Discount Rates and Asset Returns: Implications for Endowment Strategies

Transcription

1 Discount Rates and Asset Returns: Implications for Endowment Strategies 9/14/17 David T. Brown William R. Hough Associate Professor of Finance Warrington College of Business University of Florida Dan Scholz Director Investment Strategies NISA Investment Advisors LLC Abstract Many endowments and foundations set the annual dollar amount available for current operations as the product of the market value of assets and a spending rate that is relatively fixed market value spending policy. We show that a market rate spending policy adjusting the percentage of beginning of year assets available for spending with changes in discount rates/expected returns better meets the key endowment objectives of stable spending and generational neutrality. Fixed spending rates violate generational neutrality. Declining expected returns cause dollar spending to increase with the associated asset value increase beyond sustainable levels which lowers future spending. Some current spending is at the expense of future spending. The opposite occurs when expected returns rise. Market rate spending maintains generational neutrality by adjusting spending rates with expected returns. Market rate dollar spending is also more stable than market value spending when and endowment uses market rate spending and invests assets whose values are more sensitive to discount rate changes, i.e. longer duration assets. This occurs because when discount rates decline (increase) the lower (higher) spending rate offsets some of the impact of the increase (decrease) in asset values. Discount rate shocks have little impact on dollar spending when (1) the portfolio is invested in long duration assets and (2) spending is generationally neutral. Monte Carlo simulation analysis shows that market rate spending applied to an equity portfolio reduces year over year spending volatility (versus market value spending) by between 19% and 30% depending on the assumptions used in the return generating model. Year over year spending volatility using market rate spending applied to a long duration fixed income portfolio is almost 80% lower than year over year spending volatility using market value spending applied to a market duration portfolio. 1

2 1. Introduction The primary objective of Princeton s endowment spending policy is to achieve a proper balance between present and future needs of the University. Other objectives are to achieve a reasonable degree of stability and predictability in income available for current operations. 1 We show that both objectives are better met with a market rate spending (MRS) policy that adjusts the spending rate (percentage of beginning of year endowment assets available for current operations which refer to as spending ) with changes in expected asset returns versus spend a fixed percentage of beginning of year assets, the commonly used market value spending (MVS) policy. Spending rates that do not change with expected returns lead to either spending rates that exceed expected returns reducing the asset base over time and lowering future spending or spending rates below expected returns which shifts spending in the future. Thus MRS is more consistent with the legally mandated generational neutrality criteria: a proper balance between present and future needs. Spending is also generally more stable and predictable with MRS. Stable spending rates don t imply more stable spending. When asset values increase because of a discount rate/expected return decline, MVS increases proportional to the asset value increase. The MRS spending rate declines with the expected return shock that raised asset prices. When discount rates decline the larger the associated asset price increase the greater the offset against the spending rate decline. Spending is unaffected by the discount rate shock if the asset s return sensitivity to a discount rate shock (duration) equals the duration of a perpetuity. 2 MRS simply recognizes that discount rate shocks impact market values but do not impact the ability of long duration assets to support long horizon spending. 3 Asset values also change with changes in expected cash flows. Changes in asset values due to changes in expected cash flows have essentially the same impact on spending under MRS and MVS. Spending changes from expected asset cash flows cannot be avoided. Endowments and foundations are averse to year over year changes in spending: spending volatility. Spending volatility is proportional to asset return volatility with MVS. With a sustainable MRS policy, spending volatility is proportional to asset risk stemming from expected cash flow shocks. However, spending volatility from discount rate shocks declines with the duration the asset and hence with asset return volatility stemming from discount rate shocks. Investments in long duration bonds with no default risk under MRS are nearly riskless when risk is measured as spending volatility. The idea that under MRS, the 1 See Princeton University Endowment Spending Fact Sheet For simplicity of illustration inflation is ignored. Investing in an inflation indexed perpetuity would result in stable and generationally neutral real spending. 3 Tobin (1974) alludes to this idea by pointing out that spending is potentially more stable while the endowment market value is not when an endowment commits to spending investment cash flows: dividends and coupons. 2

3 impact of expected cash flow shocks is a greater source of risk than discount rate shocks is similar to the bad beta (expected cash flow risk) versus good beta (discount rate risk) distinction made by Campbell and Vuteenaho (2004). Distinguishing between expected cash flow and discount rate shocks is critical for both the appropriate level of endowment spending and the individual investor s consumption/savings choice. Campbell, Giglio and Polk (2013) show that the stock market downturn was driven by an increase in discount rates while the stock market downturn resulted from expected lower profits. They point out that whether the market down results from a discount rate or expected cash flow shock determines the optimal consumption response of a long term investor. MRS is flexible. Low spending volatility can be achieved without maintaining generational neutrality by setting the spending rate equal to the current market rate plus or minus a constant. Low spending volatility with long duration assets only requires that spending rates move with market rates. Subtracting a constant from the current market expected rate benefits future spending at the expense of current spending as the value of the endowment grows. Rather than incur a dramatic change in spending by adopting MRS, as many endowment spending rates may exceed current expected returns, Summers (2016) notes that the Harvard University spending rate has not changed over a period of time where long term real rates have declined 4%, the MRS spending rate can be set at a premium above the current expected return. The premium could be reduced over time as a gradual transition to a generationally neutral spending rate. Spending volatility is zero if an endowment invests in a riskless perpetuity and sets spending rates equal to the market perpetuity yield. Limiting a hypothetical endowment s investable bonds and spending rate to 30 year maturities, Monte Carlo simulations calibrated to match the current term structure of U.S. interest rate volatilities we estimate that the spending volatility minimizing bond maturity strategy under MRS results in year over year spending volatility as a percentage of average spending is 0.99%: about a 70% reduction versus the 4.58% year over year spending volatility from investing in the market portfolio duration bond portfolio using MVS. Estimates of the deviations from generational neutrality associated with MVS are also provided. The challenge for implementing MRS with assets like equities is that unlike bonds the discount rate is not observable. The extent that MRS reduces year over year spending volatility for equity assets depends on (1) the duration of equity and (2) the ability to accurately estimate expected returns. Using simulated equity returns we find that spending volatility is reduced by between 19% and 30% depending on the model assumptions versus a fixed spending rate (MVS) using time varying spending rates equal to 3

4 the equity dividend yield less expected dividend growth along the lines of the economically constrained coefficient approach of Ferriera and Santa-Clara (2011). The remainder of this paper proceeds as follows. A short introduction to endowment strategies and metrics for evaluating endowment strategies are presented in Section 2. The economics of MRS and the multi-period impact of a discount rate shock under MRS and MVS are provided in Section 3. Monte Carlo simulation analysis of MRS and MVS policies for fixed income and equity portfolios are reported in Sections 4 and 5. Conclusions and extensions for further research complete the paper. 4

5 2. Endowment Strategies Endowments are managed asset pools with a mandate to provide cash flows to support a charitable activity. Endowment management strategies specify both (1) a spending policy (the percentage of the asset base released annually to support the activity) and (2) an asset allocation. Effective strategies jointly determine the spending policy and asset allocation. Merton (1969, 1971) develops optimal jointly determined spending (consumption) and asset allocation strategies for individuals. However, the endowment and individual problem differ because of legal constraints on endowments described below. In addition, utility functions that specify individual preferences are not satisfactory for endowments. This section motivates metrics for ranking multi-period distributions of endowment spending. 2.1 Market Value and Market Rate Spending Policies Current endowment spending policy, MVS, distributes the product of a relatively constant spending rate and spendable base tied to the market value of the assets. 4 The spendable base is generally a three-year moving average of beginning of year asset values. Goetzmann, Griswold and Tseng (2010) evaluate asset allocation strategies distributing 4.5% of a three-year moving average of asset values. We introduce an alternative MRS spending policy adjusts the spending rate with changes in the asset portfolio expected return. The following analysis sets the annual distributions equal to the product of the expected asset returns and beginning of year asset portfolio value. 2.2 Generational Neutrality Most endowments are subject to the Uniform Management of Institutional Funds Act (UMIFA) or the revised Uniform Prudent Management of Institutional Funds Act (UPMIFA). 5 The statute guidelines require that endowments are managed to maintain perpetual support. The operational standard for maintaining perpetual support is generational neutrality: current and future generations benefit equally from the endowment. Spending rates much larger (smaller) than expected asset returns violate generational neutrality by favoring current (future) generations over later (current) generations. These statutes also include guideline limitations on spending by endowments where the market value of assets is less the amount of the donation(s) historic dollar value known as underwater 4 Some university endowments use a banded inflation methodology which grows the annual spending by the rate of higher education inflation (HEPI) or CPI. Some endowments employ a hybrid rule where the amount available for spending is a convex combination of the banded inflation and moving average methodologies: 60% to 80% based on the inflated value of last year s spending and 20% to 40% based on a moving average of asset values. Spending policies are regularly surveyed and reported by Cambridge Associates LLC. 5 Budak and Gary (2010) and Bass (2010) provide detailed discussions of these statutes and endowment practice. 5

6 endowments. Endowment policies for underwater vary but often provide a schedule of spending rates that decline as the endowment becomes further under water and in some cases eliminate spending on underwater endowments Evaluating Endowment Strategies Endowments support actives directed by individuals we refer to as end users. The asset allocation and spending policy determine both the expected flows to end users and the variability of the flows. Specifically, the endowment strategy and beliefs about asset return distributions map into distributions of flows for the next and subsequent years. Evaluating endowment strategies requires metrics for ranking the distribution of multi-year flows associated with one strategy versus another. Donors and end users may be averse to year over year changes in spending (spending volatility) as a result of having convex preferences over future expenditures: declines in spending cut into high priority or highly valued expenditures while increased spending funds lower valued expenditures. Aversion to spending volatility also arises from the real impact of spending volatility on the end user. End users are more effective when the adjustment costs of year over year spending budgets are reduced, see Dybvig (1999). The maximum committed spending statistic developed here recognizes that adjustment costs and the value to the end user of a flow of expenditures may be better captured by the distribution of expenditures over multiple periods. Before developing the maximum committed spending statistic it is important to note that there is not a mapping from year over year spending volatility to maximum committed spending. Strategy A can have lower year over year spending volatility than Strategy B but lower levels of committed spending. For example, investing in shorter duration assets with MVS is a low year over year spending volatility strategy but not a compelling maximum committed spending strategy. The practice of using a weighted average of historical asset values as the spending base, smoothing, lowers year over year spending but does not improve maximum committed spending. Appendix A shows the impact of smoothing on spending volatility. Maximum committed spending distinguishes between long horizon and short horizon expenditures. Consider the end user responsible for deciding how to spend the income from a $2 million endowment with an initial spending rate of 5%. The administrator allocates her $100,000 initial annual budget between (1) long term committed expenditures, for example hiring additional staff or equipment purchases that must be maintained over time, and (2) short run expenditures that have benefits accruing in the next year. We assume that long term expenditures provide greater benefits per dollar spent per year than short run expenditures if the level of long term expenditures are maintained sufficiently long (k years) but smaller benefits per dollar 6

7 spent per year if not. Denoting the benefit per year of a dollar spent in a long term program as a nondecreasing function of the number of years the program is maintained as BL(t) and the benefit per dollar of single year expenditures as BS, then BL(t) < BS for t less than k and BL(t) > BS for t greater than k. Our hypothetical end user choses how much of the $100,000 to commit to long run expenditures based the benefit functions (BL(t) and BS) and the expected distribution of future budgets. Larger long term expenditure commitments are beneficial up to the point where the probability the annual endowment distribution cannot meet the committed spending level chosen prior to year k is too high. The optimal long term expenditure commitment, and by assumption the value of potential benefits, are larger when spending is more stable. We define maximum committed spending as the largest amount that can be allocated to long term spending for a given horizon and probability that the annual endowment distribution falls below the amount of long run spending. Maximum committed spending, C MAX(H,p*), is a function of the horizon, H 7

8 3. The Economics of Endowment Spending Policies Asset values change with discount rate and expected cash flow shocks. The section shows that MRS and MVS result in different post discount rate shock spending trajectories. The impact of discount rates shocks on spending depends on the asset duration: sensitivity of asset prices to changes in discount rates. Spending changes increase with asset duration with MVS and decrease with asset duration with MRS. Changes in expected cash flows have essentially the same impact on spending with both spending policies The Impact of Discount Rate Shocks on Endowment Spending The salient features of the Market Rate Spending (MRS) and Market Value Spending (MVS) policies are shown by examining the impact of a discount rate shock on spending following the shock. Table 1 presents the notation. Table 1: Endowment Spending Model Notation Table 1 presents the model notation describing asset values, asset returns, and spending. MRS and MVS are modeled assuming identical endowment asset returns. Asset returns depend on D, r t, and dr. Spending and hence asset values differ with the spending policy employed and are subscripted MV for MVS or MR for MRS. Assets and Discount Rates Market Value of Assets Market Rate Spending A MRt Market Value of Assets Market Value Spending A MVt Modified Asset Duration D Expected Return/Discount Rate r t Expected Return/Discount Rate Shock dr Endowment Spending Market Value Policy Spending Rate K Market Value Policy Annual Spending S MVt = A MVt *K Market Rate Policy Annual Spending S MRt = A MRt*(r t) The percent change in the endowment asset portfolio value over the course of a year is (1) the beginning of period discount rate, plus (2) the impact of a discount rate shock, minus (3) end of year spending. The impact of the discount rate shock is the product of the discount rate shock and minus the present value weighted average life of the expected asset cash flows the asset s modified duration. 6 For a MVS policy A MVt+1 = A MVt*(1 + r t K dr*d). (1) 6 The linear relationship between discount rate and duration ignores convexity for ease of exposition. 8

9 The percentage change in spending, denoted MVS is MVS = (MVS t+1 MVS t)/mvs t = (r t K) dr*d. (2) It is clear from Equation (2) that MVS violates generational neutrality when K r t. When dr = 0, MVS t+1 > MVS t when r t > K and MVS t+1 < MVS t when r t < K. The impact of a discount rate shock on MVS increases with D through the resulting impact on A t+1. For an MRS policy A MRt+1 = A MRt*(1 dr*d). (3) With the spending rate equal to the expected return the (r t K) vanishes and spending is consistent with generational neutrality. The percentage change in spending, MRS, depends on both the asset value change and spending rate change: MRS = (MRS t+1 MRS t)/mrs t = -dr*d + dr/r t. (4) MRS depends on the extent that the asset value change -dr*d off sets the spending rate change dr. Note that MRS = 0 when D = 1/r t. An asset duration of 1/r t matches the duration of a perpetuity. The result that spending is invariant to discount rate shocks when the asset duration matches the duration of a perpetuity is intuitive. MRS is equivalent to spending the payments available from purchasing a perpetuity with a yield of r t thus an asset with a duration of 1/r t hedges the spending exposure to discount rate shocks. Endowment asset management can be thought of as an asset-liability management problem where maintaining generational neutrality implies a perpetual liability. Table 2 plots the impact of a -1% discount rate shock to MVS (Equation (2)) and MRS (Equation (4)) spending as function of asset duration. 9

10 Table 2: Endowment Spending Changes Following a Discount Rate Shock Table 2 shows the change in dollar spending ( MVS and MRS) following a decline in the discount rate from 5% to 4% for various levels of initial asset duration (D). The initial (pre discount rate shock) asset value of the hypothetical endowment is $100. K is 5% so the pre discount rate shock spending is $5 for both MVS and MRS. Difference equals MRS - MVS and represents the extent that MVS spending is not sustainable. Asset Duration MVS ($) (A) MRS ($) (B) Difference ($) (B) (A) MRS is proportional to the change in the value of the assets relative to the value of a hypothetical perpetuity rather than proportional to the asset value change as with MVS. If the asset value increase associated with a discount rate shock equals the change in the hypothetical market value of a perpetuity then spending is unchanged. 7 When the asset duration is less than 1/r t then spending declines even asset values have increased. The decline in expected returns increases asset prices but the expected cash flow from the asset arrive considerably before the end of the endowment horizon. 8 Spending is lowered in anticipation of lower reinvestment rates. Failing to lower spending results in spending levels that will leave spending at lower levels in the future. 7 The example in Table 2 calculates asset values at 4% as a linear approximation. For very small rate changes MVS is approaches zero for asset with beginning (end) of year duration of 21 (20) which equals the duration of a perpetuity at 5%. 8 Nichols (1974) notes that when asset values increase with discount rates an endowment should only increase spending when it plans to be a net seller of securities That is when endowment has a shorter horizon then considered in this analysis. 10

11 3.2 Multi-year Horizon Spending Volatility It is important to note that a discount rate change has implications for spending beyond year t+1 for MVS because r t+1 K. For an asset with a beginning of year duration of 1 and MVS, spending does not change in the year following a rate change: MVS = 0. However, following a discount rate decline E(MVS t+2) < MVS t+1 and E(MVS t+j) < E(MVS t+j-1) for j 2 because the expected return is less than the spending rate. Thus under MVS, year over year spending changes exhibit positive serial correlation for shorter duration assets. While short duration assets under MVS have small year over year spending changes the level of spending can vary substantially over a multi-year horizon. The analysis in the following section shows that the standard deviation of spending over a multi-year horizon is quite large for short duration assets under MVS and highlights the importance of evaluating endowment strategies both in terms of year-over-year and multi-year horizon spending volatility measures including maximum committed spending Impact of Expected Cash Flow Changes on Spending Asset values are also impacted by changes in expected cash flows. An asset value change resulting from a change in expected cash flows has largely the same impact on spending under MVS and MRS. When the MVS spending rate and MRS spending rate are identical then an increase in expected cash flows has exactly the same impact on spending. If the MVS spending rate exceeds the MRS spending rate then the dollar increase in MVS spending is larger the dollar increase in MRS spending. However, the change MVS spending as percent of the asset values is identical to the MRS dollar spending increase and thus the violation of generational neutrality that exists when the MVS spending rate exceeds the MRS spending rate is not clearly made worse by the change in expected cash flows. The same conclusions hold when (1) the MVS spending rate is less than MRS spending rate or (2) asset values change from a decline in expected cash flows Modified MRS Strategies Our analysis assumes that endowment preferences are to maintain generational neutrality and experience predictable spending. Some endowments may set conservative spending rates tolerate modest deviations from generational neutrality in order to grow the value of the endowment assets. Also endowment currently spending more than expected returns may want transition to MRS without a dramatic change in spending. The lower spending volatility associated with MRS can be maintained while accommodating these objectives by setting spending rates equal to expected returns minus a constant. 11

12 Suppose an endowment with a fixed spending rate of 5% and expected return on assets of 4% views the immediate transition to a 4% spending rate a 20% reduction in spending prohibitively disruptive. Setting the spending rate at the current spending rate plus 1% avoids immediate disruptions. Resetting the spending rate to reflect current expected returns and considering the each asset classes contribution to spending volatility under MRS is sufficient to obtain the benefits of MRS. The premium above expected returns added to the spending rate can be reduced over time. A faster reduction in the premium is more generationally neutral but potentially more disruptive to near to spending. The appropriate pace to transition to sustainable spending is beyond the scope of this paper. Our point is that an immediate transition to sustainable spending is not required to realize the lower spending volatility associated with MRS Underwater Endowments Some endowments lower spending rates when asset values below the original value of the endowment donation. Endowments with asset values below their original value are referred to as underwater. We do not have an opinion about the merits of policies designed with a focal point around the initial value of the endowment. However, we suggest that underwater is better defined in terms of the level of the MRS spending relative to the initial MRS spending level. Asset value declines below the initial endowment value resulting from declines in expected cash flows are underwater in terms of the current (asset value based) definition and the proposed MRS spending level definition. But if asset value declines are at least in part due to an increase in expected return then endowments that are underwater by the asset value based definition may not be underwater by the MRS spending definition: see Case D in Table 3. Likewise there are cases where asset values have increased as the result of a discount rate decline but cannot support sustainable spending at the initial spending rate and should be viewed as underwater: see Case A in Table 3. 12

13 Table 3: Underwater Endowments under MRS and MVS The example in Table 3 considers an endowment with an initial value of $100 million with an expected return of 5%. Cases A-D depict subsequent situations with different asset values and expected returns. MRS spending rates adjust with asset values and expected returns while MVS spending rates remain at 5%. The endowment is underwater under the asset value criteria when the asset value is less than $100 million and underwater under the MRS Spending criteria when spending falls below $5 million. Underwater Criteria Case Asset Value Expected Return MRS Spending MRS Spending Asset Value (millions) (millions) A $ % $4.95 Yes No B $ % $5.23 No No C $ % $4.95 Yes Yes D $ % $5.40 No Yes 13

14 4. Monte Carlo Simulation Analysis: Fixed Income Portfolios Distributions of future spending under MRS and MVS are generated from a Monte Carlo simulation of asset returns for hypothetical endowments. Applying MRS is straight forward for fixed income assets since discount rates are directly observable. For equity assets MRS relies on proxies for expected returns. Thus, Monte Carlo simulations of a hypothetical endowment invested in fixed income assets (Section 4) and a hypothetical endowment invested in equities (Section 5) are examined separately. This section proceeds with a discussion of the term structure simulation used to generate bond returns followed by simulated endowment spending under MVS and MRS Term Structure Simulation The term structure simulation generates 10, year sample paths. Specifically a term structure of interest rates with maturities out to 100 years is generated at the start of each year. The term structure paths are generated from a one factor term structure along the lines of Vasicek (1977). The model has three inputs: (1) the initial term structure of interest rates, (2) the volatility of the short term interest rate and (3) a mean reversion parameter. The initial term structure has all maturity rates at 5%, the annualized standard deviation of the short term interest rate is 0.40% and the mean reversion parameter is The short rate volatility and mean reversion parameters determine the term structure of interest rate volatility: the volatility of each maturity rate. The model parameters are chosen to approximate the recent history of U.S. Treasury bond yield volatilities. Summary statistics for simulated fixed income asset returns (mean returns in Panel A and return standard deviations in Panel B) are reported Table 4. The fixed income assets are zero coupon bonds. The returns for various maturities represent constant maturity strategies. An n year zero coupon bond is purchased at the beginning of the year and then sold at the end of year (n-1 years to maturity) with the proceeds reinvested in an n year bond. 14

15 Table 4: Summary Statistics for Simulated Bond Returns The data in Table 4 are organized by bond maturity and holding period. The mean return is the annualized return based on (1) the initial portfolio value and (2) the average end of holding period position value across 10,000 paths. The standard deviation calculation is the average across 10,000 paths of the standard deviation of returns along each path. Mean Annualized Year Returns Bond Maturity Year % 4.13% 4.25% 4.29% % 4.09% 4.11% 4.25% % 4.08% 4.08% 4.13% % 4.08% 4.08% 4.12% % 4.07% 4.07% 4.08% Annualized Return Standard Deviation Bond Maturity Year % 2.38% 4.98% 12.29% % 0.65% 1.74% 4.97% % 0.51% 0.91% 3.08% % 0.85% 0.63% 1.79% % 1.06% 0.75% 1.17% The annualized average returns (1) decline with holding period, (2) increase with maturity, and (3) the average returns converge across maturities converge for long holding periods. These patterns arise from the fact that asset values are convex functions of discount rates and the convexity of the asset value function increases with maturity. The return volatility is smallest for a holding period approximately twice the maturity (duration) of the bond portfolio and increase in the difference between the holding period and twice the maturity of the assets. 9 9 Langetieg, Leibowitz and Kogelman (1990) derive the relationship between holding period and duration for targeted duration strategy. 15

16 4.2. Spending Volatility and Asset Duration. The simulated endowment using the MVS policy provides annual amounts to end users of 5% of the beginning of year asset values. The MRS policy provides annual amounts to end users equal to beginning of year yield on a 100 year maturity bond times the beginning of year asset values. Two measures of spending volatility are presented in Table 5. Year over year spending volatility is the standard deviation of year over year changes in spending divided by average spending. Long horizon spending volatility is the standard deviation of spending over the length of the simulation divided by average spending. Both spending volatility measures are calculated for each simulated path with the mean and median spending volatility across 10,000 paths reported. 16

17 Table 5: Spending Volatility Year over year spending volatility (year over year) and long horizon spending volatility (long horizon) are reported for a hypothetical endowment invested in fixed income assets. Results are presented for various maturity (duration) portfolios. The hypothetical portfolios invest in zero coupon bonds maintaining a constant maturity. The mean across 10,000 simulated interest rate paths and median in brackets of both spending volatility measures are reported. Market Rate Spending Market Value Spending Maturity Year over Year Long Horizon Year over Year Long Horizon % [7.76%] % [6.82%] % [5.95%] % [5.15%] % [4.41%] % [3.73%] % [3.10%] % [2.52%] % [1.98%] % [1.48%] % [1.03%] % [0.62%] % [0.29%] % [0.24%] % [0.50%] % [0.80%] % [1.09%] % [1.35%] % [1.60%] % [1.83%] % [2.04%] % [2.24%] % [2.42%] % [2.74%] % [3.36%] 13.16% [12.01%] 11.56% [10.55%] 10.08% [9.18%] 8.71% [7.92%] 7.45% [6.76%] 6.28% [5.68%] 5.20% [4.67%] 4.22% [3.76%] 3.32% [2.91%] 2.52% [2.17%] 1.86% [1.59%] 1.43% [1.27%] 1.24% [1.08%] 1.20% [0.90%] 1.40% [1.02%] 1.85% [1.35%] 2.31% [1.83%] 2.75% [2.28%] 3.16% [2.68%] 3.55% [3.06%] 3.90% [3.39%] 4.23% [3.72%] 4.54% [4.00%] 5.09% [4.52%] 6.15% [5.51%] % [1.30%] 1.86% [1.82%] 2.54% [2.52%] 3.24% [3.22%] 3.93% [3.91%] 4.58% [4.56%] 5.19% [5.16%] 5.76% [5.73%] 6.29% [6.26%] 6.79% [6.75%] 7.25% [7.21%] 7.67% [7.63%] 8.07% [8.02%] 8.44% [8.39%] 8.78% [8.72%] 9.09% [9.04%] 9.39% [9.33%] 9.66% [9.60%] 9.91% [9.85%] 10.14% [10.08%] 10.36% [10.29%] 10.56% [10.48%] 10.74% [10.67%] 11.07% [10.99%] 11.70% [11.62%] 9.11% [7.67%] 8.48% [7.17%] 8.14% [6.98%] 8.05% [6.98%] 8.15% [7.24%] 8.39% [7.61%] 8.74% [8.01%] 9.15% [8.45%] 9.61% [8.93%] 10.09% [9.44%] 10.57% [9.92%] 11.06% [10.39%] 11.53% [10.84%] 11.98% [11.28%] 12.41% [11.70%] 12.82% [12.11%] 13.21% [12.49%] 13.58% [12.83%] 13.93% [13.16%] 14.25% [13.43%] 14.55% [13.71%] 14.83% [13.97%] 15.10% [14.20%] 15.57% [14.63%] 16.49% [15.51%]

18 The relationships between year over year spending volatility and duration for MVS and MRS match the analytical results in Section 3. Spending volatility increases with duration for MVS. For MRS, spending volatility decreases with duration up to 14 years and then increases with duration. The minimum spending volatility duration using MRS, 14 years, is less than the cash flow duration of a perpetuity with a 5% yield because the volatility of rates declines with maturity in our simulation. Year over year spending volatility investing in 14 year duration assets and applying MRS is 94% lower than investing in 6 year duration assets (the approximate duration of the US dollar investment grade market portfolio) and applying MVS. Long horizon spending volatility is 86% lower comparing the same strategies. It is important to note that year over year spending volatility for short duration assets using MVS is relatively low but the variability of spending over long horizons is much higher: the long horizon spending volatility investing in one year duration assets is higher than the long horizon spending volatility investing in six duration assets Spending Volatility with Limited Maturity Interest Rates The above analysis uses the yield on a 100 year annuity as the MRS spending rate. In practice, such long maturity interest rates may not be observable as there are very few actively traded bonds with maturity greater than 30 years. The following analysis uses the simulated 30 year par bond yield as the spending rate. Spending volatility under MRS is higher using the 30 year par bond yield as a spending rate than when the 100 year annuity yield is the spending rate. Using the 30 year par bond as the spending rate and constrained to a constant duration, spending volatility is minimized with an asset duration of 21 years. Year over year spending volatility is 1.56% which is higher than the 0.28% spending volatility obtained investing in a 14 year duration asset with the 100 year annuity yield as the spending rate. The spending volatility minimizing fixed income portfolio strategy under MRS adjusts the asset duration with the level of the spending rate. We refer to this strategy as the dynamic duration strategy. The asset duration choice that minimizes spending volatility under MRS is a function of the level of the spending rate and the term structure of rate volatility. If a single discount rate (Y t) priced all bonds a flat term structure of interest rates then the asset duration (D t) that minimizes spending volatility matches the duration of a perpetuity as shown in Equation (4): D t = (1/Y t). (5) Unlike the constant maturity strategies shown in Table 6, minimizing spending volatility requires a dynamic asset duration strategy: duration changes with changes in the spending rate. In our simulation, discount rate 18

19 volatility varies with maturity. Denoting the volatility of Y t as σ y and the volatility of the D year discount rate as σ D, the asset duration that minimizes spending volatility is D * t = (1/Y t)*(σ y/σ D*) (6). Table 6 provides spending volatility estimates for the dynamic duration strategy with a spending rate equal to the year 30 year par bond rate. Table 6: Dynamic Duration Spending Volatility Estimates Table 6 presents year over year spending volatility (year over year) and long horizon spending volatility (long horizon) estimates for a hypothetical endowment invested in fixed income assets. Results for (1) constant 21 year duration and 30 year par bond spending rate, (2) dynamic duration and 30 year par bond spending rate, (3) constant 14 year rate duration and 100 year yield spending rate, and (4) constant 6 year duration under MVS. The mean across 10,000 simulated interest rate paths and median in brackets of both spending volatility measures are reported. Spending Volatility Asset Duration Spending Rate Year over Year Long Horizon Dynamic 30 Year Par Bond Yield 0.98% [0.68%] 3.00% [2.56%] 21 Years 30 Year Par Bond Yield 1.56% [1.28%] 3.26% [2.63%] 14 Years 100 Year Annuity Yield 0.28% [0.24%] 1.20% [0.90%] 6 Years Fixed 4.58% [4.56%] 8.39% [7.61%] The dynamic duration strategy adjusts the asset duration with the level of the spending rate and results in lower spending volatility versus the constant duration strategy. The dynamic duration strategy with a 30 year par bond spending rate results in a 78% reduction in year over year spending volatility and a 64% reduction in long horizon volatility versus an endowment strategy that invests in the market portfolio duration and applies a fixed spending rate. 19

20 4.4. Maximum Committed Spending Estimates The maximum committed spending statistic developed in Section 2 is designed to capture the value to end users of a stable spending stream. Long term programs potentially provide greater benefits to end users if spending is maintained. The maximum committed spending statistic depends on (1) the horizon and (2) a probability threshold. For a horizon of 10 years and a probability threshold of 80%, the maximum committed spending is the level of annual spending such that there is an 80% probability that spending never falls below that level in any of the next ten years. Maximum committed spending estimates for the dynamic duration MRS and market duration MVS strategies are shown in Table 7. Table 7: Maximum Committed Spending Estimates Maximum committed spending for hypothetical bond portfolios with an initial value of $100 are presented in Table 7. Committed spending depends on the threshold probability 90%, 80% and 70% - denoted p and the commitment horizon 5, 10, 15, 20 or 30 years. A strategy that invests in market duration bonds and applies a fixed spending rate MVS and a strategy that employs the dynamic duration strategy and par 30 year spending rate MRS are compared. The value of $4.48 for the 10 year horizon, 90% threshold probability under MVS means that there is a 90% probability that the annual amount made available for spending never falls below $4.48 over ten years. Horizon (years) Market Value Spending Rule Market Rate Spending Rule p = 70% p = 80% p = 90% p = 70% p = 80% p = 90% 5 $4.78 $4.70 $4.61 $4.88 $4.86 $ $4.73 $4.62 $4.48 $4.85 $4.82 $ $4.70 $4.57 $4.40 $4.81 $4.78 $ $4.68 $4.52 $4.34 $4.79 $4.74 $ $4.65 $4.47 $4.25 $4.75 $4.68 $4.61 The estimated maximum committed spending statistics are higher under MRS. The improvement in maximum committed spending with MRS is higher for more stringent threshold probabilities: more than $0.20 for p = 90% versus $0.10 for p = 70%. The improvements in maximum committed spending under MRS obviously increase with the horizon. 20

21 4.5. Generational Neutrality Violation Estimates Market rate spending maintains spending rates equal to expected returns and therefore current spending does not come at the expense of future spending or subsidize future spending. Market value spending violates generational neutrality as expected returns deviate from the fixed spending rate. We develop two measures of violations of generational neutrality under MVS and estimate the extent of violations of generational neutrality for a hypothetical endowment invested in fixed income securities. The first measure, equal maturity generational neutrality violation denoted EMGNV, compares a strategy that invests in a market duration portfolio using MVS to a strategy that also invests in a market duration portfolio using MRS. Denoting the market value of the portfolio using MVS t periods into the simulation as MVS t and the market value of the portfolio using MRS t periods into the simulation as MRSA t, the EMGNV t statistic is the absolute value of the difference between MVS t and MRSA t: EMGNV t = MVS t MRSA t. (7) The second measure, long maturity generational neutrality violation denoted LMGNV t, is the difference between MVS t and MRSB t the market value of endowment assets using MRS and investing in long duration assets: LMGNV t = MVS t MRSB t. (8) Table 8 reports the mean and median across 10,000 paths of EMGNV t and LMGNV t. Table 8: Estimates of Generational Neutrality under MVS Table 8 reports the mean and median brackets - of EMGNV t and LMGNV t across 10,000 simulated interest rate paths for 5, 10, 20 and 30 year horizons. The initial value of the hypothetical endowment assets is $100. Horizon (years) EMGNV t LMGNV t 5 $2.67 [$2.28] $8.87 [$7.12] 10 $5.73 [$4.93] $15.77 [$12.64] 20 $11.56 [$9.73] $24.22 [$19.71] 30 $16.88 [$13.74] $32.23 [$26.62] Deviations from generational neutrality are substantial. Investing in market duration fixed income securities, the average deviation from a generationally neutral spending rate is $2.67 (or 2.67% of the initial portfolio value) after five years and almost 6% of the initial portfolio value after ten years. 21

22 5. Monte Carlo Simulation Analysis: Equity Portfolios A Monte Carlo simulation is used to generate equity returns and spending under MRS and MVS spending. Specifically, 10, year paths of equity returns and dividend yields are generated from the Binsbergen and Koijen (2010) economy. The base case simulation uses the dividend growth and discount rate process parameter estimates obtained from historical US equity market data reported in Binsbergen and Koijen (2010) and yields spending volatility estimates based on simulated returns that closely match critical features of historical U.S. equity returns. Additional simulations vary the equity model parameters providing a range of spending volatility estimates Simulated Returns The Binsbergen and Koijen (2010) economy assumes expected returns (μ t) and the expected dividend growth rates (g t) follow AR(1) processes with all values expressed as logs: μ t+1 = δ 0 + δ 1(μ t - δ 0) + ε μ t+1 (9) and g t+1 = ϒ 0 + ϒ 1(g t - ϒ 0) + ε g t+1. (10) The change in dividends from t to t+1, d t+1, is the expected dividend plus an orthogonal shock: d t+1 = g t + ε d t+1. (11) The Campbell and Shiller (1988) log-linear present value model implies that stock returns, r t, are r t+1 = κ + ρ*(p t+1 d t+1) + d t+1 (p t d t) (12) where κ and ρ are constants of the log linearization and p t d t is the log of the price-dividend ratio at time t. Iterating Equation (4) and using the processes in Equations (1) (3), the log of the price-dividend ratio follows p t d t = A B 1(μ t - δ 0) + B 2(g t - ϒ 0). (13) Binsbergen and Koijen (2010) estimate the parameters of the model with U.S. data from CRSP from 1946 to 2007 using a conditional maximum likelihood estimation. The parameters of Equation (5) are A = B 1 = , and B 2 = per Table II of Binsbergen and Koijen (2010). Simulated returns are generated from these parameter estimates and the estimated volatilities of the discount rate and dividend growth processes. The simulated data are 10,000 paths of 80 years of annual returns and dividend yields. Endowment spending for a portfolio earning the simulated returns are reported for a sixty year horizon as the first twenty years of simulated data are used to calculate the expected dividend growth rate for the SOP expected return estimate. Summary statistics for the simulated equity returns are reported in Table 9. 22

23 Table 9: Simulated Return Summary Statistics Table 9 reports the mean and standard deviation of the (1) annualized return and (2) return volatility across 10,000 paths of simulated equity returns. The annualized return is the continuously compounded return for each 80 year path, and return volatility is the annualized standard deviation of returns are calculated for each path. Mean Standard Deviation Annualized Return 9.00% 1.45% Return Volatility 15.33% 1.23% 5.2. Spending Policies Three hypothetical endowment strategies are simulated: an MVS policy and two MRS policies. The MVS policy spending rate is 9.45%. The spending is slightly above equals the long run expected stock return parameter in the equity return model and is set so that the median annual spending under MRS and MVS are approximately equal. Simulated annual MVS policy endowment spending distributes the product of the long run expected stock return parameter in the equity return model and the beginning of year asset value. The MVS policy is set with knowledge of the long run expected return. Under this assumption the average spending volatility across paths approximately equals asset return volatility. Expected equity returns for the MRS strategy are based on the constant dividend growth model of Gordon (1962) the Gordon growth model. Campbell and Shiller (1988) derive a dynamic version of the Gordon growth model in stationary economy and show the market discount rate is an affine function of the dividend yield and a weighted average of expected future dividend growth rates. Denoting the current stock price as P t, the current dividend payment as D t, and E(g t) the expected dividend growth rate, the discount rate d t is: d t = D t/p t + E(g t). (14) Following the sum of the parts approach of Ferreira and Santa-Clara (2011) we use the twenty year historical dividend growth rate up to time t, as an estimate of E(g t). The sum of the parts model of Ferreira and Santa-Clara (2011) is used in this analysis because it successfully predicts out of sample returns and is simple to implement. Ferreira and Santa-Clara (2011) show that return forecasts based on Equation (14) applied to U.S. equity returns predicts 13.4% of the variation in returns at a one year horizon using the out of sample R 2 measure developed by Welch and Goyal (2008) and Campbell and Thompson (2008). Similar out of sample R 2 for one year return forecasts are obtained combining dividend yields with alternative estimates of dividend growth: Da, Jagannathan and Shen (2014) combine dividend yields with dividend growth estimates from analyst forecasts obtain out of sample R 2 of nearly 10% for a one year 23

24 horizon over and Jagannathan and Liu (2015) obtain one year horizon out of sample R 2 over 15% using dynamic learning with data from 1976 to However, the out of sample R 2 of the sum of the parts forecasts applied to the stationary economy of Binsbergen and Koijin (2010) is 7.7% - about half the out of sample R 2 found with historical return data. The lower out-of-sample R 2 occurs because the historical average dividend growth is a poor estimate of the long run expected growth rate in a stationary economy. Thus, our estimates of reductions in spending volatility using MRS are conservative Spending Volatility Estimates Base Case Table 10 presents spending and endowment asset value statistics for the MVS and MRS strategies. Spending level is the average dollar spending along a path. Year over year spending volatility is the standard deviation of the dollar change in spending scaled by the average dollar spending along the path. The final endowment value is the value the endowment after 60 years. Table 10: Simulated Annual Endowment Spending Table 10 presents the mean, median in brackets - and standard deviation in parentheses - of (1) spending level, (2) spending volatility, and (3) ending asset value for 10,000 paths of simulated equity returns. The hypothetical endowments start with $ Spending Level Spending Volatility Final Endowment Value MVS Strategy $10.25 [$8.30] ($4.41) 16.82% [16.47%] (2.52%) $ [$61.55] ($177.95) MRS Strategy $9.11 [$8.50] ($3.33) 13.68% [13.44%] (2.19%) $86.62 [$74.23] ($53.66) The MRS strategy reduces year over year spending volatility based on the mean volatility estimates by about 19%. It is very important to note that the reduction in spending volatility is almost three times larger than the out of sample R 2 using the dividend yield and historical dividend growth rate as an estimate of future dividend growth to forecast returns. The MRS strategy reduces spending volatility to the extent that (1) our spending rate captures the long run equity discount rate and (2) discount rate volatility as opposed to cash flow shocks drive equity returns. The MRS strategy does not require that the expected return estimate accurately forecast realized returns, i.e. high out of sample R 2. 24