Optimal portfolio and spending rules for endowment funds

University endowment funds rank among the largest institutional investors. In the National Association of College and University Business Officers (NACUBO) estimated that its member organizations in the U.S. held $515 billion in endowment assets (NACUBO 2016). These funds pool donations from supporters and alumni, and are invested to support the current and future expenses of universities. Optimal investment and spending strategies for university endowments have been the subject of academic and professional scrutiny since Tobin (1974) and Merton (1993). Universities and colleges set up endowment funds to ensure a reasonably smooth earnings and consequent smooth stream of spending for current and future beneficiaries to preserve equity among generations [also called intergenerational equity, see Tobin (1974)]. To achieve this objective, many endowment funds set pre-defined spending rules. Hansmann (1990) addresses the reason why endowments would need such rules and concludes that endowments are established to fulfill some basic purposes which include: (i) ensure the support of the parent institution in its ongoing mission, (ii) protect its reputation and intellectual freedom, and (iii) hedge against financial shocks. In general, following a pre-defined spending rule is not optimal. As a result, endowment funds may invest too much in riskless assets which generally have lower returns.

Kaufman and Woglom (2005) discuss the role of spending rules in endowment funds. They analyze the spending rules based on the inflation method, banded inflation, and hybrid method, using Monte Carlo simulations in a scenario of volatile and uncertain asset returns. Ennis and Williamson (1976) present different spending rules adopted by endowment funds along with their historical spending patterns. Sedlacek and Jarvis (2010) provide an analysis of endowments’ current practices and spending policies with their relative merit and demerits.

In this paper, we examine investment strategies under two spending rules in particular: (i) the fixed consumption-wealth ratio (CW) rule, and (ii) the hybrid rule, a simplified form of the weighted average method commonly used by large endowment funds like Yale and Stanford (Cejnek et al. 2014). Under the CW rule, annual spending is a percentage of the market value of the fund, whereas the hybrid rule calculates spending as a weighted sum of the spending in the previous period and spending according to the fixed consumption-wealth ratio rule. To analyze the effect of the spending rules on risk-taking, we derive the optimal portfolio under the abovementioned spending rules for hyperbolic absolute risk aversion utility function and compare them with the classical Merton optimal investment and consumption. We also consider a detailed numerical example to investigate the effectiveness of the spending rules.

An endowment fund may exhibit intertemporal preferences where past spending generates a desire (or need) to maintain the same spending level into the future. Such a behavior can be modeled through habit formation or habit persistence, which is a more general case of subsistence level consumption. Actually, the habit can be interpreted as a time-varying subsistence consumption. In the Merton case, we consider a general form of utility function including both the cases of habit formation and subsistence level. Under the spending rules mechanism, endowments following the hybrid spending rule protect spending by investing in a less risky portfolio than Merton’s. Similarly, investment in the risky asset with habit is less than the investment without habit. Thus, this strategy is similar to portfolio insurance where the fund invests in safe assets to maintain the value needed for having smooth payouts over time.

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 held in the portfolio. Blume (2010) discusses the spending rules for endowments in volatile markets and illustrates the relation between spending rules and investment strategy under the assumption that asset classes with greater expected long-run returns display greater short-term volatility. He proposes that an endowment should choose a portfolio with a lower long run return when its value falls. 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. 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 introduces the general framework for market dynamics, preferences of endowment fund, and endowment fund investment strategies, while Sect. 3 focuses on the results of optimal investment and spending under different strategies. Section 4 presents a numerical application of the previous results, and, finally, Sect. 5 concludes. Some technical derivations are left to "Appendix A".

Endowment funds usually invest in a variety of assets. For the purpose of tractability, we consider two assets listed in a complete and arbitrage-free, continuously open financial market:The endowment fund holds \(\theta (t)\) units of the risky asset S(t), and \(\theta _{G}(t)\) units of risk-less assets G(t). Thus, at any instant in time t, the investor’s wealth R(t) is given byThe differential of wealth can be written aswhere \(c\left( t\right)\) is the instantaneous spending (or consumption) rate per unit time.

To illustrate the results of the preceding section, a simplified market structure is modeled under Assumption 1. Additionally, in order to compare and contrast the three strategies, we assume that the relative importance to the past outflow \(\alpha\) and the discount rate \(\beta\) are equal to zero, and thus there is no habit (\(h(t)=0\)). We have estimated the parameters related to the financial market and interest rate over three different time horizons: (i) January 2, 1997 to December 29, 2006 (1997–2006), (ii) January 3, 2007 to December 30, 2011 (2007–2011), and (iii) January 3, 2012 to December 30, 2016 (2012–2016). The parameters of the dynamics of the risky asset price \(S\left( t\right)\), in equation (2), are estimated from the S&P 500, and the value of constant interest rate r is estimated as the average return of U.S. 3-Month Treasury Bill (on secondary market – daily data). We assume that the risk aversion parameter \(\delta =2\) similar to the most common choice of risk aversion parameter value in the habit formation and life cycle literature (Munk 2008; Horneff et al. 2015; Gourinchas and Parker 2002). We set the subjective discount factor \(\rho\) equal to the riskless interest rate r. The estimated parameters along with some assumptions about wealth and preferences are gathered in Table 1.

We recall the general objective function under the assumption \(h(t)=0\):

In Figs. 1, 2, 3, 4, 5, the solid lines show the mean of 1000 simulations while the dashed lines with shaded areas show the confidence interval, i.e. mean plus and minus two standard deviations using the parameters estimated for the period 1997–2006. Figure 6 shows only the mean of 1000 simulations using the parameters estimated for the periods 1997–2006, 2007–2011 and 2012–2016.

Figure 1 illustrates the sensitivity of the wealth and spending to variation in weights assigned to the utility of terminal wealth and intertemporal spending in the objective function (27) under Merton strategy. The graph indicates that weights do affect the optimal portfolio and spending and a comparatively higher weight must be given to \(\phi _{R}\) as compared with \(\phi _{c}\) to maintain wealth above zero in the long run. However, higher \(\phi _{c}\) leads to higher (and increasing) spending, increasing over time, as one would expect.

Figure 2 presents the impact of the consumption-wealth ratio y on the paths of wealth and spending for the CW strategy, in the absence of subsistence wealth, i.e. when \(R_{m}=0\). We can see that, when y increases, the level of spending rises but the terminal wealth declines. For further analysis, we consider the case when the subsistence wealth \(R_{m}\) is close to initial wealth \(R_{0}\), i.e. \(R_{m}=90\), the wealth and spending become less volatile for all values of y, as shown in the Fig. 3. As expected, we can see that a rise or decline in the wealth and spending in the long run depends on the ratio y.

Figure 4 shows the wealth and spending with different values of the weight \(\omega\) for the hybrid strategy under the assumption that \(R_{m}=0\). Interestingly, we can see from the graph that the weight \(\omega\) must be chosen prudently: the higher the value of \(\omega\), the more slowly c(t) converges towards its long-term level. This is as expected since the more weight is given to past spending in the determination of future spending, the more slowly spending changes. Similar to the CW strategy, if \(R_{m}\) is close to the initial wealth \(R_{0}\), the paths of wealth and spending becomes less volatile. For this strategy, the initial spending 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 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 spending under the hybrid strategy appears to be more acceptable to endowment managers as: (i) it is less volatile in the long run compared to the Merton and CW strategies and (ii) it converges towards a long-term average level. In fact, an excessive volatility in spending is undesirable.

Figure 5 examines the sensitivity of the hybrid strategy to different values of constant y. We see that while y increases, so does the volatility in wealth and spending which implies that y must be chosen cautiously for the long term growth of wealth. The path of spending 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 spending 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 spending will remain lower even in the long run, provided that the values of other parameters remain unchanged.

Figure 6 shows the mean wealth, spending and risky portfolio (all in monetary units) for the Merton, CW and hybrid strategies using the parameters estimated for three different periods: (i) 1997–2006, (ii) 2007–2011, and (iii) 2012–2016. The graphs in the first column show the dynamic behavior of wealth. We observe that:The graphs in the second column of Fig. 6 show the dynamic behavior of spending. To effectively compare spending, we set the consumption-wealth ratio for the CW strategy and the initial spending for the hybrid strategy in order to match the initial optimal spending under the Merton strategy. We can see that:From the graphs in the third column of Fig. 6, which show the average amount held in the risky asset, we observe that

University endowment funds are long-term investors with specific objectives requiring particular investment and spending strategies. Endowment fund managers adopt different spending rules to effectively preserve the corpus of the fund and have a stable spending stream. We derived the optimal investment strategy under the fixed consumption-wealth ratio (CW) rule and the hybrid spending rule. We compared these optimal portfolios and associated spending rules with the classical Merton optimal portfolio and consumption.

A numerical analysis using market data over three different periods in the last two decades showed that the hybrid strategy leads to a less risky portfolio than both the CW strategy and conventional Merton strategy, under typical market conditions. The hybrid strategy, which explicitly smoothes inter-temporal spending, thus operates similarly to an optimal Merton portfolio with habit formation. We also found in our numerical example that spending under the hybrid strategy appears to be less volatile than under the other two strategies. The unique characteristic of the hybrid rule is that it allows fluctuation in spending in the short run but stabilizes spending in the long run regardless of the initial allocation for spending. In terms of the endowment’s wealth, the hybrid strategy performs better than the conventional Merton strategy and the CW strategy when the market is highly volatile but is outperformed in a low-volatility high-growth environment. Thus, an endowment fund must evaluate, review and modify its spending rule and investment policy periodically, depending on financial market conditions.

Future work will incorporate a stochastic interest rate environment for greater realism. Endowment fund expenditure across activities like research, teaching, estate upkeep, and administration can also be differentiated. The impact of spending and investment on these different areas may vary from one activity to another. We also intend to carry out empirical work to observe how our optimal solutions behave in stress scenarios, such as a liquidity crisis.¹