Leveraged Stock Portfolios over Long Holding Periods: A ... Wilson.pdfLeveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model Asset pricing models typically allow

Leveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model

Dale L. Domian, Marie D. Racine, and Craig A. Wilson

Department of Finance and Management Science College of Commerce

University of Saskatchewan Saskatoon, SK S7N 5A7

Canada

[email protected]; (306) 966-8425 [email protected]; (306) 966-8406

[email protected]; (306) 966-8430

(306) 966-2515 fax

May 2003

Leveraged Stock Portfolios over Long Holding Periods:

A Continuous Time Model

Abstract

We use a continuous time model to derive return and wealth distributions for leveraged portfolios over long holding periods. These theoretical distributions closely match empirical distributions obtained from a resampling procedure. The expected annualized return is a concave function of the degree of leverage. With historical parameter values, the function is maximized at 241% stock, borrowing an amount equal to 141% of net wealth. This maximal stock proportion is considerably reduced if the borrowing rate is higher than the historical lending rate.

Leveraged Stock Portfolios over Long Holding Periods:

A Continuous Time Model

Asset pricing models typically allow the use of leverage. For example, in the diagram

illustrating the traditional Sharpe-Lintner CAPM, the capital market line is tangent to the

efficient frontier at the market portfolio. Points beyond the tangency are achieved by borrowing

at the risk-free rate. The diagram can be easily modified to illustrate differential borrowing and

lending rates.

However, few academic papers examine the effects of using leverage. Numerous studies

in the asset allocation literature consider mixes between stocks and bonds (or Treasury bills),

with the most extreme case being a 100% stock portfolio. These studies do not give any insights

into the effects of using leverage.

In a study from the practitioner-oriented literature, Ferguson (1994) describes how

leverage magnifies the volatility of stock returns and leads to a positive probability of

bankruptcy. He demonstrates that long-run returns decline as leverage increases, even when

bankruptcy is not possible. These return declines are confirmed in a recent empirical study by

Domian and Racine (2002).

We use a continuous time model to derive return distributions for leveraged portfolios

over long holding periods. Ending wealth distributions are easily obtained from the return

distributions. We compare the theoretical distributions to empirical distributions obtained from a

resampling procedure.

The paper begins with a derivation of the theoretical distributions followed by the

development of the empirical distributions. The results section includes practical implications for

2

investors while the conclusion summarizes the close association found between the two

distributions.

Theoretical Distributions from a Continuous Time Model

We begin with some market assumptions:

1. We have a small investor that cannot influence the markets for the stock or the borrowing

rate and does not face transaction fees or any other market frictions.

2. The stock is infinitely divisible (for the ownership of fractional shares), and does not pay

dividends.1

3. The price or value per unit of the stock, St, has log-normal dynamics (i.e. in continuous

time it follows a geometric Brownian motion).2 Without affecting the results, we could

suppose that this price or value has been adjusted for inflation, provided we also adjust

the borrowing rate.

4. The investor faces a constant cost of borrowing — it does not change over time, nor does

it depend on the degree of leverage. This rate can also be adjusted for inflation as

required in the previous assumption. We also assume that the interest is paid at the

termination of the investment, i.e. in continuous time a unit of debt, Bt, grows

exponentially at the continuously compounded rate, r.

These assumptions allow the following characterisation of the economy in continuous

time as described by Samuelson (1965) and employed by Black and Scholes (1973) and Merton

(1973). The stock follows dynamics described by the stochastic differential equation (SDE)

,tttt dWSdtSdS σ+µ= (1)

1 The assumption of no dividends can be relaxed by assuming that they are reinvested in the stock so that the stock price is cum dividend and remains continuous. 2 This is characterized by an efficient market with constant mean and variance of returns (see Jarrow and Turnbull 2000, for instance).

3

where µ is the instantaneous expected return per unit time, σ is the return volatility per unit time,

and Wt is a standard Brownian motion.3 It is well known (see, for instance, Elliott and Kopp,

1999), that this SDE has the solution

.)2/exp( 20 tt WtSS σ+σ−µ= (2)

Since Wt has a normal distribution with mean 0 and variance t, St has log-normal distribution

with mean (conditional on knowing the current price),

).exp(]|[ 00 tSSSE t µ= (3)

The unit of debt has dynamics described by the ordinary differential equation (ODE)

,dtrBdB tt = (4)

where r is the continuously compounded borrowing rate. If we suppose that initially, at time 0,

the value of the debt is $1, then the solution to this ODE is

).exp(rtBt = (5)

We suppose that time, t, has units of years, so that the parameters µ, σ, and r, are in annual

terms. This gives us a mathematical description of the two assets in this simple economy.

The next step is to describe a portfolio, or more correctly, a dynamic investment strategy

in this world. We do this as a vector that is allowed to change through time indicating the

number of units of debt invested or borrowed and the number of stocks held, (Htb, Ht

s). A

negative number of units signifies a short position, and we are particularly interested in the case

where Htb is negative and Ht

s is positive. As this vector describes an investment rule, it is

3 We assume the existence of an underlying probability space, (Ω, F, P), large enough to support the Brownian motion, Wt.

4

imperative that it doesn’t rely on or make use of information that is not known at time t.4 The

value, Vt, of the investment at any time is

.tstt

btt SHBHV += (6)

We impose two restrictions on the investment strategy, the first being standard in this

type of problem, and the second enabling us to explore the effects of leverage in a systematic

way:

1. The investment strategy is self-financing, so that no funds are added or removed via

income or consumption. This means that changes in portfolio value come only from

changes in asset prices, and the mathematical form of this constraint is

.tstt

btt dSHdBHdV += (7)

2. The portfolio has a constant stock proportion, c:

.cV

SH

t

tst = (8)

Since we are interested in the effects of leverage, we usually take c > 1, so that more than

100% of the investor’s wealth is invested in the stock and that portion above 100% is financed

by a margin loan. However, if the borrowing and lending rate are the same and there are no

restrictions on short sales or margin ratios, then c can be any real number. It is important to

notice that in order to maintain a constant stock proportion in a continuous time setting we

require the investor to continuously rebalance her portfolio through trading and adjusting debt.

Condition 2 implies the dual condition that the portfolio has a constant debt proportion:

4 This means that we require the stochastic processes Ht

b and Hts to be adapted to the filtration generated by St. We

also require them to satisfy some technical conditions so that the SDE defined below in condition 1 is well defined.

These are ∞<∫ duHt b

u0 and ∫ ∞<

t su duH

0

2 with probability 1, for any t.

5

.1 cV

BH

t

tbt −= (9)

Solving for Htb and Ht

s give

,)1(

t

tbt B

VcH

−= and .

t

tst S

cVH = (10)

Substituting this into condition 1 gives

,)1(

tt

tt

t

tt dS

ScV

dBB

VcdV +

−= (11)

and substituting the dynamics for Bt and St gives

,)1(

ttt

tt

t

tt

t

tt dWS

ScV

dtSS

cVdtrB

BVc

dV σ+µ+−

= (12)

or simplifying

.))(( tttt dWcVdtVcrrdV σ+−µ+= (13)

As above, the solution to this SDE is

,)2/)(exp( 220 tt cWtccrrVV σ+σ−−µ+= (14)

which has log-normal distribution with mean5

.))(exp(]|[ 00 tcrrVVVE t −µ+= (15)

From this we can see that the expected wealth of an investor is increasing in both the

amount of leverage, c, and the investment horizon, t, provided the risk premium µ – r is positive.

This indicates that a risk neutral investor would optimally choose c as large as possible,

regardless of her investment horizon. But how should the investment horizon affect the leverage

5 With a solution to Vt we can explicitly write Ht

s and show that it also has log-normal dynamics; it is then easy to show that the conditions of footnote 3 are satisfied. The same is true for Ht

b.

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choice of a typical risk-conscious investor? To examine the issue we first look at the annualized

return, Rt, from the investment strategy,

.21)(log1 22

0t

tt W

tcccrr

VV

tR σ

+σ−−µ+=

≡ (16)

From this we see that the annualized return has a normal distribution with mean

,21)(][ 22ccrrRE t σ−−µ+= (17)

and variance

.]var[22

tcRt

σ= (18)

Note that this is a continuously compounded rate of return so it is sensible to allow the

possibility of the return to be less than –1 without violating a limited liability constraint. Using

the expressions for the mean and variance, Equations (17) and (18), leads to the following

density function of the annualized return:

.for 21)(

2exp

2)(

222

22 ∞<<∞−

σ+−µ−−

σ−

σπ= xccrrx

ct

ctxf (19)

Graphs of this function are discussed in the Results section below.

We observe several important points:

1. The mean of the annualized return does not depend on the investment horizon, and the

variance decreases with investment horizon on the order of t–1.

2. Furthermore, with any fixed degree of leverage, the annualized return converges with

probability 1, and in mean squared to its expected value as the investment horizon

increases to infinity. This is sometimes used (incorrectly) as an argument for advocating

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people with longer investment horizons invest more aggressively. The fallacy of this

strategy is discussed later.

3. For any fixed investment horizon, the variance of the annualized return is increasing with

the degree of leverage, which is not surprising. In particular it increases on the order of

c2, so variance is actually increasing at an increasing rate.

4. Perhaps most important is the observation that the expected annualized returns is a

concave function of the degree of leverage, c, taking a maximum at

.* 2σ−µ

=rc (20)

This indicates that the requirement of maintaining a given margin ratio has important

implications on the efficiency of highly leveraged portfolios. Indeed, a portfolio with a

stock proportion greater than c* is dominated in the Markowitz sense by the c* portfolio,

which has higher expected return and lower return variance.

Empirical Distributions

In the previous section the theoretical return distribution and the maximum degree of

leverage were derived. We now turn our attention to the empirical estimation of leveraged return

distributions for 5- and 20-year holding periods. This poses some methodological problems

because there are very few independent observations of long holding periods. For example, sixty

years of data contain just three independent 20-year holding periods.

Butler and Domian (1991) overcome this difficulty with a resampling approach to

estimate long-run distributions from monthly stock returns. Domian and Racine (2002) use this

approach for leveraged portfolios of US stocks. We use two monthly series from the CFMRC

database, value-weighted stock index returns and 91-day Government of Canada Treasury bill

8

returns. These series contain 623 values over the period February 1950 through December 2001.

We convert these to real returns using inflation data from CANSIM. The real stock returns have

a monthly mean of 0.006447 and a standard deviation of 0.044895, while the real Treasury bill

returns have a 0.001636 monthly mean and a 0.004619 standard deviation. These correspond to

continuously compounded annual returns of 7.71% on stocks and 1.96% on Treasury bills, and

an annual stock standard deviation of 15.44%.

For a 5-year holding period, the resampling procedure is implemented as follows:

1. Randomly select one of the 623 months. Record the observed real stock and Treasury bill

returns for this month.

2. Compute the portfolio returns for six asset allocations ranging from 50% stock up to

300% stock. For stock proportions below 100%, the remainder of the allocation is in

Treasury bills. Stock proportions exceeding 100% are achieved with funds borrowed at

the Treasury bill rate. Note that the current 30% margin requirement on most large stocks

would allow a Canadian investor to reach a 333% stock proportion.

3. Repeat the previous steps 60 times with replacement and compound the monthly returns

to construct one representative 5-year holding period return for each asset allocation.

4. Perform the entire procedure 100,000 times to generate 5-year holding period return

distributions from the observed history of real monthly returns.

For a 20-year holding period, the first and second steps are repeated 240 times. The resulting

empirical distributions are compared in the next section to the theoretical distributions.

9

Results

Our graphs begin with the theoretical density functions f(x) of the annualized return, as

given in Equation (19). Historical parameter values, as presented in the previous section, are

used: µ = 0.0771, r = 0.0196, and σ = 0.1544. Figure 1 uses an investment horizon t = 5 years

and six different stock proportions. The density function f becomes flatter and more spread out as

the stock proportion is increased from the lowest value c = 50% (a portfolio with half in stock

and half in Treasury bills) to the highest value c = 300%.

Figure 2 shows the relationship between expected return and the stock proportion. The

figure includes all six of the density functions that are displayed separately in Figure 1. The

darker line traces the path of expected return as the stock proportion is varied continuously from

50% to 300%. Note that only the horizontal axis is relevant for that darker line. Initially expected

return rises as the stock proportion is increased. After reaching its maximum at 241% stock (c*

from Equation 20), expected return declines. This figure could also be interpreted as a

relationship between expected return and variance, since higher stock proportions have larger

variances.

Figure 3 compares the theoretical distributions to the empirical distributions obtained by

the resampling procedure. The theoretical distributions of Figure 1 are shown with darker lines.

The shaded regions show the empirical distributions. Note that the scale has been adjusted to

reflect discrete compounding used to obtain the empirical distributions. It is apparent that the

empirical distributions have the same shape as the theoretical, although with a slight leftward

shift at higher amounts of leverage. Table 1 reports the first four moments of the distributions.

Any discrepancy between the moments of a given theoretical and empirical distribution

intensifies as the stock proportion increases. This is particularly true for skewness and kurtosis.

10

Results for the 20-year holding period are displayed in Figures 4, 5, and 6. These

densities are more concentrated than for the 5-year horizon. Nevertheless, expected return as a

function of the stock proportion follows exactly the same pattern as for the shorter holding

period. Table 2 reports the moments of the theoretical and empirical return distributions. The

results are similar to the findings for the 5-year holding period. In particular, the difference

between the theoretical and empirical third and fourth moments grows as the stock proportion

increases.

The expected return as a function of the stock proportion is independent of the holding

period. Therefore the return-maximizing stock proportion is always 241%. As Figure 7

illustrates, expected return initially increases but declines for stock proportions above 241%.

This figure begins with a 0% stock allocation at the left edge, and extends to stock proportions

well above 300%. Although margin requirements preclude such excessive leverage, it is

nevertheless interesting to see the dramatic decline in expected return.

Even when a modest amount of leverage enhances the expected return, there is still a

trade-off with risk. This can be illustrated on graphs plotting expected return and standard

deviation, analogous to the capital market line in the traditional CAPM. Figure 8 presents these

relationships for the 5-year and 20-year holding periods with stock proportions ranging from

50% to 300%. As noted in Figure 7, annualized return does not depend on the holding period, so

a 50% stock portfolio has a 4.54% expected return over both 5 and 20 years. However, the

standard deviation declines from 3.45% over 5 years to 1.73% over 20 years. The fourfold

increase in the holding period reduces the standard deviation by half for any stock proportion.

Thus, the standard deviation of the 300% stock portfolio falls from 20.72% to 10.36%, with the

expected return unchanged at 8.48%.

11

As we remarked earlier, annualized returns can be misleading when used as a criterion

relating investment horizon and aggressiveness of investment strategy (here represented by the

degree of leverage, c). This is best illustrated by examining the distribution of wealth directly.

The density function of portfolio value, with an initial investment of $1, is

( ) .0for 2/)()ln(1exp2

1)( 22222 ∞<<

σ−−µ+−

σ−

σπ= ytccrry

tccytyg (21)

The remaining graphs, Figures 9 to 14, explore the properties of this wealth density.

Figure 9 shows the wealth densities over the 5-year holding period for six stock proportions. The

wealth distributions become more highly skewed as the stock proportion is increased.

Due to the skewness of the wealth distributions, we summarize them using the median.

The median of the portfolio value is

.)2/)(exp(][ 220 tccrrVVm t σ−−µ+= (22)

The median is concave in the degree of leverage, c, as is the expected annualized return. The

median also takes its maximum at c*, given by Equation (20) and repeated here as Equation (23).

.* 2σ−µ

=rc (23)

As before, it is independent of the investment horizon, t.

Figure 10 shows the relationship between median wealth and stock proportion. The top

panel includes the six distributions illustrated in Figure 9 while the bottom panel considers a

range of stock proportions from zero to 1000%. The median increases as the stock proportion

increases to 241% and then steadily declines, consistent with the results for the mean return.

Figure 11 compares the theoretical and empirical distributions. The theoretical distributions of

Figure 9 are shown with darker lines. The shaded regions show the empirical distributions. In the

figure, the empirical distributions appear to be shifted slightly leftward, particularly at higher

12

stock proportions. Table 3, which presents the first four moments of the theoretical and empirical

wealth distributions, confirms this. Although the differences are small, the mean empirical

wealth is always less than the mean theoretical wealth. Further, the distance between the two

increases along with the degree of leverage. This is also true for the skewness and the kurtosis.

Figures 12, 13 and 14 repeat the analysis for the 20-year horizon. The lower panel of

Figure 13 shows higher wealth amounts than Figure 10, simply due to the longer holding period.

The empirical distributions in Figure 14 are more steeply peaked for the highest stock

proportions. The moments are given in Table 4. The same general patterns that were observed in

all previous comparisons of the theoretical and empirical distributions remain. The differences

between moments is positively correlated with an increase in leverage and these differences are

most pronounced for skewness and kurtosis.

Implications for Investors

Our analysis might appear to suggest that risk-tolerant investors should embrace the use

of leverage. Both expected return and median ending wealth go up as the stock proportion is

raised towards 241%, the return-maximizing stock proportion, c*, in our study. However, there

are some important practical limitations.

The standard theoretical assumption of a borrowing rate equal to the lending rate is not

satisfied in the real world. Margin borrowing rates are well above Treasury bill rates. Canadian

brokerage firms use the chartered banks’ prime rate as the benchmark for margin loans. Investors

are typically charged a borrowing rate of 1% above prime. Since the prime rate itself is normally

about 1% above Treasury bills, this implies a margin borrowing rate that is 2% above the

Treasury bill rate.

13

It is easy to determine the impact of the higher borrowing rate. The derivative of c* with

respect to r is just –1/σ2. With our parameter value of σ =.1544, the derivative is –41.92. An

increase of 2% in the borrowing rate would lower c* by about 84%, from 241% to 157%. Thus, a

small increase in the borrowing rate produces a large reduction in the maximum leverage an

investor should consider.

One interpretation of c* is that it is an upper bound for risk-averse investors. While the

most risk-tolerant investors could contemplate approaching c*, more risk-averse investors will

prefer to choose much lower stock proportions, perhaps well below 100%. Despite the lower

expected return, many investors prefer the lower variance associated with unlevered portfolios.

Conclusion

Our findings provide some useful insights into the behaviour of leveraged portfolios. In

the single-period CAPM, expected return is an increasing linear function of the degree of

leverage. But in our continuous time model, expected annualized return is a concave function of

leverage. This function achieves its maximum at 241% stock with historical parameter values.

Empirical distributions estimated by resampling with monthly rebalancing provide a

close approximation to the theoretical distributions. This is evidence that the theoretical findings

are useful for investment decisions. The wealth-maximizing degree of leverage should be viewed

as an upper bound for the most risk-tolerant investors. More risk-averse investors may prefer to

avoid leverage.

14

References

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

Butler, K., & Domian, D. (1991). Risk, diversification, and the investment horizon. Journal of

Portfolio Management, 17, 41–47. Domian, D., & Racine, M. (2002). Wealth and risk from leveraged stock portfolios. Financial

Services Review, 11, 33–46. Elliott, R., & Kopp, P. (1999). Mathematics of financial markets. New York: Springer-Verlag. Ferguson, R. (1994). The danger of leverage and volatility. Journal of Investing, 3, 52–57. Jarrow, R., & Turnbull, S. (2000). Derivative securities, 2nd edition. South-Western. Jorion, P. (2001). Value at Risk: The new benchmark for managing financial risk, 2nd edition,

New York: McGraw-Hill. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and

Management Science, 4, 141–183. Samuelson, P. (1965). Rational theory of warrant prices. Industrial Management Review, 6, 13–

31.

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Figure 1 Theoretical Return Density, 5-Year Holding Period The probability density functions are for the annualized continuously compounded real return. The probability distributions are based on the normal distribution with parameters calculated from Canadian real monthly stock and Treasury bill returns over 1950 to 2001.

50% Stock Proportion

0.000.020.040.060.080.100.12

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Figure 2 Theoretical Expected Return, 5-Year Holding Period The expected annualized continuously compounded real return is superimposed upon the return densities to illustrate the relationship between expected return and variance for varying degrees of stock proportion. The figure includes all six of the density functions that are displayed separately in Figure 1. The darker line traces the path of expected return as the stock proportion is varied continuously from 50% to 300%. Note that only the horizontal axis is relevant for that darker line. The lower points are associated with flatter distributions with higher variances. The stock proportion ranges between 50% at the top of the graph and 300% at the bottom, maximizing expected return at a 241% stock proportion.

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Figure 3 Empirical and Theoretical Return Distributions, 5-Year Holding Period The empirical annualized discretely compounded return distribution histograms are calculated from real monthly stock and Treasury bill returns over 1950 to 2001, as are the parameters for the overlain theoretical return densities. The scale for the return densities has been adjusted to reflect the discrete compounding used to obtain the empirical distribution.


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Figure 4 Theoretical Return Density, 20-Year Holding Period The probability density functions are for the annualized continuously compounded real return. The probability distributions are based on the normal distribution with parameters calculated from real monthly stock and Treasury bill returns over 1950 to 2001.


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19

Figure 5 Theoretical Expected Return, 20-Year Holding Period The expected annualized continuously compounded real return is superimposed upon the return densities to illustrate the relationship between expected return and variance for varying degrees of stock proportion. The figure includes all six of the density functions that are displayed separately in Figure 4. The darker line traces the path of expected return as the stock proportion is varied continuously from 50% to 300%. Note that only the horizontal axis is relevant for that darker line. The lower points are associated with flatter distributions with higher variances. The stock proportion ranges between 50% at the top of the graph and 300% at the bottom, maximizing expected return at a 241% stock proportion.

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Figure 6 Empirical and Theoretical Return Distributions, 20-Year Holding Period The empirical annualized discretely compounded return distribution histograms are calculated from real monthly stock and Treasury bill returns over 1950 to 2001, as are the parameters for the overlain return densities. The scale for the return densities has been adjusted to reflect the discrete compounding used to obtain the empirical distribution.


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21

Figure 7 Expected Return This graph depicts the behaviour of the theoretical annualized continuously compounded return for various stock proportions including those beyond 300%. As before, the maximum is at 241% stock proportion. This result does not depend on the particular holding period.

-50%

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-10%

0%

10%

20%

0% 100% 200% 300% 400% 500% 600% 700% 800% 900% 1000%Stock Proportion

Expe

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urn

22

Figure 8 Mean versus Standard Deviation These graphs depict the theoretical trade-off between the expected value and the standard deviation of the annualized real return on portfolios with stock proportions ranging from 50% to 300%.

5 Year Holding Period

0%

2%

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23

Figure 9 Theoretical Wealth Density, 5-Year Holding Period The probability density functions are for end-of-period real wealth from a $1 initial investment with a 5-year holding period. The probability distributions are based on the lognormal distribution with parameters calculated from real monthly stock and Treasury bill returns over 1950 to 2001. The horizontal axes measure ending wealth in dollars.


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Figure 10 Theoretical Median Wealth, 5-Year Holding Period The median end-of-period real wealth from a $1 initial investment with 5-year holding period is superimposed upon the ending wealth densities to illustrate the relationship between these variables and variance for varying degrees of stock proportion. The median ending wealth first moves slightly right, then left as the stock proportion increases. The stock proportion ranges between 50% at the top of the graph and 300% at the bottom, maximizing median wealth at a 241% stock proportion. The second graph shows the behaviour of the theoretical median end-of-period wealth with a 5-year holding period if the stock proportion were allowed to increase beyond 300%. Note that this result does depend on the particular holding period; for comparison see Figure 13.

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Figure 11 Empirical and Theoretical Wealth Distribution, 5-Year Holding Period The empirical end-of-period real wealth distribution histograms from a $1 initial investment with a 5-year holding period are calculated from real monthly stock and Treasury bill returns over 1950 to 2001, as are the parameters for the overlain wealth densities.


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Figure 12 Theoretical Wealth Density, 20-Year Holding Period The probability density functions are for end-of-period real wealth from a $1 initial investment with a 20-year holding period. The probability distributions are based on the lognormal distribution with parameters calculated from real monthly stock and Treasury bill returns over 1950 to 2001. The horizontal axes measure ending wealth in dollars.


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Figure 13 Theoretical Median Wealth, 20-Year Holding Period The median end-of-period real wealth from a $1 initial investment with 20-year holding period is superimposed upon the ending wealth densities to illustrate the relationship between these variables and variance for varying degrees of stock proportion. The median ending wealth first moves slightly right, then left as the stock proportion increases. The stock proportion ranges between 50% at the top of the graph and 300% at the bottom, maximizing median wealth at a 241% stock proportion. The second graph shows the behaviour of the theoretical median end-of-period wealth with a 5-year holding period if the stock proportion were allowed to increase beyond 300%. Note that this result does depend on the particular holding period.

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Figure 14 Empirical and Theoretical Wealth Distribution, 20-Year Holding Period The empirical end-of-period real wealth distribution histograms from a $1 initial investment with a 20-year holding period are calculated from real monthly stock and Treasury bill returns over 1950 to 2001, as are the parameters for the overlain wealth densities. Notice that the scale differs from that of the densities in Figure 12 so as to illustrate some of the finer details.


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Table 1 Theoretical and Empirical Return Distributions, 5-Year Holding Period

Stock Proportion Moment Distribution 50% 100% 150% 200% 250% 300%

Mean Theoretical 0.0454 0.0652 0.0790 0.0869 0.0888 0.0848 Empirical 0.0469 0.0696 0.0872 0.0992 0.1048 0.1031

Variance Theoretical 0.0012 0.0048 0.0107 0.0191 0.0298 0.0429 Empirical 0.0014 0.0056 0.0132 0.0246 0.0401 0.0605

Skewness Theoretical 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Empirical 0.0098 0.0847 0.1660 0.2483 0.3296 0.4071

Kurtosis Theoretical 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 Empirical 3.0311 3.0417 3.0758 3.1341 3.2151 3.3148

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Table 2 Theoretical and Empirical Return Distributions, 20-Year Holding Period






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Table 3 Theoretical and Empirical Wealth Distributions, 5-Year Holding Period






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Table 4 Theoretical and Empirical Wealth Distributions, 20-Year Holding Period



Variance Theoretical 0.8768 13.3665 132.9715 1252.58 12,901.71 157,248 Empirical 0.9191 13.4262 132.5622 1260.76 13,105.80 151,869


Kurtosis Theoretical 5.2800 19.8998 145.9964 2812.51 167,853 29,585,551 Empirical 5.3284 20.0964 121.1490 836.3902 4442.10 15,021.72

Leveraged Stock Portfolios over Long Holding Periods: A ... Wilson.pdfLeveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model Asset pricing models typically allow

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