Optimal Portfolio and Spending Rules for Endowment Funds

We investigate the role of different spending rules in a dynamic asset allocation model for an endowment fund. In particular, we derive the optimal portfolios under the consumption-wealth ratio rule (CW strategy) and the hybrid rule (hybrid strategy) and compare them with a theoretically optimal (Merton’s) strategy for both spending and portfolio allocation. Furthermore, we show that the optimal portfolio is less risky with habit as compared with the optimal portfolio without habit. Similarly, the optimal portfolio under hybrid strategy is less risky than both CW and Merton’s strategy for given set of constant parameters. Thus, endowments following hybrid spending rule use asset allocation to protect spending. Our calibrated numerical analysis on US data shows that the consumption under hybrid strategy is less volatile as compared to other strategies. However, hybrid strategy comparatively outperforms the conventional Merton’s strategy and CW strategy when the market is highly volatile but under-performs them when there is a low volatility. Overall, the hybrid strategy is effective in terms of stability of spending and intergenerational equity because, even if it allows fluctuation in spending in the short run, it guarantees the convergence of spending towards its long term mean.

the market is highly volatile but under-performs it when there is a low volatility.
The dynamic asset allocation and consumption studied in this paper is built upon the seminal work by Samuelson [1969] and Merton [1969], who present an optimal strategy for a market with constant investment opportunities with additive time-separable utility function. This preliminary work was later extended by Merton [1971], to a more general utility function which includes the income generated by non-capital gains sources. This paper is related to a number of other works in the literature. Merton [1993] applies continuous time framework to the endowment fund's problem and derives optimal expenditures and asset allocation strategy that include non-endowed funds as a part of the total university's wealth. He concludes that endowment funds prefer a safer portfolio in the presence of non-financial income risk. Dybvig [1999] suggests that an endowment fund spending can be sustainable if some TIPS (Treasury Inflation Protection Securities) are hold in portfolio. Bajeux-Besnainou and Ogunc [2006] address the asset allocation problem of an endowment fund by including, in the objective function, a minimum spending amount up-rated with inflation and obtain an explicit formula for optimal spending and portfolio allocation rules. Constantinides [1990] applies the habit formation to the equity puzzle problem and shows that the high equity premium with low risk aversion can be explained by the presence of habit formation. Gong and Li [2006] considers the general optimal investment/consumption problem for an agent with constrained consumption due to habit formation or precommitment. They found that investment in nominal bonds and equity is initially proportional to the excess wealth over the lower bound imposed due to subsistence level of consumption and then it increases nonlinearly with wealth. Munk [2008] studies the optimal strategies with general asset price dynamics under two special cases of time varying investment opportunities: stochastic interest rate, and mean-reverting stock returns. He shows that, in order to finance the habit, investment in bonds and cash is more effective than in stocks.
The rest of the paper is structured in the following way. Section 2 describes the various spending strategies commonly applied by endowment funds, while Section 3 introduces the general framework for market dynamics, preferences of endowment fund, and endowment fund investment strategies. Section 4 focuses on the results of optimal investment and spending under different strategies. Section 5 presents a numerical application of the previous results, and, finally, Section 6 concludes. Some technical derivations are left to Appendix A.
3 Categories Method Description

Simple Rules
Income-based Spend the whole current income.

Consumptionwealth ratio
Spend either the predetermined percentage of the market value of the fund or decide the percentage every year.

Inflationprotected
Spending grows at the rate of inflation.

Banded-Inflation
Same as inflation-protected but with the upper and lower bands.

Smoothing Rules
Moving Average Pre-specified percentage of moving average of market values, generally based on three-years starting market values.
Spending Reserve 5-10 percent of the market value is held in the reserve account and then invested in 90-day Treasury bills. The amount is withdrawn only when the fund's performance is below target.

Stabilization Fund
The excess endowment returns are used to make a fund which is then used to control the long term growth of the total endowment.

Hybrid Rules Weighted
Average or Hybrid (Yale/Stanford) Method Spending is calculated as the weighted average of spending adjusted for inflation and spending under the moving average method. In the following subsections we consider the following rules in detail: • Inflation rule; • Moving average method;

4
• Weighted average or hybrid method.

Inflation rule
Inflation rule is devised to acknowledge the corrosive effects of inflation and it aims to protect the purchasing power of endowment fund. The objective of endowment fund is not the mere preservation of the fund but to strive for a value addition. It can be achieved as long as the overall return from the portfolio exceeds the rate of inflation. Inflation rule increases the previous period's spending at the predetermined inflation rate.
Spending in a year equals the spending in the previous year, increased at the inflation rate λ : where the initial value of consumption is a fixed ratio of wealth, i.e. c 0 = yR 0 . The differential of c (t) is The inflation method is static and trivial, so we will not consider it separately here, however it is included in the hybrid method along with the moving average method.

Moving average method
The most popular and commonly used spending rule is the moving average. Dimmock [2012] reports that typically this rule is based on pre-specified percentage of moving average of 3-years quarterly market value.
The most desirable property of this rule is that it saves some income and reinvests it. This method is used to smooth the volatility in spending. However, the method is flawed because it uses the market value of the endowment. Therefore, when the endowment value is rising, the institution may spend more than it is prudent and when endowment values are falling sharply, this method calls for a budget cut that may curtail the institution's mission.
In discrete-time, this spending rule can be algebraically written as follows In continuous-time, instead, we can write Here, for the sake of simplicity, we take the limit of the previous rule for q which tends towards zero: 1 which is the same as the consumption-wealth ratio rule stated in Tab. I.
The goal of moving average rule is to dampen the volatility of spending. Consequently, during the period of boom, this process results in an accelerating curve of upward spending and it causes a false sense of security of sustainable spending. The smoothing effect of this rule is limited and it may give a misplaced belief that the higher spending levels are sustainable. Furthermore, the resulting shrinkage in endowment values due to market decline results into the deep cuts in spending.

Weighted average or hybrid method
The weighted average method is generally followed by the large endowment funds and it is also known as Yale/Stanford rule. It is a weighted average of the inflation method and the moving average method.
In discrete-time it is given by where ω is the weight. We can simplify it by considering q = 1 to get If we ignore inflation, i.e. λ = 0, we obtain We make the above process stationary by assuming |(1 − ω) y| < 1.
Remark 1. If we take the limit t + 1 → t, we get the same consumption-wealth ratio as in (2) c(t) = yR(t).
Under the hybrid rule, during the boom, the spending will not increase as fast as compared with the moving average rule. Conversely, this rule does not call for spending cuts as deep as the moving average method. Evidence suggests that more and more institutions are changing their spending rules to inflationbased and hybrid method from moving average method Sedlacek and Jarvis [2010].
We will only consider two spending rules, the consumption-wealth ratio rule, and the hybrid spending rule given by (2) and (5), respectively for our optimal investment and spending strategies.

General framework
Endowment funds usually invest in a variety of assets, however for the purpose of tractability we examine the aforementioned spending rules in the simplest framework. We consider two types of assets, a riskless asset and a risky asset in a complete and arbitrage free, continuously open financial market. On the financial market two assets are listed: • a riskless asset G(t) which evolves according to where G(t 0 ) = 1, and which is the numéraire and r(t) is the instantaneous nominal interest rate; • a risky asset S(t) having the price dynamics given by The endowment fund holds θ S (t) units of the risky asset S(t), and θ G (t) units of risk-less assets G(t).
Thus, at any instant in time t, the investor's wealth R(t) is given by The differential of wealth can be written as where c (t) is the consumption or spending, following one of the rules mentioned in Section 2 except when it is also a decision variable.

Endowment fund investment strategies
We consider the following three strategies: • Merton's strategy: Both investment and spending are decision variables.
• Consumption-wealth ratio (CW) strategy: Investment is the only decision variable and spending is given by the fixed consumption-wealth ratio rule.
• Hybrid strategy: Investment is the only decision variable and spending is given by weighted average or hybrid method.

General settings
According to Fraser and Jennings [2010], an endowment fund must define its investment policy statement identifying the investment beliefs, specific investment objectives, re-balancing policy and performance benchmark which are evaluated periodically. Since an endowment fund must report its performance for each accounting period, it is reasonable to consider the optimization problem for a finite time horizon [t, T ]. If we assume the objective of an endowment fund is to maximize the sum of expected utility of spending c(t) and the expected utility of the final wealth R (T ) then it can be stated as where endowment fund chooses the decision variables π(t), which may include consumption and investment depending on the strategy considered and ρ is a constant subjective discount rate. We assume the fund's preferences are defined by the following utility function where δ > 1. The Arrow-Pratt Relative Risk Aversion (ARA) index is If α X is a positive constant, then this form of the utility function belongs to Hyperbolic Absolute Risk Aversion (HARA). If α X = 0, ARA index is given by which belongs to Constant Relative Risk Aversion (CRRA).
If α X = 0, the utility functions belong to HARA and can be written as where c (t) is the instantaneous outflow or spending from the fund, the constant R m can be interpreted as the minimum subsistence level of wealth, whereas h(t) depends on the context, it is either a function representing the habit formation or a constant representing the subsistence level. Given the utility function (11), it is always optimal to have outflows higher than the threshold h(t), in fact when c(t) = h(t), the marginal utility of the outflow tends towards infinity and, accordingly, it is sufficient to increase it by an infinitesimal amount in order to have an infinite increase in the utility level.
The corresponding ARA indices of (11) are given by respectively, which implies that the higher δ the higher the risk aversion. Moreover, the higher h (t) the higher is the risk aversion. This result shows that having a higher level of minimum outflows means that it is necessary to invest bigger amounts of wealth in the riskless asset in order to guarantee the outflows, which implies a higher risk aversion. 9 4 The optimal solutions

Merton's strategy
Investment and spending are both decision variables, i.e. π(t) ≡ {c(t), θ S (t)}. We can define the value function using (9) as where φ c and φ R are constants which measures the subjective relevance of the utility obtained from intertemporal consumption and final wealth, respectively and h(t) is given by where h 0 is the initial minimum amount of outflow, α(t) is the weighting function providing the relative importance to the past outflow in computing the threshold h(t), while β (t) is a discount rate. In habit formation case, h(t) can be rewritten in continuous time as Proposition 1. Given the state variable wealth R(t) described in (8), the optimal consumption and portfolio solving problem (12) are • in the case with habit formation: where • in the case of a subsistence level: • in the classical case: where for both the subsistence level case and the classical case we have The function A(t) in the optimal solutions is the weighted sum of two discount factors: (i) the discount factor for the final date T multiplied by φ 1 δ R , and (ii) a kind of intertemporal discount factor for the intertemporal utility, multiplied by φ 1 δ c . The function B(t) in the optimal solutions, is the sum of two terms, the subsistence wealth R m appropriately discounted from time T and a sum of discount factors. We can see that habit formation has an effect on the optimal portfolio of the risky asset, as it changes the allocation due to the reason that the riskless asset (treasury) is comparatively a better investment than the risky asset (stock) to ensure that the future consumption will not decline below the habit level.
We consider the optimal investment and consumption in the habit formation case defined in (16) and (15), respectively, in detail below: Assumption 1. We assume that all the parameters α, β , µ, σ , and r are constant over time and additionally r − α + β > 0.

Corollary 1. Under Assumption 1, (i) The wealth process is itself a Markov process and the functions A(t)
and B(t) can be written as ds, and finally, if we substitute the value of B(t) into A(t), we get ds.
(ii) The optimal consumption and portfolio are where A(t) and B(t) are given by (19) and (20).
The term is positive and decreasing over time as r − α + β > 0.
(iii) The optimal portfolio is less risky with habit as compared with the optimal portfolio without habit: Proof. The optimal portfolio S(t)θ S (t) * is given by Assumption 2. We assume T → ∞.
Corollary 2. Under Assumptions 1 and 2, the optimal consumption and portfolio are In this case the dynamics of optimal wealth and habit are where we can see that the habit h(t) is a mean reverting process if β − α > 0 and is instead exploding if β − α < 0.

CW strategy
In this case, the investment θ S (t) is the only decision variable and the spending c(t) is given by therefore we can put φ c = 0 and φ R = 1 in (12), then the value function can be defined as Proposition 2. Given the state variable R(t) and c(t) described in (8) and (23), respectively, the optimal portfolio solving problem (24) is where and y is the constant defined in (23).
The function A(t) in proposition 2 is a discount factor calculated as a sum of the accumulated interest rate and Sharpe ratio diminished by the constant y. The function B(t) in the optimal solutions, is the subsistence wealth R m appropriately discounted by a discount factor.
In this case the dynamics of optimal wealth is

Hybrid strategy
In this strategy, investment is the only decision variable π ≡ θ S (t), while spending evolves according to the weighted average spending rule (5). We can write the wealth dynamics (8) in a discrete time as and the consumption in a discrete time is given by We substitute (26) into (27) to get which can be rewritten in continuous time as If we set a ≡ (1 − ω)y then we can write where we see that the consumption is a mean reverting process, whose strength of mean reversion is (1 − ω)(1 + y). The consumption reverts towards its long term mean: which depends on the portfolio choice. The higher the value of ω, the more slowly c(t) converges towards its long term mean and vice versa.
As the consumption is given by (28), we include it as an additional state variable and put φ c = 0 and φ R = 1 in (12), hence the value function can be defined as Proposition 3. Given the state variables R(t) and c(t) as in (8) and (28) and under the assumption that interest rate r is constant the optimal portfolio solving problem (29) is where and η * takes one of the following values such that −∞ < η * < 1 a and a = y (1 − ω) < 1.

Proof. See Appendix A.3.
Corollary 4. Under Assumptions 1,2 and if r > η * a (1 + r), the optimal portfolio (30) becomes In this case the dynamics of optimal wealth and optimal consumption are Corollary 5. Under Assumption 1, (i) the optimal portfolio is less risky for the Merton's strategy as compared with the optimal portfolio for CW strategy if α − β > y and r > y. 2 Under an additional assumption η = 0, we can write Proof. We suppose the negation of the given statement is true As it is reasonable to assume µ − r > 0, which implies µ−r S(t)σ 2 δ > 0. Therefore We substitute the values of the unknown functions to obtain r−α+β is also positive and decreasing over time as we have r − α + β > 0 under Assumption 1. As a consequence, the above statement can only be correct only if as r − α + β > 0, r > y and α − β > y. Therefore, we conclude that the following statement is true For part (ii), we suppose We substitute the values of the unknown functions to obtain We can clearly see that the left-hand side is smaller than the right-hand side, therefore we conclude that the statement to be proved is true.
Corollary 6. Under Assumption 1, the optimal consumption under Merton's strategy as described in (15) can be equivalent to the consumption under CW strategy as described in (2) depending on the chosen values of constant y.
Proof. As the optimal consumption for Merton's strategy is where B(t) = R m e − T t (−α(u)+β (u)+r(u))du + T t e − s t (−α(u)+β (u)+r(u))du ds, and The consumption for CW strategy is given by Consumption under Merton's strategy and CW strategy can be equivalent if y is chosen such that where φ 1 δ c can be suitably chosen to match y.

A numerical application
To illustrate the results of the preceding section, a simplified market structure is taken into account under Assumption 1 and condition that h(t) = 0. We have estimated the parameters related to the financial market and interest rate over three different time horizons: (i) January 2nd, 19972nd, and December 29th, 20062nd, (19972nd, -2006, (ii) January 3rd, 2007and December 30th, 2011(2007-2011), and (iii) January 3rd, 2012and December 30th, 2016(2012-2016. The parameters of the risky asset S (t) are obtained from the S&P 500 and the value of constant interest rate r is estimated as the average return of US 3-Month Treasury Bill (on secondary market -daily data). We assume the risk aversion parameter δ is equal to 2 similar to the most common choices of risk aversion parameter in the habit formation and life cycle literature (Munk [2008]; Gong and Li [2006] and Horneff et al. [2015]; Gourinchas and Parker [2002]). We set the subjective discount factor ρ equal to the riskless interest rate r. The estimated parameters along with some assumptions about wealth and preferences are gathered in Tab. II.
We recall the general objective function under the assumption h(t) = 0: In Fig. 1   Merton's strategy. The graph indicates that weights do affect the optimal portfolio and consumption and a comparatively higher weight must be given to φ R as compared with φ c to maintain wealth above zero in the long run.    Tab. II. Fig. 4 shows the wealth and consumption with the variation of weights ω for hybrid strategy under the assumption that R m = 0. Interestingly, we can see from the graph that the weight ω for the inflation method must be chosen prudently, the higher the value of ω, the more slowly c(t) converges towards its long term mean and vice versa. Similar to CW strategy, if R m approaches the initial wealth R 0 , the paths of wealth and consumption becomes less volatile. For this strategy, the initial consumption must not be higher than a certain threshold, otherwise the wealth will decline to zero. In fact, endowment managers face a dilemma in deciding whether to have a spending above a certain level to fund the current activities or to have spending lower than a threshold to achieve the long term objectives. Thus, in choosing the initial level of spending, the management must weigh the potential of greater spending levels in the long run against the need for current spending. The behavior of consumption under hybrid strategy appears to be more acceptable to endowment managers as: (i) it is less volatile in the long run compared to Merton's and CW strategies and (ii) it converges towards its long term mean. In fact, an excessive volatility in spending is undesirable. Fig. 5 examines the sensitivity of hybrid strategy to the variations in the constant y. We see that while y increases, so does the volatility in wealth and consumption which implies that y must be chosen cautiously for the long term growth of wealth. The path of consumption also depends on the value of y. If y is higher than the optimal value, then the wealth will not grow in the long run and eventually, the consumption will also decline to a much lower level. If y is lower than the optimal level, then the wealth will grow in the long run but the consumption will remain lower even in the long run, provided that the values of other parameters We can see that: • during the first period (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) under hybrid strategy, the wealth initially declines and then recovers while under Merton's strategy, it declines sharply.
• during the second period (2007)(2008)(2009)(2010)(2011), wealth declines for all strategies but the magnitude of decline is far less for hybrid strategy compared to the other two strategies.
• during the third period (2012-2016), wealth rises for all three strategies but it rises less strongly for hybrid strategy.
The graphs of the second column show the dynamic behavior of consumption. For the comparison of consumption we chose, the consumption-wealth ratio for CW strategy and initial consumption for hybrid strategy to match the initial optimal consumption under Merton's strategy. We can see that: • during the first period, consumption rises under Merton's strategy while it declines under hybrid strategy in a short run and converges towards its long term mean.  1997-2006, 2007-2011, and 2012-2016 as stated in Tab. II.
• during the second period, consumption remains constant under Merton's strategy, however it declines under the other two strategies.
• during the third period, consumption increases greatly for CW and Merton's strategy while it increases with less intensity for hybrid strategy.
The third column graphs shows the risky portfolio. We can see that during the second period it is optimal to short sell the risky asset whereas in the third period it is optimal to short sell the riskless asset and invest in the risky asset.

Conclusion
This paper has provided a brief overview of different spending rules applied by endowment funds. The endowment fund managers adopt these rules to effectively preserve the corpus of the fund and have a stable spending stream. We have obtained the optimal investment strategy under consumption-wealth ratio and hybrid spending rules. Furthermore, we have compared these optimal portfolios and defined spending rules with the classical Merton's optimal portfolio and consumption. We have found that the optimal Merton's portfolio is less riskier than that under consumption-wealth ratio rule, while the Merton's optimal consumption can be replicated using consumption-wealth rule by a suitable selection of the consumption-wealth ratio. The hybrid strategy, for some values of constant parameters, is less risky than both Merton's and consumption-wealth ratio, and consumption under this strategy is less volatile compared to other strategies.
The unique characteristic of hybrid rule is that it allow fluctuation in spending during the short run. Also, it converges towards its long term mean regardless of the initial allocation for spending. However, hybrid strategy comparatively outperforms the conventional Merton's strategy and CW strategy when the market is highly volatile but under-performs them when there is a low volatility. Thus, an endowment fund must evaluate, review and modify its spending rule and investment policy periodically, depending on the conditions of the financial market.

Appendix A: Mathematical proofs
For simplification of the notation, we will use the following definitions throughout this appendix:

A.1 Proof of proposition 1
The value function or indirect utility function is given by We know that the above objective function must solve the following differential equation (so-called Hamilton-Jacobi-Bellman HJB equation): The HJB equation (32) in J(t, R(t)), needs a boundary condition so that the value function coincide with the final utility function at the time T : J(T, R(T )) = U(R(T )).
We assume the following guess function where A(t) and B(t) are the functions that must be determined to solve equation (32). Both functions must satisfy boundary conditions as follows: and the first order conditions (FOCs) of (32) w.r.t. θ S (t) and c(t) are: By substituting the derivatives of the guess function into both the optimal consumption (35) and the optimal portfolio (34), we obtain Inserting the optimal consumption (36), the optimal portfolio (37) and the partial derivatives of the guess function into equation (32), we have which can be separated into two differential equations, one that consists of the terms containing (R(t) − h(t)B(t)) −1 and one without them and after few simplifications we have The above ordinary differential equations, together with their corresponding boundary conditions, have the following unique solutions: The subsistence level case, i.e. h(t) = h and the classical case, i.e. h(t) = 0 can be easily followed from the above results.

A.2 Proof of proposition 2
The value function is given by For this objective function, we can write the following HJB equation and the first order condition (FOC) of (39) w.r.t. θ S (t) is We assume the following guess function where A(t) is the function that must solve equation (39), with the boundary condition A(T ) = 1, while B(t) must satisfy the boundary condition B(T ) = R m , and the optimal portfolio process θ S (t) * in (40), for our guess function can be written as Substituting the optimal portfolio (42) and partial derivatives of the guess function into (39), we get which can be separated into two differential equations, one that consists of the terms containing (R(t) − B(t)) −1 and one without them and after few simplifications, we have The above ordinary differential equations with their corresponding boundary conditions have the following solutions:
These equations are both ordinary linear differential equations and their boundary conditions can be obtained from the boundary condition of the HJB equation: A(T, c(T )) = 1 and B(T, c(T )) = R m .
The solution of the ODE (46), together with their boundary conditions is given by .
Since the second differential equation has been obtained under the hypothesis that B c = η and B cc = 0, 30 then the only consistent functional form for B(t, c(t)) is where η (t) and h (t) may be functions of time. Thus the second ODE can be rewritten as follows (1 + r)) (η (t) c(t) + h (t)) − (η (t) (1 − ω + a) − 1) c(t) = 0, which can be separated into two ODE's, one which contains c(t) and one which contains all the terms without (1 + r + a − ω) ± (1 + r + a − ω) 2 − 4a (1 + r) where one of the two solutions (η * ) must be chosen suitably.