The optimal spending rate versus the expected real return of a sovereign wealth fund

Aase, Knut K.; Bjerksund, Petter Article The optimal spending rate versus the expected real return of a sovereign wealth fund Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Aase, Knut K.; Bjerksund, Petter (2021) : The optimal spending rate versus the expected real return of a sovereign wealth fund, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 14, Iss. 9, pp. 1-35, https://doi.org/10.3390/jrfm14090425 This Version is available at: https://hdl.handle.net/10419/258529 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Aase * and Petter Bjerksund *   Citation: Aase, Knut K., and Petter Bjerksund. 2021. The Optimal Spending Rate versus the Expected Real Return of a Sovereign Wealth Fund. Journal of Risk and Financial Management 14: 425. https:// doi.org/10.3390/jrfm14090425 Academic Editor: Daniel N. Ostrov Received: 24 June 2021 Accepted: 22 July 2021 Published: 6 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Business and Management Science, Norwegian School of Economics, 5045 Bergen, Norway *Correspondence: [email protected] (K.K.A.); petter[email protected] (P.B.) Abstract: We consider a sovereign wealth fund that invests broadly in the international financial markets. The influx to the fund has stopped. We adopt the life cycle model and demonstrate that the optimal spending rate from the fund is significantly less than the fund’s expected real rate of return. The optimal spending rate ensures that the fund will last “forever”. Spending the expected return will deplete the fund with probability one. Moreover, this strategy is inconsistent with optimal portfolio choice. Our results are contrary to the idea that it is sustainable to spend the expected return of a sovereign wealth fund. Keywords: optimal spending rate; endowment funds; expected utility; risk aversion; EIS; recursive utility JEL Classification: G10; G12; D9; D51; D53; D90; E21 1. Introduction We consider optimal investment strategies and the associated optimal extraction from an endowment fund consistent with the life cycle model. We demonstrate that the optimal spending rate is strictly smaller than the expected rate of return. The difference is far from negligible, and amounts to several percentage points in most real situations. The basic explanation is: If the fund is managed by diversification, this means that risk aversion, consumption substitution and impatience are essential in the optimal portfolio choice problem. Then, to be consistent, the spending rate must also reflect this. Accordingly, the expected real rate of return is typically not an optimal spending rate, since this criterion would normally be associated with risk neutrality. We take the security market as given—it is assumed to be in equilibrium—and introduce a price taking agent in this market. In this setting we reconsider the problem of optimal consumption and portfolio selection. In the context of an endowment fund, the results from analyzing this more general problem can immediately be utilized in order to determine an optimal spending rate. We have considered both expected utility, in which case risk aversion plays a prominent role, and recursive utility, in which case consumption substitution is separated from risk aversion and is also important. When the investment opportunity set is deterministic, there exist explicit and closedform solutions for optimal extraction. Rather than depending upon the expected rate of return, the optimal extraction rate is a convex combination of the impatience rate and the certainty equivalent rate of return. The latter quantity is significantly smaller than the expected rate of return. This is normally true also for the impatience rate, and thus for the convex combination. For a stochastic investment opportunity set, we present formulas herein which we believe are original. First and foremost, these solutions are demonstrated to result in smaller values than the expected real rate of return on the endowment fund, for plausible values of the preference parameters and the other parameters of the problem. The difference is significant in most cases. J. Risk Financial Manag. 2021,14, 425. https://doi.org/10.3390/jrfm14090425 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2021,14, 425 2 of 35 If the extraction rate is the one of expected return, this normally means that the agent is risk neutral at the level of spending, and must then, to be consistent, be risk neutral at the level of optimal portfolio selection as well. However, the consequence of such an investment strategy is rarely advocated by anyone responsible for an endowment fund, whatever its purpose. We demonstrate that a popular and much advertised extraction policy, the expected real rate, is not consistent with a sustainable spending rate, and will with probability one eventually deplete any fund that is managed by diversification. Most endowments have the perspective that they should last “forever”. Consequently, there is a trade-off between current spending and future spending opportunities. Tobin (1974) develop sustainable spending rules in a deterministic world. It can be argued that it is sustainable to spend the real interest rate within this setting. Uncertainty complicates this picture. Some would argue that it is sustainable for an endowment to spend the expected return; see, e.g., Campbell (2012), who considered university endowments. Moreover, this idea motivates the current 3% fiscal rule that applies to the 1 trillion USD Norwegian sovereign wealth fund. Our article is concerned with optimal extraction from endowment funds in general, and has in particular been motivated by the Norwegian “Government Pension Fund Global”, which in the past was called the Norwegian Oil Fund or just the Norwegian Sovereign Wealth Fund, which we consider as an example of the general theory. This is illustrated on several occasions below. Related Literature Dybvik and Qin (2019) considered a fund with normal iid log-returns. The authors found that for the fund to last “forever”, spending must not exceed the expected return subtracted by half the variance. The discrepancy between the fund’s expected return and sustainable spending is far from negligible. The two key decisions of an endowment fund that invests in the financial market are how much risk to take and how much to spend. From a theoretical point of view, the two decisions are closely related and must be determined jointly. To examine the questions one must, we claim, address the issue of the objective function by which optimality is to be measured. Merton (1971) presented optimal portfolio and consumption rules for an investor who maximizes expected, additive and separable utility with constant relative risk aversion in a continuous-time world, where risky asset returns are iid. Recursive utility is a more generalized framework where the investor’s risk aversion and consumption substitution are disentangeld; see, e.g., Epstein and Zin (1991). Campbell and Sigalov (2020) adopted the Merton model and Epstein–Zin preferences, and assumed that there is a constraint on the spending rule. The authors examined two alternative constraints: (i) spending the expected return; and (ii) the maximum sustainable spending following the assumption of Dybvik and Qin (2019). The authors found that the former constraint induces increased risk taking (referred to as “reaching out for yield”). In Merton (1990), optimal investment strategies for university endowment funds were analyzed. The objective was maximization of expected utility, related to several activities consistent with the purposes of the university. They limited the scope to how much to optimally spend in the numeraire unit of account, which is a purely financial question. How much to spend on each of several activities we consider a political issue. One purpose of this study is to compare the optimal spending with the conventional wisdom of spending the expected return, or any other ad hoc rule, under various assumptions. For this purpose, we adopt the life cycle model used by Merton (1971). We also consider the recursive utility framework in the setting of continuous time. We find that for realistic parameter values, the endogenously determined optimal spending is less than the fund’s expected return. For most cases, the discrepancy is far from negligible. We also consider a more general situation where the investment opportunity set is stochastic and derive analytical results that to our knowledge are new. We find that the insights from a J. Risk Financial Manag. 2021,14, 425 3 of 35 deterministic investment opportunity set carries over to a more general setting where the investment opportunity set is stochastic. The paper is organized as follows: The basic continuous-time model is presented in Section 2. Section 3analyzes the problem of the agent having standard additive and separable expected utility. We present examples in Sections 3.3–4. In Sections 3.5–6we look at the asymptotic behavior of the wealth as time goes to infinity under the two extraction rules. Section 3.7 deals with a stochastic investment opportunity set. In Section 4 we introduce recursive utility and analyze the problem, first with a stochastic investment opportunity set, and then (in Section 4.5) with a deterministic one. In Section 4.6 we analyze the asymptotics of the wealth for the two types of spending rates with recursive utility. Section 4.7 deals with a particular example, the Norwegian SWF Government Fund Global. In Section 5we discuss the role of a state owned sovereign wealth fund where there is additional consumption in society, and Section 6concludes. 2. The Basic Model We consider the optimal consumption and portfolio selection problem using the life cycle model. We have an agent represented by the pair (U , e) , where U(c) is the agent’s utility function over consumption processes c , and e is the agent’s endowment process. The problem consists of maximizing utility, subject to the agent’s budget constraint: supc,ϕU(c)subject to EZT 0πtctdt≤EZT 0πtetdt:=w, (1) where ϕ are the optimal fractions of wealth in the various risky investment possibilities facing the agent, and w is the current value of the agent’s wealth. The quantity πt is the state price deflator at each point in time t —i.e., the Arrow–Debreu state prices in units of probability. The horizon is T≤∞. The consumer takes as given a dynamic financial market, consisting of N risky securities and one riskless asset, the latter having rate of return rt , a stochastic process. The agent’s actions do not affect market prices of the risky assets, nor the risk-free rate of return rt. 3. Optimal Consumption and Portfolio Choice Herein we consider two different specifications of utility: (i) the standard model with separable and additive expected utility, and (ii) recursive utility of the Duffie–Epstein type with a Kreps–Porteus specification of the associated certainty equivalent, the latter being derived from expected utility. We consider a continuous-time framework. In case (i) the agent’s preferences are represented by standard expected additive and separable utility of the form U(c) = EZT 0u(ct,t)dt. (2) u(c , t) is the agent’s felicity index, which we assume to be of the CRRA-type, meaning that the real function u(x , t) = 1 1−γx1−γe−δt , where γ is the agent’s relative risk aversion and δ is the agent’s impatience rate (the utility discount rate). It follows from optimal consumption and portfolio choice theory that the optimal consumption per time unit, c∗ t , and the optimal wealth at time t , W∗ t , are connected. The starting point for this derivation is the following formula for the market value of current wealth Wt. W∗ t=1 πt EtnZT tπsc∗ sdso. (3) Et(X) = E(X|Ft) is the conditional expectation of any random variable X given the information by time t , where Ft is the information filtration, 0 ≤t≤T and πt is the state price deflator. Under the assumption of no arbitrage possibilities, it is given by πt=e−Rt 0(ru+1 2η0 uηu)du−Rt 0ηudBu(4) J. Risk Financial Manag. 2021,14, 425 4 of 35 where rt is the risk free rate of return at time t , ηt is the market price of risk and Bt is a standard d -dimensional Brownian motion, which generates the information set Ft for all t∈[0, T]. The financial market consists of N risky assets, where η0 tηt=λ0 t(σtσ0 t)−1λt , and the vector λt= (µ1(t)−rt , µ2(t)−rt , · · · , µN(t)−rt)0 represents the risk premiums of the risky assets, i.e., the excess expected returns of the risky assets over the riskless returns at any time t∈[ 0, T] . The quantity µn(t) is the rate of return on asset n at time t ; n= 1, 2, · · · , N ; and prime signifies the transpose of a vector (or matrix). The matrix σtσ0 t is the instantaneous variance/covariance matrix of the risky assets in units of prices. All these quantities may be stochastic processes. For simplicity of exposition we assume that d=N. 3.1. Optimal Consumption and Extraction with Expected Utility: A Deterministic Investment Opportunity Set The agent’s optimal consumption and portfolio choice is determined next. First we give a representation of the optimal consumption c∗ t at any time t∈[ 0, T] . By employing Kuhn–Tucker and the saddle point theorems, we find that the optimal consumption is given by c∗ t=π−1 γ t(αeδt)−1 γ, (5) where α is the Lagrange multiplier, ultimately determined by equality in the budget constraint. This gives the following dynamics for the optimal consumption. dc∗ t c∗ t =µc(t)dt +σc(t)dBt, (6) where, µc(t) = 1 γ(rt−δ) + 1 2 1 γ(1+1 γ)η0 tηt and σc(t) = 1 γηt. Let It= (rt , ηt , λt) signify the investment opportunity set. We can write the optimal wealth in Equation (3) of the agent in terms of the optimal consumption as follows. W∗ t=c∗ tEtnZT te1−γ γ[Rs t((ru+1 2η0 uηu)−δ 1−γ)du+Rs tηudBu]dso, (7) where we have used the dynamics for the state price deflator in (4) and for the optimal consumption in (6). In this expression the conditional expectation is in general a random variable (process), in which case the volatility of W∗ t is not the same as the volatility of c∗ , and the instantaneous correlation coefficient between these two processes is not unity. We want to compute the conditional expectation, and consider two cases: (i) where the investment opportunity set is deterministic, and (ii) where the set Itis stochastic. We start with (i). Clearly this assumption involves some loss of generality. We treat the situation (ii) later. In Section 3.7 (for expected utility) and in the section about general recursive utility, aside from the obvious main result of the paper which is of an applied nature, the theoretical contributions of the paper can be found. Now, by the Fubini theorem and the moment generating function of the normal distribution, we can write the above equation as follows. W∗ t=c∗ tZT te[1−γ γ(r+1 2η0η)−δ γ+1 2(1−γ γ)2η0η](s−t)ds. (8) J. Risk Financial Manag. 2021,14, 425 5 of 35 The optimal consumption to wealth ratio is then c∗ t W∗ t =kT(t)a.s. (9) where kT(t) is an estimate of the optimal extraction rate at the present time t , where 0≤t<T. The expression for kT(·)can be written as kT(t) = k 1−e−k(T−t), (10) where the kis a constant for all tby our above assumption (i), and is given by k=r−r γ+δ γ−1−γ 2γ2λ0(σσ0)−1λ. (11) Provided that k>0, the function kT(t)→kas T→∞for any fixed value of t.1 The result that k is non-random and time invariant follows from our assumption about a deterministic investment opportunity set. For example, it has as the consequence that the volatility of W is the same as the volatility of c . If the investment opportunity set is stochastic, naturally this is no longer true. However, in order to focus on the essential questions raised in this paper, we make this simplification here. We analyze the situation with a stochastic investment opportunity set in Section 3.7 below. With a very long horizon T , it is optimal for the agent to consume a fraction of the remaining wealth at any time t . In reality, this fraction is a stochastic process. Here it is a deterministic function slowly increasing in t , and when the horizon approaches, it increases sharply (see, e.g., Figure 1below). If the horizon is unbounded at the outset, the fraction k is consumed forever. 3.2. The Real Rate of Return versus the Optimal Extraction Rate Recall the dynamics of the wealth portfolio Wt . It is given by the following stochastic differential equation. dWt= [Wt(ϕ0 tλt+rt)−ct]dt +Wtϕ0 tσtdBt,W0=w. (12) We do not use control theory explicitly in our exposition. In the language of optimal control theory, in order for the wealth W to remain nonnegative, any admissible control (c , ϕ) has ϕt= 0 and Wt=ct= 0 for t larger than the stopping time inf {s:Ws= 0 } . Thus, nonzero investment and spending are ruled out once there is no remaining wealth. The problem (1) of maximizing utility subject to the agent’s budget constraint results in both the optimal fractions in the various securities, and the associated optimal consumption, see Mossin (1968); Samuelson (1969) and Merton (1969,1971) for the earliest treatments of this joint problem). With a deterministic investment opportunity set, the optimal portfolio weights at any time tare given by ϕt=1 γ(σσ0)−1λfor all t. (13) We want to compare the optimal extraction rate k given Equation (11) with the (conditional) expected real rate of return on the optimal wealth portfolio W∗ t , which is the solution to the stochastic differential equation (Equation (12)) with ct=c∗ t , the optimal consumption and the portfolio fractions given in Equation (13). The (simple) return in the time interval dt is dRt, where dRt=dW∗ t+c∗ tdt W∗ t . (14) J. Risk Financial Manag. 2021,14, 425 6 of 35 With this interpretation, Equation (14) is a standard expression for the real return with dividends. Accordingly, from (14), Equation (12) and the optimal portfolio rule in Equation (13) , the t -conditional expected real rate of return of the wealth portfolio is given by the following expression: EtdRt/dt =r+1 γλ0(σσ0)−1λ. (15) The optimal extraction rate kmay be rewritten as follows. k=δ γ+1−1 γr+λ0(σσ0)−1λ 2γ. (16) We then have the following result. Proposition 1. Assuming a deterministic investment opportunity set, the optimal extraction rate k is a constant and depends on the return from the fund only via the certainty equivalent rate of return, and can be written as k=δ γ+1−1 γr+1 2γϕ0(σσ0)ϕ. (17) Proof. Starting with the risk premium, 1 γλ0(σσ0)−1λ=1 γλ0(σσ0)−1(σσ0)(σσ0)−1λ= γ1 γλ0(σσ0)−1(σσ0)1 γ(σσ0)−1λ= γ(1 γ(σσ0)−1λ)0(σσ0)1 γ(σσ0)−1λ) = γϕ0(σσ0)ϕ, where we have used (13). From this result it follows that the quantity 1 2γϕ0(σσ0)ϕ can be recognized as relative certainty equivalent for “proportional risks”, since ϕ0σ is the volatility of the wealth portfolio (see Equation (12)).2 One basic comparison is between the expected real rate of return on the wealth portfolio given in (15) and the optimal extraction rate k . Assuming an infinite horizon for now, the inequality k≤r+1 γλ0(σσ0)−1λ(18) holds if and only if r≥δ−λ0ϕ1+γ 2γ. (19) Since the second term on the right-hand side is negative, this inequality is true for reasonable values of the parameters of this problem.3 Alternatively, using the certainty equivalent and the representation for k given in Equation (16), the inequality (19) is equivalent to 1 2γϕ0(σσ0)ϕ≥(δ−r) 1+γ. (20) J. Risk Financial Manag. 2021,14, 425 7 of 35 That is, when half the expected excess return on the fund over the risk-free rate is larger than the right-hand side of (20), then the extraction rate is lower than the expected rate of return on the wealth portfolio. Again, for reasonable values of the parameters of the problem, this can be seen to hold true. A very simple case occurs when δ≤r , in which case the inequality is obviously true, a fact which can be recognized from the inequality (19) as well. Typically, the real risk-free rate close to 1% is consistent with US-data (see Table 1 below). Additionally, a reasonable value for the impatience rate is around 1%. 4 In this case the risk premium of the fund is certainly positive, about 6% for the data of Table 1, so the inequality (20) holds with a significant margin. We conclude that for plausible values of the parameters, the optimal extraction rate is strictly smaller than the expected real rate of return on the wealth portfolio. It can be seen that when the extraction rate k equals the expected rate of return on the fund W , then the expected value E(Wt) = W0 for any horizon t , and Wt can be shown to be a martingale. Seen from time 0, the end wealth of the agent corresponds to the random variable Wt , not the sure amount W0 . Considered from the beginning of the period, a risk-averse agent would prefer the W0 to the random wealth Wt . A claim that the agent considers the random future value Wt as equivalent to the plain expected value as of time zero thus rests on an implicit assumption that the agent is risk-neutral. To use the expected return on the endowment fund as the extraction rate is, on the other hand, consistent with investing “everything” in the single risky asset, or group of assets, with the largest expected return(s) one can find, and completely ignoring risk. 5 Few responsible agents would recommend this “optimum portfolio selection strategy” for an endowment fund. This is, however, what Campbell (2012) seems to claim, where the author recommends that kis set equal to the real expected rate of return. In the author’s own words: “The sustainable spending rate of an endowment, which is the amount spent as a fraction of the market value of the endowment, must equal the expected return in order to achieve immortality.” This is called “vigorous immortality” by the author. As we have just demonstrated, this policy is a little bit too vigorous to be rational and consistent, and implies the abovementioned contradiction. This policy will eventually deplete the fund with probability 1, to be shown in the Section 3.5. Can the policy advocated by Dybvik and Qin (2019), also considered in Campbell and Sigalov (2020), be consistent with the optimal spending rule outlined in the above? A little analysis shows that this requires r=δ and γ= 0, but the latter is not allowed in our model. Accordingly, the criterion of the expected return subtracted by half the variance is not optimal for valid values of the preference parameters. 3.3. An Example of a Typical Fund We now illustrate the above theory by the use of real data. We assume that the agent takes the US market as given, where we let the risky part of our fund be represented by the S & P -500 index. This corresponds to one of the best functioning securities market in the world, and should be representative in the construction of the underlying market quantities. The relevant data are given as follows. Table 1represents the summary statistics of the data used by Mehra and Prescott (1985). 6 By σcM(t) we mean the instantaneous covariance rate between the return on the index S&P-500 and the consumption growth rate. Similarly, σMb(t) and σcb(t) are the corresponding covariance rates between the index M and government bills b and between aggregate consumption c and Government bills, respectively.7 J. Risk Financial Manag. 2021,14, 425 8 of 35 Table 1. Key US-data for the time period 1889–1978. Continuous-time compounding. Expectation Standard dev. Covariances Return S&P-500 6.78% 15.84% ˆ σMb =0.001477 Government bills 0.80% 5.74% ˆ σcb =−0.000149 Equity premium 5.98% 15.95% Consumption growth 1.81% 3.55% ˆ σMc =0.002268 3.4. Examples Based on Expected Additive and Separable Utility As an example, consider a wealth fund described by the three upper rows of Table 1. The consumption data in Table 1, the fourth row, has to do with society at large, which is not under consideration here. Let us assume a relative risk aversion of γ= 2.5, and an impatience rate δ= 0.01. For the market structure of Table 1, we obtain that the expected rate of return on the wealth portfolio is 0.065 and the certainty equivalent rate of return is 0.037, corresponding to the optimal portfolio selection rule ϕ= 0.95. The optimal extraction rate under our assumptions is k= 0.026, corresponding to T=∞ . The drawdown rate is seen to be significantly lower than the expected rate of return on the portfolio for these rather reasonable parameters of the preferences of the agent. In Figure 1we present graphs with a finite time horizon of T= 300 years using the expected utility model explained above, with the parameters of this example. The optimal long run extraction rate k is the lower horizontal (blue) line in Figure 1. The expected return on the wealth portfolio is the upper horizontal (green) line in the figure. As the horizon approaches, there is a sharp increase in the rate of consumption. After about 200 years, the rate kT(200) = 0.028, a modest increase from the steady state value of 0.026. Figure 1. The optimal drawdown rate vs. expected return. T=300. The optimal consumption in this case has the expected growth rate given by the formula µc=1 γ(r−δ) + 1 2 1 γ(1+1 γ)λ0(σσ0)−1λ. As in the proof of Proposition 1, we can alternatively write this as µc=1 γ(r−δ) + 1 2(1+1 γ)γϕ0(σσ0)ϕ. (21) This term is estimated as 0.039, and the estimate of the volatility σc is 0.1510, which equals the estimate of σW=ϕσ . According to our assumption about a deterministic investment opportunity set, this implies that these two volatilities must be equal—i.e., ϕσ =0.1510. J. Risk Financial Manag. 2021,14, 425 15 of 35 From the stochastic differential equation for the optimal wealth given in (12) we then obtain by the product rule and diffusion invariance that ϕt=1 γ(σtσ0 t)−1λt+ (σtσ0 t)−1σtEt[Dt(FT t)]/FT t. (32) The expression for the optimal portfolio has two terms. The first is identical to the optimal portfolio for a constant investment opportunity set, except that here σt and λt are allowed to be stochastic. The second term adjusts for the time and state variations of the investment opportunities, referred to as the intertemporal hedging term, and is seen from (31) and (32) to be forward-looking. The first term ignores these variations, is certainly not forward-looking and is called myopic for that reason (see Mossin (1968)). The random term Et[Dt(FT t)]/FT t can be connected to the parameters of the problem via the Malliavin derivative of FT t. We have the following. fT(t,u) = Eu[Du(FT t)] = EthZT teRs t(1−γ γrv+1 2 1−γ γ2η0 vηv)−δ γ)dv Zs t1−γ γDu(rv) + 1 2 1−γ γ2(η0 vDu(ηv) + Du(η0 v)ηv)dvdsi. (33) Therein we have used the “chain rule” and other rules of this calculus (see, e.g., Di Nunno et al. (2008)). The Malliavin derivatives Du(rv) and Du(ηv) can be further broken down by specifying the types of model for r and η . For example, if the spot interest rate follows a diffusion process of the Ornstein–Uhlenbeck model, or Vasicek type of the form rv=r0+Zv 0µr(w)dw +Zv 0σr(w)dBw where µr(w) and σr(w) are deterministic, and σr(w) = αew−v , then Du(rv) = αe−veu> 0, for αad-vector of positive constants. When the relative risk aversion γ> 1, we notice from (33) and the subsequent discussion that the second term in (32) typically is a vector of negative portfolio weights. This seems intuitive, since a risk-averse agent will invest less in the risky assets when confronted with a stochastic investment opportunity set. This term can be seen to hedge against the unanticipated changes in the variables in the investment opportunity set. The opposite conclusion follows if γ< 1, but as we have indicated before, this case is not very intuitive with expected utility because of the two different interpretations of the parameter. With an infinite horizon, the extraction rate is smaller than the real rate of return when the inequality kt≤rt+ϕ0 tλtholds, which is equivalent to kt≤rt+1 γλ0 t(σtσ0 t)−1λt+1 F∞ t f∞(t,t)0ηt, (34) assuming that σtis invertible. This inequality holds for all tif and only if rt γ≥δ γ−1 γλ0 t(σtσ0 t)−1λt1+γ 2γ−1 F∞ t f∞(t,t)0ηt. (35) We then have the following result: Proposition 3. With a stochastic investment opportunity set, provided the inequality |1 F∞ tf∞(t , t)0ηt| <1 γλ0 t(σtσ0 t)−1λt1+γ 2γ holds, the optimal extraction rate is strictly smaller than the real rate of return on the fund, unless the impatience rate δis unreasonably large. J. Risk Financial Manag. 2021,14, 425 16 of 35 Proof. From the inequality (35) we notice that the second and third terms on the right-hand side add up to something negative under the condition of the proposition. Thus, if rt≥δ the inequality then holds, and the conclusion follows. How reasonable is the assumption of the proposition in practice? Unless the inequality holds, the investment policy more or less prescribes short-selling most of the risky assets in the portfolio, which is unheard of in real life portfolio choices of the type that we are studying here. Further insights from the analysis involving a stochastic investment opportunity set can be gained from inspection of the expression in Equation (33). For example, the optimal portfolios are seen to depend on the impatience rate δ and the horizon T , neither of which is present in the standard expression with a deterministic investment opportunity set. In other words, impatience has a direct impact on the optimal portfolio, and the dependence on T has a potential to address the horizon problem. There is, however, nothing in the model that indicates that the investments in the risky assets should decrease when t approaches the horizon T(see, e.g., Aase (2017) for a treatment of this problem). The application of the results of this section to the data in Table 1is by and large similar to the illustrations given in Section 3.4, since the data, like the data in Table 1, are based on estimates, assuming stationarity (or some kind of ergodicity), and are therefore estimates of the expected value ¯ kt . However, the margin between the optimal extraction rate and the expected rate of return may have diminished, depending upon the stochastic structure of rtand ηt. 4. Recursive Utility This preference structure is known to give far more reasonable results than the expected utility model when it comes to calibrating to real data; see, e.g., Aase (2016a,2016b), where the celebrated equity premium puzzle is solved using recursive utility, among other things. We use the framework established by Duffie and Epstein (1992a,1992b) and Duffie and Skiadas (1994) which elaborates the foundational work by Kreps and Porteus (1978) of recursive utility in dynamic models. Recursive utility leads to the separation of risk aversion from the elasticity of intertemporal substitution in consumption, within a timeconsistent model framework. The recursive utility U:L→R is defined by two primitive functions: f:R×R→R and A:R→R . The function f(ct , Vt) corresponds to a felicity index, and A corresponds to a measure of absolute risk aversion of the Arrow–Pratt type for the agent. In addition to current consumption ct , the function f also depends on future utility Vt at time t , a stochastic process with volatility ˜ σV(t):=Ztat each time t. The utility process V for a given consumption process c , satisfying VT= 0, is given by the representation Vt=EtnZT tf(cs,Vs)−1 2A(Vs)˜ σV(s)0˜ σV(s)dso,t∈[0, T]. (36) If, for each consumption process ct , there is a well-defined utility process V , the stochastic differential utility U is defined by U(c) = V0 , the initial utility. The pair (f , A) generating Vis called an aggregator. The utility function U is monotonic and risk-averse if A(·)≥ 0, and f is jointly concave and increasing in consumption. As for the last term in (36), recall the Arrow–Pratt approximation of the certainty equivalent of a mean zero risk X . It is −1 2A(·)σ2 , where σ2 is the variance of X , and A(·) is the absolute risk aversion function. In the discrete time world the starting point for recursive utility is that future utility at time t is given by Vt=g(ct , m(Vt+1)) for some function g:R×R→R , where m is a certainty equivalent at time t (see, e.g, Epstein and Zin (1989)). If h is a von Neumann– J. Risk Financial Manag. 2021,14, 425 17 of 35 Morgenstern index, then m(V) = h−1(E[h(V)]) . The passage to the continuous-time version in (36) is explained in Duffie and Epstein (1992b), and in a direct form from the discrete time analog, by Svensson (1989). Unlike expected utility theory in a timeless situation, i.e., when consumption only takes place at the end, in a temporal setting where the agent consumes in every period, derived preferences have been claimed not to satisfy the substitution axiom (e.g., Mossin (1969); Kreps (1988)). In Aase (2021) this claim is demonstrated to be incorrect. 4.1. The Specification We work with the Kreps–Porteus utility, where the aggregator has the following CES specification: f(c,v) = δ 1−ρ c(1−ρ)−v(1−ρ) v−ρand A(v) = γ v. (37) The parameter δ≥ 0 is the agent’s impatience rate; ρ≥ 0, ρ6= 1 is what we referred to earlier as the marginal utility flexibility parameter; and γ≥ 0, γ6= 1, is the relative risk aversion. The parameter ψ= 1 /ρ is the elasticity of intertemporal substitution in consumption, referred to as the EIS parameter. The higher the value of the parameter ρ , the more aversion the agent has towards consumption substitution across time in a deterministic world. The higher the value of γ , the more aversion the agent has to consumption fluctuations, due to the different states of the world that can occur. Clearly these two properties of an individual’s preferences are different. In the conventional Eu-model, however, ρ=γ. It can be shown that this specification is the continuous-time analogue of the one used by Epstein and Zin (1989,1991) in discrete time. Using the notation Z(t) = VtσV(t), the dynamics of the utility process are dVt=−δ 1−ρ (c∗ t)1−ρ−V1−ρ t V−ρ t +1 2γVtσ0 V(t)σV(t)dt +VtσV(t)dBt, (38) for 0 ≤t≤T , where VT= 0. This is the backward stochastic differential equation, where a solution consists of the pair (V , Z) . For the particular Kreps–Porteus version that we consider, the standard Lipschitz condition in Duffie and Epstein (1992b) is not satisfied, but existence and uniqueness are shown for this version in Duffie and Lions (1992) under certain conditions. See also Schroder and Skiadas (1999) for uniqueness and existence of solutions of such equations, in particular their Theorem A2. 4.2. The Optimal Consumption and Portfolio Rule As with the standard EU-model, we will need the optimal consumption of an agent. Here the agent is one with recursive utility (U , e) who takes the market as a given and shifts his endowment e in each period from the given et to the optimal one c∗ t using the financial markets. In each period the agent decides how much to consume and how much to invest in the given opportunity set for future consumption. Thus these two problems are intimately connected. 4.2.1. The First Order Conditions By properly extending Pontryagin’s maximum principle to a stochastic environment, we can solve for the basic version of recursive utility as follows. 10 The first-order conditions can be written as α πt=Y(t)∂f ∂c(c∗ t,Vt)a.s. for all t∈[0, T]. (39) Here Y(t) is an adjoint variable. Notice that the first-order condition depends on the future utility Vt . This means, among other things, that the agent is in general not myopic (in the sense of Mossin (1968)). J. Risk Financial Manag. 2021,14, 425 18 of 35 4.2.2. The Optimal Consumption It has been shown in Aase (2016b) that the answers to these two problems are given as follows: The stochastic representation for the optimal consumption growth rate is given by dc∗ t c∗ t =µc(t)dt +σc(t)dBt, (40) where, µc(t) = 1 ρ(rt−δ) + 1 2 1 ρ(1+1 ρ)η0 tηt−(γ−ρ) ρ2η0 tσV(t) +1 2 (γ−ρ)γ(1−ρ) ρ2σ0 V(t)σV(t), (41) and σc(t) = 1 ρηt+ (ρ−γ)σV(t). (42) VtσV(t) = ˜ σV(t) . The latter appears in the definition (36) of recursive utility. Both σV and Vt exist as solutions to a backward stochastic differential equation for V . The quantity ηt is the market price of risk vector, and η0 tηtcorresponds to our previous λ0 t(σtσ0 t)−1λt. For recursive utility in discrete time it is known that the consumption to wealth ratio is equal to ct Wt =1−β (Vt ct)1−ρ, where β=e−δ . It is seen that this ratio is a constant only when ρ= 1, in which case our model is not valid. Thus the consumption to wealth ratio is, in general, a stochastic process. However, in the continuous-time model this is a bit different, as a constant consumption to wealth ratio is possible without requiring that ρ= 1. We treat this special case below. With a stochastic investment opportunity set as we have assumed here, this ratio is not constant. This means that the volatility of consumption σc is not equal to the volatility of wealth σW=ϕσ. It is shown in Aase (2016a) in the context of equilibrium that the wealth of the agent may be internalized as follows: σW(t) = (1−ρ)σV(t) + ρσc(t). (43) This is really an equilibrium result. We may “invert” this relationship when ρ6= 1 to obtain σV(t) = 1 1−ρσW(t)−ρσc(t). (44) where the volatility of utility, one of the primitives of the model, is connected to “observable” quantities. Combining this with (42), we find that σc(t) = 1−ρ ρ(1−γ)ηt−γ−ρ ρ(1−γ)σW(t). (45) With expected utility γ=ρ , and σc(t) = 1 γηt . This formula is, however, not possible to reconcile with aggregated data in society, unless γ is disproportionately large. The result (45) on the other hand, can be used to explain market and consumption data with reasonable values for the two preference parameters ρand γ(see Aase (2016b)). By taking account of the expression (45) for σc(t) , the formula for σV(t) in (44) can alternatively be written as σV(t) = 1 1−γϕ0 tσt−ηt. (46) J. Risk Financial Manag. 2021,14, 425 19 of 35 4.2.3. The Conditional Optimal Portfolio Selection Strategy Turning to optimal portfolio choice in the life cycle model, where the agent is not necessarily the “representative agent”, this problem has been treated in detail by Schroder and Skiadas (1999). Here we pursue a slightly different route. Given that the covariance rate between of the optimal consumption and the market is known, the optimal portfolio fractions in the risky assets associated with this are given by the following formula: ϕ(t) = 1−ρ γ−ρ(σtσ0 t)−1λt−ρ(1−γ) γ−ρ(σtσ0 t)−1(σtσ0 c∗(t)), (47) assuming γ6=ρ . 11 This formula follows from (42) and (44) by noticing that σW(t) = ϕ0 tσt , and must be interpreted as a consistency result, given the covariance rate (σtσ0 c∗(t)) . Formula (47) can not be directly compared to the result for expected utility in Equation (32) , but is nevertheless useful in what follows. 12 In the next sections we establish an optimal spending rate based on (47), and also a formula for ϕ(t), which can indeed be compared to (32). 4.3. The Optimal Spending Rate versus the Real Rate of Return with Recursive Utility With these preparations, we now turn to the spending rate with general recursive utility in a model with a stochastic investment opportunity set. The analysis is analogous to the analysis in Section 3.7 for the standard EU-model, with the exception that we utilize the expression in (47) for the optimal portfolio weights, which calls for some care in interpreting the results. We have found the optimal consumption path c∗ t given an optimal portfolio strategy, and consequently the optimal wealth, is given by the formula W∗ t=1 πt EtnZT tπsc∗ sdso. First observe that the optimal consumption can be represented as c∗ s=c∗ texpnZs t(µc(u)−1 2σ0 c(u)σc(u))du +Zs tσc(u)dBuo, where µc is given in (41) and σc in (42). With the expression for the state price deflator π given in (4) we can write πsc∗ s in terms of πtc∗ t for any t≤s≤T , since we can write the state price deflator as follows: πs=πtexpn−Zs t(ru+1 2η0 uηu)du +Zs tηudBuo. With these preparations, the expression for the optimal wealth can be written as W∗ t=c∗ tEtnZT texphZs t(µc(u)−1 2σc(u)0σc(u))du −Zs t(r(u) + 1 2η0 uηu)du (48) +Zs t(σc(u)−ηu)dBuio. Next we condition on σc and η , and obtain the following wealth to consumption ratio (see Section 3.7 for details): W∗ t c∗ t =ZT tEtnexphZs t(µc(u)−1 2σc(u)0σc(u))du −Zs t(r(u) + 1 2η0 uηu)du +1 2Zt s(σc(u)−ηu)0(σc(u)−ηu)duio. (49) J. Risk Financial Manag. 2021,14, 425 20 of 35 Denoting the integrand in the exponent by −k(u), we have that k(u) = r(u)−µc(u) + η0 uσc(u). (50) Using (41) and (42), this can be written as a convex combination with weight 1 ρ k(u) = δ ρ+1−1 ρru+ρ−γ ρη0 uσV(u)+ 1 2 γ(γ−ρ) ρσV(u)0σV(u) + 1 2ρη0 uηu. (51) By Jensen’s inequality we then have W∗ t c∗ t =ZT tEte−Rs tk(u)duds ≥ZT te−Et(Rs tk(u)du)ds. (52) We now assume first-order stationarity of the investment opportunity set. By Fubini’s theorem we then get W∗ t c∗ t ≥ZT te−Rs tEt(k(u))duds =ZT te−Rs t¯ ktduds =ZT te−¯ kt(s−t)ds, where ¯ kt=Et(k(u)) does not depend on u≥t by our stationarity assumption. This gives that c∗ t W∗ t ≤¯ kT(t)(53) where ¯ kT(t) = ¯ kt 1−e−¯ kt(T−t). This result we illustrate below for the data given in Table 1. The comparison of interest is still between the optimal expected spending rate and the expected real rate of return on the wealth portfolio. With an infinite horizon, the former is the smaller of the two whenever k(u)≤r(u) + ϕ0 uλufor all u≥0, (54) where the optimal portfolio weights ϕt are given in (47), consistent with the above optimal consumption. With a little algebra, we can see that this inequality can be written as −µc(u)≤1−ρ γ−ρη(u)0η(u)−γ(1−ρ) γ−ρσc(u)0η(u). (55) We argue that for reasonable values of the quantities µc , σc and η , and for reasonable values of the preference parameters δ,γand ρ, this inequality holds. Concerning the latter, we restrict attention to the following two situations: (i) γ>ρ and ρ< 1, (ii) γ<ρ and ρ> 1. The former corresponds to the preference for early resolution of the uncertainty (γ>ρ ) and EIS > 1; the latter corresponds to a preference for late resolution of uncertainty (γ<ρ ) and EIS < 1. Both these sets correspond to plausible values of the parameters, and were also found to explain empirical puzzles well (e.g., Aase (2016a)). Consider (i): Then the first term on the right-hand side of (55) is positive, and the second is negative, provided σc(t)0η(t)> 0. Since the consumption growth rate µc can safely be thought of as strictly positive, the inequality will certainly hold provided η(u)0η(u)≥γσc(u)0η(u) for all u . Just to illustrate numerically, using the data in Table 1 , the left-hand side of the latter inequality is about 0.14, and the right-hand side is γκc,η· 0.01125. Here κc,η is the correlation coefficient between the consumption growth rate J. Risk Financial Manag. 2021,14, 425 21 of 35 and the market price of risk, and thus − 1 ≤κc,η≤ 1. Suppose γ= 2. Then the inequality is 0.14 ≥κc,η·0.0225, so the inequality holds with a very good margin. Similarly to case (ii), the signs of the coefficients on the right-hand side of (55) are again the same as just considered, and the comparison is similar, except that the difference between the optimal expected spending rate and the expected rate of return is now larger, since γhas decreased. The application of the results of this section to the data in Table 1is by and large similar to the illustrations given in Section 3.4, since the data, like the data in Table 1, are based on estimates, assuming stationarity (or some kind of ergodicity), and are therefore estimates of the expected value ¯ kt . However, the margin between the optimal extraction rate and the expected rate of return may have changed, depending upon the stochastic structure of rtand ηt. Unit EIS : From (51) it follows that when ρ= 1, then k(t) = δ for all t , a constant. Here the inequality (55) is reduced to µc(t)≥0, or rt+ηtσW(t)≥δ, for all t. Since σtηt=λtand σW(t) = ϕ0 tσt, this can be written as rt≥δ−λ0 tϕt, for all t(56) Since λ0 tϕt≥ 0 for all t , this requirement constrains the impatience rate δ from being too large.13 The Optimal Portfolio Selection Rule ϕ(t) We can find the an expression for the portfolio weights ϕt when the investment opportunity set is stochastic. Although it may not be a simple task to interpret this formula, it will give some additional insights, and its derivation also addresses a problem of independent interest. Towards this end write Equation (49) as W∗ t=c∗ tGT t, where GT t=EthZT teRs t(µc(v)−r(v)−η0 vσc(v))dvdsi. (57) When ρ= 1 we notice that GT t is deterministic, in accordance with the discrete time model, but recall that this is strictly speaking not an allowed value for ρin our treatment. The function GT t is seen to be Ft -measurable by definition, and by Itô’s representation theorem there exists a process gT(t,u)with u≤tsuch that GT t=E(GT t) + Zt 0gT(t,u)dBu. By the Clark–Ocone formula we know that gT(t,u) = Eu[Du(GT t)], where Du(GT t)is the Malliavin derivative of GT tat u≤t. From the stochastic differential equation for the optimal wealth given in (12) we then obtain by the product rule and diffusion invariance that ϕt=1−ρ ρ(1−γ)(σtσ0 t)−1λt−γ−ρ ρ(1−γ)(σtσ0 t)−1σtσW + (σtσ0 t)−1σtEt[Dt(GT t)]/GT t. (58) J. Risk Financial Manag. 2021,14, 425 22 of 35 Using that ϕ0 tσt=σW, this equation can be written more compactly as ϕt=1 γ(σtσ0 t)−1λt+ρ(1−γ) γ(1−ρ)(σtσ0 t)−1σtEt[Dt(GT t)]/GT t. (59) The expression for the optimal portfolio has again two terms, where the first is equal to the optimal portfolio for a constant investment opportunity set, except that now σt and λt are allowed to be stochastic. The second term is forward-looking, and depends upon both the horizon T and the impatience rate δ . This term is can be interpreted as a hedge against the unanticipated changes in the variables in the investment opportunity set. The premise for interpreting (59) as a formula for ϕt rests on the concept of a solution to the basic backwards stochastic differential Equation (38). This gives the pair ( Vt , Zt ), and consequently the volatility of utility σV(t) = Zt/Vt(see Section 4.5 below). Alternatively, since σV(t)=(ϕ0 tσt−η)/( 1 −γ) , (59) may also be considered as an equation in ϕ. The random term Et[Dt(GT t)]/GT t can again be connected to the parameters and the primitives of the model via the Malliavin derivative of GT t . Using the rules of Malliavin calculus, involving the chain rule and the product rule, we have the following. gT(t,u) = Eu[Du(GT t)] = EthZT teRs t(µc(v)−r(v)−η0 vσc(v))dv Zs t 1−ρ ρDu(rv) + ρ−γ ρ(Du(η0 v)σV(v)) + η0 vDu(σV(v))+ 1 2 γ(γ−ρ) ρ(Du(σ0 V(v))σV(v) + σ0 V(v)Du(σV(v))+ 1 2ρ(Du(η0 v)ηv+η0 vDu(ηv))dvdsi. (60) The Malliavin derivatives Du(rv) , Du(ηv) and Du(σV(v)) for u≤t can, as explained, be further broken down by specifying the types of stochastics for rη,σc(t)and σW(t). When γ=ρ we do get the analogous formula for expected utility given in Section 3.7 , so unlike the formula (47), the expression (59) reduces to the standard formula when γ=ρ . With an infinite horizon, the extraction rate is smaller than the real rate of return when the inequality kt≤rt+ϕ0 tλtholds, which is equivalent to kt≤rt+1 γλ0 t(σtσ0 t)−1λt+ρ(1−γ) γ(1−ρ) 1 G∞ t g∞(t,t)0ηt, (61) assuming that σtis invertible. This inequality holds for all tif and only if rt γ≥δ γ−1 γλ0 t(σtσ0 t)−1λt1+ρ 2ρ−ρ(1−γ) γ(1−ρ) 1 G∞ t g∞(t,t)0ηt. (62) We then have the following result: Proposition 4. With a stochastic investment opportunity set and recursive utility, provided the inequality |ρ(1−γ) γ(1−ρ)1 G∞ tg∞(t , t)0ηt|<1 γλ0 t(σtσ0 t)−1λt1+ρ 2ρ holds, the optimal extraction rate is strictly smaller than the real rate of return on the fund, unless the impatience rate δis unreasonably large. Proof. From the inequality (20) we notice that the second and third terms on the right-hand side add up to something negative under the condition of the proposition. Thus, if rt≥δ, the inequality then holds, and the conclusion follows. How reasonable is the assumption of the proposition in practice? Unless the inequality holds, the investment policy more or less prescribes short-selling most of the risky assets J. Risk Financial Manag. 2021,14, 425 23 of 35 in the portfolio, which is not what fund managers do when considering a long-term perspective for the types of funds that we are discussing here. We round off the treatment of this model with recursive utility by showing some numerical illustrations from the data in Table 1. 4.4. Numerical Illustrations We now illustrate the above with numerical examples based on the data in Table 1 . First we consider the situation where the agent has a preference for early resolution of uncertainty γ>ρ , and the EIS = 1 /ρ> 1. First we assume that the agent holds the market portfolio, and find the preference parameters consistent with the Formula (47) and the other expressions for the consumption growth rate and consumption volatility given above, consistent with this. Here this means that ϕt= 1, and this is consistent with γ= 1.07, ρ= 0.95. Additionally, we set δ= 0.04. The optimal expected extraction rate is then ¯ k0= 0.01. The expected rate of return on the wealth portfolio is 0.07. Furthermore, σc is smaller than σW, where σW=0.16. The time horizon T=300 years. This is illustrated in Figure 5. We notice that the difference between the optimal spending rate and the real rate of return is large. If the impatience rate is lowered, so is the spending rate. For example, the value of ¯ k0 becomes negative when δ= 0.01, but with a finite horizon the actual spending rate is of course strictly positive for all values of t≤T. Figure 5. Optimal spending rate vs. expected return; γ>ρ. By increasing the relative risk aversion, ceteris paribus, the expected real rate of return decreases and the optimal expected spending rate increases. By increasing the parameter ρ , ceteris paribus, the optimal expected spending rate increases and the expected real rate of return decreases (when γ> 1). By increasing the impatience rate δ , ceteris paribus, the optimal expected spending rate increases, and the expected return is unaffected. Next we consider the situation where the agent has a preference for late resolution of uncertainty γ<ρ, and the EIS =1/ρ<1. Suppose γ=0.95, ρ=1.02 and δ=0.03. The upper horizontal line in Figure 6is the expected real rate of return er(γ , ρ) ; the lower horizontal line is the optimal expected spending rate ¯ k0 corresponding to the perpetual case; and the curve corresponds to optimal spending with a finite horizon of 300 years. J. Risk Financial Manag. 2021,14, 425 24 of 35 Figure 6. Optimal spending rate vs. expected return; γ<ρ. Here ¯ k0= 0.01. The expected rate of return on the wealth portfolio is 0.049, following from an optimal portfolio strategy of ϕ= 0.108. Additionally, σW= 0.11, strictly larger than σc. By increasing the relative risk aversion (when ρ> 1), ceteris paribus, the expected real rate of return increases, and the optimal expected spending rate decreases. By increasing the parameter ρ , ceteris paribus, the optimal expected spending rate increases and the expected real return also increases. By increasing the impatience rate δ , ceteris paribus, the optimal expected spending rate increases, and the expected return is again unaffected (see Equation (51) and the inequality in (54)). For reasonable market quantities and plausible sets of preference parameters, the main conclusion of the paper holds with general recursive utility: the optimal spending rate is significantly smaller than the expected rate of return. When the investment opportunity set It is deterministic and preferences are represented by recursive utility, this conclusion can be given with the same kind of precision as for expected utility. This is treated in Section 4.6 below. Next a digression on optimal portfolio selection. 4.5. The Basic Backward Stochastic Differential Equation (Bsde) Returning to the BSDE given in Equation (38), suppose we have a method to solve this equation for the optimal consumption c∗ t . 14 Then we can find closed-form solutions for both the optimal consumption to wealth ratio and the optimal portfolio rule in the general case with a stochastic investment opportunity set. This problem is addressed in Schroder and Skiadas (1999). They considered the parameterization of an ordinary equivalent version of recursive utility. This parameterization was different from the one we used here, so it cannot be directly compared to our version, except, perhaps, for the most general case. The problem of solving the BSDE was addressed, which determined jointly Vt and Zt . By doing so, an auxiliary process Jt was introduced for Vt such that Vt=g(Jt , c∗ t) for some given function g . As in our approach, the consumption growth rates µc(t) and σc(t) were expressed by the parameters of the problem and by the two processes Zt and Vt —in our case by σV(t) , or equivalently, by Zt and Jt . In particular, they found that σc(t) = 1 γηt+ρ−γ 1−ρ Zt Jt expressed in our parameter version. In order to demonstrate consistency between this method and our approach, let us equate this volatility with our corresponding expression for σc(t)given in (42), which is σc(t) = 1 ρηt+ (ρ−γ)σV(t), J. Risk Financial Manag. 2021,14, 425 31 of 35 Figure 10. Spending rate when kis zero; γ>ρ, 1/ρ=5. If we increase the EIS further, the value of k becomes negative. Still, the optimal spending with a finite horizon is strictly positive, and increasing as the horizon comes closer, as in Figure 10. In this situation we can calculate the conditional expected time till the fund leaves a given interval at a specified level for the first time, treated in Section 3.6. Consider the interval (a , b) where a= ( 1 / 10 )W0 and b= 2 W0 . In this scenario and with the optimal spending rate, the parameters are µW= 0.02315, σW= 0.1156 and c=− 2.46 (the constant). The first exit probabilities are p+(W0 , J) = 0.9972 and p−(W0 , J) = 0.0028, so it is far more likely that the first exit takes place at the upper level b than at the lower level a . From the results of Section 3.4 we obtain that EW0{τ∗(J)|X(τ∗(J)) = b}= 41.35 years, and EW0{τ∗(J)|X(τ∗(J)) = a}=121.42 years. Here EW0{τ∗(J)}=41.58 years. In the situation where the spending rate is the expected rate of return, µW= 0 and c= 1; σW= 0.1156 remains the same. The first exit probabilities have changed to p+(W0 , J) = 0.86 and p−(W0 , J) = 0.14, so it is still more likely that the first exit takes place at the upper level b than at the lower level a , but less so than in the optimal case. Now we get EW0{τ∗(J)|X(τ∗(J)) = b}= 74 years, and EW0{τ∗(J)|X(τ∗(J)) = a}= 184 years. Here EW0{τ∗(J)}= 132 years, yet we know that in this situation Wt will eventually end up in zero, although it may take a long time. In the former case with optimal extraction in place, this does not ever happen with probability 1. There are several important lessons we can draw from this example. First, for reasonable parameter values, it is optimal to consume considerably less than the expected rate of return of the fund. Second, if the utility impatience rate and the certainty equivalent fund return are equal, the optimal consumption rate equals the two regardless of EIS. Third, if the utility impatience rate is less than the certainty equivalent fund return, the latter is an upper bound for the optimal consumption rate. 5. Additional Consumption in Society The analysis in the preceding sections took place under the assumption that the fund can be considered in isolation from consumption in the rest of society. For a fund established by society for the benefits of its inhabitants, it may be of interest to investigate whether the above analysis is general enough, since the ownership and purpose of the fund may be more complex. If the fund is owned by a state, the rest of the wealth in society may matter. Typically, for a sovereign wealth fund owned by the state, the government could perhaps be inclined to compare the extraction from the fund with consumption in society that originates from other more common sources. This we now address. Let us assume that there is a consumption stream in society that does not originate from the fund, denoted cS t , and the consumption that originates from the fund is denoted cF t , so that total consumption ct=cF t+cS t at any time t . The objective is to maximize utility U(c) subject to the relevant budget constraint. Here we assume U(c) = E(RT 0u(ct , t)dt) where u(x,t)is power utility of the kind used in Sections 2and 3of the paper. J. Risk Financial Manag. 2021,14, 425 32 of 35 In order to discuss this problem, let us return to Equation (3) for the market value of the optimal wealth. This equation can be expressed as follows under our present assumptions. Wt=1 πt EtnZT tπscF sdso+1 pt EtnZT tpscS sdso. (76) Here Wt is the total wealth in society at time t and pt is the state price deflator related to the consumption cS that stems from other sources than the fund, so we can write Wt=WF t+WS t for all t , where WF t is the optimal wealth from the fund, and where WS t is the wealth stemming from other sources than the fund, at any time t∈[0, T]. The central planner’s problem is then to solve the following. supcU(c)subject to EnZT 0(πtcF t+ptcS t)dto≤w, where wis the present value of wealth in the society. The Lagrangian of this problem is L(cF,cS;µ) = EZT 0u(cF t+cS t,t)dt −µZT 0(πtcF t+ptcS t)dt −wdt where µ is the Lagrange multiplier. Using directional derivatives, the first-order conditions are (cF t+cS t)−γe−δt=µπt, and (cF t+cS t)−γe−δt=µpt,∀t∈[0, T] where γ is the relative risk aversion and δ is the impatience rate. As a direct consequence of this, πt=ptfor all t, so the two state price deflators must be identical (a.s.). In the same vein we consider the two wealths. Here we make the bold assumption that all assets in society are marketed, so that, for example, we can consider labor as a shadow asset contained in WS. We then get dWt=dWF t+dWS t=WF t(ϕF tλF t+rt)−cF tdt +WF tϕF tσFdBt +WS t(ϕS tλS t+rt)−cS tdt +WS tϕS tσSdBt. The first-order conditions of optimal portfolio selection, using either dynamic programming or otherwise, lead in the same manner to the following. ϕF t=1 γ(σFσ0F)−1λFfor all t, and ϕS t=1 γ(σSσ0S)−1λSfor all t. The conclusion of this is that an endowment fund, whether owned by the state, by a university or otherwise, should be managed optimally as a fund, separated from the rest of the consumption problem in society. This separation principle is also rather intuitive. A Real Case Let us discuss a concrete case, and consider again the Norwegian Government Pension Fund Global, formerly simply the Norwegian Oil Fund, from the perspective of the last section. The idea of the origins of this fund is that also future generations are supposed to benefit from the oil exploration of the present generation, not only those who live in Norway at the present. J. Risk Financial Manag. 2021,14, 425 33 of 35 Consider, for example, a situation where the pension liabilities increase in the future for some limited amount of time, and then returns to a more normal state after this period. If the fund is supposed to take care of this particular problem, one can simply use actuarial methods to calculate the relevant extraction rates in the future. This problem is not connected, or at the best, just vaguely related to the problem analyzed above. In principal, no utility function is needed for the actuarial calculations involved. Thus we must make assumptions about both ownership of the fund and the intended purpose of the fund. Despite the change in the name of the former Norwegian Oil Fund, the fund is still normally referred to by its former name (or simply “the oil fund”). The conclusion from the last section is then to use the separation principle and treat this fund in isolation; an optimal extraction policy must be consistent with the portfolio selection strategy used. Since this fund is diversifying its assets through acquiring various government bonds and real estate, it is clear that this implies risk aversion on the investment side. Consistently with this, the extraction rate should also take into account risk aversion, consumption substitution and impatience, as explained in this paper. This is contrary to the current state of affairs of the Norwegian Government Pension Fund Global, where the extraction from this fund is determined by a mandate from Parliament (Stortinget) to be equal to the expected real return on the fund. As we have shown, this is not the sustainable spending rate and will deplete the fund in the future with probability one. 6. Conclusions We have derived concrete formulas for optimal extraction from an endowment fund consistent with risk aversion, and demonstrated that it can be written as a convex combination of the impatience rate and the certainty equivalent rate of return on the fund, in the most basic form of the models considered. As a consequence, the optimal extraction rate is strictly smaller than the expected rate of return, provided that the impatience rate is reasonable. The difference is far from negligible, and amounts to several percentage points in most real situations. The explanation has to do with the strategy at the portfolio selection stage: If the fund is managed by diversification, this means that risk aversion, consumption substitution and impatience are all essential in the optimal portfolio choice problem. Then, to be consistent, the spending rate must also reflect all these three properties, which is what we have shown. We have taken a security market as given, assumed to be in equilibrium, and introduced a price taking agent into this market. In this setting we have reconsidered the problem of optimal consumption and portfolio selection. In the context of an endowment fund, the results from analyzing this more general problem can immediately be utilized in order to determine an optimal spending rate. We have considered expected, additive and separable utility, in which case risk aversion plays a prominent role together with impatience; and recursive utility, in which case consumption substitution is separated from risk aversion. When the investment opportunity set is deterministic, there exist explicit and closedform solutions for optimal extraction, which we have rewritten in a form that is easy to interpret. For a stochastic investment opportunity set, we have developed formulas in the paper, which we claim to be original. First and foremost, these solutions are demonstrated to be smaller than the expected real rate of return on the endowment fund, for plausible values of the preference parameters and the other parameters of the problem. The difference is significant in most cases. If the extraction rate is the same as the expected return, this usually goes along with agent risk neutrality at the level of extraction, and should then, to be consistent, go along with risk neutrality at the level of optimal portfolio selection as well. However, the consequence of such an investment strategy is rarely advocated by anyone responsible for an endowment fund, whatever its purpose. J. Risk Financial Manag. 2021,14, 425 34 of 35 We demonstrate that a popular and much advertised extraction policy, the expected real rate, is not consistent with a sustainable spending rate, and will with probability one eventually deplete any fund that is managed by diversification. Author Contributions: Methodology, K.K.A. and P.B.; formal analysis, K.K.A. and P.B.; writing— original draft preparation, K.K.A. and P.B.; writing—review and editing, K.K.A. and P.B. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Acknowledgments: Special thanks to Jostein Lillestøl. Any remaining errors are ours. Conflicts of Interest: The authors declare no conflict of interest. Notes 1 The result in (11) can alternatively be derived by dynamic programming, assuming that the horizon is infinite at the outset. A transversality condition must then be satisfied, which holds if k>0 (see Merton (1971) for this approach). 2 It is really the Arrow–Pratt approximation of this quantity. In continuous-time models with Brownian-driven uncertainty, this approximation is in fact exact. 3 Based on about 100 years of US-data, an estimate of the real short rate r is around 1 per cent, which is also the usual suggestion for the impatience rate δ. 4Tobin (1974) suggests, in the situation of university endowments, that δis set equal to 0. 5 “Everything” here includes borrowing risk-free as much as possible. This problem has, of course, no mathematical solution unless there is a borrowing constraint. 6The data were adjusted from discrete-time to continuous-time compounding. 7 These quantities were “estimated” directly from the original data obtained from R. Mehra; we used an underlying assumption about ergodicity, and estimates are denoted by ˆ σM,c, etc. 8This is the value that is recommended by an expert panel for the Norwegian Government Pension Fund Global. 9 This is analogous to the result that the price of a zero-cupon bond is a process of bounded variation in T , but an Itô-process in t . 10 With this method we do not need to go via the ordinally equivalent version with the corresponding A=0. 11 In this formula, and otherwise throughout, a term such as (σtσ0 c∗(t)) is to be interpreted as the covariance rate between the market for risky securities and the optimal consumption, and not as a mere multiplication of volatilities, which would imply an instantaneous correlation coefficient of 1. 12 The relationship (47) cannot be put into an equation in ϕt through the use of (42) and (44), since these have already been used once. 13 Notice that the criterion (56) is identical to the two corresponding criteria (19) and (35) for expected utility when γ=1. 14 There is a fairly large amount of literature on this topic, which began in the early 1990s. 15 This formula was first derived by Svensson (1989) in his special model of recursive utility in continuous time; he restricted attention to a deterministic investment opportunity set only. 16 When discussing whether EIS is larger or smaller than 1, many economists implicitly seem to be taking the standard expected utility model as the “truth”, in which case EIS < 1, since γ> 1 is considered most reasonable ( γ= 1 /EIS for expected utility). 17 The report uses geometric returns. 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