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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.

REFERENCES

Bodie, Zvi. “On the Risk of Stocks in the Long Run,”

Financial Analysts Journal, 51(1995), 18-22.

Bodie, Z., R. Merton, and W. Samuelson, “Labor

Supply Flexibility and Portfolio Choice in a

Lifecycle Model,” Journal of Economic

Dynamics and Control, 16(1992), 3, 427-450.

Eves, H. An Introduction to the History of

Mathematics, Holt, Rinehart, and Winston:

Toronto, 1976.

Fama E. F. and K. R. French. “Permanent and

Temporary Components of Stock Prices,” Journal

of Political Economy, 96(1998), 2, 264-273.

Harlow, W. V. “Asset Allocation in a Downside Risk

Framework,” Financial Analysts Journal,

47(1991), 28-40.

Huang, C., and R. H. Litzenberger. Foundations for

Financial Economics, North-Holland: New York,

1998.

Fall 1999 13

Kritzman, M. “What Practitioners Need to Know ...

... About Time Diversification,” Financial

Analysts Journal, 50(1994), 14-18.

Leibowitz, M. L., and W. S. Krasker. “The

Persistence of Risk: Stocks versus Bonds over the

Long Term,” Financial Analysts Journal,

44(1988), 40-47.

Merton, R., and P. A. Samuelson. “Fallacy of the Log-

Normal Approximation to Portfolio Decision-

Making over Many Periods,” Journal of

Financial Economics, (1974), 67-94.

Peavy, J. W., and M. J. Vaughn-Rauscher. “Risk

Management through Diversification,” Trusts and

Estates, 133(1994), 42-46.

Poterba, J. and L. Summers. “Mean Reversion in

Stock Returns: Evidence and Implications,”

Journal of Financial Economics, 22(1988), 27-

59.

Samuelson, P. A. “Risk and Uncertainty: A Fallacy of

Large Numbers,” Scientia, (April/May,1963), 1-6.

Samuelson, P. A. “The Judgment of Economic

Science on Rational Portfolio Management:

Timing and Long-Horizon Effects,” Journal of

Portfolio Management, 16(1989), 4-12.

Samuelson, P. A. “Asset Allocation could be

Dangerous to your Health: Pitfalls,” Journal of

Portfolio Management, 16(1990), 5-8.

Samuelson, P.A. “Long-Run Risk Tolerance when

Equity Returns are Mean Regressing:

Pseudoparadoxes and Vindication of

‘Businessmen’s Risk’”, in W.C. Brainard, W.D.

Nordhaus, and H.W. Watts, eds., Money,

Macroeconomics, and Economic Policy, The

MIT Press (1991), 181-200.

Samuelson, P. A. “The Long-Term Case for Equities

and how it can be Oversold,” Journal of Portfolio

Management, 21(1994), 15-24.

Thorley, S. R. “The Time-Diversification Controversy,

Financial Analysts Journal, 51(1995), 68-76.