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Fall 1999 1
Time Diversification: Fact or Fallacy
David A. Stangeland and Harry J. Turtle
We offer a simple analysis to demonstrate that arguments suggesting time diversification
benefits either always obtain, or never obtain are equally incorrect. In a unifying example we
show that time diversification is affected by utility parameters and by the process governing
risky asset returns. We summarize how other important externalities may affect the choice
between proportions invested in riskless and risky assets. We also introduce the idea that
changes in investment knowledge over an investor’s life may affect the investor’s portfolio
choice.
INTRODUCTION
The popular strategy of time diversification is
recommended by many investment advisors and some
academics (see, for example, Peavy and Vaughn-
Rauscher [1994], or Thorley [1995]). The crux of the
strategy is that a longer time horizon for an investor
implies a greater portfolio proportion allocated to risky
assets such as equities. The reason cited is time
diversification -- that fluctuations in security returns
tend to cancel out through time, thus more risk is
diversified away over longer holding periods. It
follows then that apparently risky securities like
equities are potentially less risky than previously
thought if held for long time periods, yet their average
returns are superior to low-risk securities like T-bills.
Other academics argue that time diversification is
a fallacy. Merton and Samuelson [1974], Samuelson
[1963, 1989, 1990, 1994], Kritzman [1994], and others
conclude that expected utility does not increase with a
time diversification strategy. Harlow [1991], Leibowitz
and Krasker [1988], and Bodie [1995] use option
based equity-insurance arguments to show that longer
holding periods imply increased, not decreased, risk for
equities and thus time diversification does not occur.
______________________________________
University of Manitoba, Winnipeg, Manitoba, Canada R3T
5V4 and Washington State University, Pullman, WA 99164,
respectively. We thank Gilbert Bickum, Paul Brockman, Rick
Sias, Steven Thorley, Max Zavanelli and an anonymous
referee for helpful comments.
The only clear conclusion that can be drawn at this
point is that the time diversification controversy
continues. We reduce this controversy by showing
that the optimality of a time diversification strategy
depends critically on a number of important and highly
context-dependent factors. We present a simple
unifying example to show the effects of time
diversification in practical situations and we stress six
reasons why it is inappropriate to either always reject,
or always recommend, the strategy of time
diversification. The large cross-sectional variability in
investors suggests that a large number of investors will
be well-advised to increase their risky asset holdings
with horizon length, and at the same time, a large
number of investors will be well-advised to decrease
their risky asset holdings with the length of their
investment horizon. Because of the extensive
variability across investors, there is little motivation to
debate the general merit of time diversification for a
typical investor (unless we have a very clear
understanding of a typical investor). The general
argument in support of time diversification suggests
that a longer investment horizon by itself will induce
investors to devote relatively more of their portfolio to
equity instruments. From a practical viewpoint,
investment goals and the resulting investment strategy
depend upon an investor’s objectives for long term
capital appreciation, income needs, and other important
constraints that must be met. Important constraints
might include future cash flow needs (such as
retirement needs, or needs related to children’s
education) and nontraditional investment criterion
2 Journal of Financial Education
(such as restrictions to include only green investments
in a portfolio). It has long been recognized that these
factors may vary over an investor’s life, or as an
investor’s preferences, knowledge, consumption and
income change. We describe factors that may be
correlated with the investment horizon and show how
such factors may induce strategies that parallel or
contradict the time diversification strategy.
In the next section of the paper, we consider the
use of expected utility maximization in a simple, yet
generalizable, example. We then present a list of
external factors and show how they may also
influence portfolio decisions given different investment
horizons. Finally, we offer concluding comments and
practical investment implications.
APPLYING EXPECTED UTILITY
TO INVESTMENT DECISIONS
It is widely accepted in the economic, finance, and
insurance literature that individuals seek to maximize
expected utility, not expected wealth. In the appendix,
we present Bernoulli’s St. Petersburg Paradox to
illustrate this fact. In this section, we show how
expected utility maximization can be used in
investment decisions. We find that time diversification
benefits are dependent on the investor’s preferences
regarding risk and the asset return process.
1
Consider
a simple investment context that allows a careful
examination of expected utility maximization and time
diversification.
2
Assume the existence of a single
riskless asset and a single risky asset. Further, assume
that investors face no additional constraints, that
income and consumption flows are negligible, and
therefore investors are concerned solely with the utility
of future wealth (say at retirement).
3
We now demonstrate that the issue of time
diversification cannot be completely resolved by
resorting to an expected utility framework. As a
simple example consider an extended power utility
function that allows us to consider a variety of
differences in tastes across individuals.
4
In this case,
utility of wealth, z, is represented by,
(1)U z
b
a bz
b
( ) ( )
/
=
−
+
−
1
1
1 1
where b > 0 and
z Max
a
b
> −
,
.
0
Two useful economic measures of risk aversion are
absolute and relative risk aversion. The general
definitions of these measures and their particular
values in this case are,
(2)
R z
U z
U z a bz
A
( )
''( )
'( )
=− =
+
1
and
(3)R z
U z z
U z
z
a bz
A
( )
''( )
'( )
= − =
+
respectively. For the utility specification chosen here,
it is straightforward to observe from equation (2) that
absolute risk aversion decreases as wealth increases.
It is generally accepted that investors possess
decreasing absolute risk aversion because the
importance of an extra dollar of wealth is smaller for
higher levels of wealth -- so for higher levels of
wealth, an investor is more willing to put an extra
dollar into a risky investment. Decreasing absolute
risk aversion therefore implies that as wealth
increases, more dollars of wealth will be allocated to
risky assets.
Relative risk aversion has implications for the
proportion of wealth an investor will commit to risky
assets. Decreasing relative risk aversion implies that
an investor will devote a larger proportion of wealth
to the risky asset as wealth increases. Constant
relative risk aversion implies that the percentage of
wealth allocated to risky assets remains constant
regardless of the overall wealth level. Increasing
relative risk aversion implies that an investor would put
a lower percentage of wealth at risk as the overall
wealth level increases. For convenience, economists
often assume constant relative risk aversion for
investors. We examine all three possible relative risk
aversion contexts to demonstrate that time
diversification strategies will be positively influenced,
negatively influenced, or unaffected by decreasing,
increasing, or constant relative risk aversion,
respectively.
In our specification, changes in relative risk
aversion depend on the value of the parameter a. If a
is positive then relative risk aversion increases with
wealth. If a is zero, then relative risk aversion is
Fall 1999 3
Figure 1. A Riskless Portfolio Over Time
The future value of ending wealth and the utility of ending wealth are reported at the end of each successive 5-year period.
__________________________________________________________________________________
constant regardless of wealth. Finally, if a is negative,
then relative risk aversion decreases with wealth.
5
We consider two processes for the risky asset’s
returns. First, we consider the case of a risky asset
that displays no time dependencies in its returns. That
is, previous deviations from expected return are of no
value in predicting future deviations from expected
returns. Next, we consider a general risky-asset
process that may display either positive or negative
autocorrelation. The prevalence of momentum and
mean-reversion strategies makes this analysis
important.
For presentation, we report wealth and utility values
for four successive five-year periods.
6
We assume
the risky asset’s expected return is constant and equal
over each five-year period. We then determine the
riskless rate that equates the utility of the riskless asset
and the risky asset over the first five-year period and
examine whether expected utility of ending wealth
(after twenty years) is larger for the risky asset or the
riskless asset.
Case 1. A Memoryless Risky-Asset Process
The choice between the riskless and risky asset as
the investment horizon changes is demonstrated with
the example presented in Figures 1 and 2.
7
First,
consider an amount, $8,000, invested in the riskless
asset. If we assume that the five-year riskless rate is
35.45 percent (or approximately 6.25 percent per year
compounded for five years), we can determine the
future values available after 5, 10, 15, and 20 years.
Figure 1 shows a time line of these wealth values and
corresponding utility values with utility parameters
a = 10,000 and b = 2. After one five-year period the
wealth will grow to $10,836.28 with corresponding
utility of 177.97. At the end of twenty years, the
investment in the riskless asset provides $26,930.89,
with a corresponding utility of future wealth of 252.71.
To model the risky-asset process, we use a
binomial tree and assume that for each five-year
period the asset will either increase in value by 100
percent (approximately 15 percent per year) or
decrease by twenty percent (approximately four
percent per year) from its current level. Given both
outcomes are equally likely the expected risky-asset
return is 40 percent (over one five-year period). This
is represented in the first branch of Figure 2. For each
possible wealth flow, we calculate the utility of wealth
for given utility parameters, a and b. With an initial
investment of $8,000 in the risky asset, the value either
rises to $16,000.00 or falls to $6,400.00 at the end of
the initial five-year period. Given utility parameters of
a = 10,000 and b = 2, the $16,000 ending value results
in utility of 204.94 versus utility of 151.00 for the
$6,400 ending value. Notice that the expected utility
for wealth after the first five-year period is 177.97 --
so an investor with a time horizon of only 5 years
would be indifferent to the risky and riskless assets (as
stated above, the risk-free rate is chosen to ensure this
result).
4 Journal of Financial Education
Figure 2. An Evolving Risky Portfolio
Fall 1999 5
Table 1. Expected Utility of Ending Wealth for Various Investments
Given Different Assumptions About Relative Risk Aversion*
____________________________________________________________________________________
Risky Asset Process
Riskless Asset Independent Returns Mean Reversion Momentum
Increasing Relative
Risk Aversion a=10,000 252.71 249.35 256.79 241.74
Constant Relative
Risk Aversion a=0 224.58 224.58 234.53 213.16
Decreasing Relative
Risk Aversion a=5,000 205.14 209.58 222.07 190.65
____________________________________________________________________________________
*The extended power utility function is used with b=2 and a as specified in the row headings. Expected utility is calculated for
possible wealth levels after four successive five-year investment periods. The initial investment is assumed to be $8,000. The
risky asset is assumed to have an expected return of 40% per five-year period. The riskless asset’s expected return is set such
that, given the parameters of the utility function, the investor is indifferent between the risky and riskless investments after one
five-year investment period.
For each possible outcome of the risky asset there
may again be an increase of 100 percent or decrease
of twenty percent over the next five-year period. This
process repeats itself over each five-year period; thus
at the end of twenty years (four five-year periods)
there are equally likely outcomes for the risky
investment. Evaluating these possible outcomes we
find that the expected future value (after twenty
years) for an investment of $8,000.00 in the risky asset
is $30,732.80. For utility parameters of a = 10,000 and
b = 2 expected utility of wealth at the end of twenty
years is 249.35. Comparing 249.35 to 252.71 (the
utility received from an investment in the riskless
asset) we find that an investor in this simplified context
with a twenty-year horizon will prefer investing in the
riskless asset rather than the risky asset.
8
Because an
investor with a five-year horizon is indifferent between
the two assets, we conclude that time diversification
benefits do not occur in this instance.
In Table 1 we report expected utility values for
twenty-year investments in the risky and riskless
assets. (Also reported are utility values for investments
in risky assets with mean-reversion or momentum;
these are discussed under Case 2). In the same
manner discussed in Figures 1 and 2, we calculate the
expected utility given different values for the
parameter a of the investor’s utility function.
9
The
results of Figures 1 and 2 are presented in the first two
cells of Table 1, row one (i.e., where a = 10,000 and
b = 2).
Utility rankings are directly dependent on the
coefficient of relative risk aversion (determined by the
parameter a). Notice that with constant relative risk
aversion (Table 1, row two, a = 0) the investor is
indifferent between the risky and riskless investment
opportunities. When there is decreasing relative risk
aversion (Table 1, row three, a = -5,000) the risky
investment strategy yields the higher expected utility.
In general, we confirm that increasing relative risk
aversion leads to a preference for the riskless asset as
the investment horizon lengthens while decreasing
relative risk aversion leads to a preference for the
risky asset.
10
Case 2. A Time Dependent Risky-Asset Process
The previous example demonstrates how the
relative risk aversion parameter affects the choice
between investing in riskless or risky assets. In this
example we allow the risky asset’s return process to
follow a time-dependent pattern (i.e., asset returns
may exhibit non-zero correlation through time). We
show that an investor’s preference for riskless or risky
assets may be strengthened or reversed depending on
the correlation of risky-asset returns. For tractability
we only allow the risky-asset return process to display
6 Journal of Financial Education
time variability. The unconditional risky-asset expected
return is held fixed at 40 percent in all cases.
Figure 3 shows an example of a time dependent
risky-asset process. In this example we again begin
with a risky-asset return that may increase by 100
percent or fall by twenty percent with equal
probability; however, unlike the earlier example,
subsequent risky-asset values depend on previous
values. The probability of any given occurrence is
represented by the probability value labeled on each
branch of the tree. In this example, risky-asset returns
display mean-reversion or negative serial correlation.
If the risky-asset value rises in the previous period, it
is more likely to fall in the next period (with probability
of 0.7), rather than rise again (probability of 0.3). If
the risky-asset value falls in the previous period, it is
more likely to rise in the next period (with probability
of 0.7) rather than fall again (probability of 0.3). For
example, in the second last column and third row of
the decision tree, we observe a wealth value of
$29,010.18. Given that the previous risky-asset value
change was positive, we assume an upward move in
value will occur with probability of 0.3, and a negative
value change will occur with probability of 0.7. All
entries in each column of Figure 2 are multiplied by a
constant (i.e., the 100 percent gain and 20 percent loss
no longer apply) to ensure the unconditional expected
return is still 40 percent given the new probabilities in
Figure 3.
The remaining columns of Table 1 display the
effects of time dependence in the risky-asset process.
In column three of Table 1 we report the expected
utility of ending wealth assuming the risky-asset
returns follow a mean reverting process. In column
four we report expected utilities given risky-asset
returns that possess momentum. For mean-reversion
we use the process shown in Figure 3. We model
momentum in a similar manner, except we reverse the
time-dependence to exhibit persistence. Specifically,
we assume a positive wealth change is more likely to
be followed by another positive change (with
probability of 0.7) instead of negative wealth change
(probability of 0.3) and a negative wealth change is
more likely to be followed by another negative wealth
change (with probability 0.7).
We see that the expected utilities given mean-
reversion in risky-asset returns are always higher than
the expected utilities given independent asset returns;
in contrast, the expected utilities given momentum are
always lower. In fact, given the degree of mean-
reversion in the example, the risky asset is always
preferred to the riskless alternative even when there is
increasing relative risk aversion (a = 10,000). With our
momentum process for the risky asset’s returns, the
riskless asset is always preferred even when there is
decreasing relative risk aversion (a = -5000).
12
To understand how mean-reversion is beneficial to
investors (or alternatively how momentum is harmful)
reconsider Figure 3. Mean-reversion leads to a
dampening of the volatility of asset returns over time
without a commensurate change in the expected
returns for the risky asset. This reduction in extreme
wealth values leads to a larger expected utility value,
due to concavity of the utility function. From a
practical viewpoint, many marginal investors will be
affected. Momentum affects investors in the opposite
direction. Increases in momentum, increase the asset's
riskiness (for a given expected return) and lead to a
reduction in the benefits of time diversification.
Our example can also be used to determine what
level of momentum or reversion will give rise to
indifference between the risky and riskless asset
choices. As in the example presented in Figure 3, we
assume that the probability of an increase in wealth
followed by another increase is the same as the
probability of a decrease in wealth followed by another
decrease. Again we assume a binomial tree (i.e., one
increase and one decrease from each wealth level)
and we maintain the assumption of a constant
expected return of 40 percent for the risky asset for
each five-year period.
As an example, we consider two special cases of
relative risk aversion: an investor with increasing
relative risk aversion (i.e., utility parameters of a =
10,000 and b = 2 in our example), and an investor with
decreasing relative risk aversion (i.e., utility
parameters of a = -2,000 and b = 2). For each set of
utility parameters we first solve for a risk-free rate
that makes the investor indifferent between the risky
and riskless asset over one five-year investment. We
then solve for the temporal-persistence probability
values that will make the investor indifferent between
the risky and riskless assets at the end of a twenty-
Fall 1999 7
Figure 3. An Evolving Risky Portfolio Assuming Mean Reversion
The future value of ending wealth and the utility (in parentheses) of ending wealth are reported at the end of each successive
5-year period for all possible outcomes. The probabilities of an upward and downward move conditional on the prior node being
realized are indicated on the branches of the tree.
8 Journal of Financial Education
year investment horizon. We find that if the probability
of a wealth increase followed by another increase is
reduced from .5 to an amount below .4051, then there
is enough mean-reversion to make the investor with
increasing relative risk aversion switch from preferring
the riskless asset to preferring the risky asset. That is,
there is enough mean-reversion to make time
diversification a utility enhancing strategy even though
the investor has increasing relative risk aversion. In
contrast, if the probability of a wealth rise followed by
another wealth rise is increased from .5 to an amount
above .5284, then there is enough momentum to
eliminate the benefits of time diversification so that
even the investor with decreasing relative risk aversion
prefers the riskless asset over the long time horizon.
The important point to note is that both the temporal
patterns in asset returns and the utility parameters are
important in determining whether a time diversification
strategy is beneficial or harmful.
OTHER FACTORS RELATED TO
INVESTMENT HORIZON THAT AFFECT
THE INVESTMENT CHOICE
In the preceding section we demonstrate that as the
investor’s time horizon increases, the preference for
risky investments may increase, remain constant, or
even decrease. In effect, whether time diversification
holds depends on the asset return process and
investor’s preferences (in particular, relative risk
aversion). If one still concludes at this point that time
diversification holds, the temptation is to recommend
that investors with longer time horizons assign greater
portfolio weights to risky securities like equities.
Before such a recommendation can be made,
however, other factors that may be correlated with
investment horizon must be examined to determine
whether they also imply such a strategy or whether
they actually offset any benefits obtained from time
diversification. In this section, we consider other
factors related to the investment horizon and show
how these factors can cause investors with longer time
horizons to allocate more, or less, of their portfolio to
risky securities.
Table 2 provides a review of six factors -- five
from the literature and a new factor -- that may affect
portfolio choice. Column one provides a brief
description of each factor. In columns two we
describe aspects of each factor that lead investors
with longer time horizons to allocate more of their
portfolio to risky securities (i.e., aspects that would
complement a time diversification strategy). In column
three we discuss aspects of each factor that could
offset possible benefits of time diversification. For
completeness, the first two factors of Table 2
summarize our previous examples.
The first factor in Table 2 is relative risk aversion.
As we demonstrate in our example and as shown in
Samuelson [1994] and Thorley [1995], the general
result is that when relative risk aversion is decreasing,
time diversification benefits obtain. That is, investors
should invest proportionally more in risky assets, the
longer is their investment horizon. In contrast, if utility
displays decreasing relative risk aversion, a time
diversifying strategy will reduce investor well being.
Only if the utility function being considered displays
constant relative risk aversion will optimal investment
proportions be independent of the investment horizon.
Factor two in Table 2 is the pattern of risky-asset
returns through time. There is substantial
documentation of temporal patterns in asset returns
(see, for example, Fama and French [1988], or
Poterba and Summers [1988]. Trading strategies
based on temporal patterns in asset returns sometimes
presume a mean-reverting process (negative serial
correlation). Intuitively, these strategies follow the
prescription that, ‘what goes up must come down’.
Momentum based strategies follow from positive
correlation in risky-asset returns. As developed in
Samuelson [1991] and as shown in our example,
negative correlation in the risky-asset return implies
that variability declines over a longer horizon and thus
there are benefits to time diversification. Positive
correlation implies that variability increases over a
longer horizon -- the opposite of time diversification.
The ability to change work habits is the third factor
presented in Table 2. Bodie, Merton and Samuelson
[1992] argue that investors can place higher
proportions of their portfolio in risky securities when
they are young because they are able and willing to
work harder and give up leisure should they suffer any
shortfalls from expectations. In latter stages of life,
investors do not have the time or human capital to
work harder to recover investment losses and thus
Fall 1999 9
Table 2. The Effect of Factors on an Investor’s Portfolio Weight Assigned to Risky Securities
Recommendation
Increase the weight assigned Decrease the weight assigned
Factor* to equities if.... to equities if...
1. Investor’s Preference utility displays decreasing utility displays increasing
toward risk relative risk aversion relative aversion
2. Risky-asset security returns display security returns display
return process mean reversion momentum
3. Ability of investor to investor can work more when investor cannot work more when
change work habits risky asset returns are low risky asset returns are low
4. Frequency of required one future withdrawal is periodic withdrawals are required
withdrawals from the required to finance future to finance everyday consumption
the investor’s portfolio consumption (e.g., to
Purchase an annuity for
Retirement)
5. Existence of nontradable human capital depletes human capital increases over time
assets over time
6. Potential for changing investment knowledge investment knowledge increases
knowledge of investments decreases with age with age
_____________________________________________________________________________________
* Each factor is considered individually as if it were added to a base case of independent risky-asset returns and constant
relative risk aversion for the investor. Recommendations are made for an investor with a long investment horizon relative to when
the investor will have a short investment horizon.
they are less willing to commit a high proportion of
their portfolio to risky assets. Alternatively, if an
investor’s ability to work is reduced in times of poor
risky-asset returns (for example in a depression) then
shortfalls are magnified. A longer investment horizon
allows for more of these types of extreme shortfalls
and thus an investor is less willing to take risky
investments.
Samuelson [1989] proposes the frequency of
required withdrawals from an investor’s portfolio will
affect time diversification benefits. If an investor
requires at least a certain minimum wealth
accumulation by retirement, say W
min
, a riskless fund
may be established to meet this need. The investor
will then choose a combination of the riskless asset
and the risky asset with remaining wealth. As
retirement approaches, the riskless fund set up to
provide W
min
will grow and become a larger proportion
of total invested funds. The observed investment
strategy may appear to be driven by time
diversification benefits but it is actually determined by
the required riskless fund. Constraints on periodic
consumption needs have the opposite effect. If an
investor requires a minimum amount of funds for
consumption needs each period, then a large riskless
fund will be set aside initially. The riskless fund will
shrink as time passes (because there will be fewer
future consumption periods to fund) giving the
appearance of a strategy opposite to time
diversification.
The fifth factor in Table 2 is the importance of non-
traded assets. If an investor has non-traded assets,
10 Journal of Financial Education
The fifth factor in Table 2 is the importance of non-
traded assets. If an investor has non-traded assets,
such as human capital, that vary through time, then
proportions of traded securities (risky and riskless
assets) must be rebalanced through time to maintain
the desired proportion in risky assets for the total
portfolio of traded and non-traded assets. Samuelson
[1994] presents an argument that young professional
investors often have a large portion of total wealth in
the form of low-risk human capital that will deteriorate
over time. Early in their working careers these
professionals allocate a large portion of their
measurable wealth (or traded assets) to risky assets.
As time passes and their low-risk human capital is
reduced, they make corresponding shifts away from
higher-risk equities in their financial portfolio.
Although the share of total wealth in risky assets may
in fact be constant, this strategy will be observationally
equivalent (in traded assets) to a time diversification
strategy.
We propose that changes in investment knowledge
that occur over an investor’s life (factor 6 in Table 2)
will also affect portfolio proportions. For the typical
investor, investment knowledge increases through life
as investment experience accumulates. If an investor
becomes sufficiently skilled in asset management late
in life, there may appear to be an increase in tolerance
for risky assets with age. Thus, with age (and a
shorter investment horizon), a greater proportion of the
portfolio may be allocated to risky securities. In this
case the changing investment knowledge may offset
time diversification arguments. In contrast, some
investors’ access to investment information decreases
through time. For example, consider an MBA
graduate who switches to a corporate position
following burn-out from an initial job on Wall Street. In
this case, information effects may cause investment
patterns to behave as specified in time diversification
arguments -- again, without any time-diversification
benefits necessarily occurring. The important point is
that an investor’s conditional investment opportunity
set will be dependent on the investor’s knowledge. To
the extent this knowledge is negatively correlated with
age, time diversification will appear to obtain.
CONCLUDING COMMENTS
The general conclusion of this discussion is that the
effectiveness of time diversification should not be
naively accepted or discarded. We show that time
diversification benefits do occur with decreasing
relative risk aversion or mean-reversion in risky-asset
returns.
We summarize other factors that may induce the
same strategy recommended by time-diversification
proponents and show how these factors may at other
times offset time-diversification benefits. In addition
to the factors previously discussed in the literature, we
also consider the effects that may occur as a result of
changes in investment knowledge over an investor’s
life. Relative changes in an investor’s investment
knowledge may directly affect the choice between the
riskless and risky assets.
A clear understanding of an investor’s preferences,
constraints, goals and knowledge is necessary before
an investment strategy can be formulated. Blind
application of the strategy recommended by time
diversification proponents may be quite harmful for
some investors. Further research is warranted to
examine the extent of relative risk aversion displayed
by a wide variety of investors. With reference to the
time diversification issue, direct empirical evidence
examining relative risk aversion through investors’
lives would clarify the role of risk preferences in asset
allocation choices.
APPENDIX: THE ST. PETERSBURG
PARADOX
In 1725, at the age of 25, Daniel Bernoulli (1700-
1782) became professor of mathematics at St.
Petersburg. Among other accomplishments, he
proposed a solution to the St. Petersburg paradox
posed by his elder brother Nicolaus (1695-1726)[c.f.,
Eves (1996)]. In his resolution of the paradox, Daniel
coined the term moral expectation, as the appropriate
measure of worth or value (as opposed to
mathematical expectation). Bernoulli proposed a
valuable counter-example to demonstrate that wealth
maximization is not in general optimal.
The St. Petersburg paradox can be described in the
context of a simple coin flip game. A coin is tossed
until the first head appears. The payoff received when
the first head occurs is given by,
Fall 1999 11
Table A1. St. Petersburg’s Paradox, Calculating the Expected Value of the Risky Gamble
_____________________________________________________________________________________
Number of tails Probability of n tails Payoff of n tials, Probability (n)
before the first head Probability (n) Payoff (n) × Payoff (n)
0 ½ $1.00=$(2.00)
0
$.50=½×$1.00
1 ¼=½×½ $2.00=$(2.00)
1
$.50=¼×$2.00
2 c=(½)
3
$4.00=$(2.00)
2
$.50=c×$4.00
3 1/16=(½)
4
$8.00=$(2.00)
3
$.50=1/16 ×$8.00
.
.
n (½)
n+1
$(2.00)
n
$.50=(½)
n+1
×$(2.00)
n
.
.
_____________________________________________________________________________________
Column Totals
1
2
1 1
1
+ =
=
∞
∑
n
n
( )
200
1
.
n
n=
∞
∑
1
2
00
1
2
1
2
1
2
1
1
1
= + +
=
∞
+
=
∞
=
∞
∑
∑
n
n
n
n
$2.
...
___________________________________________________________________________________
first head appears. To maintain a link with a simple
investment setting, consider the payoff as the close-out
value for a stock portfolio, where all distributions are
reinvested. We wish to know how much any player
would be willing to pay to play the game (or to
purchase the portfolio).
The typical approach to analysis begins by
considering the expected winnings from playing the
game. We can calculate the expected payoff after
constructing a table of the possible payoffs with their
respective probabilities. Table A1 demonstrates the
necessary calculations. In the first column of the
table, we list the number of times a tail is flipped
before the first head appears (at which time the game
ends). The second column lists the probability of each
possible occurrence of n tails. The third column
shows the resultant payoff received for every possible
occurrence of n, as determined by the rules of the
game. Finally, the last column multiplies the second
and third columns to determine the expected cash flow
for each possible occurrence. For example, the fourth
row shows that the probability of three tails occurring
followed by the first head is 1/16, with resultant payoff
of $8.00.
The important point noted by Bernoulli is that the
expected value (or mathematical expectation) of the
gamble is infinite (notice that all entries in the final
column are always $.50). The paradox is now
immediately apparent. Although the mathematical
expectation is infinite, no investor is willing to pay an
arbitrarily large sum of money to play the game.
Bernoulli resolves the paradox by suggesting the use of
a moral expectation, rather than a mathematical
expectation. In modern terms, Bernoulli recognizes
that utility, or wellness, does not increase linearly with
wealth. Instead, initial changes in wealth are worth
relatively more than later units. In introductory
economics terminology, a well defined utility function
always increases at a decreasing rate. In modern
financial economics, gambles can be compared using
expected utility calculations.
ENDNOTES
12 Journal of Financial Education
1
We assume that an expected utility representation
exists for all gambles under consideration. Technical
conditions necessary for the existence of an expected
utility representation can be found in many sources
(c.f., Huang and Litzenberger [1988]).
2
In the following section, we extend this simple
context to consider a more complicated real-world
situation.
3
As in much of the time diversification literature,
we do not consider the more general case of
intertemporal consumption and income flows. This
framework is consistent with the notion that time
diversification is typically posited as advice that is
dependent solely on an investor's age and time until
retirement, with little concern for future cash flows.
4
The extended power utility function is also
interesting because it may be specialized to consider
the familiar narrow power utility and log-utility
functions.
5
Because the derivative of relative risk aversion
with respect to wealth is given by [a/(a+bz)
2
], and b
is restricted to be strictly positive, b will have no effect
on the sign of this derivative.
6
The use of successive five-year intervals allows us
to examine a realistic long-term investment context
without an unnecessarily complex binomial tree.
7
We generalize and extend the pedagogy developed
in Kritzman [1994] to show how time diversification
depends on relative risk aversion and the underlying
asset processes.
8
The relatively small difference in expected utility
values should not be emphasized. Because utility
functions are unaffected by a positive linear
transformations, we could without loss of generality,
make this difference in expected utility values
arbitrarily large if we so desired.
9
We can also calculate expected utility values given
different values for the parameter b of the investor’s
utility function. As b increases there is less relative
difference between the utilities from the riskless and
risky assets. (As clarified in the previous footnote, any
difference in utility may be arbitrarily magnified
through a suitable transformation of the utility function,
with no affect to the investor's asset allocation
decisions.) This is because as b grows without bound,
the extended power utility function simplifies to ending
wealth.
10
These results correspond to Samuelson [1994]
and Thorley[1995]. They show that the optimal
proportion invested in the risky asset increases with
investment horizon under decreasing relative risk
aversion (or increasing relative risk tolerance).
11
This finding is quite general and can be readily
modified to reflect any economic agent’s decision
problem (assuming the general investment context and
the ability to capture utility with an extended power
function). For example, an economic agent with
extended power utility represented by a=-5,000, 0, or
5,000 and b=1.1, beginning wealth of $80,000, and
independent increases in wealth of 100 percent or
independent losses of 20 percent every period for four
successive periods, produces the same qualitative
finding that time diversification benefits exist only if
utility displays decreasing relative risk aversion (a=-
5,000, in this case).
12
These results correspond with Samuelson’s
[1991] finding that negative serial autocorrelation
causes an investor to be more tolerant of risk when
young than when old. Whether the effects of mean-
reversion will reverse our conclusions with respect to
relative risk aversion depends of course on the level of
risk aversion and persistence in the investor’s portfolio.
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