Optimal Investing after Retirement Under Time-Varying Risk Capacity Constraint
OOptimal Investing after RetirementUnder Time-Varying Risk Capacity Constraint
Weidong TianUniversity of North Carolina at Charlotte Zimu ZhuUniversity of Southern California Corresponding author: Weidong Tian, Belk College of Business, University of North Carolina at Char-lotte. Email: [email protected]. Zimu Zhu, Department of Mathematics, University of Southern California.Email:[email protected]. We thank Prof. Jianfeng Zhang for stimulating discussions on this paper. We alsothank Dr.Xiaojing Xing for her assistance in numerical implementation. a r X i v : . [ q -f i n . P M ] J un ptimal Investing after RetirementUnder Time-Varying Risk Capacity Constraint Abstract
This paper studies an optimal investing problem for a retiree facing longevity risk and living stan-dard risk. We formulate the investing problem as a portfolio choice problem under a time-varyingrisk capacity constraint. We derive the optimal investment strategy under the specific conditionon model parameters in terms of second-order ordinary differential equations. We demonstrate anendogenous number that measures the expected value to sustain the spending post-retirement. Theoptimal portfolio is nearly neutral to the stock market movement if the portfolio’s value is higherthan this number; but, if the portfolio is not worth enough to sustain the retirement spending, theretiree actively invests in the stock market for the higher expected return. Besides, we solve anoptimal portfolio choice problem under a leverage constraint and show that the optimal portfoliowould lose significantly in stressed markets. This paper shows that the time-varying risk capacityconstraint has important implications for asset allocation in retirement.
Keywords : Risk Capacity, Retirement Portfolio, Longevity Risk, Leverage Constraint
JEL Classification Codes : G11, G12, G13, D52, and D90
Introduction
Investing during retirement is significantly different matter from investing for retirement. Economistshave found profound challenges of investing in retirement due to aging and health shocks, risk-taking, and retirement adequacy. Compared to investing before retirement, retirees invest in anunknown but finite length of time because of longevity risk.They worry about the balance betweenspending and leaving wealth as an inheritance. In addition, as stated by Kenneth French, theseindividuals will also face capacity risk if a market downturn were to occur, leading to a substantialdecline in their standards of living. Despite its importance of investing problem post-retirement,its theoretical studies, particularly in a continuous-time framework, are very few. To fill the gap inthe literature, this paper studies this optimal investment problem for retirees, taking both longevityrisk and living standard risk into consideration in a continuous-time framework.Specifically, we study an optimal portfolio choice problem by incorporating the following severalsignificant investment features post-retirement. First, we consider the mortality risk that the retireehas an uncertain investment time-horizon. The length of each individual’s retirement may differfrom the statistical life expectancy, and the mortality risk is virtually independent of the marketrisk in the financial market. A second feature we consider is that the retiree has a cash inflowfrom the social security account and retirement account. In most countries, retirees are forcedto withdraw from the retirement account, which provides a fixed income stream for spending.For example, since its inception, Bengen’s “four percent rule” has been recognized as a standardin retirement professionals (Bengen, 1994). Third, the retiree may hope to leave an inheritance.Finally, the absence of labor income results in individual risk capacity, therefore, strong risk-aversebehaviors. With a massive market risk exposure in the portfolio, the retiree faces a risk to sacrificethe standard of living when the market declines as in the 2008-2009 financial crisis or 2020 COVID-19. We formula and solve an optimal portfolio choice problem with these investment features forretirees.For an analytical purpose, we consider two assets in the economy. One asset is a risky assetthat provides a risk premium, and another asset is risk-free. To focus on investment decisions, wefix a withdrawal rate in the consumption policy, which is consistent with market practices. Theretiree has a CRRA utility function on consumption flows and the bequest of wealth. To capture There are many approaches to construct the retirement portfolio before retirement. See Gustman and Steinmeier(1986), Roozebt and Shourideh (2019) for its structural and optimal reform approach of the retirement model. Theoptimal portfolio choice approach with labor income includes Bodie, Merton, and Samuelson (1992), Cocco, Gomes,and Maenhout (2005), Viceira (2001). For the empirical studies of post-retirement, see Coile and Milligan (2009), Goldman and Orszag (2014), Gustman,Steinmeier and Tabatabai (2012), Poterba (2016). Yogo (2016) develops a dynamic discrete-time model with healthshocks. As Kenneth French presented at the Annual Conference for Dimensional Funds Advisors, 2016, “It is livingstandard risk you should know about the risk. It is what your exposure is to a major change in your standard ofliving during the entirely uncertain numbers of years you remain alive.” time-varying risk capacity constraint on the investment. Thatis, the dollar amount invested in the risky asset is always bounded from above by a predeterminedpercentage of the retiree’s portfolio wealth at the retirement date (the initial wealth).We show that, under the specific condition on model parameters, the value function (expectedutility function) of the optimal portfolio choice problem is a C smooth solution of the correspond-ing Hamilton-Jacobi-Bellman equation (Proposition 4). If the value function is C smooth, weexplicitly characterize the region in which the risk capacity constraint is binding. We further derivethe optimal investment policy by using a well-defined second-order ordinary differential equation(Proposition 2 and Proposition 3). The second-order ordinary differential equation can be numer-ically solved efficiently to illustrate the results. For a general utility function, we characterize thevalue function as the unique viscosity solution of the HJB equation (Proposition 1).Our results have several important properties and implications for retirement investment. First ,the investment strategy is not a myopic one. The risk capacity constraint on the future investmentdecisions affects the investment at any instant time. Therefore, the investing strategy is not merelycutting the benchmark strategy, which has no risk capacity constraint.
Second , the optimal investment strategy displays a remarkable wealth-cycle property, in contrastto the life-cycle feature suggested in both academic and practice. Specifically, when the portfoliovalue is unsustainable for the entire retirement period, the retiree should invest in the market,because the dollar investment in the stock will increase the expected portfolio value. Nevertheless,the percentage of wealth in the market declines when the portfolio is worth more. The decliningpercentage of the wealth invested in the risky asset is due to the retiree’s standard living concernto protect the portfolio value. This decreasing feature of the percentage becomes significant whenthe portfolio worths sufficiently high.Since the dollar invested in the stock is always a constant L when the portfolio wealth is higherthan a threshold W ∗ , this threshold W ∗ measures the expected lump sum of the spending in theretirement period. Intuitively, when the portfolio worths more than this threshold, the retiree aimsto protect the portfolio by investing only a fixed amount of L in the stock market without losingthe living standard. By implementing this contingent constant-dollar strategy , the retiree sellsthe stock when the stock market moves high, which is consistent with the retirement portfolio’sdecumulation process. In contrast, investing for retirement is an accumulating asset process. According to Modigliani (1986), individual’s investment and consumption decision has a life-cycle feature. SeeBenzoni, Dufresne and Goldstein (2007), Viceira (2001), Cocco, Gomes, and Maenhout (2005), Gomes and Michaelides(2005), Bodie et al (2004), and Bodie, Detemple and Rindisbacher (2009) for life-cycle theoretical and empiricalstudies. The life-cycle hypothesis is also used for preparing the retirement portfolio and in the retirement portfolio.For instance, a conventional rule for an agent of age t is to invest (100 - t)/100 percent of wealth in the stock market.See Malkiel (1999). In a classical constant-dollar strategy, the dollar invested in the risky asset is always fixed. In contrast, by acontingent constant-dollar strategy in this paper we mean a fixed dollar is invested in the risky asset if and only ifthe portfolio value is higher than a threshold. hird , the portfolio is nearly independent of the stock market when the retiree’s portfolio worthssufficiently to embrace the living standard; therefore, reducing the retiree’s living standard risk.As a comparison, we solve an optimal portfolio choice problem by imposing a standard leverageconstraint on the percentage of the wealth in the stock market. We demonstrate that the optimalportfolio under the leverage constraint moves precisely in the stock movement direction, which is asevere concern of the standard living risk in a stressed market period.This paper contributes to the optimal portfolio choice literature by solving a stochastic controlproblem with a new objective function and a new risk capacity constraint. At first glance, thisrisk capacity constraint seems to be a particular case of leverage constraint or collateral constraint, X t ≤ f ( W t ), where X t represents the dollar instead in the risky asset and W t the wealth at time t . However, earlier literature on the leverage constraint does not study the situation that f ( W t ) isindependent on W t . Therefore, our results provide new insights into the portfolio choice problemunder an extreme leverage constraint, a dynamic constraint X t ≤ L for all time t . The objectivefunction is also new to the best of our knowledge. Since we consider the uncertain time-horizon(Yaari, 1965, Richard, 1975, Blanchet-Scaillet et al. 2008), both the consumption process and thewealth process are involved in the objective function to capture the retiree’s mortality risk.The structure of the paper is organized as follows. In Section 2, we introduce the model andpresent the retiree’s optimal investment problem to capture his mortality risk and living standardrisk. In Section 3, we present a characterization of the value function for a general utility function.In Section 4, we derive the value function and the optimal investment strategy for a CRRA utility.We present several properties of the optimal portfolio, and numerically illustrate these properties inSection 5. We solve and make a comparison with another relevant optimal portfolio choice problemunder a leverage constraint in this section. The conclusion is given in Section 6, and technicalproofs are given in Appendix A - Appendix B. In this section, we introduce the model and an optimal portfolio choice problem for a retiredindividual (retiree). See, for instance, Zariphopoulou (1994), Vila and Zariphopoulou (1997), Detemple and Murthy (1997). Studieson portfolio choice and asset pricing under other dynamic constraints on the control variable c t or the state variable, W t , include Dybvig (1995), El Karoui and Jeanbalnc-Picque (1998), Detemple ad Serrat (2003), Elie and Touzi (2008),Dybvig and Liu (2010), Chen and Tian (2016), Ahn, Choi and Lim (2019), and reference therein. .1 Investment Opportunity There are two assets in a continuous-time economy. Let (Ω , F t , P ) be a filtered probability spacein which the information flow in the economy is generated by a standard one-dimension Brownianmotion ( Z t ). The risk-free asset (“the bond”) grows at a continuously compounded, constant r . Wetreat the risk-free asset as a numeaire so we assume that r = 0. F ∞ is the σ -algebra generated byall F t , ∀ t ∈ [0 , ∞ ).The other asset (”the stock index”) is a risky asset, and its price process S follows dS t = µS t dt + σS t dZ t (1)where µ and σ are the expected return and the volatility of the stock index. We consider an individual right after his retirement. We simply name “he” for this retiree. Theretirement date is set to be zero. The retiree’s initial wealth is W at the retirement date. Theretiree is risk-averse and his utility function is denoted by a strictly increasing and concave function u ( · ) : (0 , ∞ ) → R and u ( · ) satisfies the Inada’s condition: lim W ↑∞ u (cid:48) ( W ) = 0, and lim W ↓ u (cid:48) ( W ) =0. Since the retiree faces his mortality risk, the investment time-horizon is uncertain, neithera fixed finite time nor infinity. We assume that the investor’s death time τ has an exponentialdistribution with mean λ , that is, P { τ ∈ dt } = λe − λdt . Therefore, the probability of the retireesurvives in the next t years is e − λt . The investor’s average life time is λ and the variance of his lifetime is λ . For example, if λ = 0 .
05, it means that a normal retiree who retires at 65 is likely diedat 85 years old. We assume that τ is independent of the information set F ∞ . Compared with a standard investor before retirement, there are several distinct features in theretiree’s portfolio choice problem. (1) The retiree has a fixed cash flow from his social securityaccount post-retirement. (2) He is either able to withdraw without penalty or enforced to withdrawfrom his retirement account. (3) He has no labor income anymore. (4) He has a mortality risk,and (5) he becomes more risk-averse than when before retirement because he has concerns on the In U.S.A, people are able to withdraw around 60 years old (and enforced to withdraw the minimal distributionsnearly 70 old) from the retirement account. Moreover, a standard withdrawal rate is between 4% to 5%. See Bengen(1994). ( X ) E (cid:20)(cid:90) τ e − δs u ( cW s ) ds + Ke − δτ u ((1 − α ) W τ ) (cid:21) (2)where δ is the retiree’s subjective discount factor, α is the inheritance tax rate of the wealth, K is anumber that determines the strength of the bequest (to heirs), and his wealth process W t satisfies dW t = X t ( µdt + σdZ t ) − c t dt, ∀ ≤ t ≤ τ, (3)and X t represents the dollar amount in the risky asset, and c t is the consumption rate. By assump-tion, both ( X t ) and ( c t ) are adapted to the filtration F t .Because of the social security safety net and income from his retirement portfolio, we chooseand fix the withdrawal rate, c t = cW t , c ∈ (4% , Therefore, theretiree focuses on the investment decision X t to maximize his expected utility. Moreover, there isno labor-income flow in the budget equation (3). To model the risk-averse preference of the retired,we assume that X t ≤ L, ≤ t ≤ τ. (4)It states that the investor’s dollar amount in the risky asset is bounded above by a fixed constant.We call it a time-varying risk capacity and L a capacity level. Equivalently, X t ≤ lW , where l = L/W , then the dollar amount is bounded above by a percentage of his initial wealth. Forexample, when l = 30% , W = 1 , , risk capacity constraint ” to controlthe retiree’s living standard risk. Since the retiree might encounter a terrible market downturn,such a risk capacity constraint prevents the portfolio from a huge loss.Lastly, we assume the non-negative wealth for no-arbitrage condition W t ≥ , ≤ t ≤ τ. (5) A constant percentage consumption rate, c t = cW t , is standard in literature to find the optimal spending rule.See, for instance, Dybvig (1995), Campbell and Sigalov (2019). It is also consistent with Modigliani’s life-cycle theoryof consumption. τ , and the independent assumption between τ and F ∞ , by the Fubini’stheorem, we have E (cid:20)(cid:90) τ e − δs u ( cW s ) ds (cid:21) = E (cid:20)(cid:90) ∞ e − δs u ( cW s )1 s ≤ τ ds (cid:21) = E (cid:20)(cid:90) ∞ e − δs E [ u ( cW s )1 s ≤ τ |F ∞ ] ds (cid:21) = E (cid:20)(cid:90) ∞ e − δs u ( cW s ) P ( s ≤ τ ) ds (cid:21) = E (cid:20)(cid:90) ∞ u ( cW s ) e − ( λ + δ ) s ds (cid:21) where we make use of the fact that P ( τ ≥ s ) = (cid:82) ∞ s λe − λt dt = e − λs . Similarly, we have E [ e − δτ u ((1 − α ) W τ )] = λ E (cid:20)(cid:90) ∞ e − ( λ + δ ) t u ((1 − α ) W t ) dt (cid:21) . Therefore, the retiree’s optimal retirement portfolio problem (2) is reduced to a Merton-Richardtype problem as follows (Merton, 1971; Richard, 1975), J ( W, L ) = max ( X ) E (cid:20)(cid:90) ∞ e − ( λ + δ ) t ( λKu ((1 − α ) W t ) + u ( cW t )) dt (cid:21) (6)subject to constraints (3) - (5). The function inside the integral of J ( W, L ), λKu ((1 − α ) W t ) + u ( cW t ), represents the combination of the preference on the consumption as well as the terminalwealth by incorporating the tax-rate, longevity risk. The general value function at any time t is J ( W t , L ) = max ( X ) E (cid:20)(cid:90) ∞ t e − ( λ + δ )( s − t ) ( λKu ((1 − α ) W s ) + u ( cW s )) ds (cid:21) (7)subject to (3) - (5). For a retiree, to invest solely in the risk-free asset is one admissible strategy for the retiree. Letting X t = 0 for all time t , then W t = W e − ct , and the value function is bounded below by (cid:90) ∞ e − ( λ + δ ) t (cid:0) λKu ((1 − α ) W e − ct ) + u ( cW e − ct ) (cid:1) dt. The portfolio choice problem under uncertain time-horizon is studied widely in literature. For instance, Karatzasand Wang (2001) for a general stopping time, Blanchet-Scaillet, et al. (2008) considered the case when the randomtime τ has continuous conditional probability distribution conditional on the market prices. J ( W ; 0) when L = 0. In general, J ( W ; L ) ≤ J ( W ; L ) ≤ J ( W ; ∞ ) for any 0 ≤ L ≤ L , where J ( W ; ∞ ) = max ( X ) E (cid:20)(cid:90) ∞ e − ( λ + δ ) t ( λKu ((1 − α ) W t ) + u ( cW t )) dt (cid:21) denotes the value function for a retiree without the risk capacity constraint (4). The stochastic control problem (6) can be viewed as a special case of the following general stochasticcontrol problem, E (cid:20)(cid:90) T f ( t, C t , W t , X t ) dt (cid:21) , and there are extant studies on this kind of problem in stochastic control literature back to Bismut(1973). In this section, we characterize the value function J ( W ; L ) in terms of the viscosity solutionof the Hamilton-Jacobi-Bellman (HJB) equation. Proposition 1
The value function J ( W ) is the unique viscosity solution in the class of concavefunction, of the following HJB equation: ( λ + δ ) J ( W ) = max ≤ X ≤ L (cid:20) σ X J (cid:48)(cid:48) ( W ) + µXJ (cid:48) ( W ) (cid:21) − cW J (cid:48) ( W ) + λKu ((1 − α ) W ) + u ( cW ) , ( W >
0) (8) with J (0) = λKλ + δ u (0) . By Proposition 1, the value function is uniquely characterized by the viscosity solution of theHJB equation. However, this characterization is not strong enough to derive the explicitly solutionof the optimal portfolio. The main insight in Proposition 1 is that, without knowing the smoothproperty (“ex-ante”) of the value function of a portfolio choice problem, the value function can stillbe uniquely characterized in the framework of viscosity solution. Building on this characterization, Briefly speaking, a viscosity subsolution V of a second-order equation F ( x, u, u x , u xx ) = 0 if for any smoothfunction ψ and a maximum point x of V − ψ , the inequality F ( x , V ( x ) , ψ x ( x ) , ψ xx ( x )) ≤
0. Similarly, V is a viscosity supersolution if for any smooth function ψ and a minimum point x of V − ψ , the inequality F ( x , V ( x ) , ψ x ( x ) , ψ xx ( x )) ≥
0. A viscosity solution is both a viscosity subsolution and supersolution. We referto Fleming and Soner (2006) for the theory of viscosity solution. Extant studies on the twice differentiability in optimal control problems, including Ren, Touzi and Zhang (2014),and Strulovici and Szydlowski (2015) and reference therein, rely on certain conditions on the model and the controlprocess. These results cannot be applied in our problem directly since the utility function u ( · ) does not satisfy theglobal Lipschitz condition.
7e will derive further smoothness properties of the value function and the optimal strategies for aparticular class of utility function.
From this section, we consider the following CRRA utility function u ( W ) = W − R − R , R > , R (cid:54) = 1 . By its scaling property u ( cW t ) = c − R u ( W t ), we have λKu ((1 − α ) W t ) + u ( cW t ) = (cid:0) λK (1 − α ) − R + c − R (cid:1) u ( W t ) . Then J ( W ) = (cid:0) λK (1 − α ) − R + c − R (cid:1) V ( W ) , where V ( W ) = max ( X ) E (cid:20)(cid:90) ∞ e − ( λ + δ ) t u ( W t ) dt (cid:21) . For a CRRA utility, the HJB equation (8) of the value function can be written as( λ + δ ) V ( W ) = max ≤ X ≤ L (cid:20) σ X V (cid:48)(cid:48) ( W ) + µXV (cid:48) ( W ) (cid:21) − cW V (cid:48) ( W ) + u ( W ) , ( W > . (9)The contribution of this section to demonstrate that the value function is a C smooth solutionof the HJB equation under certain assumptions on model parameters. We also derive the optimalstrategy analytically by well-defined second order ordinary differential equation. For simplicity, weassume that R < and focus on the function V ( W ). We start with a benchmark situation, L = ∞ , that is, there is no constraint on the risky investment.For this purpose, we assume that Most arguments can be applied for other risk aversion parameter
R >
1. The choice of
R < u (0) < ∞ and V (0) < ∞ . ssumption A. λ + δ > ρ − c (1 − R ) , (10)where κ = µ σ , ρ = (1 − R ) κR . Lemma 4.1
Under Assumption A, the value function in the absence of the risk capacity constraint, V ( W ; ∞ ) , is V ( W ; ∞ ) = 1 λ + δ − ρ + c (1 − R ) W − R − R .
The risky asset investment amount is X t = µRσ W t . (11)When c = 0, Lemma 4.1 is essentially given in Liu and Loewenstein (2002), Lemma 1. Fora general positive consumption rate c , the optimal investment strategy is independent of the con-sumption rate. Blanchet-Scaillet et al. (2008), Theorem 3 and Proposition 5 show that the optimalstrategy is the same if τ is independent from F ∞ and its conditional probability density functionconditional on F ∞ is continuous. By Lemma 4.1, the portfolio value process satisfies dW t = W t (cid:16) ( µRσ − c ) dt + σ µRσ dZ t (cid:17) . Since the portfolio W t is a lognormal process, there is a positive probability that µRσ W t > L forany positive number L . Moreover, there is a positive probability of W t < (cid:15) for any t > (cid:15) . Therefore, the retirement wealth portfolio has a substantial living standard risk. In our approach, we first characterize the region in which the risk capacity constraint is binding byassuming the C smooth property of the value function V ( W ) (in this subsection). Then, relyingon the characterization of the constrained and unconstrained region, we will verify the smoothproperty by an explicit construction of the candidate smooth function as the value function (inSection 4.3 and Section 4.4) below.Since the optimal dollar amount X is X = min (cid:26) µσ V (cid:48) ( W ) − V (cid:48)(cid:48) ( W ) , L (cid:27) , (12)9he unconstrained region U is U = (cid:26) W > µσ V (cid:48) ( W ) − V (cid:48)(cid:48) ( W ) < L (cid:27) . Then, over the unconstrained region, the value function V ( · ) satisfies( λ + δ ) V ( W ) = u ( W ) + κ ( V (cid:48) ( W )) − V (cid:48)(cid:48) ( W ) − cW V (cid:48) ( W ) . (13)Similarly, the constrained region B is given by B = (cid:26) W > µσ V (cid:48) ( W ) − V (cid:48)(cid:48) ( W ) > L (cid:27) , in which the constraint (4) is binding. Over this region, V ( · ) satisfies a second-order linear ODE( λ + δ ) V ( W ) = u ( W ) + ( µL − cW ) V (cid:48) ( W ) + 12 σ L V (cid:48)(cid:48) ( W ) . (14) Proposition 2
Assume V ( W ) is C smooth, then there exists a positive number W ∗ such that U = (0 , W ∗ ) and B = ( W ∗ , ∞ ) . Proposition 2 characterizes the region in which the risk capacity constraint is binding by onepositive number W ∗ . By Lemma 4.1, the risk-capacity constraint (4) is not binding for all W > W ∗ is a finite number. The characterization of U and B is important to deriveexplicitly the value function in our subsequent discussion. From an economic perspective, if theretiree starts to put the maximum $L in the stock when his portfolio can sustainable his retirementspending, he will not invest a smaller dollar amount in the market when his wealth growths more.Therefore, the constrained region is an open interval. In this regard, the number W ∗ measuresthe expected value to sustain the spending with comfortable living standard. Consequently, theinvesting strategy will be different when W > W ∗ or W < W ∗ . In this section, we present the optimal solution explicitly for a CRRA utility, assuming the valuefunction is C . The smooth property will be verified in the next subsection under certain condition. Proposition 3
Assume V ( W ) is C smooth and Assumption A holds, then in the region (0 , W ∗ ) , V ( W ) = V (0) + (cid:90) G − ( W )0 g − R G (cid:48) ( g ) dg, here G ( g ) is strictly increasing, G (0) = 0 , and it satisfies the following ODE κR gG (cid:48)(cid:48) = (cid:16) λ + δ + c − κR (1 − R ) (cid:17) G (cid:48) − cR ( g − G ) − g R G − R G (cid:48) . (15) The value function V ( W ) satisfies ODE (14) in the region ( W ∗ , ∞ ) . Moreover, there exists twopositive numbers C and C such that C ≤ V ( W ) W − R , V (cid:48) ( W ) W − R ≤ C , ∀ W > . Proposition 2 plays a crucial role in this characterization. By Proposition 2, the value functionand the optimal investment strategy can be examined into two separate regions. If the portfoliovalue is high enough,
W > W ∗ , the value function is a solution of a second-order linear ODE(14). Assuming W ∗ is given, then the value function can be characterized by the conditions that V ( W ) W − R ≤ C and V (cid:48) ( W ) W − R ≤ C, ∀ W ∈ ( W ∗ , ∞ ).On the other hand, in the region W ∈ [0 , W ∗ ) and assuming W ∗ is known, we reduce thenonlinear ODE (13) to a second-order ODE by using the well-known transformation: V W = g − R , and W = G ( g ) for an increasing auxiliary function G ( · ). The HJB equation of the value functionis reduced to the second-order ODE (15), and the function G ( g ) is characterized by appropriateboundary conditions. Finally, the smooth-fit property of the value function determines the number W ∗ uniquely by Proposition 1 and Proposition 2.As shown in both Proposition 2 and Proposition 3, the number W ∗ is essential in the explicitcharacterization of the value function. Indeed, the next result demonstrates that, if there is anumber W ∗ to separate the unconstrained and constrained region in general, the value function issmooth. Lemma 4.2
Assume V ( x ) is a continuous viscosity solution of a second-order (HJB) equation F ( x, u, u x , u xx ) = 0 and the region of x is D = (0 , ∞ ) . Moreover, there exists x ∗ such that V ( x ) issmooth in both (0 , x ∗ ) and ( x ∗ , ∞ ) , then V ( x ) must satisfies the smooth-fit condition at x ∗ , that is, V (cid:48) ( x ∗ − ) = V (cid:48) ( x ∗ +) . Lemma 4.2 can be viewed as a converse statement of Proposition 2. If the value function issmooth in each region (0 , W ∗ ) , ( W ∗ , ∞ ), then the value function must be smooth as long as thevalue function is continuous and a viscosity solution of a HJB equation. This result is interesting inits own right and it can be used to verify the smooth property as will be shown in the next section. This transformation is well-known to solve the optimal consumption-portfolio choice problem since g is theendogenous consumption rate in the HJB equation. See Karatzas and Shreve (1998), Chapter 5, and its references.Even though there is no optimal consumption rate in our model, this transformation is also essential to characterizethe optimal solution in this paper. We thank Prof. Jianfeng Zhang for providing this lemma to simplify our previous arguments. .4 A special case In this section, we consider a particular case that c = 0 and derive the C smooth property ofthe value function under certain condition on model parameters. Our analysis is motivated byProposition 3 and Lemma 4.2. A zero consumption rate occurs when the retiree is willing totransform entire wealth to his heirs, or social security safety net and other incomes are sufficientfor spending during his retirement period.Define two real numbers β = − µ + (cid:112) µ + 2( λ + δ ) σ σ L , β = − µ − (cid:112) µ + 2( λ + δ ) σ σ L . (16) β and β are two roots of the following quadratic equation12 σ L β + Lµβ − λ − δ = 0and β > > β .Define V ( W ) = 2( β − β )(1 − R ) σ L × (cid:26) e β W (cid:90) W x − R e − β x dx − e β W (cid:90) W x − R e − β x dx (cid:27) . The function V ( W ) is a well-defined smooth function for W >
0. We recall the expression ofGamma function, Γ( x ) = (cid:90) ∞ s x − e − s ds, which is well-defined for all real number x > Given a W ∗ >
0, we define two real numbers C ∗ and W ∗ by C ∗ = − β − β )(1 − R ) σ L β R − Γ(2 − R ) e β W ∗ [ µ + Lσ β ] − µV (cid:48) ( W ∗ ) − σ LV (cid:48)(cid:48) ( W ∗ ) σ L ( β ) e β W ∗ + µβ e β W ∗ , (17)and g ∗ = (cid:26) β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W ∗ + C ∗ β e β W ∗ + V (cid:48) ( W ∗ ) (cid:27) − R . (18) We refer to Appendix B for basic properties of the Gamma function. roposition 4 Under Assumption A, and assume the existence of a positive solution W ∗ of thefollowing equation u (0) λ + δ + (cid:90) g ∗ g − R G (cid:48) ( g ) dg = 2( β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W ∗ + C ∗ e β W ∗ + V ( W ∗ ) , (19) where G ( g ) satisfies the second-order ordinary differential equation (ODE), ≤ g ≤ g ∗ , G (cid:48)(cid:48) ( g ) = Rκ g − G (cid:48) ( g ) (cid:2) λ + δ − ρ − g R ( G ( g )) − R (cid:3) (20) with boundary condition G (0) = 0 , G ( g ∗ ) = W ∗ and G (cid:48) ( g ∗ ) = LRσ µ ( g ∗ ) − . Then, the number W ∗ is unique and the value function V ( W ; L ) is C smooth and given by V ( W, L ) = u (0) λ + δ + (cid:82) G − ( W )0 g − R G (cid:48) ( g ) dg, W ≤ W ∗ β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W + C ∗ e β W + V ( W ) , W > W ∗ . (21)Proposition 4 presents a closed-form expression of the value function and the optimal strategyin terms of W ∗ and the auxiliary function G ( · ). By its construction, V ( W ; L ) is the smooth functionof the HJB equation. Then, by the unique characterization of the value function in Proposition 1,the value function is given by the expression (21) in Proposition 4 .In particular, if L = ∞ , then G ( · ) is a linear function and W ∗ = ∞ . In general, the function G ( · ) is non-linear, and its non-linearity is equivalent to the non-myopic property of the optimalstrategy, as will be explained in the next section. In this section, we present several properties and implications of the optimal strategy. We alsosolve an optimal portfolio choice problem under a leverage constraint.
We start with the optimal investing strategy.
Proposition 5
The optimal portfolio strategy is X ( W ) = µRσ gG (cid:48) ( g ) , W ≤ W ∗ = G ( g ∗ ) L, W > W ∗ (22)13 his optimal portfolio strategy is not the myopic strategy. By Proposition 4, the optimal portfolio strategy is explicitly given by the auxiliary function G ( · ) in the unconstrained region. If the optimal strategy is the myopic strategy in the sense that X t = min (cid:8) µRσ W t , L (cid:9) , then G ( g ) must be a linear function of g , and W ∗ = LRσ µ . In this case,Equation (19) fails in general because the left side is a polynomial function while the right handis virtually an incomplete Gamma function. Intuitively, the risk-capacity constraint affects theinvestment decision even though the constraint is not binding instantly.As a numerical illustration, we plot the auxiliary function G ( · ) and the investing strategy X ( W )with the wealth W . We choose the risk premium µ = 0 .
10 that is consistent with the market data ofS & P 500 between 1948 to 2018. The parameter λ is chosen as 0.07 to consistent to approximately15 years of life after retirement. We choose the initial dollar amount of 700 ,
000 in a portfolio of 1million as the maximum dollar amount in the stock market. Equivalently, 70 percent of the wealthinvested in the stock market, as suggested in Vanguard (2018), for the construction of retirementportfolios. We let σ = 30%.This number σ is slightly higher than the calibration of the marketindex since our purpose is to highlight the high likelihood of the market downturn, which is a bigconcern for the retiree. Other parameters are R = 0 . , c = 0. By calculation, the expected valuefor retirement level is W ∗ = 492 , G ( · ) is not a linear function, the strategy is not amyopic one. By the same reason, X ( W ) as a function of the wealth is not C since ∂X ( W ) ∂W | W = W ∗ − = µRσ ( gG (cid:48) ( g )) (cid:48) G (cid:48) ( g ) = µRσ (cid:18) gG (cid:48)(cid:48) ( g ) G (cid:48) ( g ) (cid:19) (cid:54) = 0 . The percentage of wealth in the risky asset, X ( W ) W , can be analyzed similarly. In the constrainedregion, W ≥ W ∗ , the percentage of wealth is LW . The more the wealth, the smaller the percentageof wealth. On the other hand, in the unconstrained region, X ( W ) W = µRσ gG (cid:48) ( g ) G ( g ) . This functionalso decreases with respect to the wealth as shown in Figure 3. Both Figure 2 and Figure 3 showthat the optimal portfolio strategy displays a strong risk-aversion feature, by comparing with thebenchmark model without the risk capacity constraint. Given the optimal strategy in Proposition 5, the optimal wealth process is uniquely determined by dW t = min (cid:110) µRσ gG (cid:48) ( g ) , L (cid:111) ( µdt + σdZ t ) , W = W > . (23)14t can be shown that the stochastic differential equation (23) has a strong solution. Therefore, wecan directly analyze the portfolio by the stochastic differential equation (23).The portfolio dynamic is as follows. Assuming wealth W t = W ∗ at a time t from below, then inthe instant time period, [ t, t + δt ], W t,t + δt = W t + L ( µδt + σ √ δtζ ) , and S t + δt = S t + S t (cid:16) µδt + √ δtζ (cid:17) , where ζ is a standard normal variable. In a good scenario of the stock market, S t + δt ≥ S t , thatis, µδt + σ √ δtζ >
0, then W t + δt ≥ W t , so the same dollar amount L is still invested in the stockmarket. If the market drops in the period [ t, t + δt ], S t + δt < S t , then W t + δt < W ∗ , the portfoliovalue reduces and is smaller than the threshold W ∗ , then a new dollar amount, µ − Rσ V (cid:48) ( W t + δt ) V (cid:48)(cid:48) ( W t + δt ) , isinvested in the stock market. The process continuous between the unconstrained region and theconstrained region.The retirement portfolio’s return process is dW t W t = min (cid:8) µRσ gG (cid:48) ( g ) , L (cid:9) W t ( µdt + σdZ t ) . Therefore, the instantaneous variance,
V ar (cid:104) dW t W t (cid:105) converges to zero when W → ∞ . When thewealth is sufficiently high, the risk of the portfolio is very small so the retiree is able to resolvethe living standard risk, regardless of possible market downturn. Moreover, the instantaneouscovariance between dW t W t and dS t S t is Cov (cid:18) dW t W t , dS t S t (cid:19) = X ( W t ) W t σ → , as W t → ∞ . (24)Hence, the portfolio is virtually independent from the stock market if the portfolio value is largeenough. We have demonstrated several properties of the proposed retirement portfolio and the portfoliodynamic under the risk capacity constraint. In both theory and practice, an alternative and mightbe a more popular strategy is to impose a maximum percentage of the wealth invested in the stockmarket. Namely, X t ≤ bW t . This kind of constraint is a special case of leverage X t ≤ f ( W t ) initiallystudied in Zariphopoulou (1994), and Vila and Zariphopoulou (1997). In this section, we solve thecorresponding optimal portfolio choice problem and compare it with the risk capacity constraint.Our contribution in this section is to show that the leverage constraint cannot resolve the livingstandard risk for the retiree.We use a predetermined number of b to represent the highest possible percentage of wealthinvested in risky asset. For instance, b = 0 . V b ( W ) = sup ( X ) E (cid:20)(cid:90) ∞ e − ( λ + δ ) t u ( W t ) dt (cid:21) (25)where the risk capacity constraint is replaced by X t ≤ bW t , ∀ t . By Lemma 4.1, we assume that b < µRσ . Otherwise, V b ( W ) is solved by Lemma 4.1 for all b ≥ µRσ . Proposition 6
Under the constraint that X t ≤ bW t , the value function is V b ( W ) = 1 λ + δ + (1 − R )( σ b R − µb + c ) u ( W ) (26) and the optimal strategy is X t = bW t . Proposition 6 states that any constant percentage strategy X t = bW t is an optimal policy undera leverage constraint. Given this strategy, the wealth portfolio satisfies dW t = W t (( bµ − c ) dt + σdZ t )).Therefore, W t is a GBM, and there is a positive probability that the retirement portfolio is less thanany a positive number. Same as in the benchmark model in Lemma 4.1, the portfolio is perfectlycorrelated to the stock market; thus, a downturn market could wipe out the retirement portfolio.Therefore, by implementing the optimal investment strategy in Proposition 6, the retiree is subjectto a substantial living standard risk if the stock market declines significantly. In this section, we explain several implications of our model.First, the retiree needs to invest in the stock market since the all-safe strategy is too conser-vative to sustain the spending given longevity risk. Second, we demonstrate that the risk capacityconstraint captures the retiree’s living standard risk, and the optimal portfolio under the risk capac-ity constraint is a reasonable retirement strategy. Specifically, if the retirement portfolio value is nothigh enough, the retiree should invest some money in the stock market to increase the growth rate.However, when the portfolio value is high enough, the retiree implements a “contingent constant-dollar amount strategy” by only placing L dollar of the portfolio in the stock market as long as theportfolio value is higher than W ∗ . Third, under the risk capacity constraint, the higher the port-folio value, the smaller percentage of the wealth in the stock market. The portfolio can reduce theliving standard risk because its return is asymptotically independent of the stock market for a highlevel of the portfolio value. Fourth, the risk-capacity constraint and the leverage constraint yielddifferent investment strategies. By implementing a leverage constraint, the generating retirementportfolio is perfectly correlated to the stock market, so the retiree faces a substantial market risk.These properties are illustrated in Figure 2 and Figure 3.16ow to choose the maximum dollar amount of L or l = L/W is of practically interesting.Given the risk capacity level L , and define W L by the following equation µRσ W L = L, (27)then W L ≡ LRσ σ . W L is the portfolio wealth level at which the benchmark’s strategy in Lemma4.1 provides the exact dollar amount of L in the stock market. Since the portfolio is not myopic,the threshold W ∗ is higher than W L in Proposition 4. It implies that X ( W ) W is always boundedby µRσ in the constrained region. This point is illustrated in Figure 2 and Figure 3. If L < L ,the invested dollar amount in the stock market under the constraint X t ≤ L is bounded by thecorresponding money invested in the stock market for the level L . While an increasing level of L invests the expected return of the portfolio, the portfolio becomes riskier, as shown in Figure 4.Therefore, a suitable level of L depends on its counter-effect to the expected return and risk. In this paper, we solve an optimal investing problem for a retiree facing longevity risk and livingstandard risk. By imposing a time-varying risk capacity constraint on the portfolio choice problem,the corresponding optimal strategy enables the retiree to resolve the concern on the standard ofliving in retirement. We also compare the risk capacity constraint with a leverage constraint.We show that the leverage constraint yield a popular constant-percentage strategy, introducinga substantial living standard risk. By contrast, the risk-capacity constraint implies a contingentconstant-dollar strategy to reduce the standard living risk.17 ppendix A. Proofs
To simplify the notations, we use J ( W ) , J ∞ ( W ) to represent J ( W ; 0) , J ( W ; ∞ ). Lemma 6.1
The boundary condition for Proposition 1.1 is J (0) = λKλ + δ u (0) . Proof:
It suffices to show that if W = 0, then W t ≡ , ∀ t > T >
0, a simple calculation leads to (for W = 0) W t e ct = (cid:90) t e cs X s σd ˜ Z s ∀ < t < T where ˜ Z s = Z s + µσ s, ≤ s ≤ T . { ˜ Z s } ≤ s ≤ T is a Brownian Motion under ˜ P T by Girsanov’s theorem, d ˜ P T dP = exp (cid:20) − µσ Z T − (cid:16) µσ (cid:17) T (cid:21) . Since W t is non-negative, the local martingale (cid:82) t e cs X s σd ˜ Z s is always non-negative, hence, a su-permartingale under ˜ P T . Therefore, ˜ E [ W t e ct ] ≤ ∀ < t < T. This implies W t ≡ , ∀ The value function J ( W ) is (strictly) continuous, increasing and concave. Proof: Let A W be the admissible set of ( X t ) for the control problem starting at W = W . Clearly, A W ⊂ A W if W ≤ W . The increasing property follows. Next, we show that the J ( W ) isconcave. For X ∈ A W and X ∈ A W , it is easy to verify that λX + (1 − λ ) X ∈ A λW +(1 − λ ) W .Therefore, J ( W ) is concave by the concavity of u ( · ). Since J ( W ) is concave in (0 , ∞ ), hence it iscontinuous in (0 , ∞ ).For the continuity at 0, we observe that J (0 , L ) ≤ J ( W, L ) ≤ J ( W, ∞ ) and J (0 , L ) = J (0 , ∞ ). Now sending W to 0, the desired result follows.Finally, we prove J ( W ) is strictly increasing by a contradiction argument. Assume not, since J ( · )is concave, then there exists ˆ W , such that J ( W ) is constant on [ ˆ W , ∞ ). However, this is impossiblebecause its lower bound J ( W ) goes to infinity as W → ∞ . (cid:3) Lemma 6.3 Dynamic Programming Principle : If ˆ τ is a stopping time of the filtration F t ,then J ( W ) = sup A W E (cid:20)(cid:90) ˆ τ e − ( λ + δ ) t ( λKu ((1 − α ) W t ) + u ( cW t )) dt + e − ( λ + δ )ˆ τ J ( W ˆ τ ) (cid:21) roof: The proof is standard, see Fleming and Soner (2006), Chapter 3. (cid:3) Proof of Proposition 1. We show that J ( x ) is the viscosity solution of (8) and such asolution is unique. The existence part is standard in the theory of viscosity solution. See Flemingand Soner (2006), Chapter 3. It can be also modified with a similar argument in Zariphopuulou(1994), Theorem 3.1, which studies a relevant leverage constraint. To prove the uniqueness partit suffices to prove the following comparison principle: if J ( W ) is the viscosity supersolution and J ( W ) is the viscosity subsolution and satisfies J (0) ≥ J (0), then J ( W ) ≥ J ( W ) for all W ∈ (0 , ∞ ).In this situation, since the function u ( W ) is not Lipschitz, we cannot apply the standardcomparison principle directly in our situation. For this purpose, we will separate (0 , ∞ ) into twoparts: (0 , δ ) and ( δ, ∞ ) for some proper positive number δ , then show that ∀ (cid:15) > J ( W ) + (cid:15) ≥ J ( W ) , ∀ W > . Since J (0) ≥ J (0), there exists δ > 0, such that J ( W ) + (cid:15) ≥ J ( W ) , ∀ W ∈ (0 , δ ] . (A-1)On the region W ∈ ( δ, ∞ ), u ( W ) is Lipchitz so is the function λKu ((1 − α ) W ) + u ( cW ). Since ψ ( W ) + (cid:15) is the test function for J ( W ) + (cid:15) , J ( W ) is also a supersolution of (8), then we utilize thestandard comparison principle in Fleming and Soner (2006), Chapter 5 to obtain J ( W ) + (cid:15) ≥ J ( W ) , ∀ W ∈ ( δ, ∞ ) (A-2)Now, combine (A-1) and (A-2), we have J ( W ) + (cid:15) ≥ J ( W ) , ∀ W > . Since (cid:15) is arbitrary, the comparison principle holds and the proof is now complete. (cid:3) Proof of Lemma 4.1. By Proposition 1, it suffices to verify that the function V ( W ) = au ( W )is a C function of the HJB equation (9), in the absence of constraint X t ≤ L , for a positive number a . Given the specification of the value function, X = µRσ W . Then, the HJB equation becomes a (cid:18) λ + δ − R − κR + c (cid:19) = 11 − R , then a = λ + δ − ρ + c (1 − R ) . By Assumption A, a > 0, then V ( W ) = a W − R − R is the value function ofthis optimal portfolio choice problem. (cid:3) We start with several lemmas before proving Proposition 2.19 emma 6.4 Assume V ( W ) is C smooth, then there exists two positive numbers C , C such that C − W − R ≤ V (cid:48) ( W ) ≤ C − W − R , ∀ W > . In particular, lim W → V (cid:48) ( W ) = ∞ and lim W →∞ V (cid:48) ( W ) = 0 . Proof: By a direct calculation, V ( W ) = u ( W ) λ + δ + c (1 − R ) and Lemma 4.1 states that V ∞ ( W ) = u ( W ) λ + δ − ρ + c (1 − R ) . Then, by using the concave property of the function V ( · ) (Lemma 6.2), for anypositive number W > E > 0, we have V (cid:48) ( W ) ≥ E [ V ( W + E ) − V ( W )] ≥ E [ V ( W + E ) − V ∞ ( W )]= 1 E (cid:20) − R λ + δ + c (1 − R ) ( W + E ) − R − − R λ + δ − ρ + c (1 − R ) W − R (cid:21) . Choosing E = kW , we have V (cid:48) ( W ) ≥ k (cid:20) − R λ + δ + c (1 − R ) ( k + 1) − R − − R λ + δ − ρ + c (1 − R ) (cid:21) W − R Let C − = sup k> k (1 − R ) (cid:20) λ + δ + c (1 − R ) ( k + 1) − R − λ + δ − ρ + c (1 − R ) (cid:21) + , (A-3)where x + = max( x, C is positive.By the same reason, for any E = βW, β ∈ (0 , V (cid:48) ( W ) ≤ βW [ V ( W ) − V ( W − βW )] ≤ βW (cid:2) V ∞ ( W ) − V ( W − βW ) (cid:3) ≤ β (1 − R ) (cid:26) λ + δ − ρ + c (1 − R ) − λ + δ + c (1 − R ) (1 − β ) − R (cid:27) W − R . Let C − = inf <β< β (1 − R ) (cid:20) λ + δ − ρ + c (1 − R ) − λ + δ + c (1 − R ) (1 − β ) − R (cid:21) + . (A-4)The proof is finished. (cid:3) emma 6.5 Assume V ( · ) is C smooth, then there exists ˜ W such that the open interval (0 , ˜ W ) isincluded in U , and X ∗ ( ˜ W ) = L. Proof: Assume not, then there exists a sequence W n → X ∗ ( W n ) = L . Apply thedefinition of B of the corresponding HJB equation. We have λV ( W n ) ≥ − R W − Rn − cW n V (cid:48) ( W n ) + 12 µLV (cid:48) ( W n )= 11 − R W − Rn + ( 12 µL − cW n ) V (cid:48) ( W n ) . Since V ( W ) is continuous (Lemma 6.2), as n → ∞ , the left hand side of the last inequalityapproaches to λV (0) = 0. However, µL − cW n → µL , so the term (cid:0) µL − cW n (cid:1) V (cid:48) ( W n ) willtends to + ∞ (By Lemma 6.4) on the right hand side of the last inequality, which is a contradiction. (cid:3) Proof of Proposition 2. For simplicity, we present the proof for c = 0. By assumption, V ( · )is C smooth. We define a function Y ( W ) = µV (cid:48) ( W ) + σ LV (cid:48)(cid:48) ( W ) , W > . (A-5)Then, Y ( W ) < , ∀ W ∈ U , and Y ( W ) > W ∈ B . Step 1. In the unconstrained region, the value function V ( · ) satisfies the ODE (13). Bydifferentiating the ODE equation once and twice, we obtain( λ + δ ) V (cid:48) = u (cid:48) ( W ) − κV (cid:48) + κ ( V (cid:48) ) V (cid:48)(cid:48)(cid:48) ( V (cid:48)(cid:48) ) , and ( λ + δ ) V (cid:48)(cid:48) = u (cid:48)(cid:48) ( W ) − κV (cid:48)(cid:48) + κ ( V (cid:48) ) V (cid:48)(cid:48)(cid:48)(cid:48) ( V (cid:48)(cid:48) ) + 2 κV (cid:48) V (cid:48)(cid:48)(cid:48) ( V (cid:48)(cid:48) ) (cid:8) ( V (cid:48)(cid:48) ) − V (cid:48) V (cid:48)(cid:48)(cid:48) (cid:9) . By the definition of Y ( W ), the last two equations imply( λ + δ ) Y = µu (cid:48) ( W ) + Lσ u (cid:48)(cid:48) ( W ) − κY + κ ( V (cid:48) ) ( V (cid:48)(cid:48) ) Y (cid:48)(cid:48) + 2 κV (cid:48) V (cid:48)(cid:48)(cid:48) ( V (cid:48)(cid:48) ) (cid:26) V (cid:48)(cid:48) σ L Y − V (cid:48) σ L Y (cid:48) (cid:27) . We then define an elliptic operator on the unconstrained region by L U [ y ] ≡ − κ ( V (cid:48) ) ( V (cid:48)(cid:48) ) y (cid:48)(cid:48) − κV (cid:48) V (cid:48)(cid:48)(cid:48) ( V (cid:48)(cid:48) ) (cid:26) V (cid:48)(cid:48) σ L y − V (cid:48) σ L y (cid:48) (cid:27) + ( λ + δ + 2 κ ) y − µu (cid:48) ( W ) − Lσ u (cid:48)(cid:48) ( W ) . Therefore, L U [ Y ] = 0 in U . 21 tep 2. In the constrained region B , by differentiating the ODE (14) of V ( W ) once and twice,we have ( λ + δ ) V (cid:48) = u (cid:48) ( W ) + µLV (cid:48)(cid:48) + 12 σ L V (cid:48)(cid:48)(cid:48) and ( λ + δ ) V (cid:48)(cid:48) = u (cid:48)(cid:48) ( W ) + µLV (cid:48)(cid:48)(cid:48) + 12 σ L V (cid:48)(cid:48)(cid:48)(cid:48) . Then, ( λ + δ ) Y = µLY (cid:48) + 12 σ L Y (cid:48)(cid:48) + µu (cid:48) ( W ) + σ Lu (cid:48)(cid:48) ( W ) . Similarly, we define an elliptic operator L B [ y ] = − σ L y (cid:48)(cid:48) − µLy (cid:48) + ( λ + δ ) y − µu (cid:48) ( W ) − σ Lu (cid:48)(cid:48) ( W ) . Then L B [ Y ] = 0 in B . Step 3. By Lemma 6.5, there exists W > , W ) ⊆ U and Y ( W ) = 0. It sufficesto show that ( W , ∞ ) ⊆ B by a contradiction argument. Assume not, then there exists W > W such that ( W , W ) ⊆ B and Y ( W ) = 0. Moreover, there exists W (possibly infinity) such that( W , W ) ⊆ U . It remains to derive a contradiction to finish the proof.First, since Y ( W ) > W , W ) ⊆ B and Y ( W ) = Y ( W ) = 0, we show that the constantfunction y = 0 is not the supersolution for L B [ y ] = 0 in ( W , W ). The reason is as follows.By Proposition 1, the function Y is the solution of the equation L B [ Y ] = 0. If y = 0 is thesupersolution, then y = 0 ≥ Y in ( W , W ) by the comparison principle, which contradicts to thefact that Y ( W ) > , ∀ W ∈ ( W , W ). Since y = 0 is not the supersolution, then there exists some W ∈ ( W , W ) such that L B [0] = − [ µu (cid:48) ( W ) + σ Lu (cid:48)(cid:48) ( W )] = − W − R − ( µW − σ LR ) ≤ . Therefore, µW − σ LR ≥ W ∈ ( W , W ); thus, µW − σ LR ≥ 0. It implies that µu (cid:48) ( W ) + σ Lu (cid:48)(cid:48) ( W ) ≥ . (A-6)Second, in ( W , W ) ⊆ U , by calculation, we have L U [0] = − W − R − ( µW − σ LR ) . 22y Equation (A-6), we have L U [0] ≤ . (A-7)Since Y ( W ) = Y ( W ) = 0, then the constant function y = 0 is the subsolution for L U [ y ] = 0. Bythe comparison principle, we obtain Y ( W ) ≥ , ∀ W ∈ ( W , W ) (A-8)which is impossible since ( W , W ) belongs to the unconstrained region. We notice that if W = ∞ ,then we apply the comparison principle for the unbounded domain ( W , ∞ ). See Fleming and Soner(2006) for the comparison principle. The proof is thus completed. (cid:3) Proof of Proposition 3. Under Assumption A and Lemma 4.1, V ∞ ( W ) < ∞ , then the valuefunction V ( W ) is well-defined. By Proposition 1, the value function V ( W ) is the unique viscositysolution of the HJB equation (8). Moreover, the unconstrained region and the constrained regionare given by [0 , W ∗ ) , ( W ∗ , ∞ ) for a positive number W ∗ , by Proposition 2.First, in the constrained region, the general solution of the homogeneous second-order linearODE L f ( W ) = ( λ + δ ) f, L f = 12 σ L f (cid:48)(cid:48) ( W ) + ( µL − cW ) f (cid:48) ( W )is smooth (See Borodin and Salminen, 1996, Chapter 2). Therefore, by the method of variation ofparameters method (King, Billinghan and Otto, 2003, Chapter 1), the general solution to the ODE(14) is smooth in the constrained region.Second, for the constrained region, we use the variable g such that V W = g − R . This variableis well-defined because of the strictly concavity of the value function. Then we have W = G ( g )for an increasing function G ( · ). By Proposition 2, the unconstrained region (0 , W ∗ ) correspondsone-one to a region (0 , g ∗ ). Moreover, Lemma 6.4 implies G (0) = 0. By differentiating both sidesof equation (13), the HJB equation for the value function is reduced to the following second-orderODE for the function G ( · ), G (cid:48)(cid:48) ( g ) = Rκ (cid:0) ( λ + δ + c − ρ ) g − − g R − G − R (cid:1) G (cid:48) − c R κ g − G. (A-9)Then G ( · ) is smooth inside the region (0 , g ∗ ). Lemma 4.1 and Lemma 6.4 provide boundaryconditions to solve the ordinary differential equations in the unconstrained and constrained region.Because V ( W ) is the unique smooth solution of the HJB, then Proposition 2 implies a unique W ∗ such that the function V ( W ) presented above is smooth in the region (0 , ∞ ). (cid:3) Proof of Lemma 4.2: Without lost of generality, we assume that V (cid:48) ( x ∗ − ) < < V (cid:48) ( x ∗ +) andderive a contradiction. Since there is no available test function, the subsolution holds automatically.23e next check the supersolution. Let the test function in the form of ψ ( x ) ≡ V ( x ∗ ) + 12 (cid:2) V (cid:48) ( x ∗ − ) + V (cid:48) ( x ∗ +) (cid:3) ( x − x ∗ ) + α ( x − x ∗ ) We claim that α can take any real value: To make ψ ( x ) the valid test function, we need to guaranteethat ψ ( x ) ≤ V ( x ) when x is in a small neighborhood of x ∗ . However, when x → x ∗ , the linearterm [ V (cid:48) ( x ∗ − ) + V (cid:48) ( x ∗ +)] ( x − x ∗ ) will dominate the quadratic term α ( x − x ∗ ) . Therefore, when x and x ∗ are close enough, we could choose sufficiently large α such that ψ ( x ) ≤ V ( x ). It is nowclear that α can take any value.Now, apply the viscosity property at x ∗ , we have F (cid:18) x ∗ , V ( x ∗ ) , (cid:2) V (cid:48) ( x ∗ − ) + V (cid:48) ( x ∗ +) (cid:3) , α (cid:19) ≥ , which is impossible by the free choice of the parameter α . (cid:3) Proof of Proposition 4. We construct explicitly a candidate function of the value function byassuming its smooth property, and verify it is indeed the smooth value function under assumptions.The proof is divided into several steps. Step 1. We derive candidate solution of equation (14) in the constrained region, assuming W ∗ is known. To simplify notation we still use V ( W ) to represent the feasible solution of the valuefunction, being a solution of a corresponding ODE.We notice that the solution of the homogeneous ODE, σ L V W W + LµV W − ( λ + δ ) V ( W ) = 0,can be written as C e β W + C e β W . By the method of partial integral, one particular solution forthe non-linear ODE (13) is V ( W ) = − (cid:90) W σ L u ( x ) (cid:26) e β x e β W − e β W e β x W ( e β x , e β x ) (cid:27) dx (A-10)where W ( f, g ) = f g (cid:48) − f (cid:48) g is the Wronskian determinants of two solutions { f, g } of a homogeneoussecond-order ODE. By a straightforward calculation, V ( W ) = 2( β − β )(1 − R ) σ L × (cid:26) e β W (cid:90) W x − R e − β x dx − e β W (cid:90) W x − R e − β x dx (cid:27) . Therefore, the function V ( W ) is well-defined and it can be expressed in terms of the incompletegamma function. The general solution of the ODE (13) is V ( W ) = C e β W + C e β W + V ( W ) . (A-11)24 tep 2. We show that C = β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) in equation (A-11).By Lemma 4.1, V ( W ) W − R is bounded above by a constant. Therefore, V ( W ) /e β W → W → ∞ in the constrained region. On the other hand, by (A-11), we have as W → ∞ C + C e ( β − β ) W + V ( W ) e β W → V ( W ) e β W = 2( β − β )(1 − R ) σ L × (cid:26) e ( β − β ) W (cid:90) W x − R e − β x dx − (cid:90) W x − R e − β x dx (cid:27) . (A-13)For the the first term in the bracket of (A-13), since β < 0, we have e ( β − β ) W (cid:90) W x − R e − β x dx = e − β W (cid:90) W x − R e β ( W − x ) dx ≤ e − β W (cid:90) W x − R dx = e − β W W − R − R which tends to 0 as W → ∞ .For the second term in the bracket of (A-13), change of variable y = β x leads to (cid:90) W x − R e − β x dx = ( β ) R − (cid:90) β W y − R e − y dy. Therefore, it is an incomplete gamma function γ (2 − R, β W ), By the property of incompleteGamma function (B-3) in Appendix B,( β ) R − (cid:90) β W y − R e − y dy → ( β ) R − Γ(2 − R ) . Then, we obtain C = 2( β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) . Step 3. We characterize the feasible solution in the unconstrained region.Following the classical method in Karatzas and Shreve (1998), Villa and Zariphopoulou (1997),we introduce a new variable g by V (cid:48) ( W ) = g − R . By Lemma 6.2, V (cid:48) ( W ) is a decreasing function.Then, W = G ( g ) for an increasing function G . We characterize the function G ( g ) and derive thefeasible function in terms of the auxiliary function G ( · ).25ince W = G ( g ( W )), then 1 = G (cid:48) ( g ) g (cid:48) ( W ), yielding G (cid:48) ( g ) = 1 /g (cid:48) ( W ). By using V (cid:48)(cid:48) ( W ) = − Rg − R − G (cid:48) ( g ) , the HJB equation becomes( λ + δ ) V ( G ( g )) = 11 − R [ G ( g )] − R + κR g − R +1 G (cid:48) ( g ) . We differentiate both sides of the above equation again with respect to W , obtaining G (cid:48)(cid:48) ( g ) = Rκ g − G (cid:48) ( g ) (cid:2) λ + δ − ρ − g R ( G ( g )) − R (cid:3) (A-14)Since G ( · ) is strictly increasing, the unconstrained region of W corresponds one-one to a region of g . Moreover, by Lemma 6.1, we have, for any W ≤ W ∗ , V ( W ) = u (0) λ + δ + (cid:90) W V W dW = u (0) λ + δ + (cid:90) G − ( W )0 g − R G (cid:48) ( g ) dg. Therefore, the feasible value function in the unconstrained region is uniquely determined by theauxiliary function G ( · ). Step 4. We derive the boundary condition for ordinary differential equation (A-14), assumingthe existence of W ∗ . Since V (cid:48) (0) = + ∞ (Lemma 6.4), we have G (0) = 0. Second, at W = W ∗ , G ( g ∗ ) = W ∗ . Moreover, the constraint − µσ V (cid:48) ( W ∗ − ) V (cid:48)(cid:48) ( W ∗ − ) = L implies that G (cid:48) ( g ∗ ) = LRσ µ ( g ∗ ) − . By the characterization of the feasible value function in Step 3, the required smooth-fit conditionis ( g ∗ ) − R = 2( β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W + Cβ e β W ∗ + V (cid:48) ( W ∗ )Therefore, the boundary condition of the ODE (A-14) are G (0) = 0, G ( g ∗ ) = W ∗ and G (cid:48) ( g ∗ ) = LRσ µ ( g ∗ ) − . Step 5. We determine the parameter C in terms of W ∗ and the parameter W ∗ . The smooth-fitequation can be written as − µσ V (cid:48) ( W ∗ +) V (cid:48)(cid:48) ( W ∗ +) = L . Then, the feasible function in Step 2 implies that − µ (cid:20) β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W + Cβ e β W ∗ + V (cid:48) ( W ∗ ) (cid:21) = σ L (cid:20) β − β )(1 − R ) σ L ( β ) R Γ(2 − R ) e β W + Cβ e β W ∗ + V (cid:48)(cid:48) ( W ∗ ) (cid:21) g ∗ and C are determined uniquely by W ∗ as in (18) and (17). It remains to solvethe parameter W ∗ . The value-matching equation, V ( W ∗ − ) = V ( W ∗ +), can be written as u (0) λ + δ + (cid:90) g ∗ g − R G (cid:48) ( g ) dg = 2( β − β )(1 − R ) σ L ( β ) R − Γ(2 − R ) e β W ∗ + Ce β W ∗ + V ( W ∗ ) . (A-15)By assumption, there exists such a positive number W ∗ in equation (19).By the above discussions in Step 1 - Step 6, the presented function is a smooth solution of theHJB equation. Then, W ∗ is unique and V ( W, L ) is given by (21) in Proposition 1. (cid:3) Proof of Proposition 5. In the unconstrained region, V W = g − R . Since V (cid:48)(cid:48) ( W ) = − Rg − R − G (cid:48) ( g ) ,the optimal strategy is X ( W ) = µRσ gG (cid:48) ( g ). G ( · ) is not a linear function in general. Otherwise, W ∗ = Rσ µ L . Then equation (19) is viewed as an equation of of L , in which both sides are analyticalfunction of the variable L . By the analytical function property, it cannot hold for a general choiceof the capacity level L . (cid:3) Proof of Proposition 6. By using the same argument in proving Proposition 1, we can provethat the value function is the unique viscosity solution of the HJB equation (for V ( W ) = V b ( W ))( λ + δ ) V ( W ) = max ≤ X ≤ bW (cid:20) σ X V (cid:48)(cid:48) + µXV (cid:48) (cid:21) + u ( W ) − cW V (cid:48) ( W )with initial value V (0) = 0. We next find a C solution of the form V ( W ) = a W − R − R to the aboveHJB equation for a positive number a .By a straightforward computation in the HJB equation, and since X ∗ = bW , we have( λ + δ ) a W − R − R = 12 σ b W a ( − R ) W − R − + µbW aW − R + 11 − R W − R − cW aW − R , yielding a = 1 λ + δ + (1 − R )( σ b R − µb ) + c (1 − R ) . Since b < µ − rRσ , then X ∗ = bW is the solution in max ≤ X ≤ bW (cid:2) σ X V (cid:48)(cid:48) + µXV (cid:48) (cid:3) . The proof iscompleted. (cid:3) ppendix B: Incomplete Gamma function The lower incomplete gamma function and the upper incomplete gamma function are defined byby Γ( s, x ) = (cid:90) ∞ x t s − e − t dt ; γ ( s, x ) = (cid:90) x t s − e − t dt. (B-1)For any Re ( s ) > 0, the functions Γ( s, x ) and γ ( s, x ) can be defined easily. Each of them can bedeveloped into a holomorphic function. In fact, the incomplete Gamma function is well-defined forall complex s and x , by using the power series expansion γ ( s, x ) = x s Γ( s ) e − x ∞ (cid:88) k =0 x k Γ( s + k + 1) . (B-2)The following asymptotic behavior for the incomplete gamma function are used in the proof ofProposition 4. lim x →∞ γ ( s, x ) = Γ( s ) , (B-3)and lim x → γ ( s, x ) x s = 1 s . (B-4)See N.M. Temme, “The asymptotic expansion of the incomplete gamma functions” , SIAM J. Math.Anal. 10 (1979), pp. 757 - 766.It can also be connected with Kummer’s Confluent Hypergeometric Function, when Re ( z ) > γ ( s, z ) = s − z s e − z M (1 , s + 1 , z ) (B-5)where M (1 , s + 1 , z ) = 1 + z ( s + 1) + z ( s + 1)( s + 2) + ... (B-6)Therefore, the incomplete Gamma functions can be computed effectively.28 eferences [1] Ahn, S., K. Choi., and B. Lim., 2019. Optimal Consumption and Investment under Time-Varying Liquidity Constraints. Journal of Financial Quantitative Analysis , forthcoming.[2] Bengen, W.,1994. 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Parameters are µ = 0 . , σ = 0 . , R = 0 . , l = 0 . . By calculation, the wealth threshold W ∗ = 492 , abovewhich the retiree invests 700,000 in the stock market. When the wealth portfolio issmaller than W ∗ , the optimal strategy is µRσ gG (cid:48) ( g ) where the auxiliary function G ( · ) isillustrated in Figure 1. “Benchmark” denotes the optimal dollar amount in Lemma 5.1in the absence of the constraint on the risky asset investment. Finally, “BPC” denotesa bounded percentage constraint that X t ≤ µRσ . igure 3: This figure displays the optimal percentage of wealth, X ( W ) W , invested in thestock market. The parameters are the same as in Figure 2. As shown, the percentageis decreasing in the entire region of W . We also notice that the percentage curve issteeper in the beginning of the retirement time when the wealth is closes to initialwealth than that when the wealth closes to the threshold W ∗ . As a function of W , X ( W ) W is not C smooth in contrast to the standard model (Richard, 1965, and Liu andLowenstein, 2002) or the model under leverage constraint (Proposition 6 in this paper,and Vila and Zariphopiulou (1997)). igure 4: This figure displays the effect of the risk capacity level, L , on the investingstrategy. The parameters are the same as in Figure 2. As shown, the higher thecapacity level L , the higher the dollar amount in the risky asset. The figure alsodemonstrates that the threshold, W ∗ , positively depends on L . The risk capacity level L affects both the expected level of spending and the investing strategy even whenthe portfolio value is smaller than this threshold.affects both the expected level of spending and the investing strategy even whenthe portfolio value is smaller than this threshold.