is negative, then G must e negative. Condition 20) is equivalent to Γy) = A + β 1 1)K 1 y β 1 + β 2 1)K 2 y β 2 < 0. As shown in the sequel, we must have ΓX) = 0. Since Γ y) = β 1 β 1 1)K 1 y β1 1 + β 2 β 2 1)K 2 y β2 1, for K 1 > 0 and K 2 < 0 to e justified in the sequel), the function Γ is strictly increasing and has at most one root on [Y, X]. Hence, ΓY ) 0 is a necessary and sufficient condition to guarantee that Γ < 0 on Y, X). Boundary condition at u = 1. First of all, from relationship 12) we have 1 = Af 1)) 1 + K1 f 1)) β 1 1 + K 2 f 1)) β 2 1. 21) 35

Then, for h > 0, over the interval of time [t, t + h], the HJB equation is [ ] F t, M t ) = max E t uc t )h + e θh F t+h, M t+h ), c t,z t) so using Ito lemma for semi-martingales 0 = max c t,z t) uc t)h + E t [ t+h [ t+h ] +E t F 2 dm s As derived in Grossman and Zhou 1993) t t ) ] θf + r s c s + z s µ r)) F 1 + σ2 2 z2 sf 11 ds 22) E t [M t+h M t t = M t ] = 2 π σ z t h + Oh). hen h is small, h dominates h so in order for the Bellman equation to hold at = M, we must have F 2 M, M) = 0 or z 1) = 0. Since HJB involves a maximization over z, whenever feasile, it is optimal to choose z 1) = µ r)f 1), instead of z 1) = 0. It remains to investigate the impact of the σ 2 f 1) term of order h from the maximum wealth process on the HJB equation. Recall that F 2 M, M) 0. hen z 1) = 0 is optimal and F 2 M, M) < 0, suppose that we choose a strategy at = M for the consumption process such that dm s > 0, thus we have F 2 dm s < 0. This implies that the maximum value achievale for the HJB in relationship 22) must e strictly lower than in the case of a strategy that yields dm s = 0. Proposition 1. of the paper estalishes that there exists a strategy at = M for the consumption that indeed achieves dm s = 0 and therefore it must e the optimal one. Thus, for the optimal strategy dm s = 0. e conclude that in oth cases F 2 dm s = 0, and the expression for reduced) HJB given y relationship 12) is also valid at u = 1. Case 1: F 2 M, M) = 0. This is equivalently to f 1) = 1 )f1). Using relationship 11) leads to θ = 1 1 + r + 1 µ r) 2 ) 2 σ 2 )A f 1)) 1 + r β 1 1 µ r) 2 ) 2 σ 2 K 1 f 1)) β 1 1 + r β ) ) 2 1 µ r 2 K 2 f 1)) β 2 1. 2 σ Using the definition of A, relationship 21) and the fact that β 1 and β 2 are the roots of the quadratic 6), it follows that ) β 1 β 2 Af 1)) 1 1 = 1 ) β 1 K 1 f 1)) β 1 1 ) + β 2 K 2 f 1)) β 2 1. Case 2: z 1) = 0. Imposing z 1) = 0 leads to A = β 1 1)K 1 f 1)) β 1 + β 2 1)K 2 f 1)) β 2. 36

Boundary condition at u = α. e zero, so from relationship 15), we must have At = αm, as in the fixed floor prolem, risky investment must A = β 1 1)K 1 f α)) β 1 + β 2 1)K 2 f α)) β 2. Then, at u = α, relationship 12) provides the following condition α = Af α)) 1 + K1 f α)) β 1 1 + K 2 f α)) β 2 1. To summarize, the oundary conditions are: αx = A + K 1 X β 1 + K 2 X β 2 A = β 1 1)K 1 X β 1 + β 2 1)K 2 X β 2 Y = A + K 1 Y β 1 + K 2 Y β 2 β 1 β 2 Y A) = 1 ) β 1 K 1 Y β 1 + β 2 K 2 Y β 2) if F2 M, M) = 0 β 1β 2 A A = β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2 if z 1) = 0. since A 0 = 1 β 1 β 2, the system can e rewritten as stipulated in the core of the paper. Appendix D Appendix D1 Proof of existence and uniqueness of X, Y, K 1, K 2 ) when F 2 M, M) = 0 is imposed. to show existence and uniqueness for system S e want αx = A + K 1 X β 1 + K 2 X β 2 23) A = β 1 1)K 1 X β 1 + β 2 1)K 2 X β 2 24) Y = A + K 1 Y β 1 + K 2 Y β 2 25) ) β 1 β 2 Y A) = 1 ) β 1 K 1 Y β 1 + β 2 K 2 Y β 2. 26) Comining relationships 25) and 26) leads to β 1 1 β 2 )K 1 Y β 1 + β 2 1 β 1 )K 2 Y β 2 = 0. Since oth β 1 1 β 2 ) and β 2 1 β 1 ) are positive, it must e the case that K 1 and K 2 have opposite signs. Then K 1 Y β 1 = β 21 β 1 ) Y A β 1 β 2 1, which implies that K 1 has the same sign as Y A 1. Eliminating K 1 and K 2 from relationship 24) yields ) X β1 ) ) X β2 Y A A = β 2 β 1 1)1 β 1 ) β 1 β 2 1)1 β 2 ) Y Y 1)β 1 β 2 ). 37

Since β 2 β 1 1)1 β 1 ) > 0 and β 1 β 2 1)1 β 2 ) > 0, we find that Y A 1 > 0. Hence K 1 > 0 and K 2 < 0 and Y A Y A) exactly when 1 1). Comining 23) and 24) leads to αx = β 1 K 1 X β 1 + β 2 K 2 X β 2, and eliminating K 1 and K 2 using relationships 25) and 26) yields ) X β1 ) ) X β2 Y A αx = β 1 β 2 1 β 1 ) 1 β 2 ) Y Y 1)β 1 β 2 ). Set ϖ = X Y 1, we have β 1 β 2 A 1 β 1 )ϖ β 1 1 β 2 )ϖ β ) 2 X = α β 2 β 1 1)1 β 1 )ϖ β 1 β1 β 2 1)1 β 2 )ϖ β, 27) 2 ) and ϖ is implicitly defined y Φ α ϖ) = 0 where for x 1, the auxiliary function Φ α is defined y Φ α : x α β 2 β 1 1)1 β 1 )x β 1 β 1 β 2 1)1 β 2 )x β ) 2 β 1 β 2 1 β1 )x β1 1 1 β 2 )x β 2 1 ) α1 )β 1 β 2 ). e want to show that for α < 1, Φ α has a unique root ϖ > 1. Φ α is continuously differentiale and Φ α 1) = 1 α)β 1 β 2 β 1 β 2 ) < 0. Then, we show that Φ α 1 α ) < 0. A little it of algera yields Φ α 1 ) 1 β1 1 ) 1 β2 1 α ) = β 21 β 1 ) + β 1 1 β 2 ) α1 )β 1 β 2 ). α α For x > 1, define an auxiliary function Θ : x β 2 1 β 1 )x β 1 + β 1 1 β 2 )x β 2. Again Θ is continuous and differentiale and lim 1 Θ = 1 )β 1 β 2 ). Clearly, Θ is decreasing, which implies that for all x in 1, ), Θx) < 1 )β 1 β 2 ) and in particular Φ α 1 α ) < 0. Then Φ αx) = β 1 β 2 x β 2 2 αx 1)Ψx), where Ψx) = β 1 1)1 β 1 )x β 1 β 2 + β 2 1)1 β 2 ). Since β 1 1)1 β 1 ) > 0 and β 1 β 2 > 0, Ψ is strictly increasing and Ψ1) = β 1 1)1 β 1 ) + β 2 1)1 β 2 ). Note that Ψ1) 0 β 1β 2 A 1 β 1 β 2 A 0 1 r r < θ. β 1 β 2 A β 1 1)β 2 1) Case 1: Ψ1) 0. Ψ > 0 and therefore Φ α is decreasing on [ 1, α] 1 and increasing on [ 1 α, ). Since lim Φ α = we conclude that Φ α has a unique root ϖ [ 1 α, ). Note that Ψϖ) > 0. Case 2: Ψ1) < 0. Then define x such that Ψx ) = 0, i.e. x = ) 1 β2 1)1 β 2 ) β 1 β 2. β 1 1)1 β 1 ) 38

It follows that Ψ < 0 on [1, x ] and Ψ > 0 on [x, ). Case 2.1: 1 α < x. Φ α is increasing on [ 1, 1 α], decreasing on [ 1 α, x ] and increasing on [x, ). Since Φ α 1 α ) < 0, we conclude that Φ α has a unique root that elongs to [x, ). Again, note that Ψϖ) > 0. Case 2.2: x < 1 α. Φ α is increasing on [1, x ], then decreasing on [ x, 1 α] and increasing on [ 1 α, ). It remains to show that Φ α x ) < 0 to conclude that Φ α has a unique root in [ 1 α, ). Using the definition of x, one can show that Φ α x ) = β 1 β 2 )β 1 + 1) αβ 1 1)x ) β 1 β ) 1β 2 β 2 1 x ) β 1 1 α1 )β 1 β 2 ). For x > 1, define an auxiliary function Ξ : x αβ 1 1)x β 1 β 1β 2 β 2 1 xβ 1 1. Ξ is continuous and differentiale and Ξ x) = β 1 β 1 1)x β1 2 αx β 2 [ ] [ decreasing on 1, and increasing on β 2 αβ 2 1) β 2 αβ 2 1), 1 α β 2 1 ). It follows that Ξ is ]. Since β 1 β 2 )β 1 + 1) > 0, it must e the case that Φ α x ) max {Φ α 1), Φ α 1 α )}, so in particular Φ αx ) < 0 and the desired result follows. Again, note that Ψϖ) > 0. To summarize, there is a unique real numer ϖ > 1 α such that Φ αϖ) = 0. In addition, we have Φ αϖ) > 0. From the definition of ϖ, we have 1 α Φ αϖ) ϖ = β 1β 2 ϖ β2 1 ) α 2 1 β 1 )ϖ β 1 β 2 1 β 2 ). Since ϖ > 1, we have 1 β 1 )ϖ β 1 β 2 1 β 2 ) < β 1 β 2 ) < 0, we otain that ϖ < 0. Setting α c = Y c X, this implies that hen = 1, Y = A and ϖ is defined y < 0. The existence and uniqueness of X, Y, K 1 and K 2 follow. ϖ β 1 β 1 αβ 1 1)ϖ) = ϖ β 2 β 2 αβ 2 1)ϖ). 28) Finally, we check that r 1 αx > 0. Using relationship 27), elementary algera shows that condition r 1 αx > 0 is equivalent to β 1 1)1 β 1 )ϖ β 1 + β 2 1)1 β 2 )ϖ β 1 > 1. This condition is the same as Ψϖ) > 0, which is satisfied. Appendix D2 Proof of existence and uniqueness of X, Y, K 1, K 2 ) when z 1) = 0 is imposed. e want to 39

show existence and uniqueness for system S αx = A + K 1 X β 1 + K 2 X β 2 29) A = β 1 1)K 1 X β 1 + β 2 1)K 2 X β 2 30) Y = A + K 1 Y β 1 + K 2 Y β 2 31) A = β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2. 32) Note that K 1 and K 2 must have opposite sign otherwise the function Λ : y β 1 1)K 1 y β 1 + β 2 1)K 2 y β 2, is monotonic and therefore the equation Λy) = A cannot have two distinct roots. Then we have αx = β 1 K 1 X β 1 + β 2 K 2 X β 2, and since X > 0, β 1 > 0 and β 2 < 0, it must e the case that K 1 > 0 and K 2 < 0. Then comining relationships 31) and 32) yields β 1 β 2 )K 1 Y β 1 = β 2 A β 2 1)Y β 1 β 2 )K 2 Y β 2 = β 1 A β 1 1)Y. Once again, define ϖ = X Y 1. Eliminating K 1 and K 2 leads to β 1 1)β 2 A β 2 1)Y )ϖ β 1 β 2 1)β 1 A β 1 1)Y )ϖ β 2 = β 1 β 2 )A β 2 A β 2 1)Y )ϖ β 1 β 1 A β 1 1)Y )ϖ β 2 = β 1 β 2 )αx A). Eliminating Y leads to X = A β 1 β 2 )ϖ β 1+β 2 + β 1 β 2 1)ϖ β 1 β 2 β 1 1)ϖ β ) 2 αβ 1 1)β 2 1)ϖ β 1 ϖ β 2), 33) and ϖ is implicitly defined y Φ α ϖ) = 0 where for x 1, the auxiliary function Φ α is defined y Φ α : x αβ 2β 1 1)x β 1 + β 1 β 2 1)x β1 1 + β 1 β 2 )x β 1+β 2 1 β 2 β 1 1)x β2 1 + αβ 1 β 2 1)x β 2 + αβ 1 β 2 ). e want to show that for α < 1, Φ α has a unique root ϖ > 1. Φ α is continuously differentiale and Φ α 1) = 0. Then Φ αx) = αβ 1 β 2 β 1 1)x β 1 1 + β 1 β 2 1)β 1 1)x β 1 2 + β 1 β 2 )β 1 + β 2 1)x β 1+β 2 2 β 2 β 1 1)β 2 1)x β 2 2 + αβ 1 β 2 β 2 1)x β 2 1. 40

Φ α1) = β 1 β 2 β 1 β 2 )α 1) < 0. In addition, for all x 1, Φ αx) has the same sign as Θx) where Θx) = αβ 1 β 2 β 1 1)x 1 β 2 + β 1 β 2 1)β 1 1)x β 2 + β 1 β 2 )β 1 + β 2 1) β 2 β 1 1)β 2 1)x β 1 + αβ 1 β 2 β 2 1)x 1 β 1. e have Θ1) = β 1 β 2 β 1 β 2 )α 1) < 0 and Θ x) = αβ 1 β 2 β 1 1)β 2 1) αx 1) x β 1 1 x β 1 β 2 1). Hence, the function Θ is strictly decreasing on [1, 1 α ] and strictly increasing on [ 1 α, ). Since lim Θ =, we conclude that Θ has a unique root y in 1 α, ). Thus, Φ α is decreasing on [1, y ] and increasing on [y, ) with lim Φ α =. This shows that Φ α has a unique root ϖ > 1 α and Φ αϖ) > 0. From the definition of ϖ, we have Φ αϖ) ϖ = β 2β 1 1)ϖ β 1 β 1 β 2 1)ϖ β 2 β 1 β 2 ). Since x β 2 β 1 1)x β 1 β 1 β 2 1)x β 2 β 1 β 2 ) is decreasing and ϖ > 1, we have β 2 β 1 1)ϖ β 1 β 1 β 2 1)ϖ β 2 β 1 β 2 ) < 0. Hence ϖ < 0. Setting α c = Y c X, this implies that and uniqueness of X, Y, K 1 and K 2 follow. Finally, we check that r 1 αx elementary algera shows that condition r 1 αx > 0 is equivalent to < 0. The existence > 0. Using relationship 33), β 1 β 2 )ϖ β 1+β 2 β 1 ϖ β 1 + β 2 ϖ β 2 > 0. 34) For x 1, define the auxiliary function Ξ : x β 1 β 2 )x β 1 β 1 x β 1 β 2 + β 2. Condition 34) is equivalent to Ξϖ) > 0. Then Ξ x) = β 1 β 1 β 2 )x β1 1 1 x β 2 ) > 0, for x > 1 since β 2 < 0. Ξ is an increasing function with Ξ1) = 0, so we must have Ξϖ) > 0. Appendix E Proof of proposition 1 Step 0: The value function F to e well defined iff β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2 A. 35) If F 2 M, M) = 0 is imposed, since A β 1 1)K 1 Y β 1 β 2 1)K 2 Y β 2 = 1 β 1β 2 A 0 Y ), 1 41

we must have Y A 0 Y A 0 ) if 1 1). Conversely, if the solution of system S is such that Y A 0 Y A 0 ) whenever 1 1), then condition 35) is satisfied, and the value function F is well defined; therefore F 2 M, M) = 0 is optimal. Step 1: Around α = 0, F 2 M, M) = 0 is always optimal. The solution K 1 of system S is such that K 1 > 0 and K 1 a finite non-negative) limit when α goes to 0. Since K 1 Y β 1 = β 21 β 1 ) β 1 β 2 Y also has a finite non-negative limit when α goes to 0. This implies that K 2 also a finite nonpositive limit when α goes to 0. Then recall that β 1 β 2 )K 2 Y β 2 > 0 see appendix F), so K 1 must have Y A 1, we can conclude that β 1 β 2 )K 1 Y β 1 = β 2 1)αX + β 2 A > 0, so β 2A β 2 1 αx β 1A β 1 1. Finally, writing β 1 1)αX β 1A β 1 1 ) = β 1 β 2 )K 2 αx) β 2 α β 2. Since β 2 < 0 and αx is ounded, we have lim α 0 αx) β 2 α β 2 = β 1 1)αX β 1 A < 0, and = 0, which implies that lim α 0 αx = β 1A β 1 1. It follows easily that lim K 2 Y β 2 = lim K 1 Y β 1 = 0, and therefore lim Y = A. Since for > 1 < 1), α 0 α 0 α 0 A A 0 A A 0 ), we can conclude that the condition of proposition 1. is satisfied so around α = 0, F 2 M, M) = 0 is optimal. Step 2: Behavior of systems S and S around α = 1. A 0, if r θ For system S, lim Y = α 1 > A 0 < A 0 ), when > 1 < 1), if r < θ For system S, lim α 1 Y = 1 r Proof: System S. For 0 < α < 1, from appendix D1. the equation Φ α x) = 0 for x 1 has a unique solution ϖ = X Y > 1. hen α = 1, Φ 1x) = 0 has at most two solutions, including value 1. e want to show that lim ϖ = 1 and lim Y = A 0. Set k = β 1 1)1 β 1 ) α 1 α 1 β 2 1)1 β 2 ) > 0. It is easy to verify that k 1 1) iff r θ θ). Then, for x 1, recall that Φ 1 x) = β 2 β 1 1)1 β 1 )x β 1 β 1 β 2 1)1 β 2 )x β 2 Φ 1 is a smooth function with Φ 1 1) = 1 and Φ 1x) = β 1β 2 x β 1 β 2 1 β 1 )x β 1 1 1 β 2 )x β 2 1 ) + 1)β 1 β 2 ). ) β 1 1)1 β 1 )x β 1 x β1 1 ) β 2 1)1 β 2 )x β 2 x β2 1 ) Again, we have Φ 1 1) = 0 and note that Φ 1 x) has the same sign as Γx) = kxβ 1 β 2 1 x β 1 β 2 2 ) x 1). Γ is a smooth function with Γ1) = 0 and Γ x) = kβ 1 β 2 1)x β 1 β 2 2 β 1 β 2 2)x β 1 β 2 3 ) 1.. 42

Γ 1) = k 1 and it is easy to see that for k > 1, Γ x) > 0 for all x > 1. Case 1: r θ. It follows that Γ 0, so Φ 1 0 and for x > 1, Φ 1x) > 0. This implies that x = 1 is the unique solution of Φ 1 x) = 0, and thus lim X = lim Y. If α = 1, System S ecomes a 3 y 3 α 1 α 1 system A = β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2 Y = A + K 1 Y β 1 + K 2 Y β 2 0 = β 1 1 β 2 )K 1 Y β 1 + β 2 1 β 1 )K 2 Y β 2. Eliminating K 1 and K 2, it is easy to verify that the solution is such that lim α 1 Y = A 0. Case 2: r < θ. On some interval 1, 1 + ε), with ε > 0 independent of α, Γ is negative, so is Φ 1, which implies that Φ 1 is negative on 1, y) for some y > 1 independent of α. Since lim Φ 1 =, function Φ 1 must have a root that is greater than y. e conclude that lim α 1 ϖ > y > 1. Next, we show that lim α 1 Y > A 0 < A 0 ), whenever > 1 < 1). For x 1, define P x) = β 2 β 1 1)1 β 1 )x β 1 β 1 β 2 1)1 β 2 )x β 2 Qx) = β 1 β 2 1 β 1 )x β 1 1 1 β 2 )x β 2 1 ). Note that P 1) = β 1 β 2 )1 β 1 β 2 ), Q1) P 1) = 1)β 1 β 2 ) and for x > 1, P x) > 0 and Qx) > 1. From the definition of Φ 1 in Appendix D1. we have Φ 1 x) = P x) P 1) Qx) Q1)). Since Φ 1 ϖ) = 0, we have P ϖ) P 1) = Qϖ) Q1), and Recall that A 0 = A1 + Y = X ϖ = AQϖ) P ϖ) = A1 + 1)β 1 β 2 ) ). P ϖ) 1 1 β 1 β 2 ), so in order to show the result, it is enough to show that β 1 β 2 P ϖ) > 1 1 β 1 β 2 P 1) > P ϖ). From Appendix D1. we know that on 1, ϖ), Φ 1 x) < 0, and for x > x, Φ 1 x) > 0, where 1 < x < ϖ is defined in Appendix D1. It is also easy to check that for x > x, P x) > 0 and Q x) > 0. e now show y contradiction that for all x x, ϖ), Q x) Q1) < 0. Assume it is not the case. Since Q is decreasing on [1, x ], there exists y in x, ϖ), such that Q y) = Q1). e want to show that P y) P 1) > 0, 43

which contradicts the fact that Φy) < 0. e have P y) > P 1) if and only if ) β 2 β 1 1)1 β 1 ) y β 1 1 > β 1 β 2 1)1 β 2 )y β 2 1) Since y satisfies 1 β 1 )y β 1 1 1) = 1 β 2 )y β 2 1 1), it is equivalent to show that β 2 β 1 1) yβ 1 1 y β 1 1 1 > β 1β 2 1) 1 yβ2 1 y β 2 1, and it is enough to show that for x > 1, the function Υ is positive where Υx) = β 2 β 1 1)x β 1 + β 1 β 2 1)x β 2 + β 1 β 2 )x β 1+β 2 1 β 1 β 2 1)x β 1 1 β 2 β 1 1)x β 2 1 + β 1 β 2. Υ x) = β 2 β 1 β 1 1)x β 1 1 + β 2 β 1 β 2 1)x β 2 1 + β 1 β 2 )β 1 + β 2 1)x β 1+β 2 2 β 1 β 2 1)β 1 1)x β 1 2 β 2 β 1 1)β 2 1)x β 2 2. Υ 1) = 0 and Υ x) has the same sign as Σx) where Σx) = β 2 β 1 β 1 1)x 1 β 2 + β 2 β 1 β 2 1)x 1 β 1 + β 1 β 2 )β 1 + β 2 1) β 1 β 2 1)β 1 1)x β 2 β 2 β 1 1)β 2 1)x β 1. Then for x > 1, Σ x) = β 2 β 1 β 1 1)1 β 2 )x 1)x β1+1) x β 1 β 2 1) > 0. The function Σ is strictly increasing with Σ1) = 0 so for x > 1, Σx) > 0 and Υ x) > 0. Since Υ1) = 0, we find that Υx) > 0 for x > 1 and the desired contradiction follows. System S. In Appendix D2., for 0 < α < 1, we estalish the existence and uniqueness the root ϖ > 1 of the equation Φ α x) = 0, where Φ α is defined in Appendix D2. hen α = 1, the previous equation has at most two solutions, including value 1. e want to show that actually x = 1 is the unique solution of Φ 1 x) = 0, so that lim ϖ = 1 and lim X = lim Y. In Appendix D2., we have seen α 1 α 1 α 1 that Φ 1 1) = 0 and Φ 1 has the same sign as Φ 1 x) has the same sign as Θx) and in the case where α = 1 we have Θ x) = β 1 β 2 β 1 1)β 2 1)x 1)x β 1 1 x β 1 β 2 1) > 0 if x > 1. Θ is increasing with Θ1) = 0, so for x > 1, we have Θx) > 0, so Φ 1 x) > 0 and thus finally Φ 1x) > 0. This shows that x = 1 is the only root. Let us now determine the limit of Y when α goes to 1. For α 44

given, let us write K 1 α), K 2 α), Xα), Y α)) the solution of system S. If α = 1, X1) = Y 1) and system S ecomes a 2 y 2 system A = β 1 1)K 1 1)Y 1) β 1 + β 2 1)K 2 1)Y 1) β 2 Y 1) = A + K 1 1)Y 1) β 1 + K 2 1)Y 1) β 2. Next, we assume the following asymptotic expansion in α for the solution around α = 1 K 1 α) = K 1 1)1 + k 1 h + oh)) K 2 α) = K 2 1)1 + k 2 h + oh)) Xα) = Y 1)1 + xh + oh)) Y α) = Y 1)1 + yh + oh)), where k 1, k 2, x, y) are numers independent of α, h = 1 α. Plugging ack these asymptotic expressions into system S and taking a first order Taylor expansion yields x 1)Y 1) = k 1 + β 1 x)k 1 1)Y 1) β 1 + k 2 + β 2 x)k 2 1)Y 1) β 2 0 = β 1 1)k 1 + β 1 x)k 1 1)Y 1) β 1 + β 2 1)k 2 + β 2 x)k 2 1)Y 1) β 2 yy 1) = k 1 + β 1 y)k 1 1)Y 1) β 1 + k 2 + β 2 y)k 2 1)Y 1) β 2 0 = β 1 1)k 1 + β 1 y)k 1 1)Y 1) β 1 + β 2 1)k 2 + β 2 y)k 2 1)Y 1) β 2. Manipulating these relationships we find that β 1 β 2 )K 1 1)Y 1) β 1 = β 2 1)Y 1) + β 2 A β 1 β 2 )K 2 1)Y 1) β 2 = β 1 1)Y 1) β 1 A β 1 β 1 β 2 )y x)k 1 1)Y 1) β 1 = β 2 1)1 + y x)y 1) β 2 β 1 β 2 )y x)k 2 1)Y 1) β 2 = β 1 1)1 + y x)y 1). Eliminating K 1 1)Y 1) β 1, K 2 1)Y 1) β 2, y x leads to β 1 β 2 A β 2 1)Y 1)) β 2 β 1 1)Y 1) β 1 A) = β 2 1 β 1 1). It follows easily that Y 1) = β 1 β 2 A β 1 1)β 2 1) = 1 r. 45

Step 3: Proof of proposition 1. Case θ r. Given step 2, for the solution of system S, we know that lim X = lim Y = A 0. hen F 2 M, M) = 0 is optimal, we show in Appendix F that α 1 α 1 solution Y satisfies Y < 0 Y > 0) whenever < 1 > 1), and when = 1, Y = A 0. Since lim α 1 Y = A 0, this implies that for all α 1, when > 1 < 1) we have Y A 0 Y A 0 ). Condition 35) is satisfied, and the desired result follows. Case θ > r. Given step 2, for the solution of system S, we know that lim Y > A 0 < A 0 ), whenever α 1 > 1 < 1), and the solution of system S satisfies lim Y = 1 α 1 r > A 0. From step 1, for small values of α, F 2 M, M) = 0 is optimal. hen > 1 < 1), as long as F 2 M, M) = 0 is optimal, we know > 0< 0). Since for > 1 < 1) the solution Y of system S is such that lim Y > A 0< A 0 ), α 1 around α = 1, condition Y A 0 A 0 ) is violated. By continuity of the solution in α, there must that Y exist a critical value α 0, 1] such that for all α < α, for > 1 < 1) we have Y < A 0 > A 0 ) and F 2 M, M) = 0 is optimal. At α = α, Y = A 0, we have F 2 M, M) = 0 and z 1) = 0. Then, for α > α, we must have z 1) = 0. In Appendix F, when z 1) = 0 is imposed, we show that Y > 0. Hence, for all α α, 1), Y < 1 r and therefore at = M, we have d t = r 1 ) Y t dt < 0: ealth can never achieve a new maximum. Appendix F Proof of proposition 2 Case 1: F 2 M, M) = 0. hen α increases f1) must decrease. Since f 1) = 1 )f1), we deduce that when < 1 > 1), Y < 0 Y > 0). Hence 1 Y Y A Y = 0. Then K 1 Y β 1+1 = β 1 K 1 Y β 1 + β 21 β 1 )Y β 1 β 2 ) 1) ) Y = K 1Y β 1 Y Y A β 1A + 1 β 1 )Y ). > 0, 1. hen = 1, then Y = A, so Recall that z 1) 0 so A Y + β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2. Since Y = A + K 1 Y β 1 + K 2 Y β 2, it follows that β 1 A + 1 β 1 )Y β 2 β 1 )K 2 Y β 2 > 0. Therefore K 1 12) with respect to α leads to > 0. Differentiating relationship f u) ) 1 A f u) ) 1 β 1 1 K 1 f u) ) β 1 1 β 2 1 f u) ) β 1 1 + K 2 f u) ) β 2 1. = K 1 K 2 f u) ) ) β 2 1 f u) 36) 46

The sign of the LHS is the same as K 1 f u)) β1 β 2 + K 2 and u K 1 f u)) β1 β 2 + K 2 its minimum at u = 1 and its maximum at u = α. Then K 1 Y β 1 β 2 + K 2 = Y 1+β 2) Y A β 1 1)K 1 Y β 1 β 2 1)K 2 Y β 2 ). Since A β 1 1)K 1 Y β 1 β 2 1)K 2 Y β 2 addition, when = 1, K 1 Y β 1 β 2 + K 2 c < 0. Conversely, when < 1, we have K 1 Y β 1 β 2 + K 2 K 1 Xβ 1 β 2 + K 2 = = X β 2 Y A)Y X β 2 Y A)Y Y > 0, when > 1, Y > 0, so K 1 Y β 1 β 2 + K 2 achieves > 0. In = 0. Hence, if 1, f u) > 0. As c = M f u)) 1, < 0. Then K 1 X β 1 Y 1 β 1 ) + β 1 A) + K 2 X β 2 Y 1 β 2 ) + β 2 A) Y AαX Y ) > 0. ) Hence, there exists u α in α, 1), so that f u) that c < 0 on [α, u α) and c > 0 on u α, 1]. > 0 on [α, u α) and f u) < 0 on u α, 1]. e conclude Case 2: z 1) = 0. In this case, we have Y Y = β 1K 1 Y β 1 + β 2 K 2 Y β 2 ) Y + K 1 Y β 1+1 + K 2 Y β 2+1. Recall that Y = β 1 K 1 Y β 1 + β 2 K 2 Y β 2, so K 1 Y β 1 + K 2 Y β 2 = 0, and therefore, K 1 and K 2 have opposite signs. Similarly X + α X = K 1 Xβ 1 + K 2 Xβ 2 + 1 X β 1K 1 X β 1 + β 2 K 2 X β 2 ) X. Since αx = β 1 K 1 X β 1 + β 2 K 2 X β 2, we find that K 1 Xβ 1 + K 2 Xβ 2 = X > 0. It remains to investigate the sign of K 1 f u)) β 1 β 2 + K 2. Set y = f u)) 1 : y K 1 yβ 1 β 2 + K 2, must and for y in [Y, X] define the auxiliary function is a continuous monotonic function with Y ) = 0 and X) > 0. Hence, we must have K 1 > 0 and K 2 Appendix G c < 0 and is positive on [Y, X]. e conclude that < 0, and in particular Y > 0. Proof of proposition 3. Note that ) z = µ r)yβ 1 1 Aβ 2 1 K 1 + β2 2K 2y β 2 β 1 ) + K 1 K 2 β 1 β 2 ) 2 y β 2 y σ 2 A + K 1 y β 1 + K2 y β 2 ) 2 For y in [Y, X], define the auxiliary function ). Ψ : y Aβ 2 1 K 1 + β 2 2 K 2y β 2 β 1 ) + K 1 K 2 β 1 β 2 ) 2 y β 2. 47

Ψ is strictly increasing so it has at most one root. At u = 1, we have z 0, so either Ψ has no root and is strictly positive or, Ψ has one root so it is first negative and then positive. e examine the sign of ΨY ). Case 1. hen z 1) = 0, since z α) = 0, Ψ indeed must have a root. z is hump-shaped in u. Case 2. hen F 2 M, M) = 0, we have ΨY ) = Aβ1K 2 1 + β2k 2 2 Y β 2 β 1 ) + K 1 K 2 β 1 β 2 ) 2 Y β 2 = β 1β 1 β 2 )K 1 Aβ 1 + β 2 + 1) + 1 β 2 )1 β 1 ) Y A ) β 1 + 1 1 ) β 1 β 1 β 2 )K 1 1 = 1 β 2 ) 1) θ Y, β 1 β 2 A as β 1 + 1)1 β 2 ) = 1 θ. Hence ΨY ) > 0 iff Y < 1 θ Y > 1 θ ) whenever > 1 < 1). Case = 1. e have Y = A and ΨY ) = K 1 β 1 β 2 )Aβ 1 + β 2 β 1 β 2 )K 1 Y β 1 ) = AK 1β 1 β 2 ) ϖ β 1 where ϖ is defined y relationship 28). ΨY ) 0 0). Appendix H z ) β 1 + β 2 )ϖ β 1 β 2 αβ 2 1)ϖ) is strictly increasing hump shaped) in u exactly when, Proof of proposition 4. Let y = f u) 1. From relationship 36), we have Since z = µ r σ 2 z = µ r σ 2 y 1 y y = y β1 K1 + yβ 2 K 2 A β 1 1)K 1 y β 1 β2 1)K 2 y β. 2 Ay 1 β 1 1)K 1 y β1 1 β 2 1)K 2 y β 2 1 ), it follows that β 1 1) K 1 yβ 1 + β 2 1) K 2 yβ 2 + A + β 1 1) 2 K 1 y β 1 + β 2 1) 2 K 2 y β 2 ) y ) = µ r)yβ 1+β 2 σ 2 y K Aβ 1 1 y β 2 K + β 2 2 y β 1 K ) + β 1 β 2 )β 1 1)K 1 1 β K 2 1)K 2 2 A β 1 1)K 1 y β 1 β2 1)K 2 y β. 2 The denominator of the aove fraction is positive. To investigate the sign of its numerator, let us for y in [Y, X], define the auxiliary function Θ : y Aβ 1 K 1 y β 2 + β 2 K 2 y β 1 ) + β 1 β 2 )β 1 1)K 1 K 1 β 2 1)K 2 K 2 ). Θ is differentiale and Θ y) = β 1 β 2 A K 1 y β2 1 + K 2 y β1 1 ). Since K 1 > 0 and K 2 < 0, Θ is strictly increasing and either ΘY ) 0 or Θ achieves its minimum at y such that K 1 y ) β 1 + 48

K 2 y ) β 2 = 0. If Θ achieves its minimum at y, then Θy ) = β 1 β 2 )y ) β 2 K 1 > 0, and in this case, Θ > 0 on [Y, X]. Otherwise, Θ > 0 and Θ is strictly increasing. To prove that Θ 0 on [Y, X], it is enough to show that ΘY ) 0 or equivalently that z 1) is a decreasing function of α. Case 1: F 2 M, M) = 0. then Y = A and z 1) If 1, we have z 1) = µ r σ 2 In this case, z 1) = µ r σ 2 Y = µ r σ 2 β 1 1) K 1 A 1 + β 1β 2 Y A) z 1) 1)Y = µ r σ 2 A β1 1)K 1 Y β 1 β 2 1)K 2 Y β ) 2. If = 1, Aβ 1 + β 2 1) K 2 ). Therefore β 1β 2 A Y 2 Y 1 ) < 0. Aβ 2 ) < 0. Case 2: z 1) = 0. e have z 1) Appendix I = 0 and ΘY ) = 0. The proof is complete. Representation of c M, c and as stochastic processes Process c M. c For u in α, 1), recall that u = G M ) so denoting H = G 1, we have H u) = 1 and H u) = G c M ). Applying Ito lemma, we find G c ))3 M ) 2 zt H u t )dt Then ) c d t M t = H u t )du t + σ2 2 = rg c t M t ) c t M t M t + µ r)2 σ 2 ) c t G c t M t ) 1 µ r) 2 c 2 t 2 2 σ 2 M t G c t G c t M t ) M t ) rg c t ) c t 1 µ r) 2 ) c 2 t M t M t 2 2 σ 2 G c t ) M t M t = r 1 A ) r β 1β 1 1) µ r) 2 ) c 1 β1 2 2 σ 2 )K t 1 M t + r β 2β 2 1) µ r) 2 ) ) c 1 β2 2 2 σ 2 )K t 2 + A c t M t M t = r 1 A ) c t M t G c t M t ). dt + µ r σ c t M t dw t. G c M ) So ) c d t = c t r M t M t 1A + µ r)2 σ 2 )dt + µ r ) σ dw t. 49

For u in [0, 1], we have 1 X Xf 0 M 0 )) 1 and A representation of the process c M c M 1 Y. Define the geometric Brownian motion v such that v 0 = dv s = v s r 1 A + µ r)2 σ 2 see Harrison 1985) p 22) is c t M t = v te Lt Ut X, )ds + µ r ) σ dw s. where the processes L and U are increasing and continuous with L 0 = U 0 = 0 and L t = sup Consumption and wealth processes. 0 s t [log v s U s ] [ U t = sup log X ] 0 s t Y log v s L s. Case 1: F 2 M, M) = 0. Using Ito lemma for semimartingales, we find that ) ) t 0 log H = log H + r θ µ r)2 + M t M 0 2σ 2 )t + µ r σ w t H 1) H1) log M t, M 0 ) Note that H 1) H1) = Y > 0. As explained in Grossman and Zhou [21], the quantity log H t G 1 Y ) M t is ounded from aove y log H1) = log Y and H 1) H1) log Mt M 0 ) 0 serves as a regulator to keep the arithmetic Brownian motion from exceeding log Y. Define [ ) 0 l t = sup log H + r θ µ r)2 + 0 s t M 0 2σ 2 )s + µ r ] + σ w s + log Y. It follows that M t = M 0 e for the wealth and consumption processes is Y G t = M 0 e Y 1 ) lt G H ) c 0 t = H e r θ Case 2: z 1) = 0. wealth processes are given y Y G 1 Y ) lt. Recall that t = M t G c t M t ) and c t = M t H t M t ), so a representation M 0 0 M 0 + µ r)2 2σ 2 ) e r θ + µ r)2 2σ 2 µ r )t+ σ wt lt. ) µ r )t+ σ wt lt The wealth process can never exceed its all time high M 0. Consumption and c t = M 0 X v te Lt Ut t = M 0 G v te Lt Ut X ). 50

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Tales Tale I: Disentangling hedging and ratchet effects α u 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.925 0.95 0.975 1 0.6 ẑ 0 0.283 0.383 0.450 0.499 0.537 0.568 0.581 0.592 0.603 0.613 z 0 0.279 0.375 0.436 0.478 0.506 0.526 0.532 0.537 0.540 0.541 0.8 ẑ - - - - 0 0.248 0.339 0.372 0.402 0.428 0.450 z - - - - 0 0.230 0.299 0.318 0.331 0.336 0.334 0.9 ẑ - - - - - - 0 0.168 0.234 0.283 0.322 z - - - - - - 0 0.143 0.184 0.199 0.190 55

Footnotes 1. In the US, trustees and charity professionals who run foundations are only oliged to spend as little as 5% a year of the capital. In many foundations, capricious and poorly thought out projects or programs were undertaken to fulfill the interests of trustee managers not the wishes of the founder 2005). 2. The high-water mark is a target value that can depend on the current asset value of the fund. It is periodically adjusted due to withdraws, allocated expenses and a contractual growth rate. In the simplest case, the high-water mark is the highest level the asset has reached in the past. 3. This constraint was first introduced y Grossman and Zhou 1993) to examine the prolem of maximizing the long term growth rate of expected utility of final wealth. Their analysis is quite insightful ut they do not allow for endogenous withdraws from the fund to finance intermediate consumption. Cvitanic and Karatzas 1995) extend their work to a more general class of stochastic processes. More recently, Sekine 2013) and Cherny and O lój 2013) generalize the analysis to the case of a non-linear drawdown constraint when the riskfree rate is equal to zero. 4. See for instance such Sundaresan 1989), Constantidines 1990), Detemple and Zapatero 1991), Campell and Cochrane 1999). 5. For instance see Baski and Chen 1996) and Smith 2001). 6. Cvitanic and Karatzas 1992) and Cuoco 1997) develop a general martingale approach to cope with convex contemporaneous constraints on trading strategies which includes the case of incomplete markets and prohiited short sales. Cuoco and Liu 2000) analyze the optimal consumption portfolio choice prolem under margin requirements and evaluate the cost of the constraint. He and Pages 1993) and El Karaoui and JeanBlanc-Picqué 1998) treat the case of non-negative wealth in presence of laor income. Grossman and Villa 1992) followed y Villa and Zariphopoulou 1997) study the consumption-portfolio prolem for a CRRA investor facing a leverage constraint. 7. This assumption is only needed for the logarithmic preferences case. hen uc) has a constant sign, Leesgue Monotone Convergence Theorem can e applied directly to prove step 4 of the Verification Theorem provided in section 4 of the paper. 56

8. The strict concavity of F comes from the fact that the utility function is strictly concave and the constraint is linear so that if and are admissile wealth processes, then for all λ in [0, 1], λ + 1 λ) is also admissile. 9. The case r = 0 has een extensively studied y Elie and Touzi 2008) and leads to optimal allocations that exhiit distinct properties than the ones derived in this paper. 10. In the Appendix, we show that the value function F is gloally concave iff β 1 1)K 1 Y β 1 + β 2 1)K 2 Y β 2 A. 11. This property is actually necessary for a well defined prolem. In the sequel, when the drawdown constraint 3) is imposed, restrictions on the parameters of the model are made so that this reflecting condition is satisfied. 12. In the limit case α = 1, when F 2 M, M) = 0 is optimal, we have X = Y = A 0. Since for < 1, 1 A 0 > 1 A, y continuity in α, for α large enough, it follows that αx < A, which implies that c > 1 A, for all u in [α, 1]. 13. For the definition of a regulated Brownian motion, see Harrison 1985), p14. 14. The University of Newcastle upon Tyne study y Parker, Pearce and Tiffin 2005) indicates that women are twice as likely to e downwardly moile. The study involved men and women orn in 1947 in Newcastle and followed them from childhood to age 50. Researchers noted the findings might e explained y the fact that men orn during that era gained much of their self-esteem from their careers, whereas women found fulfillment from social pursuits outside of work, such as children and friendships. 15. The choice of the functional form of the utility function is motivated y tractaility reasons. 57