Determining the return-maximizing portfolio leverage and
its limitations

Robert A. Ott, CFAa,*, Timothy E. Zimmera

aAssistant Professor, School of Business, 1400 East Hanna Avenue, Indianapolis, IN 46227, USA

Abstract

Leverage in the risk allocation of an investment portfolio can be an effective strategy in achieving
overall portfolio goals. While the literature on portfolio leverage is robust, quantifying the amount and
discussion of its limitations are often minimized. This article focuses on the limitations by explicitly
including the volatility drag from leveraging the expected portfolio returns. Maximizing the expected
portfolio returns with respect to leverage results in a return-maximizing condition that balances the
gains from leverage with the losses in the volatility drag. The return-maximizing condition is
graphically illustrated over a range of investment returns to produce a return-maximizing leverage
curve. © 2016 Academy of Financial Services. All rights reserved.

JEL classification: D14 Microeconomics: Personal Finance; G00 Financial Economics: General

Keywords: Investment; Leverage; Investment returns; Return-maximizing investment leverage

1. Introduction

Behavioral finance recognizes the limitations of individual decision making when the
process is complicated by an overwhelming amount of information. Rather than facilitating
better decisions, the profusion of information can potentially lead to suboptimal outcomes by
intimidating individual investors and acting as an impediment to the decision-making
process. In this manner, information availability can be its own moral hazard that restricts
individual access. To compensate and simplify the decision-making process, individuals

* Corresponding author. Tel.: �1-317-788-3317; fax: �1-317-788-3586.
E-mail address: ottr@uindy.edu (R. A. Ott)

Financial Services Review 25 (2016) 415–425

1057-0810/16/$ – see front matter © 2016 Academy of Financial Services. All rights reserved.



frequently implement rules-of-thumb decision devices that approximate value-maximizing
decisions.

Individuals increasingly find themselves in the role of investment decision makers,
especially regarding retirement funds. As defined contribution retirement vehicles grow in
number, there is an ever increasing need for individuals to gain the required knowledge to
appropriately manage these accounts. According to the Employment Benefit Research
Institute (2011), the use of 401(k) and Individual Retirement Accounts (IRAs) among
working individuals 21 to 64 has increased from 1996 through 2009 in the United States. In
2009, about 33% of workers utilized a 401(k) and over 20% used IRAs. Providing invest-
ment tools and research in portfolio management are essential if the public are to become
better stewards of their retirement accounts.

A key concern for individuals managing retirement accounts is the preservation of
retirement funds while also providing for sufficient growth to maintain future purchasing
power. While people wish to be conservative with retirement funds, there is a real risk of
capital exhaustion during retirement. Excess caution with retirement funds management can
be as disastrous to a retirement account as excessive weighting towards risk. An investment
portfolio should appropriately balance risk and risk-free allocations to ensure current income
and asset growth. Using leveraged investments within the risk allocation of an investment
portfolio can help provide the desired asset growth. Current research assesses the validity of
adding leverage to the risk allocation of an investment portfolio. This article adds to current
research by deriving an algebraic equation for the return-maximizing level of leverage.

2. Review

As individuals are increasingly responsible for management of investment portfolios,
particularly retirement portfolios, the volume of research examining the methods of man-
agement has increased. Given limited interest and time availability in respect to the volume
of research, simplified approaches which approximate optimized strategies are desired and
well received by the public. One of the most cited of these approaches is the 4% rule
provided by William Bengen (1994). According to this rule, from a well-constructed
portfolio, 4% annually can be removed for consumption. It is argued that this approach will
provide the portfolio owner with sustainable spending and limit the possibility of portfolio
exhaustion. In the years since publication, numerous studies have attempted to enhance the
approach (Bengen 1997, 2001; Cooley, Hubbard, and Walz, 2003; Guyton, 2004; Guyton
and Klinger, 2006; Stout, 2006, 2008) in response to critiques. The strategy still maintains
significant interest in academic literature and public use.

Examinations of utility models are some of the earliest forms of investment portfolio
research. Seminal works on utility approaches include Merton (1969) and Samuelson (1969).
These approaches advocate constant risk portfolios with a mix of fixed return assets
combined with risk assets. The risk assets such as stocks allow for growth, while fixed return
assets such as bonds ensure a positive income stream even in the event of investment
volatility. However, requiring a fixed ratio between bonds and stocks can be problematic,
especially during periods of market volatility. In periods of a severe stock market correction,

416 R.A. Ott, T.E. Zimmer / Financial Services Review 25 (2016) 415–425



the income generated from the fixed asset portion of the portfolio may be unsustainable as
bonds are sold to purchase stocks to maintain the fixed asset ratio. Attempts to modify this
approach and create a sustainable structure during stock market declines are introduced by
Perold and Sharpe (1988). They propose selling stocks and buying bonds in market down-
turns to reduce risk and increase sustainability. The approach is called constant proportion
portfolio insurance (CPPI).

Dybvig (1995, 1999) uses an expected utility maximization model to determine the
optimal investment strategy that includes the preference for sustainable income. However,
adoption of this type of strategy is often rigorous, complex, and time-consuming. Scott et al.
(2009) successfully extend this line of research by simplifying the application. The authors
propose a rule-of-thumb for managing retirement funds while maintaining a given level of
spending which is dubbed the floor-leverage rule. Under this rule, 85% of the retirement
assets is allocated to risk-free assets such as bonds to maintain current spending levels. The
remaining 15% is allocated to a risk portfolio such as stocks that is designed to build wealth
to increase future consumption. The authors demonstrate that this method is at least as
efficient as other compensation strategies in maximizing utility.

To increase the growth potential of the risk allocation Scott and Watson (2013) recom-
mend a stock portfolio which is leveraged. The authors suggest investing in stocks leveraged
three times through exchanged-traded funds (ETFs), despite the limitations of volatility drag
as noted by Jarrow (2010) and Sullivan (2009). The choice of leverage in the Scott and
Watson model is exogenously provided by the authors. This article builds on this work by
endogenously deriving an equation to determine the return-maximizing level of leverage
unique to each investment scenario. The equation integrates the presence of volatility drag,
which had been previously noted but not incorporated. The proposed method is generalized
and available to multiple applications.

3. Methodology

The approach to managing retirement assets suggested by this paper follows that of Scott
and Watson (2013) by first setting aside a risk-free portfolio to guarantee a minimum
standard of living before using leverage to optimize the residual portfolio. The advantage of
this approach is that the level of risk-taking in the residual portfolio will have no impact on
the risk-free portfolio. The 85% risk-free allocation they suggest is a useful rule-of-thumb but
may not be appropriate for all portfolio holders. The specific allocation should account for
the size of the portfolio and the consumption patterns of the portfolio owner. The larger the
portfolio and/or the smaller the minimal consumption needs, the smaller the optimal risk-free
portfolio requirement. An assumption of the split-account portfolio model is that sufficient
income can be generated from the risk-free assets to sufficiently cover consumption needs.
In instances where this assumption does not hold, either alternate strategies or reassessment
of consumption needs may be required.

In establishing an appropriate allocation mix to achieve both the current income needs and
growth, a model should minimally consider the consumption needs of the portfolio owner
and the size of the portfolio. If the goal of the risk-free portfolio is to maintain a minimal

417R.A. Ott, T.E. Zimmer / Financial Services Review 25 (2016) 415–425



level of real purchasing power, a conservative approach would simply capitalize the mini-
mum annual consumption needs (S) by the real risk-free rate (Rf). Dividing the resulting
value by the size of the portfolio (P) provides a risk-free allocation (Z) of the portfolio as
shown in the following equation.

Z � �S/Rf�/P (1)

This simple formulation illustrates that the lower the minimum consumption needs or the
larger the portfolio, the smaller the allocation to a risk-free portfolio.

The risk-free allocation of the portfolio would be invested in real risk-free assets, such as
TIPS, to maintain its real purchasing power. The remaining percentage of the portfolio (1 –
Z) could be invested in riskier assets that focus on maximizing expected returns. As an
example, an individual with an investment portfolio of $3 million, minimal annual spending
needs of $50,000, and a real risk-free rate of 2% will have a risk-free allocation of
approximately 83%.1 The remaining 17% of the portfolio would be available to be invested
in riskier assets that can be leveraged to maximize expected return.

Eq. (1) provides a simple formula for deriving a risk-free allocation of a portfolio.
However, the assumption of an infinite time horizon, thus guaranteeing the preservation of
the portfolio’s principle in perpetuity, is overly restrictive. A less stringent assumption would
allow for a limited time horizon and permitting the principle to be spent over time. Eq. (1)
can be modified as a fixed annuity adjusted for inflation. The adjustment allows for principle
exhaustion at the end of the time horizon (T), as shown in the following equation.

Z � �S � �1 � �1 � Rf�
�T�/Rf�/P.2 (2)

For portfolio planning over a finite number of years, Eq. (2) provides a reasonable allocation
mix. As an example, by incorporating a 40-year time horizon into Eq. (2), the portfolio
allocation of risk-free assets needed is reduced from 83% to 46%.3 As 46% will now generate
sufficient income to minimally cover the consumption needs, the larger proportion of the
remaining assets can be invested in riskier assets to produce maximum future purchasing
power. Clearly, the decision to allow for possible principle exhaustion has a significant
impact on the allocation decision between risk and risk-free assets. Given the long-term
application of this method, the allocation mix can be examined annually using Eq. (2) and
the portfolio modified when appropriate. Annual adjustment ensures long term sustainability
of the portfolio in line with spending rules adjustments as proposed by Waring and Siegel
(2015).4

The second step in this approach to managing retirement assets is to maximize the
expected return by leveraging the risk portion of the portfolio. Using leverage in the risk
allocation of the portfolio can better achieve the long term capital preservation aim of the
portfolio holder. A leveraged investment provides benefit to the holder if the investment
return (RI) exceeds the cost of borrowing (RB). The leverage and spread between return and
borrowing costs can add expected incremental returns to the portfolio that is expressed in the
following equation:

RP � RI � �RI � RB� �n � 1�, (3)

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where RP is the expected return on the portfolio using a leverage multiple of n, represented
as the ratio of the portfolio to the equity investment. The expression n-1 in this equation is
the debt-to-equity ratio. As an example, an investment yielding 8% that is leveraged three
times, with borrowings costing 2%, yields an expected portfolio return of 20% [8% �
(8%–2%) (3–1)].

As long as the expected investment return exceeds the cost of borrowing, Eq. (2) suggests
that investors could benefit from ever-increasing levels of leverage. In a world of constant
returns, increasing levels of leverage generate infinite returns on the leveraged portfolio.5

However, investment returns are not constant and portfolio returns from ever-increasing
leverage are bounded. A limiting factor on investment leverage is volatility drag, as implied
in the Scott and Watson (2013) analysis. The application of leverage within a portfolio
increases the potential of higher returns. However, the downside of increased portfolio
leverage is cost of volatility drag as noted by many including Booth and Fama (1992),
Messmore (1995), Sullivan (2009), and Jarrow (2010).

Volatility drag is the difference between the arithmetic and the geometric average returns.
As an illustration, Scott and Watson (pp. 55–56) used the example of an investment that
over the course of 250 days randomly earns 1% half the days while losing 1% on the other
half. While this investment yields an arithmetic return of zero, the geometric return is
�1.24%. In this example, the volatility drag is the 1.24%. For normally distributed returns,
the relationship between the arithmetic and geometric returns can be expressed more
generally as6

Rn � RP � 1⁄2��n
2�, (4)

where RP represents the arithmetic return, Rn the geometric return, and �n
2 the variance of

the returns. In the previous example, the arthritic return is zero and the variance is 2.50%.7

If the returns were normally distributed, the geometric return on the investment would be a
�1.25%, instead of the actual �1.24%. The volatility drag is one-half the variance of the
returns.

The aim of a leveraged investment is to increase the expected portfolio return. However,
a leveraged investment also increases the volatility drag because the increase in volatility of
leveraged assets carries more risk. The general expression for the variance of an investment
leveraged n times is the product of the variance of the investment and the squared multiple
as shown below.

�n
2 � �2n2. (5)

As more leverage is applied, the return volatility of the asset increases as well as the resulting
volatility drag. For example, if the investment example of Scott and Watson illustrated above
is leveraged three times, the arithmetic return remains at zero percentage, while the geo-
metric return is estimated to lose 11.25%.8 The increasing volatility drag functions as a
binding constraint to increases in the portfolio return from the deployment of portfolio
leverage.

The return on an investment with leverage n can be derived by combining Eqs. (3) through
(5) as shown in the following expression.

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Rn � RI � �RI � RB� �n-1� � 1⁄2��2n2�,9 (6)

where one-half the variance of the leveraged investment of Eq. (5) represents the volatility
drag. When no borrowings are used, n equals one and Eq. (6) simplifies to Eq. (4). Using the
assumptions RI � 8%, RB � 2%, and � � 18%, the leveraged return of Eq. (6) can be
expressed as:10

Rn � 8% � 6% �n-1)�1.62% n2, (7)

and depicted in Fig. 1 as the leveraged return curve. The straight line represents the expected
return from leverage with no volatility drag. This leveraged return curve shows that the
expected gain from leveraging eventually is dominated by the volatility drag. In this case the
maximum return occurs at a leverage multiple of 1.85 times.11

When estimating the volatility drag in Eq. (6), the time horizon of the investor
becomes an important factor. Blume (1974) showed that the geometric average return
R(T) for a particular time horizon T can be approximated as a weighted average of the
arithmetic average return and a geometric average return estimated over N years as
follows:

R�T�n � RP �N-T�/�N-1� � Rn �T-1�/�N-1). (8)

If T equals one, the average return is the arithmetic return and the leveraged return curve is
the straight line in Fig. 1. If T equals N, the average return is the estimated geometric return
and the leveraged return curve is the curved line in Fig. 1. Depending on the time horizon
of the investor, the average return will lie between the arithmetic and geometric returns, with
the leverage return curve lying between the two curves in Fig. 1. More important, the
volatility drag will lessen as the investor’s time horizon shortens, influencing the leverage
that achieves the maximum return.

The return on an investment with leverage n shown in Eq. (8) can be modified for the
investor’s time horizon T by substituting Eqs. (3) and (6) into Eq. (8).

R�T)n � RI � �RI � RB� �n-1� � 1⁄2��2n2��T-1�/�N-1�. (9)

Fig. 1. Leveraged return curve.

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The investor’s time horizon must lie between 1 and N. If T is equal to one, the volatility drag
is zero and the portfolio return is the arithmetic return. As T approaches N, the volatility drag
becomes one-half the variance of the leveraged investment expressed in Eq. (6).

The volatility drag of leverage is a constraint on the portfolio’s performance, which is now
realized in the equation for the return on a leveraged investment. The maximum expected return
with respect to the level of leverage is determined by taking the first order partial derivative of
Eq. (9) with respect to the leverage multiple, n, and setting the equation equal to zero. Solving
for n provides the following expression for the leverage multiple that maximizes the expected return.12

n � �RI � RB�/��2�T-1�/�N-1)). (10)

An interpretation of Eq. (10) can be better understood by rearranging the equation as the
following expression.

�RI � RB� n � �2n2 �T-1)/(N-1). (11)

The left hand side of the equation represents the return on the leveraged portfolio (RP) less
the borrowing cost while the right hand side is the variance of the leveraged return for a time
horizon of T. An investor will continue to benefit from leveraging as long as the gain from
leveraging is greater than the variance of the leveraged return.

Substituting Eq. (10) into Eq. (9) gives the maximum return with leverage as the following
expression.

R�T�n � RB � 1⁄2 ���2��n2���T-1)/(N-1))].13 (12)

The maximum return equals the borrowing cost plus one-half the variance of the leveraged
returns. This equation is the straight line depicted in Fig. 2 as the leveraged curve. In examining
the application of leverage within ETF’s, Cooper (2010) found a consistent relationship between
leverage and returns over multiple time periods and indices as that observed in Fig. 2.

4. Leveraging the S&P 500

To test the return maximizing leverage of Eq. (10), we examined the monthly returns to
the S&P 500 index, including dividends, over 65 years from November 1950 through

Fig. 2. The leveraged curve (risk measured with variance).

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October 2015.14 The average monthly return was 0.688%, representing an annual percentage
rate (APR) of 8.25%. The variance of these monthly returns was 0.173% or 2.08% annual-
ized. Using Eq. (4), the estimated geometric return is 0.601%, representing an APR of 7.21%.
The actual geometric return over the 65 years was 0.600% or an APR of 7.20%. In this data
series, Eq. (4) slightly overestimates the actual geometric return.15

To calculate the return maximizing leverage, a borrowing rate must be specified. We
chose to simply use the 3-month Treasury bill rate as a proxy for the borrowing rate. Over
the 65 years, the 3-month Treasury bill rate averaged 0.366% monthly or 4.40% APR. Using
the S&P 500 average return and the variance, the return-maximizing leverage over the 65
years implied by Eq. (10) was 1.85 times. To examine the accuracy of this measure, we
calculated the leverage that maximizes the actual return. Calculating the actual monthly
return on a leveraged portfolio using Eq. (3), we solved for the leverage multiple that would
have generated the greatest return over the 65 years. This optimal leverage multiple was 1.77
times, 4.3% percent less than our estimated value.

5. Discussion

The derived return-maximizing leverage in Eq. (10) allows for easy application within the
risk allocation of an investment portfolio. Using the assumptions from Eq. (7)16 in addition
to N � 85, and T � 40, the leverage multiple that maximizes the portfolio return is 3.99
times. As the investor’s time horizon lengthens, the leverage multiple is reduced toward 1.85
times, the leverage multiple if T � N. Scott and Watson suggest a leverage multiple of 3
implies a 53-year time horizon. Rather than an exogenous selection of leverage, Eq. (10)
provides an investor the means to tailor the leverage to the specific circumstances. The
introduction of the volatility drag explicitly in the portfolio return differentiates this study
from previous studies. Without volatility drag, ever-increasing levels of leverage increase
portfolio returns. Volatility drag limits the portfolio returns from leveraging, eventually
outweighing the gains. Continuing the same example, the influence of volatility drag on
portfolio returns is displayed in Fig. 3 which illustrates the expected leveraged return across
the variances of the leveraged portfolio.17 As the expected investment return increases, the

Fig. 3. Risk/return with leverage (risk measured with variance).

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leverage multiple that maximizes the portfolio return also increases, as expressed in Eq. (12)
and depicted in Fig. 2.

Once the return-maximizing level of leverage is determined, Scott and Watson (2013)
suggest using leveraged exchange-traded funds (ETFs) to achieve the desired level of
leverage. An investor can simply choose the ETF that uses leverage closest to the return-
maximizing derived leverage. If the exact leverage required is not available, or more
precision is sought, it is possible to invest in multiple ETFs with different leverages,
adjusting the investment weight between them to achieve the desired aggregate leverage.

The approach assumes a normal long-term distribution of market returns in an attempt to
simplify the model. Acknowledging the non-normal long-term market returns, and incorpo-
rating terms to account for distribution skewness and kurtosis within market returns would
add to the basic model. This area would be open to further research and certainly expand the
application of the approach presented in this article.

6. Conclusion

This article provides endogenously derived equations designed to assist in managing a
retirement investment portfolio. The approach is to allocate retirement funds between a
risk-free portfolio to support minimum annual consumption spending and a risk portfolio
leveraged to provide long-term purchasing power. While leveraging the risk portfolio can
enhance expected portfolio returns, this article emphasizes the limitations by explicitly
including the volatility drag of leveraging. If the spread of the leveraged portfolio return over
the borrowing costs is greater than the variance of the leveraged return, then the investor can
enhance expected portfolio returns by increasing leverage. Otherwise, the volatility drag of
leveraging diminishes the expected portfolio returns. This limitation occurs at higher lever-
age multiples if an investor has a shorter time horizon.

While this article focuses on using leverage in a retirement portfolio, the approach is
applicable to other issues involving the use of leverage to enhance portfolio returns. In
particular, hedge fund managers and managers of financial institutions typically operate with
high levels of leverage and may experience the limitations from volatility drag. This topic is
also relevant in addressing the optimal capital structure for a firm in corporate finance.

Notes

1 This allocation is close to the 85% suggested by Scott and Watson (2013).
2 As T approaches infinity, the influence of the annuity diminishes back to 1/Rf and Eq.

(2) reverts back to Eq. (1). In such a case, the principle never depletes and the portfolio
will exist in perpetuity.

3 Scott and Watson (2013) used 40 years in their example.
4 Waring and Siegel (2015) propose a spending rule that constantly adjusts the annuity

through time so as to never deplete the portfolio.
5 This is Modigliani-Miller’s (1958) proposition II.

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6 Bodie, Kane, and Marcus (2011) p. 132.
7 �n

2 � 250 � 0.01% � 2.5%.
8 RI � 0% and �2 � 22.5% [or 250 � 0.01% � 9]. In the Scott and Watson example,

the actual loss from 3 times leverage is �10.64%, p. 56.
9 This equation can also be expressed as: Rn � RB � (RI � RB) n � 1⁄2(�2n2).

10 These are the assumptions used by Scott and Watson (2013, p. 49).
11 When n � 0, the portfolio is leveraged �1 times. This is equivalent to selling the

investment portfolio and investing it at the borrowing rate.
12 Rn � RI � �RI � RB� �n-1)�(1⁄2�2)(n2���T-1)/(N-1)),

�Rn/�n � �RI � RB� � �2n��1⁄2�2���T-1)/(N-1)),

0 � RI � RB � �n���2���T-1)/(N-1)),

n � �RI � RB�/��2� ��T-1)/(N-1)).

13 Eq. (12) can also be expressed as a function of the spread between the investment yield
and the borrowing cost as R(T)n � RB � 1⁄2 (RI � RB)2/�2 (T-1)/(N-1)).

14 Data series is from Yahoo Finance.
15 The precise relationship is derived from continuously compounded rates that are

lognormally distributed. See Jacquier, Kane, and Marcus (2003).
16 RI � 8%, RB � 2%, and � � 18%.
17 Figure 2 is the same curve as Figure 1, but with the variance as the dependent variable

instead of leverage.

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