Download The Optimal Draw-down for a Charity

The Optimal Draw-down for a Charity.
S.E.SATCHELL
Faculty of Economics,
University of Cambridge,
Cambridge, UK
And
S.THORP
School of Finance and Economics
University of Technology Sydney,
Australia
7 March, 2007
Abstract
We determine the optimal draw-down for a charity. The top 500 charitable trusts in
the UK have assets in excess of £33 billion but there is little published quantitative
research in this area. We treat the problem as one of consumption for an infinitelylived entity and derive the optimal draw-down available to beneficiaries. We allow a
general form for the returns density. In particular we examine and estimate the
income/substitution effects whereby a change in future returns influences our
current draw-down rate. Optimal consumption rates are sensitive to small changes in
future returns distributions, particularly as relative risk aversion diverges from one.
Our analysis can explain puzzling increases in spending when prospects worsen, as
well as very large increases in draw-down rates by conservative foundations when
prospects improve.
1
I. Introduction
The problem we wish to address is how to determine the optimal draw-down for a
charity. We shall use the terms foundations, endowed institutions and charities
interchangeably. Endowed institutions in the U.K. include universities, schools,
research institutions, and grant-making charities. The top 500 charitable trusts in the
U.K. have assets in excess of £33 billion, and investment income of more than £4
billion per annum with a similar amount being raised by donation. Surprisingly,
there is little quantitative research published in this area.
The general problem of an entity choosing optimal consumption and an investment
portfolio over a finite or infinite horizon subject to uncertainty is a huge literature
originating with Merton (1969).1 The typical model involves power or log utility
where uncertainty is generated by diffusions. In some cases, it is possible to derive
explicit solutions for consumption and asset portfolios.
Merton (1969) analyses an infinitely-lived entity (charity) with a constant relative risk
aversion utility function. In the case where all asset returns are log-normally
distributed and some regularity conditions on the rate of discounting of future utility
are satisfied, the optimal rate of consumption is constant, and optimal wealth is lognormal and bounded below. All calculations are done continuously rather than
discretely. Although the key features of the Merton solution, a constant draw-down
and strictly positive wealth, are interesting, and the solution is relatively easy to
compute, it only partially addresses the problem at hand. Furthermore, the
continuous time framework is hardly an advantage in describing the decisionCampbell and Viceira (2002) is a standard work in the area. In chapter 6, they consider the
case where the investor has labour income which could proxy for future endowment income
or donations to a charity. See also Korn and Korn (2001).
1
2
making of a charitable institution, and joint log-normality seems excessively
restrictive, given the asset classes invested in by endowments, which include hedge
funds and private equity.
Early papers on the related issue of the economics of university endowments
concentrate on identifying a draw-down rule consistent with ‘intergenerational
equity’ over an infinite horizon, without explicitly deriving an optimal portfolio
allocation. (See, for example, Tobin 1974, Litvack, Makiel and Quandt 1974, and
Nichols 1974.)
Tobin’s main concern, for example, is to improve on arbitrary policies limiting
consumption from an endowment to either annual cash income (dividends, interest
and rent) or to annual cash income plus all capital gains. He sets out a more flexible,
far-sighted draw-down rule that distinguishes between changes to the long-run rate
of return on assets and temporary revaluation effects. He proposes consuming out of
permanent income instead of exploiting temporary revaluation effects caused by
changes in the discount rate, effects that may in fact signal lower returns in the
future. However Woglom (2003) shows that Tobin’s definition of intergenerational
equity, which aims for fixed real consumption through time, implies a zero rate of
inter-temporal substitution (and consequently infinite risk aversion in the CRRA
model). Using a deterministic, continuous-time model, Woglom confirms that
endowments should consume from recurrent capital gains, but he relaxes the
intergenerational equity constraint to allow optimal real consumption to vary over
time.
In a later paper on university finance, Merton (1990) addresses the more general
question of optimal consumption and portfolio allocation at the whole university,
3
rather than the endowment, level, when income streams (gifts, bequests etc.) and the
costs of university activities covary with investment returns. He argues that
university portfolio managers can hedge against future cost changes and adjust to
non-tradable income sources by employing replicating strategies.
All these studies provide insight into related problems without addressing the
special features currently faced by charities. Many charitable endowments are set up
to provide funding perpetually, but work with particular clients or projects within
quite short time-frames so we choose to define the problem as choosing annual
spending rates over an infinite horizon. A typical charitable trust makes decisions in
discrete time and faces non-normally distributed returns. So, without using
continuous time or making specific distributional assumptions we derive the optimal
spending plan for a charity earning risky investment returns, and go on to examine
the way this draw-down responds when the density of investment returns is
transformed. By shifting probability mass from the lower to the upper tail of the
returns density, and by creating mean-preserving spreads, we can examine the tradeoff between income and substitution effects and the ensuing changes to spending
rates. Consequently our analysis is more robust to the real-world peculiarities of
financial data than existing models.
Superficial intuition might predict that spending out of an endowment will be
positively related to an optimistic investment outlook and negatively linked to
pessimism, but we show that this is true only for a sub-set of preferences and the
reverse reaction can be optimal. Our estimation of the these effects indicate that
optimal consumption rates are remarkably sensitive to small changes in future
returns distributions, particularly as relative risk aversion diverges from one. We
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show that the optimal constant draw-down for a charity with power (constant
relative risk aversion CRRA) utility is set by the simple rule:
1
m  1  ( ) ,
where m is the proportion of wealth spent each year,  is the charity’s rate of time
preference,  is relative risk aversion and  is the expected value of Z 1 , where Z
is the gross return to the charity’s portfolio.2 We estimate that, for plausible
parameter values, such a draw-down might lie between 1 and 3 per cent in real
terms, depending on risk preferences, and based on historical estimates of
investment returns.
II. Defining the Problem
The standard problem is to find the optimal functional form of consumption and the
set of asset-allocations that maximise the expected multi-period utility of wealth
functional through time. Over a finite horizon, this problem can be solved by
dynamic programming, however in what follows we shall use a simplifying but
reasonable assumption, namely that charities are infinitely lived, making calculations
easier, rendering any finite-horizon bequest function irrelevant.
Indeed, in many cases, endowments state in their charter that they are endowed to
provide some sort of support in perpetuity. It is interesting that whilst the charity
can take a very long-term approach, the needs of their clients, at least in a research
context, may be very short-lived indeed, so we have a supply of charitable services
provided from an infinite horizon perspective, whilst the demand could be based on
2
Early analysis of a related problem in growth is due to Phelps (1962).
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a two to three year horizon. One large UK foundation, for example, offers funding to
charity projects over a two to three year horizon subject to an option to withdraw
funding at six months’ notice. On the other hand, the option to withdraw from the
project can also be exercised by the beneficiary after six months’ warning.
We cast the problem in the framework described by Ingersoll (1987, chapter 11). The
multi-period utility of the charity over its infinite life-time, Uˆ (C0 , C1 , ... ) , is
additively separable and concave, and is given by,
Uˆ (C0 , C1 , ... ) 

U (C
t
) ,
t=0
where Ct is consumption in the form of payments to worthy causes and costs.
The amount of money available to the charity for investment, I t , is given by,
It  Wt  Yt - Ct ,
(1)
where Wt is the wealth at time t, Yt is the income from donations or bequests. If I t is
invested in n assets, buying N i ,t shares in the ith asset at a price Pi ,t , then,
It 
n
N
i ,t
Pi ,t .
(2)
i=1
If one defines the return to the ith asset as the random variable
zi ,t  Pi ,t 1 / Pi ,t ,
6
then it is possible to write an expression for the stochastic wealth of the charity at
time t+1,
n
Wt  1  I t  wi ,t zi ,t .
(3)
i 1
where wi ,t  Ni ,t Pi ,t / I t , represents the relative weights of the assets, and which sum
to one.
And we define the utility of wealth functional,
J [Wt , t ]  max Et [

U (C, s) ] .
(4)
s=t
The only impact of time on the U function is via exponential discounting.3 We
assume a constant rate of time preference, or subjective discount rate,  , such that
U (C, t )   tU (C ) . The parameter  brings together utilities in different periods in
the same way as a discount rate brings together income in different periods. Its value
is assumed known to the charity and must be elicited by the modeller. Further
parameters necessary for analysis are those that determine the U (C ) function; these
parameters measure the charity’s risk aversion. As with any application of utility
theory to institutions, it is not exactly clear whose risk aversion is being measured; it
could be the trustees, it could be the beneficiaries, or it could be an amalgam of both.
III. Some Explicit Solutions
3
Time separable utility is convenient but perhaps not realistic where charities smooth stochastic wealth
to decrease the fluctuations in consumption. We leave this issue to future research.
7
We now consider how to progress our problem in the case of a charity. We assume
that Ct  mWt
t , that is a constant proportion of wealth, m, is “consumed” every
period, the technical definition of draw-down. Setting aside questions of portfolio
allocation, and assuming that no donation income is received, Yt  0 , then
Wt 1  (Wt  Ct ) Z t
where Z t is the growth in investments from t to t+1. If Ct  mWt ,
Wt 1  (1  m)Wt Z t .
(5)
This gives us a difference equation,
t 1
Wt  (1  m)t W0  Z i
i 0
 (1  m) W0Vt 1
t
where Vt 1 
t 1
Z .
i 0
i
Vt 1 is the accumulated value of one unit of wealth invested at
time 0 and held until time t; it is random and assumed non-negative. If Z i are also
 
non-negative and independent and identically distributed (iid), then Vt 1



has a
1
constant mean and  E Vt 1  t is constant for all t, if the mean exists. We shall

return to this point later.4
4
One especially simple solution to the problem applies when the charity chooses a draw-down subject
to the constraint that wealth
Wt  is fixed in real terms. If wealth is random, the constraint is
E0 (Wt )  v, some constant. Since Wt  (1- m)t W0Vt -1 , we see that the relevant condition is that
8
We now consider the current value of expected utility of consumption as an explicit
function of the draw-down rate:
 

L0  E0    tU (Ct )  , where 0    1,
 t 0




 E0    tU (mWt ) 
 t 0




 E0    tU (m(1  m)t W0Vt 1 )  .
 t 0

(6)
The first-order-condition for the optimal draw-down rate is
L0
 E0
m
   U '(.) (1  m)  mt (1  m)
t
t
t 1

 Vt 1  0,
or

(m(1  m))t E0 (U '(.)Vt 1 )
(7)
 m ( t (1  m)t 1 t ) E0 (U '(.)Vt 1 ).
Equation (7) can only be solved by specifying the functional form for the utility of
~
consumption, U (C ) and the distribution of Vt . We shall investigate in detail for
some well-known choices of U (C ) to see how to proceed.
1. Log utility: U (Ct )  ln( Ct ).
Then, equation (6) becomes
E0 (1- m)t Vt -1  v is constant. Suppose that Zi are iid for all i, E0 ( Z i )   Z and defining v  W0 ,
1
1
our required draw-down is  Z 
or m  1 
.
1 d
Z
9

L0    t E0 (ln(m(1  m)t W0Vt 1 )
t 1
 ln(mW0 )

 ln(mW0 )    t ln(m(1  m)t W0 )
t 1

  E0 (ln Vt 1 ).
t 1
 ln m  ln W0 

1 
ln W0 


t 1
t 1

1 
ln m
   t t ln(1  m)   E0 (ln Vt 1 )
L  L ( m) 

1
1

ln W0 
ln m  ln(1  m)

 E0 (ln Vt 1 )),
1 
1 
(1   ) 2 t 1

1
 1 
L '(m)  
0
 m 
2
(1   ) 1  m
 1  
1

1

m 1  1 m
(1   )(1  m)  m
(8)
or m    1 or m  1   .
Thus for the case of logarithmic utility, we get m  1   irrespective of our

assumption about

t 1
~
E0 (ln Vt 1 ) . This result is simple but not terribly useful as the
critical parameter in the model is based solely on the unknown rate of time
preference,  . The optimal draw-down is independent of asset allocation, although
the amount of wealth drawn down is not. However, as  increases m decreases; this
means that as we value future utility more and the extent to which we want to
consume now decreases.
We now digress to examine the properties of the investment returns.
10
Proposition 1. If Z i is positive iid and Z 1i  is a well defined random variable such
1
that. E(Zi
)   exists for 0     , it follows that E0 (Vt11 )   t for all integer t ≥
0.
Proof: Since Z i is iid, Z 1i  is iid and
1
t 1
E0 (V
1
 t 1
 
)  E   Z i  
 
 i 0
 t 1 1 
 E  Z i 
 i 0

t 1
  E ( Z i1 )
i 0
t 1
    t.
i 0
Furthermore, we restrict  
1

, meaning that the average expected growth rate of
utility via investment is less than the implied growth from inverting the subjective
discount rate. This is to stop the problem becoming unbounded and is a technical
issue embedded in such operations as interchanging infinite sums and expected
values.
2. Power utility: U (Ct ) 
1
Ct1 , 0    .
1
For all positive  , the power utility function represents the preferences of an
individual with constant relative risk-aversion, however the limits of the function as
C  0 and C   vary with  . In the region where 0    1 utility over
consumption is bounded below, approaching zero as consumption goes to zero, but
is unbounded above. For the log utility case (which is the limiting form of the power
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utility function as   1) utility is unbounded below and above, and whereas for
1     utility is unbounded below, but approaches zero as C   . Intuitively,
this functional form implies that very large increases in wealth are highly valued by
low-risk aversion agents, whereas the more risk-averse gain most by avoiding low
consumption states. These properties are key to the discussion of income and
substitution effects in section IV below.
For the power utility case, the objective function of the charity at time zero is:
 mW0 

1
L0
1

   t E0 [
t 1
(m(1  m)t W0Vt 1 )1
]
1
m1W01  mW0 


1
1
1


t 1
( (1  m)1 )t E0 (Vt11 ).
W01
~
We re-normalise L0 by
and replace E0 (Vt 11 ) by  t , so
1

L0 (1   ) / W01  m1 (1   ( (1  m)1 )t )
t 1
1
m
1
[1   (1  m)
/(1   (1  m)1 )].
Note that in the above we have used the condition that  (1  m)1  1 , which
follows from  
1

.
L0 (1   ) / W01  m1 /(1   (1  m)1 )
or ln( L0 (1   ) / W01 )  (1   ) ln m  ln(1   (1  m)1 ).
Maximizing,
12
 ln( L0 (1   ) / W01 ) 1   (1   ) (1  m)


 0.
m
m
1   (1  m)1
Solving
1   (1  m)1  m (1  m) 
1  ( (1  m)  m )(1  m) 
  (1  m)
(9)
1
or m  1  ( ) .

It is worth noting that if we let  tend to 1, then  also tends to 1, in which case we
recapture the logarithmic solution, but unlike the log utility case, when   1 both
the distribution of investment returns and relative risk aversion are crucial to fixing
the ideal draw-down.
Whilst in principle we could solve numerically for optimal asset allocation, we may
not wish to do so. This is for two reasons: first, the analysis would be complex and
dependent on distributional assumptions, or, if based on sample data, dependent on
making assumptions that the future would be similar to the past and secondly, asset
allocation is in practice determined by committee and consultation, and it seems
likely that such a calculation would antagonise.
However, for any given asset allocation, we can calculate the impact on the optimal
draw-down of changes in assumptions about the distribution of future returns, of
changes in risk attitudes and changes in portfolio weights. We proceed to this
problem in section IV below.
13
We could also generalise the problem to the case where “income” is included, by
which we mean donations. Donations Yt are received during the period time t – 1 to t
but invested at the end of the period. (Income received from donations during the
period cannot be invested in this discrete time framework until the market opens in
integer time.) This means that the wealth equation needs to be adjusted.
Wt 1  Wt (1  m) Z t  Yt .
Then
Wt  W0 (1  m)t Vt 1
(10)
t 1
 Y j (Vt 1 / V j )(1  m)t 1 j
j 0
where V0 is assumed to equal 1.
It is apparent that no closed-term solution exists for additive income for general
distributions. However, using the fact that donation income must be positive, we can
use instead a multiplicative addition to wealth, thus we get, defining yt 1 
Wt  W0 (1  m)t Vt 1 yt 1.
t 1
Y ,
i 0
i
(11)
In this case we can derive a solution exactly as before. The new interpretation of  is
now
14



E Z 1 t yt1  E ( Z 1 t ) E yt1

 cov Z 1 t yt1


(12)
 Z1 Y 1   Y 1 Z1 .
The necessary assumptions are the same as before. We might expect the covariance
term to be positive (as in the case studied by Merton 1992) but there may be reasons
why it could be negative. An increase in financial market returns may be co-existent
with a fall in donations as the population shifts from altruism to greed.
Empirical Illustration
To illustrate the explicit solutions, we create a representative portfolio for a
charitable trust, simulate returns and compute optimal draw-down rates for some
feasible parameter ranges. Here we design the portfolio of our artificial entity to
approximate the asset allocation of an independent U.K. biomedical researchfunding charity, the Wellcome Trust.
The Wellcome Trust describes its principal investment objective as ‘total return in
inflation-adjusted terms over the long term in order to provide for real increases in
annual expenditure while preserving at least the Trust’s capital base in real terms’
(Wellcome 2005). Our returns data are constructed using an asset allocation similar
to the Trust’s (Figure 1). Consistent with the aim of maximising total returns, more
than 90 per cent of capital is held in public and private equity, hedge funds and
property, with a small proportion in gilts and cash.
Figure 1. Asset allocation of simulated charity
This figure shows the proportions of total funds invested in each asset class for simulated
portfolio returns. A consistent series of returns to hedge funds are not available prior to
January 1994, so from January 1990-December 1993, the allocation to each of U.K., Global,
Emerging and Private Equity was increased by 0.9% and Hedge Funds set to zero. Appendix
A lists data sources for each returns series.
15
5.0%
11.5%
7.5%
U.K. Equity
32.2%
3.6%
2.8%
Global Equity
Emerging Equity
Private Equity
Property
5.4%
Hedge Funds
U.K. Gilts
Cash
32.0%
We calculate monthly real portfolio returns over the period January 1990 to June
2006, (198 observations) deriving individual asset class returns from standard
indexes, and deflating using consumer prices and earnings data.5 We expect that
wages are the majority of costs for most beneficiary projects, and deflation using
consumer prices alone will overstate the real spending power of the charity, so we
treat inflation is 50 per cent consumer-price-driven and 50 per cent due to wage
increases.6
The mean (log) real return to this portfolio is 4.75 per cent annualised with a
volatility of 13 per cent. Summary statistics in Table 1 show that the data are
significantly non-normal: negatively skewed and leptokurtic. However the
autocorrelation structure of the de-meaned returns and squared de-meaned returns
supports an assumption that real portfolio returns are iid. (Ljung-Box Q statistics, not
5
See Appendix A for data sources and calculations.
We use an arbitrary rule here, but most research into endowment spending (see, for example, Tobin
1974 and Woglom 2003) recognises the importance of an institution-specific deflation procedure, while
Merton (1990) goes further and suggests hedging strategies for future cost changes.
6
16
reported here, are insignificant to at least 50 lags for the de-meaned returns. The
squared residuals have one significant autocorrelation at lag ten.)
Table 1: Summary Statistics, real annualised portfolio returns
Note: This table shows summary statistics for monthly log returns to the portfolio constructed
using weights and asset classes from Figure 1. Portfolio returns are the weighted sum of the
log changes in indexes for each asset class, less the log change in inflation. See Appendix A
for a complete description of data sources. Data are monthly from January 1990 to June 2006:
198 observations.
Mean
4.75%
Standard Deviation
13.02%
Skewness
-0.69
Kurtosis
3.88
Jarque-Bera
21.87
(p-value)
(0.000)
Equation (9) above is the optimal rate of draw-down for an infinitely-lived charity
under a fixed asset allocation, rate of time preference,  , and relative risk aversion,
 . The key determinant is the mean, E ( Z i1 )   . To estimate  , we bootstrap the
monthly portfolio returns using 120,000 random draws, and sum to get 10,000 annual
real (gross) returns. We then use these to compute the sample mean
ˆ 
1 10,000 1
 Zi ,
10, 000 i 1
(13)
for   0.97 and 0    15 , subject to the condition that ̂ 
1

.
The estimated optimal draw-down rate m̂ is shown in Figure 2 for values of relative
risk aversion from 0.6 to 5.0. The fine grey curves give an approximate 95 per cent
17
error range for the estimate of m: using the condition 0  ˆ  1, we fit a beta
1
distribution to 1000 bootstrapped estimates of v̂  ˆ  by maximum likelihood.
From the estimated beta parameters, we can back out 1  vˆ ,0.025  1  F 1  0.025 and
1  vˆ ,0.975  1  F 1  0.975 as a guide to the accuracy of m̂ . Consistent with the
solution for log utility in equation (6), the optimal draw-down rate is 3 per cent per
ˆ  1   . ) As relative risk aversion rises above
annum when   1 . (Since ˆ  1, m
two, the optimal draw-down falls, reaching about one per cent when   5 . The
error range widens rapidly as risk aversion diverges from one.
18
Figure 2: Optimal draw-down rate and error range
Note: This Figure graphs the estimated optimal rate of draw-down for a charity with assets
1
invested as for Figure 1, where the optimal draw-down is given by
ˆ 
mˆ  1  (ˆ )  ,
1 10,000 1
 Zi where the gross portfolio returns Zi are computed using 10,000
10, 000 i 1
random draws from the historical portfolio returns series, the rate of time preference is
  0.97 and relative risk aversion 0.6    5 . The fine grey lines represent beta
distribution approximations to a 95% error range for m̂ .
14%
draw down per annum
12%
10%
8%
6%
4%
2%
0%
0.6
1
1.4
1.8
2.2
2.6
3
3.4
3.8
4.2
4.6
relative risk aversion
These estimates illustrate the consequence of both risk aversion and returns volatility
for charity spending policies.
IV. Income and substitution effects
Now consider a change in  , where we keep  fixed but vary distributional
parameters:
1
m
1 1 1
      0.


19
5
As before we assume payoff, Z i , is a discrete iid positive process but that
Zi  exp(    xi ) for some random variable xi , with moment generating function
given by hx  t  .7 This means that we do not assume log-normality, although it is a
particular case, where hx  t   exp
 2t 2
2
. Then
  E ( Z i1 )  exp((1   )  )hx ( (1   )) and

 (1   ) .

It now follows that
1
m
1 1
     (1   ).


Equation
and
(8)
tells
us
(14)
that
m
0

if
  1 and
m
 0 if 0    1 ,

m
 0 if   1 , which is exactly the generalised income/substitution effect

discussed by Merton (1969).8 Individuals with low relative risk aversion will choose
to consume less now and save more of the increased expected geometric return   
(substitution effect dominates income effect). High risk averters (   1) will increase
m now as  increases (income effect dominates substitution effect).
Likewise,
1 1
1
m
(1   ) '

hx ( (1   )) exp((1   )  )    ,


7
(15)
This analysis follows Merton (1969) except that Merton assumes continuous time log-normal wealth.
See also Phelps (1962) for early work on income and substitution effects in a discrete-time model.
Phelps assumes uniformly distributed returns to risky capital.
8
20
so that if
hx '  is positive (as it is for x normal) we have the same
income/substitution effects as for  . However, if hx '  is negative, the effects
reverse.
The above income and substitution effects are too restrictive in that we assume that
returns are iid logarithmic location-scale. We now generalise.
Proposition 2. If Z i first order stochastic dominates (FSD) Zi then m is decreased if
0    1 and increased if   1.
Proof. Note that if Z i FSD Zi then F ( Z )  F (Z ) where F ( Z ) and F ( Z ) are the
respective distribution functions. Denote expectations with respect to them by E ( )
and E ( ) . First order stochastic dominance implies that E (U ( Z ))  E (U ( Z )) for
U () , any increasing function.9 If U  Z   Z 1 , 0    1, then U  Z  is increasing
in Z, hence under F ( Z ) ,  is increased and m is decreased. If   1 we have U () a
decreasing function in Z and under F ( Z ) the reverse happens,  decreases and m
increases.
Accordingly, we consider more transformations of distributions that can be
described by FSD. For an arbitrary positive continuous density, pdf(x); we consider
two points xl and xu and the probabilities
xl

0
xu
Pl   pdf ( x)dx, Pu 
9
 pdf ( x)dx
See Appendix B for proof of these results.
21
and
Pmd 
xu
 pdf ( x)dx; clearly P  P  P
u
l
md
 1.
xl
We can construct a new density by the following shift,
Pu'  Pu  
Pl '  Pl  
(16)
Pmd'  Pmd
where 0    min( Pu , Pl ), and
pdf '( x) 
Pu' pdf ( x)
Pu
 pdf ( x)

Pl ' pdf ( x)
Pl
xu  x  
xl  x  xu
(17)
0  x  xl
It is easy to check that pdf '( x ) is still a well-defined density although no longer
continuous at x  xl or x  xu . Furthermore the above transformation can be called
optimistic in that it transfers probability from the lower tail to the upper tail of the
density. A pessimistic transformation can be defined similarly. Note that since we
assumed a continuous density with zero probability mass at any point, the
discontinuities induced by our transformation will not affect the existence of the
integrals.
Corollary. If h( x) is a positive increasing function then
22


0
0
 h( x) pdf '( x)dx  () h( x) pdf ( x)dx
for pdf '( x ) the result of an optimistic (pessimistic) transformation. An opposite
result applies to positive decreasing functions. If we now apply the result for
  E ( Zi1 ) , we see that Z i1 is positive increasing for 0    1 hence    , and
positive decreasing for   1 so that    , where  is a positive transformation.
Proof: This follows immediately from Proposition 2.
If we now consider m  1  ( )  ; we shall investigate the change in m due to an
1
optimistic (pessimistic) transformation in  , the changed value of  we call   .
Thus,
1
1


m  1  ( )  (1  ( ) )
1
1
(18)
1
   (     ).
It is clear that m  0, if
1
1
    , and we get the following result:
Proposition 3. If our transformation of Z i is optimistic (pessimistic) then our draw1
1
down (m) decreases/increases if 0    1 , since FSD in this case implies     
1
1
(substitution effect dominates income effect). If   1 , FSD implies      (income
effect dominates substitution effect) and we get the opposite effect where m
increases/decreases in response to an optimistic/pessimistic transformation of the
density.
23
Empirical Illustration
Figures 3a and 3b show the impact on optimal draw-down of a range of
transformations of Z i , the portfolio return. For highly risk tolerant charities, when
0    1 , pessimistic transformations of the portfolio returns distribution markedly
raise the optimal draw-down, the charity shifts consumption forward to avoid future
disappointment despite the prospect that income will be much less. Extremely
pessimistic values for expected returns such as when the whole upper tail mass is
removed in favour of lower returns (   0.2 ), combined with high risk tolerance
(   0.4 ) imply draw-down rates over 30 per cent. We see that a very risk-tolerant
foundation facing bleak prospects does best by massively increasing spending rates.
Even for risk aversion approaching one, (   0.8 ), a pessimistic shift of   0.1
warrants an increase in draw-down from 2.8 per cent of wealth to 5.8 per cent of
wealth. The same foundation making an optimistic revision of the same scale,
  0.1, will reduce spending from 2.8 per cent to less than 0.1 per cent. Then again,
positive transformations in this region combined with low values for risk aversion
frequently produce values of  that fail the convergence condition,  
1

, at the
chosen level of time preference (i.e.,   0.97 ), indicating an implausible willingness
to accumulate wealth when investment prospects are improving.
Hence we can conclude that an escalation of spending in bad times by charities or
foundations, even where this means a large deduction from principal, is not
necessarily irrational or irresponsible. On the contrary, such episodes could be
interpreted as the right response to worsening circumstances given the preferences
24
and expectations of the trustees: evidence for a mix of pessimism and high risk
tolerance.
Figure 3a: Optimal draw-down under transformations of the portfolio returns
distribution, 0    1 .
Note: This Figure shows optimal rates of draw-down, m, when the distribution of portfolio
returns is re-weighted by the factor  , and relative risk aversion is less than one, 0    1 .
We sort the original 198 real returns into percentiles and divide the dataset into three
sections: d1 = 40 lowest observations representing the 1-20th percentiles, d2 = 118
observations representing the middle 20-80th percentiles, and d3=40 observations
representing the 80-100th percentiles. We rescale the probability of low (high) returns by
increasing
the
probability
of
an
extreme
event
so
that
pi 
di
, pu '  pu  , pl '  pl   and the probability of draws from the middle stays
d j
j
constant. We draw returns from the low, middle and high range randomly with replacement
1
and in proportion to the assigned probability, and compute Z i
,
  , and m, for a range of
draw-down per cent per annum
risk aversions.
40%
35%
30%
alpha=0.4
25%
alpha=0.6
20%
alpha=0.8
15%
alpha=1
10%
5%
0%
-0.20
-0.15
-0.10
-0.05
0.00
0.05
rescaling factor
On the other hand, conventional belt-tightening is the counter response of more riskaverse trustees to poor prospects. Figure 3b gives us an idea of how optimal
spending rates react to shifts in the distribution of portfolio returns for relative risk
aversion (  ) from one to five. (This time, the convergence condition becomes
binding for pessimistic changes which shift probability mass from the upper to the
25
lower tail.) If we consider the case of   2 , the optimal rate of consumption before
re-weighting the probability distribution is close to 3 per cent, but a moderate shift
downwards (   0.05 ) will shrink spending to around 0.2 per cent. An expectation
of better times raises the rate of consumption steeply so that a similar shift towards
optimism (   0.05 ) increases draw-down from 3 per cent to 5.9 per cent. An
extremely positive outlook (   0.2 ) means spending rates as high as 13 per cent of
wealth for   2 . As risk aversion increases towards 5 and the income effect
increasingly dominates the substitution effect, the prospects of much higher income
pushes draw-down towards 18 per cent: the same-sized positive change in the
returns distribution elicits a higher rate of spending from the more risk averse.
While for small improvements (   0.05 ) the spending rates of the less risk averse
are still higher than for the more risk averse, we conclude that optimal policy when
the future looks much more promising is for the more cautious to spend at a faster
rate than the less cautious. The analysis does not support a ‘commonsense’ view that
there is a universally positive correlation between saving rates and caution. If ‘rainy
days’ become more likely, it is true that the risk averse will cut back more rapidly,
but when the outlook is for a very long summer, the rationally cautious foundation
outspends the more risk-tolerant.
26
Figure 3b: Optimal draw-down under transformations of the portfolio returns
distribution, 1    5 .
Note: This Figure shows optimal rates of draw-down, m, when the distribution of portfolio
returns is re-weighted by the factor  , and relative risk aversion is greater than one,
1    5 . We sort the original 198 real returns into percentiles and divide the dataset into
three sections: d1 = 40 lowest observations representing the 1-20th percentiles, d2 = 118
observations representing the middle 20-80th percentiles, and d3=40 observations
representing the 80-100th percentiles. We rescale the probability of low (high) returns by
increasing
the
probability
of
an
extreme
event
so
that
pi 
di
, pu '  pu  , pl '  pl   and the probability of draws from the middle stays
d j
j
constant. We draw returns from the low, middle and high range randomly with replacement
1
and in proportion to the assigned probability, and compute Z i
,
  , and m, for a range of
draw-down per cent per annum
risk aversions.
20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
-0.050
alpha =1
alpha =2
alpha =3
alpha =4
alpha =5
0.000
0.050
0.100
0.150
0.200
rescaling factor
We now consider changes in risk.
Proposition 4. If Z i second order stochastic dominates (SSD) Zi then m is decreased
if 0    1 and increased if   1.
Proof. Note that if Z i SSD Zi then

Z
0
Z
F (s)ds   F (s)ds for all Z [0, ] where
0
F ( Z ) and F ( Z ) are the respective distribution functions. Denote expectations with
respect to them by E ( ) and E ( ) . Second order stochastic dominance implies that
27
E (U (Z ))  E (U (Z ))
for
U () ,
any
increasing,
concave
function.10
If
U  Z   Z 1 , 0    1, then U  Z  is increasing and concave in Z, hence under
F ( Z ) ,  is increased and m is decreased. If   1 we have U () a decreasing and
convex function in Z and under F ( Z ) the reverse happens,  decreases and m
increases.
Here we consider a mean-preserving spread of the distribution as a special case of
SSD.
For
an
arbitrary
positive
continuous
density,
pdf(x)
where
xi  x   i ,  i ~ iid (0,  2 ) , we can construct a mean-preserving spread by the
following transformation of xi ,
x 'i  x  (1   ) i , 0<  
(?)
We can see that the mean of both distributions is
E ( xi )  E ( xi ')  x ,
(?)
and that for 0<   , the variance of the transformed variable x 'i is greater than
the variance of xi ,
var( xi ')  (1   ) 2  2  var( xi )   2 ,
(?)
which are sufficient conditions for second order stochastic dominance of pdf ( x )
over pdf '( x ) .
10
Proof for these properties is in appendix B.
28
We can also shrink the variance of pdf ( x ) by choosing an optimistic transformation
such that -1<  0 . A pessimistic transformation can be defined as an increase in
risk when 0<   .
Corollary. If h( x) is a positive increasing, concave function then


0
0
 h( x) pdf '( x)dx  () h( x) pdf ( x)dx
for pdf '( x ) the result of an optimistic (pessimistic) transformation. An opposite
result applies to positive decreasing, convex functions. If we again apply the result
for   E ( Z i1 ) , we see that Z i1 is positive increasing and concave for 0    1
hence    , and positive decreasing and convex for   1 so that    , where
 is an optimistic transformation.
If our transformation of Z i shrinks the variance, then our draw-down (m) decreases
1
1
if 0    1 , since SSD in this case implies     (substitution effect dominates
1
1



income effect). If   1 , SSD implies   
(income effect dominates substitution
effect) and we get the opposite effect where m increases as risk shrinks and decreases
as risk rises (for a constant expected return).
Empirical Illustration
Figures 4a and 4b illustrate these results for mean-preserving spread transformations
of the portfolio returns distribution that imply linear transformations of the standard
deviation from zero to 200 per cent. The standard deviation of the bootstrapped
portfolio returns is 12.6% p.a., so re-scaling values creates a volatility range from 0 to
29
25.2% as 1    1 . The mean of the distribution stays at a constant real return of
4.75% p.a. Each increase in  implies a distribution function that is dominated by
the previous one, and the changes in draw down rates confirm again that risk
aversion of one divides preferences.
Increasing risk leads charities for which 0    1 to increase current draw-down,
but not very rapidly. For   0.8 , a doubling of volatility leads in a growth in
optimal consumption rates of less than 0.5 per cent, while decreases in risk do not
encourage much decrease in drawdown.
Figure 4a: Optimal draw-down under mean-preserving spread transformations of
the portfolio returns distribution, 0    1 .
Note: This Figure shows optimal rates of draw-down, d, when the variance of the distribution
of portfolio returns is re-weighted by the factor  , and relative risk aversion is less than one,
0    1 . We draw 120,000 of the original 198 real returns with replacement to compute
10,000 annual gross portfolio returns
Z i . We then compute the mean-zero errors
 i   Zi  Z  , resample these without replacement and compute Zi1 '   Z  (1   ) i 
1
1    1 ,   , and m, for a range of risk aversions.
3.5%
draw-down per annum
for
3.0%
2.5%
alpha = 0.4
2.0%
alpha = 0.6
1.5%
alpha = 0.8
alpha = 1
1.0%
0.5%
0.0%
-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
re-scaling factor
30
For risk aversion in the positive range, increases in volatility discourage spending. If
a foundation or charity has risk aversion of 2, a 50% increase in volatility almost
halves the optimal consumption rate, and a doubling of volatility discourages
spending almost entirely. These results confirm the importance of risk as well as
return forecasts to planning in organisations with very long horizons: even if
expected returns are constant, increasing uncertainty warrants a more conservative
draw-down plan.
Figure 4b: Optimal draw-down under mean-preserving spread transformations of
the portfolio returns distribution, 1    5 .
Note: This Figure shows optimal rates of draw-down, d, when the variance of the distribution
of portfolio returns is re-weighted by the factor  , and relative risk aversion is less than one,
1    5 . We draw 120,000 of the original 198 real returns with replacement to compute
10,000 annual gross portfolio returns
 i   Zi  Z  ,
resample
Zi1 '   Z  (1   ) i 
1
these
for
Z i . We then compute the mean-zero errors
without
replacement
and
and
compute
1    1 ,   , and m, for a range of risk aversions.
draw-down per annum
4.5%
4.0%
3.5%
alpha = 1
3.0%
alpha = 2
2.5%
alpha = 3
2.0%
alpha = 4
1.5%
alpha = 5
1.0%
0.5%
0.0%
-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
re-scaling factor
Conclusion
Charities whose trust deeds specify a very long (infinite) horizon and who generate
independent and identically distributed returns from investment portfolios can
31
operate optimally using simple, constant draw-down policies. While the specific
topic of our study, charitable trusts, has not been widely researched, the general
tenor of these results has been known to economists for some time. Using returns to a
representative portfolio based on a leading U.K. charitable trust, we estimate that
such an optimal draw-down will range between one and three per cent each year
assuming power utility preferences, risk aversion between 0.6 and 5 and a constant
rate of time preference of 0.97.11 Then again, we demonstrate that the degree of
statistical uncertainty associated with these optimal draw-down rates is large.
Our contribution is to investigate the responsiveness of these draw-down policies to
changes in the shape of very general returns distributions and to tease out the
empirical implications of such changes. We identify the effects of optimistic and
pessimistic transformations of the returns distribution using the properties of
stochastic dominance. Without assuming a specific functional form for the
probability density, we derive the effects on optimal draw-down due to a transfer of
probability mass from the lower to the upper tail, and vice versa, and the effects of
mean-preserving spread. This allows us to incorporate, into both analysis and
estimation, the important idiosyncratic features of actual returns distributions.
In addition, we do not solve for the charity’s optimal portfolio allocation, treating the
asset weights as fixed, and returns as independent and identically distributed. While
charitable trusts obviously do make investment decisions, it is not clear that
investment choices are joint with choice over spending rates. By treating charities
with different preferences as if they actually hold the same portfolio, we clarify the
11
Some empirical estimates of the risk aversion parameter suggest that it is greater than one and almost
certainly greater than 2. The literature on the risk-premium puzzle indicates that risk aversion may
actually by much higher (and/or time varying). See Friend and Blume (1975) for early results and
Cochrane (2001) for a more recent discussion. However other studies of household spending find much
lower levels of risk aversion, close to or less than one, see Laibson et al. (1998).
32
trade-off between income and substitution effects and demonstrate that an
equivalent change in expectations can produce very different, but nevertheless
equally optimal, reactions from trustees.
Commonsense suggests that a charity facing a bleak investment outlook should
‘tighten its belt’ and reduce spending. And we confirm this view for cautious types
with risk aversion parameters greater than one. However a less risk-averse charity
will optimally increase their draw-down in response to the same news, even dipping
into capital in an effort to make the most of the present before the bad times arrive.
Similarly, when prospects improve, the more cautious charities with risk aversion
greater than one will react by increasing consumption. For very large expected
improvements, however, risk aversion has a surprising impact: a charity with risk
aversion of four will outspend a charity with risk aversion of two. In other words,
similar organisations can react to plausible changes in the returns density by shifting
spending rates in differently, or even in opposite directions, and still be making
rational, indeed optimal, decisions.
It is apparent that the solution of this problem is actually much broader than it
appears. For example, consider a pension fund, normally is modelled via an asset
liability model, where liabilities are represented by pension payments and costs and
assets are the funds available for investment. Moreover, liabilities are composed of
guarantees, which must be met by the pension funds, and surpluses and bonuses,
which are paid if the investments are deemed to have exceeded the appropriate
target. For a wide range of pension funds an infinite horizon seems a reasonable
assumption and such a treatment would suggest that liabilities would correspond to
“consumption” in that the expected utility of the fund increases with higher
33
liabilities. This interpretation raises the possibility of demographic effects changing
the utility function. So the exit from the fund of a cohort of people who are, on
average, less risk averse, will raise the average level of risk-aversion among the
remainder of the group, with the relevant implications for future policy.
It is also clear that in practise many endowments and pension funds smooth wealth
and compute draw-downs as a fraction of average wealth taken over k years, where k
is typically three; the reason they do this is to reduce the volatility of wealth so that
the money value of the draw-down is less volatile. Our calculations above have
solved the k=1 problem but an explicit solution for arbitrary k seems very
challenging. We shall pursue this in future work.
34
Appendix A
Portfolio returns data are monthly from January 1990 to June 2006. A consistent series of returns to hedge funds are not available prior to
January 1994, so from January 1990-December 1993, the allocation to each of U.K., Global, Emerging and Private Equity was increased by 0.9%
and Hedge Funds set to zero. Total portfolio return is the weighted sum of log changes in each returns index and the cash rate (expressed on a
monthly basis), less the log change in the inflation rate: p  0.5ln(
Asset
Class
U.K.
Equity
CPI t
Earningst
)  0.5ln(
).
CPI t 1
Earningst 1
Data
Source
Portfolio weight
FTSE All Share
returns index
DataStream
FTALLSH(RI)
32.2%
Global
Equity
MSCI ex UK
32.0%
Overseas
Equity
MSCI
Emerging
Markets
DataStream
MSWFUK$(RI)~U$,
translated to BPN using
BBGBPSP(ER)
DataStream
MSEMKF$(RI) ~U$,
translated to BPN using
BBGBPSP(ER)
U.K. Gilts
FTA
DataStream
FTBGTTF(RI)~£
2.8%
FTA British Government fixed 10-15 years total returns
index
Property
IPD: Total
return index
DataStream
UKIPDRI.F
7.5%
UK Investment Property Databank
Index measures total returns to investment in
commercial property investment.
5.0%
35
Notes
Capitalisation-weighted index, for London Stock
Exchange representing at least 98% of eligible U.K.
companies.
The MSCI World Index is a free float-adjusted market
capitalization index consisting of the 22 developed
market country indices where the UK market is
excluded.
The MSCI Emerging Markets Index is a free floatadjusted market capitalization index consisting of 25
emerging market country indices.
Hedge
Funds
CSFB/Tremont
hedge fund
index, net asset
value
DataStream; CSFB/Tremont:
www.hedgeindex.com
CSTHEDG~£
3.6%
Private
Equity
UK-DS
Investment
Trusts Private
Equity total
returns index.
DataStream
ITVCAPT(RI)~£
11.5%
Cash
3-month CD
rate
Average of CPI
and Earnings
Bank of England
5.4%
Inflation
DataStream
CPI: UKCPHARMF
Wages: UKWAGES.E
36
Credit Suisse/Tremont Hedge Fund Index is an assetweighted hedge fund index and includes only funds, as
opposed to separate accounts. The Index uses the
Credit Suisse/Tremont database, which track over 4500
funds, and consists only of funds with a minimum of
US$50 million under management, a 12-month track
record, and audited financial statements. It is calculated
and rebalanced on a monthly basis, and shown net of
all performance fees and expenses.
The index is constructed by DataStream to measure the
performance of all UK listed investment trusts in the
private equity sector.
End month sterling certificate of deposit 3 month rate,
mean of bid-offer.
Equally weighted log change in UK CPI- Harmonised
European Union basis, 2005=100 and UK average
earnings index, whole economy, seasonally adjusted.
Appendix B
1. Let F ( Z ) and F (Z ) on [0, ] be two continuous distributions functions such that
F  (0)  F (0)  0, and F ( Z ) first order stochastically dominates F ( Z ) so that
F (Z )  F (Z ) for all Z.
Proposition:
For any increasing function U(Z), E (U ( Z ))  E (U ( Z )) .
Proof:
Integration by parts gives:
E (U ( Z )) =  U ( Z )dF ( Z )  [U ( Z ) F ( Z )]0   U '( Z )F ( Z ) dZ
(B1)
E (U ( Z )) =  U ( Z )dF ( Z )  [U ( Z ) F ( Z )]0   U '( Z )F ( Z )dZ .
(B2)
Therefore, since the terms in square brackets in each expression are equal,
E (U ( Z ))  E (U ( Z ))   U '( Z )[ F ( Z )  F ( Z )]dZ ,
(B3)
and since F (Z )  F ( Z ) for all Z, and U '( Z ) >0, E (U ( Z ))  E (U ( Z )) .
2. Let F ( Z ) and F (Z ) on [0, ] be two continuous distributions functions such that
F  (0)  F (0)  0, and F ( Z ) second order stochastically dominates F ( Z ) so that

Z
0
Z
F (s)ds   F (s)ds for all Z [0, ] .
0
37
Proposition
For any increasing, concave function U(Z), E (U ( Z ))  E (U ( Z )) .
Proof
From above we have
E (U ( Z ))  E (U ( Z ))   U '( Z ) F ( Z ) dZ   U '( Z ) F ( Z )dZ .
(B4)
Integrating by parts,
 U '(Z ) F (Z )dZ  U '(Z ) F (Z )dZ   F (Z )dZU ''(Z )dZ ,
(B5)
And
 U '(Z ) F (Z )dZ   U '(Z ) F (Z )dZ





= U '( Z )[ F ( Z )dZ  F ( Z )dZ ]  U ''( Z )[ F ( Z ) dZ  F ( Z ) dZ ]dZ .
(B6)
Since U '( Z ) >0 and U ''( Z ) <0, and from the properties of SSD, E (U ( Z ))  E (U ( Z )) .
If U(Z) is decreasing and convex, the reverse holds.
38
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40