Optimal Portfolios for the Long Run
David Blanchett, CFA, CFP
®
Head of Retirement Research
Morningstar Investment Management
[email protected]
Michael Finke, Ph.D., CFP
®
Professor and Ph.D. Coordinator at
the Department of Personal Financial
Planning at Texas Tech University
[email protected]
Wade D. Pfau, Ph.D., CFA
Professor of Retirement Income
at the American College
[email protected]
November 15, 2013
Abstract
There is surprisingly little agreement among academics about the existence of time diversification,
which we define as the anomaly where equities become less risky over longer investment periods. This
study provides a thorough analysis of time diversification conducted, using 113 years of historical
data from 20 countries (over 2,000 years of total return data). We construct optimal portfolios for 20
different countries based on varying levels of investor risk aversion and time horizons using both
overlapping and distinct historical time periods.
We find strong historical evidence to support the notion that a higher allocation to equities is
optimal for investors with longer time horizons, and that the time diversification effect is relatively
consistent across countries and that it persists for different levels of risk aversion. We also
note that the time diversification effect increased throughout the 20th century despite evidence of
a declining risk premium. Although time diversification has been criticized as inconsistent with
market efficiency, our empirical results suggest that the superior performance of equities over longer
time horizons exists across global equity markets and time periods.
The authors thank Alexa Auerbach, Roger Ibbotson, Paul Kaplan, Lubos Pastor, and Hal Ratner
for helpful edits and comments.
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 2 of 27
Optimal Portfolios for the Long Run
The primary critique of time diversification is theoretical and the primary defense empirical.
Samuelson (1963) and Bodie (1995) point out that if stocks are less risky in the long run there
is a free lunch for long-run equity investors. Bodie (1995) emphasizes the fact that time diversification
violates the Black-Scholes option pricing model. If time diversification exists, then options
hedging long-run equity risk should reflect a decreasing likelihood of loss over longer time periods
(they don’t) .
1
Historically, stocks in the United States haven’t been very risky over long holding periods. Campbell
and Viceira (2003) argue that empirical evidence shows that stock returns are not independent
and identically distributed over time (they tend to mean revert), which implies that long-run stock
returns may be predictable and that long-run investors should overweight equities . The authors
find that the annualized standard deviation of stocks is actually lower than annually reinvested T-bills
after fewer than 30 years. This leads to the counterintuitive conclusion that risk-averse investors
should demand stocks as a hedging strategy against a long-term drop in real consumption.
2
Other empirical studies of optimal equity allocation over longer time horizons show equally strong
support for time diversification. Dolvin, Templeton, and Rieber (2010) find that a 100% stock
portfolio dominated other strategies for a retiree with a 40-year time horizon over historical rolling
periods in the United States. Estrada (2013) finds evidence that stocks consistently outperformed
bonds in an international sample over longer holding periods. Barberis (2000) shows how return
predictability observed in U.S. data means that all but the most risk-averse investors would hold 100%
equities for a 10-year or longer investment horizon. However, uncertainty over the persistence
of time diversification will reduce the optimal allocation to equities.
The most convincing explanation for time diversification (and the mean reverting equity prices that
cause time diversification) is perhaps sentiment-driven investor valuation (Barberis, Shleifer and Vishny,
1998). Investors require a larger risk premium during recessions and are more risk tolerant during
an expansion. This creates short-run volatility through regular swings in prices, but these prices even
out over the long-run as valuations move toward their long-run average. Time diversification relies
on these regular but irrational swings in values that follow business cycles. Our analysis is similar in
scope to work by Hanna and Chen (1997), although they focus entirely on U.S. returns using the Ibbotson
data series and their focus is primarily on subjective and objective risk tolerance.
Black-Scholes could also be inconsistent with time diversification if security returns don’t follow a Weiner or Brownian process
Campbell and Viceira (2003) also note that the mean reverting returns, which are necessary for time diversification, imply that a buy and hold strategy
is suboptimal. Those who practice time diversification should be aware that the logic which underlies time diversification is also consistent with
valuation-based market timing.
1
2
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 3 of 27
International Evidence of Time Diversification
In order to judge the reliability of time diversification, we conduct an empirical analysis of
different portfolio combinations among 20 of the largest global equity markets. If time diversification
exists outside the United States, then we may be more confident that behavioral factors exist
which consistently produce mean-reverting equity returns and provide opportunities for long-run equity
investors. If they do not exist or are inconsistent, this may call into question previously held
beliefs about the importance of time horizon when determining portfolio allocations. Our objective
in this paper is not to determine the reasons why time diversification may or may not exist in
the future; it is to estimate empirically the existence of global time diversification and optimal portfolio
allocation for those who wish to engage in time diversification strategies.
Equities have historically yielded higher average returns than bonds or cash, something commonly
referred to as the equity risk premium (ERP). For example, between 1900 and 2012, stocks outperformed
bonds in the United States by 6.24% per year, on average. Globally, the average ERP has been a
slightly lower 4.42% over the same time period (approximately 2.0% less). The ERP has been positive
for each of the 20 countries in the Dimson, Marsh and Staunton (DMS) dataset (obtained from
Morningstar Direct) from 1900 to 2012.
In Table 1 we show the historical ERPs for stocks against cash and bonds, as well as the geometric
annual real returns of cash (bills), bonds, and stocks. We include t-statistics for the ERPs for individual
countries as well as the aggregate average ERPs across all the countries. While all of the ERPs
are positive, most of the individual country ERPs do not have t-statistics greater than 2, especially when
the ERP is proxied against bonds versus cash, which is likely a better proxy for longer term investors.
This suggests that while stocks have outperformed cash and bonds on average, the outperformance is
far from certain. The average ERPs across the 20 countries is positive (~4%) with significant t
statistics (greater than 12). While we can say that stocks have outperformed cash and/or bonds with
some degree of certainty, there is significant variation across countries regarding the level and
the consistency of this outperformance.
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 4 of 27
Table 1: Historical Country Returns: 1900 to 2012
Equity Risk Premium (ERP) (%)
vs Cash Austria
Australia
Belgium
Canada
Denmark
Finland
France
Germany
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
South Africa
Spain
Sweden
Switzerland
United Kingdom
United States
Average
t-statistic
vs Bonds
Geometric Average Real
Annual Return (%)
Cash
Bonds
9.63
4.87
-8.21
4.04
6.55* 5.60* 0.701.60
2.73
2.25
-0.26
0.20
4.09*
3.38
1.54
2.23
2.79
1.78
2.16
3.18
5.75*
5.32
-0.51
-0.10
5.95*
2.96
-2.81
0.01
5.56
4.85
-2.38
-1.71
3.16
2.61 0.67
1.20
5.59
3.36
-3.64
-1.55
5.75*
4.83
-1.88
-1.03
4.20
3.26
0.62
1.54
4.19* 3.69* 1.662.15
2.94
2.24
1.16
1.85
6.27* 5.38* 0.981.84
3.12
2.05
0.28
1.33
3.63
2.91
1.90
2.61
3.41
1.99
0.81
2.21
4.25* 3.66* 0.931.51
5.31* 4.17* 0.902.01
4.74
3.56-0.27
0.85
12.45 12.84 n/a
n/a
Stocks
0.63
7.30
2.46
5.70
5.01
5.21
2.98
3.05
3.85
1.75
3.76
4.85
5.93
4.13
7.32
3.41
5.60
4.25
5.23
6.26
4.43
n/a
* t-statistic > 2
Source: Dimson, Marsh, and Staunton
The majority of past empirical research on time diversification has been based on U.S. stock returns
using historical periods either from 1926 to present (Ibbotson data) or 1802 to present (Siegel
data). For our analysis we use historical real stock returns from the DMS dataset. The DMS dataset
consists of historical annual returns from 20 different countries from 1900 to 2012 (113 years
of data). This results in a total of 2,260 years of return data, which are roughly 10 times the annual
returns reviewed by Siegel (2008) and approximately 25 times the annual returns available
in the Ibbotson data series.
3
The total market capitalization of the 20 countries included in the DMS dataset represented 94.6%
of the total market capitalization of listed companies in 1988 based on data obtained from the
World Bank website . This percentage declined to 70.5% by 2012, yet still represents the majority
of global publicly traded securities, especially over the full historical period considered.
4
We focus on the annual real returns earned by local investors in bills (cash), bonds, and stocks
for the 20 respective countries in the DMS dataset. We use real returns under the assumption that
investors in each country seek to maintain some level of inflation-adjusted wealth within that country.
Austria, Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, South Africa, Spain,
Sweden, Switzerland, the United Kingdom, and the United States.
3
http://data.worldbank.org/indicator/CM.MKT.LCAP.CD
4
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 5 of 27
We use rolling returns for our analysis, both overlapping and distinct periods. For overlapping
analysis we use the maximum number of return years available for each test period that roll forward
through time. For example, our one-year return model would include each year from 1900 to
2012; however, for the 20-year period the last rolling set of returns would be assumed to begin in
1993. The use of overlapping periods results in underweighting the earliest and latest returns
in the dataset, since, for example, the years 1900 and 2012 will only be used in a single 20-year
simulation while the middle years (for example 1950) would be used in 20 different rolling periods.
This is important given the poor relative performance of 2008 since it will show up less
frequently than other periods.
For the distinct period analysis we assume the first year of available data (1900) is the first
available investing year, where future periods roll forward through time. For example, a five-year
investment period would be between 1900 to 1904, 1905 to 1909, etc. Unlike the overlapping
analysis, we limit the distinct analysis to investment periods from 1 to 10 years (versus 1 to 20 years
for the overlapping analysis). We do this to ensure some minimum number of test periods,
which would be 11 given the data available.
We use the cumulative real growth of the portfolio value (i.e., final inflation-adjusted wealth
over the period) to represent the “return” of the portfolio. Instead of using a definition of risk such
as standard deviation, which treats outcomes above and below the target goal as equally risky,
we use a utility function, which we believe better approximates how investors feel about good and
bad outcomes. A utility function also allows us to consider cumulative wealth as the outcome
versus annualized return dispersion. More specifically, we use a Constant Relative Risk Aversion
(CRRA) utility function, as depicted in equation 1.
U
=
t
[1]
Equation 1 allows us to estimate the utility (U) received for a given value of wealth (W) for
a given investment period (t). We estimate the ending inflation-adjusted (real) value of the portfolio
for a given asset allocation at the end of the period for a given risk aversion coefficient (γ).
The risk aversion coefficient measures the degree to which the investor (or decision-maker) is averse
to taking risks. The larger the coefficient, the more risk averse the investor. We test levels
of risk aversion from 1 to 20 in increments of 1 for this analysis.
5
We determine the optimal allocation to cash, bonds, and stocks for each scenario using a nonlinear
optimization routine. The goal of the optimizer is to solve for the allocation that maximizes the
resulting utility in equation 1, for a given investment period, risk aversion, and historical country returns.
The two constraints for our optimization are that the total allocation across the three asset classes
must be equal to 100% and that none of the asset class weights can be negative.
5
Technically the lowest value is 1.001, since a value of 1 would result in an infinite negative utility in our utility function.
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 6 of 27
Results
For our first test we examine how the optimal allocation to equities changes across varying
test periods for each of the different countries. We do this by determining the optimal allocation
to equities for each country over each investment period (which is defined as the portfolio
with the highest utility) and then running an ordinary least squared (OLS) regression where we
regress the optimal equity allocations (Eq%) against the respective time periods (t), as
noted by equation 2. For example, we find
[2]
The intercept (α) in equation 2 can be interpreted as the optimal equity allocation if there is
neither an advantage nor a disadvantage in long holding periods. It can also be interpreted as the
optimal equity allocation for a single period since the first observation of the independent
variable t = 1. A positive slope (β) indicates a positive relationship between investment horizon
and utility and a negative slope, the opposite. A negative slope would be consistent with
the work of Pastor and Stambaugh (2012).
Figure 1 includes an example of how we use equation 2 to analyze the results obtained from
equation 1 for a given country and level of risk aversion. With equation 1 we solved for the optimal
equity allocation that maximizes the utility of final real wealth for different potential portfolio
combinations of cash, bonds, and stocks given an investor’s risk aversion. Figure 1 includes these
optimal equity allocations (the blue diamonds) from years 1 to 20 for the overlapping analysis
for a risk aversion (γ) of 4. Within Figure 1 we have run a simple linear regression (equation 2) on
the resulting equity equations against the investment period to describe how the allocations
change over time. As we can see in Figure 1, the optimal equity allocation increases for
longer holding periods (as t increase), and therefore the slope is positive (2.99%), suggesting
time diversification has existed historically in the United States.
Figure 1: Optimal Equity Allocation for Different Investment Periods for U.S.
Historical Overlapping Return Periods
Optimal Equity Allocation (Eq%)
100
y = -0.0299x + 0.4502
R = 0.95264
2
80
60
40
20
0
13 5 7 911 13 151719
Investment Period (t)
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 7 of 27
Figure 1 also demonstrates how to interpret the results we include later in Tables 2 and 3.
In Figure 1 we note an intercept (α) of 45.02% (which we will assume is 45% for simplicity purposes)
and a slope (β) of .0299 (which for simplicity purposes we will assume is .03). Therefore the
optimal historical allocation to equities for an investor with a five-year holding period would be 60%
stocks, which would be determined by: 45% + 5(3%) = 60%.
We display the intercept and slope values for the regressions for four different levels of risk
aversion (2, 4, 8, and 16) for each of the results for each of the 20 different countries for the overlapping
analysis in Table 2 and for the distinct-period analysis in Table 3. We include t-statistics for
the average slope values in both Tables 2 and 3 to convey the statistical significance of the average.
We do not include t-statistics for the individual countries for the overlapping analysis (Table 2)
because the periods are not independent. While there are techniques to potentially adjust for this, our
primary concern is not whether time diversification exists for a given individual country, but its
overall international applicability. We are also able to include t-statistics for individual country slopes
for the distinct-period analysis in Table 3.
Table 2: Regression Equity Allocation Results: Overlapping Periods
y = 2
y = 4
y = 8
y = 16
Intercept Slope
(%)(%)
Intercept Slope
(%)(%)
Intercept Slope
(%)(%)
Intercept Slope
(%) (%)
Austria
100.000.00
62.222.30
24.373.85
15.72 3.68
Australia
81.69
90.06
93.96
94.37
Belgium
61.372.45
45.762.78
40.632.26
Canada
97.260.19
60.122.66
25.894.31
7.13 5.33
Denmark
74.401.74
46.523.48
26.724.59
22.50 4.45
Finland
82.441.20
81.431.27
89.500.60
84.78 0.83
France
81.381.28
77.291.59
72.961.84
69.14 2.07
Germany
94.37
0.40
98.04
-0.35
79.38
0.51
73.94
Ireland
58.40
2.50
19.47
4.00
9.39
2.18
-3.39
1.93
Italy
72.45
1.87
78.19
0.66
86.31
-0.27
91.44
-0.79
Japan
72.761.86
43.223.91
35.804.32
30.39 4.54
Netherlands
65.121.98
44.171.64
26.042.17
13.94 2.95
New Zealand
85.570.96
54.662.87
30.603.97
5.76 5.31
Norway
54.03
2.1323.98
2.56
South Africa
100.00
0.0072.41
1.90 35.25
3.86 25.36
3.66
Spain
37.53
2.14
10.28
2.04
0.98
1.69
3.31
2.15
Sweden
66.82
0.53
30.66
0.96
1.49
1.92
-8.17
2.00
Switzerland
51.38
2.66
24.41
1.16
1.68
1.25
-3.06
0.94
United Kingdom
91.750.58
42.963.48
16.874.72
12.87 5.20
United States
85.550.98
45.022.99
24.002.96
11.44 3.10
Average
75.711.29
n/a6.35
52.542.10
n/a7.45
36.202.48
n/a6.84
29.37 2.67
n/a 6.44
t-statistic
0.38
0.07
-0.11
-0.07
51.46 2.07
0.64
2.08
3.03-11.54
3.44
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 8 of 27
For the risk aversion levels of 2, 4, and 16 in Table 2, 90% of the slopes are greater than 0
(18 of 20), while 95% of the slopes are greater than 0 (19 of 20) for a risk aversion level of 8. Those
countries with negative slopes or slopes that are zero tend to be relatively small in absolute
terms and already have relatively high intercepts (i.e., base equity allocation where t=0). The intercept
and slope values vary significantly by country and by level of risk aversion; however, the average
slope for each of the four levels of risk aversion is positive and the minimum t-statistic is 6.35 (where
γ =2). This suggests the optimal allocation to equities has historically been higher over longer
time periods.
Our results for the distinct-period analysis (Table 3) are very similar to Table 2, especially in terms
of the positive average slopes across the countries and the relatively large t-statistic. The average
slope values are higher for the distinct analysis (Table 3) when compared to the overlapping analysis
(Table 2); however this can be attributed to the different time periods for the two studies (the distinct
analysis considers 1 to 10 years versus 1 to 20 years for the overlapping analysis).
Table 3: Regression Equity Allocation Results: Distinct Periods
y = 2
Intercept (%)
y = 4
Slope
(%)
y = 8
Intercept Slope
(%)
(%)
Intercept (%)
y = 16
Slope
(%)
Intercept (%)
Slope
(%)
Austria
100.00 0.00** 60.09 3.0**
26.10 5.4
10.403.8*
Australia
71.73
1.9*
74.56
2.7*
80.42
2.4
83.44
0.0**
Belgium
54.93
3.5*
33.24
5.0*
16.43
8.1
16.48
2.8*
Canada
100.00 0.0**
41.92 6.2
12.26 8.6
-3.574.9
Denmark
61.51
4.7
32.79
6.7
24.34
5.8
20.14
4.1
Finland
72.34
3.4*
58.67
4.9
84.60
1.5
92.80
0.8**
France
68.06
3.9
64.21
4.8*
60.05
5.3
55.27
2.2
Germany
88.20
1.5
99.61
0.1**
100.00
0.0
100.00
Ireland
63.52
2.4**
32.71
3.3*
21.41
1.0
10.64
1.9
Italy
51.21
5.9
38.64
7.3
46.10
6.2
54.36
-0.2**
Japan
48.61
4.0**
-0.81
9.8*
-1.23
9.8
-1.52
4.4
Netherlands
65.24
2.7*
37.80
3.5
16.86
4.7
5.63
2.8
New Zealand
87.71
1.4**
51.39
5.3
27.01
6.1
11.70
5.2
Norway
51.33
4.3*
24.73
5.1**
12.77
4.9
9.09
3.3
South Africa
100.00 0.0**
Spain
50.99
Sweden
61.23
Switzerland
United Kingdom
0.7**
55.16 5.8*
14.78 9.6
-0.4**
29.60
-1.6**
19.49
-1.8
13.74
1.9**
31.63
1.6**
8.97
2.3
1.01
2.0
55.51
3.0**
25.48
2.3**
6.69
2.2
0.00
0.9
85.49
1.6*
36.36
5.5
8.41
7.3
-10.79
4.9
United States
79.21
2.1*
38.06
5.6
18.01
5.8
11.20
3.1
Average
70.84
n/a
2.39
6.03
43.27
n/a
4.34
7.45
30.17
n/a
4.76
6.44
24.07
n/a
2.62
6.80
t-statistic
1.343.6
1.4*
** t-statistic > |2|
* t-statistic > |4|
©2013 Morningstar. All rights reserved. This document includes proprietary material of Morningstar. Reproduction, transcription or other use, by any means,
in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
wholly owned subsidiaries of Morningstar, Inc. The Morningstar name and logo are registered marks of Morningstar.
Page 9 of 27
For both tests we find that the slopes, i.e., the relative attractiveness of equities over longer investing
durations, increases for higher levels of risk aversion, while the average intercept decreases.
This suggests stocks become increasingly attractive to risk-averse investors for longer investment
periods, consistent with Campbell and Viceira (2003).
In Figure 2 we average the optimal equity allocation across the 20 countries included in our
overlapping analysis for the same risk aversion levels in Table 2 in Panel A and for the distinct periods
in Table 3 in Panel B. We add the average results for γ =1 for comparison purposes . Consistent
with Tables 2 and 3, we see that the optimal equity allocation is lower for shorter investment periods
and for higher levels of risk aversion. The optimal allocation increases significantly, though, for
longer investment horizons, and it increases faster for investors with higher levels of risk aversion. Even
for a highly risk-averse investor with a risk aversion coefficient of 16, the equity allocation has
been approximately 60% for an investment period of 10 years on average across the 20 countries,
versus approximately 25% for an investment horizon of a single year for either the overlapping
or distinct-period test.
6
Figure 2: Average Equity Allocation by Risk Aversion Coefficient and Investing Period
Panel A: Overlapping Periods
Panel B: Distinct Periods
100
100
y=1
y=1
y=2
Equity Allocation (%)
80
y=4
y=8
60
y = 16
40
Equity Allocation (%)
y=2
80
y=4
y=8
60
y = 16
40
20
20
0
0
1
4
7
10
Year
13
16
19
1 2 3 4 5 6 7 8 9 10
Year
Figure 3 provides a more detailed perspective of the optimal equity allocations across each
of the different time periods and levels of risk aversion considered for the analysis (a total of 400
scenarios) for the overlapping analysis. We use the overlapping periods because it contains
a greater number of test periods (up to 20 years versus only up to 10 years for the distinct analysis).
Figure 3 includes the optimal equity allocation for each of the U.S. scenarios (Panel A), as
well as the average equity allocation for each scenario across the 20 countries (Panel B). Panel B
of Figure 3 contains a cross section of the results in Panel A of Figure 2.
As a reminder, the actual risk aversion coefficient is 1.001, since a value of 1 would result in an infinite negative utility in our utility function.
6
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Page 10 of 27
Figure 3: Optimal Equity Allocation by Risk Aversion Coefficient and Investment Period
Panel A: Unites States
Panel B: 20 Country Average
13
16
13
10
7
4
13
16
1
19
20
16
Investment Period (Years)
13
10
7
4
Very High
19
20
10
Very High
16
7
0–10
11–20
21–30
31–40
41–50
51–60
61–70
71–80
81–90
91–100
Level of Risk Aversion
10
Level of Risk Aversion
7
4
Very Low
4
1
Very Low
1
Equity
Allocation (%)
1
Investment Period (Years)
Optimal U.S. equity allocations have been relatively similar to the 20-country average. While U.S.
portfolios tend to have slightly lower equity allocations for shorter investment periods and for higher
levels of risk aversion, and slightly higher equity allocations for longer investment periods and
for lower levels of risk aversion, the international results are similar. These results suggest that the
empirical historical time diversification effect is robust in international data and similar in effect
to U.S. data.
While historical optimal equity allocations in the United States are similar to the 20-country
average, some countries differed significantly. We demonstrate these differences in Figure 4 (again
for the overlapping analysis), which include the optimal equity allocation for Australia (Panel A)
and Switzerland (Panel B). Optimal Australian portfolios are considerably more aggressive
than the optimal Swiss portfolios, but in both cases there is some degree of time diversification
where the optimal equity allocations tend to increase across an investment period for the
same level of risk aversion, even if only slightly.
Figure 4: Optimal Equity Allocation by Risk Aversion Coefficient and Investment Period
Panel A: Australia
Panel B: Swizterland
13
16
13
10
7
Investment Period (Years)
4
1
16
19
20
16
13
10
7
4
Equity
Allocation (%)
0–10
11–20
21–30
31–40
41–50
51–60
61–70
71–80
81–90
91–100
Very High
19
20
13
Very High
16
10
Level of Risk Aversion
10
7
4
Very Low
7
1
Level of Risk Aversion
4
Very Low
1
1
Investment Period (Years)
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 11 of 27
The considerable differences in the optimal allocation to equities for different levels of risk
aversion and over different investment periods (e.g., in Figures 3 and 4) could potentially be a result
of the different level of historical returns experienced by each country. It may be that those
countries that experience higher historical ERPs are more likely to have positive estimated slope
values (β in equation 2). We note, for example, that Australia has had one of the higher
historical ERPs while Switzerland has had one of the lower ERPs. If this relationship were to
persist across countries, it could be that the time diversification effect could be explained
entirely as a result of higher returns (potentially either stock returns, ERP, or some combination
of the two).
We test this theory by regressing the estimated slope values (β in equation 2), which are proxies
for how much the equity allocation should increase based on the length of the holding period (i.e., time
diversification), against the average historical annual geometric equity risk premium (versus
bills and bonds individually), real cash return, real bond return, and real stock return for the individual
countries over the entire historical time period (1900 to 2012), which are included in Table 1.
We find no significant or consistent relationship between any of the five return series and the
estimated slope values. As an example we display the results for the ERP versus bonds and slopes (β)
for the overlapping analysis where γ = 4. Note the t-statistic for slope in Figure 5 is only -1.22).
In other words, we cannot explain the apparent historical existence of time diversification at the
individual country level through historical returns alone, since there is no meaningful relationship
between the optimal level of time diversification and the historical returns across countries.
Figure 5: The Relationship Between ERP and Time Diversification
5.00
y = -0.2542x + 0.0294
R = 0.07681
2
Slope Where y = 4 for Overlapping
Period Analysis (%)
4.00
3.00
2.00
1.00
0.00
-1.00
0.00
1.002.003.004.005.006.00
Equity Risk Permium (versus Bonds)
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Page 12 of 27
Capturing the Serial Correlation
Our finding that equities have been more attractive investments for investors with longer investing
durations suggests that there is some level of serial correlation (or autocorrelation) between
returns across time. In other words, the return experienced this year is somehow (generally negatively)
related to the return in the previous year (and potentially additional years before that). It is
possible to capture this relationship by running regressions against the return for stocks (S) in
a given year (t) versus the return in previous years (t –1, t – 2, etc ), as noted by equation 3.
[3]
Equation 3 is an autoregressive model with five lags, which is also referred to as an AR (5) model.
We acknowledge our selection of an AR(5) model is somewhat subjective and we selected
it without attempting to select an AR model that optimally fits the data to reduce the potential bias
in our analysis (e.g., we did not run any type of correlograms to fit the data). We run the
AR(5) model (equation 3) against the annual real historical equity returns for each of the 20
countries. The results for these regressions are included in Table 4.
Table 4: AR(5) Coefficients by Country
Intercept (%) t-1
t-2
t-3
t-4
t-5
Austria
11.73
-0.077-0.124 0.002 -0.045 -0.084
Australia
4.77
0.090
-0.106
-0.018
Belgium
5.63
0.107
-0.174
0.000
Canada
10.54
0.102 -0.230**-0.043 -0.126
Denmark
10.81
-0.169-0.093-0.047-0.139 -0.116
Finland
8.99
0.258**
-0.180
0.046
-0.131
France
5.19
0.179
-0.047
0.068
-0.205**
0.087
Germany
8.28
-0.108
0.168
-0.005
0.084
0.020
Ireland
6.80
-0.009
-0.084
0.081
-0.167
0.161
Italy
7.19
0.020-0.147-0.046-0.077 0.009
Japan
8.52
0.233**
-0.1490.0840.022 -0.206**
Netherlands
6.96
0.057
-0.0940.1110.012-0.053
New Zealand
8.92
-0.060
Norway
11.40
-0.095-0.201 -0.176 0.049 -0.146
South Africa
12.74
0.030-0.141-0.084-0.078 -0.074
Spain
4.49
0.280**
0.057
0.002
-0.256**
Sweden
8.09
0.116
-0.134
0.027
-0.063
0.023
Switzerland
7.33
0.179
-0.144
-0.067
0.039
-0.210**
United Kingdom
9.15
-0.086
-0.112
-0.075
0.045
-0.011
United States
11.50
-0.011 -0.230**0.009 -0.070 -0.116
Average
8.45
0.052
-0.013
-0.109
0.223**
0.004
0.215**
-0.041
-0.262**
-0.068**
-0.127
0.045
-0.154
0.044
-0.065
0.097
-0.044
** t-statistic > |2|
Unlike Tables 2 and 3, where the individual country results were generally consistent, the AR(5)
results are considerably less so, and the average results lack the same level of statistical significance.
We see considerable differences across countries with respect to the relationship between past
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 13 of 27
year returns and subsequent returns. We do see that the single lag period (t-1) tends to be
positive, which suggests the existence of momentum (a relationship as noted by Jegadeesh and
Titman (1993) among others), although the coefficient is not-statistically significant. In contrast,
the second and fourth lagged periods are usually negative and the average coefficient for
the 20 countries is significant at the 5% level (with values of -5.11 and -2.66, respectively). The
fact the second and fourth lags are negative is important within the context of time
diversification, since it implies mean reversion, and is likely a key reason for the results
noted in the previous section.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 14 of 27
Monte Carlo Simulation
The strong international evidence of time diversification has important implications for financial
modeling; in particular Monte Carlo simulations, which are commonly used to estimate the range of
potential portfolio outcomes for different types of clients. The majority of estimates used in
Monte Carlo simulations do not incorporate any type of autoregressive model (i.e., assume that stock
returns are independent and identically distributed, or iid), and implicitly assume time diversification
doesn’t exist. This has important implications for equity investors over longer time horizons.
In order to better approximate the historical risk of equities over time, we use the results for
the United States in the AR(5) model in Table 4. We select U.S. equity returns since they are relatively
similar to the 20-country average, although in reality any country could be selected. We use
equation 3 to model equity returns, where εR is an independent white noise that follows a standard
normal distribution with a mean of 0% and a standard deviation of 20% (which is the approximate
standard deviation of the error terms), and the regression coefficients are based on the values
in Table 4.
To capture the real return of bonds (rrb), we use equation 4, where the intercept (ib) equals
1.75%, the slope coefficient (βs) to the return on stocks (rrs) equals .1, and the standard deviation
for the error term (εB) equals 10.2%. When combined, this yields a model that has return,
standard deviation, and correlations that are nearly identical as the historical real return on stocks
and bonds over our analysis period.
[4]
We conduct a 10,000 run Monte Carlo simulation and apply the same type of analysis we did
previously where we test investment periods from 1 to 20 years (in 1-year increments) and risk aversion
coefficients from 1.001 to 20 (again, in increments of 1) for the 21 different stock and bond
combinations where the allocations differ in 5% increments. The optimal allocation by risk aversion
and investment period are included in Figure 6.
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Page 15 of 27
Figure 6: Optimal Equity Allocation by Risk Aversion Coefficient and Investment Period
Panel A: AR(5) Return Model
Panel B: No AR Model
1
(Very
Low)
1
(Very
Low)
(Very
Equity
Equity
0–10
Low)
Allocation
Allocation
11–20
11
4
4
Level of Risk Aversion
Very High
Very High
# 21–30
0%-10%
0%-10%
# 31–40
11%-20%# 41–50
11%-20%
#
21%-30%# 51–60
21%-30%
# 61–70
31%-40%# 31%-40%
# 71–80
41%-50%# 81–90
41%-50%
#
51%-60%# 91–100
51%-60%
#
61%-70%# 61%-70%
#
(Very
71%-80%# 71%-80%
#
High)
81%-90%# 81%-90%
#
91%-100%
# 91%-100%
Level of Risk Aversion
Level of Risk Aversion
Level of Risk Aversion
Level of Risk Aversion
Level
of Risk Aversion
#4 4
#
#7
77
7
7
#7
#
#
10
10
10
10
#10
10
#
#
13
13
13
13
#13
13
#
#
16
16
16
16
#16
16
(Very
(Very ##
(Very
19
19
19
19
High)
High) #19
High)
19
#
20 16 1320 1016 713 410 17
1320 16
1016 13
713 10
410 17
4
1 20 16 20
7
44
1#
1
20 16
13
10
7
4
1
#
Investment
Period
(Years)
Investment Period
(Years)
Investment
Period (Years)
Investment
Period
Investment Period
(Years)
Investment
Period(Years)
(Years)
44
Very Low
(Very
Low)
Very Low
11
Equity
Allocation (%)
In Panel B, with the no AR model, we see very little evidence of time diversification. Panel B is
what the previous figures would look like if time diversification did not exist, whereby
the optimal portfolio equity allocation is static across holding periods. In reality, the relative
benefit of equities tends to increase over longer holding periods, which is captured to some degree
in Panel A (which is based on the AR(5) model). A Monte Carlo simulation that is not based
on an autoregressive model would yield similar results as Panel B, leading to the conclusion that
holding equities over longer time periods does not reduce their relative risk.
While our AR(5) model (Panel A of Figure 6) does a much better job capturing the historical relationship
exhibited by U.S. returns in Panel A of Figure 3, the resulting equity allocations are slightly less
aggressive than historical estimates. We attempted to explain this by conducting an additional analysis
to see how fast equity markets in different countries recovered from the five largest historical
annual market declines. We find that the recoveries are not any faster than predicted by our AR(5)
model for each country. This is an important consideration since recoveries do not occur more
rapidly, on average, than we would expect given our AR(5) model even for the worst historical returns.
Therefore, we accept that the remaining differences may flow from the simplicity of our model.
For example, we assume that returns are normally distributed (since the error terms are normally
distributed), while equity returns have been noted to have non-normal characteristics in past
research (Xiong and Idzorek, 2011, for example).
Although an AR(5) model is not essential when running a Monte Carlo simulation, it does
reasonably capture the historical level of time diversification associated with U.S. equities. Not using
an autoregressive return for a Monte Carlo simulation makes the implicit assumption that time
diversification does not exist. While this is inconsistent with historical international stock returns, there
is no guarantee time diversification will continue to persist in the future. An ideal recommendation
would be to temper historical findings with the reality of future uncertainty, although the empirical
evidence supports an increasing allocation to equities with longer time horizons.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 16 of 27
Time Diversification Across Time
Our primary analysis looked at the entire period of available historical returns to explore the potential
existence of time diversification across 20 countries. In order to investigate how the potential
benefits of time diversification may have changed historically, we rerun our overlap analysis for smaller
time periods (41 years each) that are assumed to “roll forward” by a decade. For example, the initial
period is 1900 to 1940 (41 years inclusive), while the second period is 1910-1950, etc. The overlap
across each test period is 31 years, which is approximately 75% of the test years. We do not conduct
a distinct analysis given the relatively short period for each test.
In this analysis we test different allocations to cash, bonds, and stocks in 5% increment combinations
for a total of 231 portfolios, and then select the combination that generates the highest utility
versus solving for the true optimal allocation. We summarize the regression coefficients in Table 5,
which is similar to the average results across all countries in Tables 2 and 3. We find that the
intercept, which is effectively the optimal allocation for an investor with a one-year investing horizon,
decreases in more recent time periods. The slope, which is the potential benefit from investing
in equities over longer time periods, increases. This suggests the optimal equity allocation for shortterm investors has been decreasing historically while the potential benefits of time diversification
have been increasing.
Table 5: Regression Equity Allocation Results Across Time
y = 2
y = 4
y = 8
y = 16
Intercept Slope
(%)
(%)
Intercept Slope
(%)
(%)
Intercept Slope
(%)
(%)
Intercept Slope
(%)
(%)
1900–1940
65.13-0.61
47.061.49
37.191.84
33.27 1.95
1910–1950
79.96-0.10
62.890.94
47.301.75
36.96 2.24
1920–1960
87.330.50
69.591.41
56.052.05
48.48 2.40
1930–1970
92.990.38
75.661.40
56.982.29
46.56 2.72
1940–1980
86.130.63
62.151.72
40.432.52
29.18 2.89
1950–1990
77.390.11
40.381.65
14.152.60
1.26 3.00
1960–2000
59.870.83
29.221.95
8.742.65
-2.28 3.02
1970–2010
44.121.98
20.122.82
5.413.18
-3.17 3.45
Intercept
89.34-0.10
75.420.85
63.151.51
56.48 1.83
Slope
-3.380.16
-5.450.18
-6.640.19
-7.27 0.19
Slope t stat
-1.421.94
-2.223.47
-2.999.30
-3.4512.95
Full Period Avg
76.501.10
52.502.00
35.802.40
27.10 2.60
Test Period
We include the results by country for investors with moderate risk tolerance (γ = 4) in Appendix 1, for
those readers interested in a more detailed perspective about how the intercept and slope changes
over time. While the aggregate results are relatively stable across all countries (Table 5), the individual
country results are considerably less so (Appendix I). In general, the optimal initial equity allocation
was much higher between 1920 and 1960 (from 34% to 100%) than in more recent time periods (e.g.
-7% to 61% between 1970 and 2010). The intercept and slopes values in Table 5 should be compared to
the average results in Table 2 (which are included in Table 5) instead of the individual country
results, since they are the average across all countries.
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Page 17 of 27
The slope of the annual time diversification benefit to equities appears to have risen consistently
over time. In the most recent time period (1970–2010), an investor should have increased his or her
optimal equity allocation by at least 2.5% for each additional year for 70% (14) of the countries
studied. Between 1910 and 1950, only two countries saw such a large time diversification benefit. This
suggests that while the equity premium in general may have declined in the late 20th century,
the time diversification benefits from holding a greater portfolio share in equities have increased for
longer-term investors.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 18 of 27
Implementing a Time Diversification Portfolio
While our results are consistent with the majority of the existing empirical research on time
diversification, it is inconsistent with the Black-Scholes option pricing model as well as the recent
work by Pastor and Stambaugh (2012). Pastor and Stambaugh note that while we can treat
historical parameters such as the ERP as certain, an investor today faces far more uncertainty into
the future. Uncertainty becomes increasingly important over longer investment periods
because estimation errors will compound over time.
In a previous analysis, we found no meaningful statistical relationship between the average
historical ERPs (versus cash or bonds) by country and the level of time diversification exhibited by
individual countries (defined as the slope resulting from the regression in equation 2). However,
the future expected return on equities, especially relative to bonds, is likely to be a key driver in an
investor’s decision to adopt a time diversification strategy. For example, while the average
annual compounded historical ERP across the 20 countries has been 3.56% (as noted in Table 1),
an investor may feel some smaller number (such as 2%) may be a more reasonable estimate
for the future.
A lower expected ERP may reduce the potential future benefit of time diversification. In this section
we determine how low the ERP would have to have been historically to eliminate the benefit of
time diversification for long-run investors. In other words, we seek to determine how much we must
reduce (or potentially increase, in a few cases) the ERP to eliminate the benefit of higher equity
allocations over longer holding periods.
Previously, we used equation 2 to approximate the relationship between the optimal equity allocations
and the investment time period. The resulting slope (β) from equation 2 was positive for the
majority of countries historically (~90%), as noted in Tables 2 and 3, which suggests the existence of
time diversification (i.e., the optimal allocation to equities increased over longer historical holding
periods). If the estimated slope in equation 2 was zero, which would mean the optimal equity allocation
did not change over the investment period, we would conclude that time diversification has not
existed. We can extend this line of thinking to determine how much the historical return of equities
would have had to change to eliminate the positive slope (or to call the existence of time
diversification into question).
In this section we perform an analysis where we first determine the optimal allocation to equities
over 1-, 5-, 10-, and 20-year investment horizons using overlapping historical data for all 20 countries for
risk aversion levels of 2, 4, 8, and 16. Unlike each of the previous tests that include cash, bonds,
and stocks, this analysis includes only bonds and stocks in the optimization opportunity set. We choose
to exclude cash to simplify the analysis and because bonds are a more likely investment option for
longer-term investors. We include the resulting optimal equity allocations for each of the simulations
in Appendix 2 for reference purposes.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 19 of 27
Next, we determine the annual return change (i.e., alpha) required so that the optimal equity allocation
for the 5-, 10-, and 20-year investment periods result is the same equity as the 1-year investment
period using an optimization routine. We test alpha values from -10.0% to +10.0%. For example, in Appendix 2 the optimal equity allocation for a risk aversion level (γ) of 4 for a U.S. investor with
a 1-year investment period would have been 47% versus 82% for the same investor with a 10-year
investment period.
Our goal is determine how much less attractive we must make the historical return on equities,
relative to bonds, so that an investor with a 10-year investment period will only invest 47% in equities
versus 82%. In other words, we determine how much we have to reduce the return on equities
to make bonds more attractive over a longer investment period. This “alpha” context tells the reader
how different historical returns would have had to have been in order to eliminate the benefits
of time diversification.
7
We include the results of our analysis in Table 6 as the ERPs (versus bonds). The values in Table
6 are effectively the historical ERP required by country, for a given investment period and risk aversion,
to make the optimal equity allocations for 5-, 10-, and 20-year investment periods the same as
a 1- year investment period. This is constant risk over time, not increasing risk as suggested by Pastor
and Stambaugh (2012), which would result in even lower ERP values.
Table 6: Historical Average Equity Risk Premium versus Bonds to Eliminate Time
Diversification for Varying Levels of Risk Aversion and Investment Periods
Austria
Australia
Belgium
Canada
Denmark
Finland
France
Germany
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
South Africa
Spain
Sweden
Switzerland
United Kingdom
United States
Average
t-statistic
7
y = 2
y = 4
y = 8
5yr10yr 20yr
2.9 0.90.9
0.2-4.4 0.6
0.9-0.2 -0.2
2.4 1.4 1.4
0.7-0.2 -0.3
3.51.3 0.6
0.9-1.0 -1.4
3.81.8 3.8
1.4 0.4 -0.4
-1.2-3.3 -3.0
-5.2-5.2 -5.2
2.8 1.9 1.1
2.9 1.1 0.6
1.1 0.3 0.4
3.4 2.4 1.4
3.1 2.4 0.8
2.62.3 2.1
1.91.3 1.0
2.7 1.7 0.7
3.32.2 1.3
1.7 0.4 0.3
3.64 0.72 0.75
5yr10yr 20yr
5yr10yr20yr
3.80.4 -0.7
-2.7-4.4 -2.1
-0.7-2.0 -2.6
0.8-1.0 -1.1
-0.6-1.8 -1.5
0.80.0 -0.1
-3.9-5.5 -4.5
2.81.8 -5.2
0.2-0.4 -0.6
-6.6-6.6 -6.6
-5.2-5.2 -5.2
1.7 0.7 -0.8
1.1-0.9 -1.5
-0.3-1.1 -0.8
2.0 1.3
0.6
-7.9 5.5
0.4
2.62.3 0.2
2.92.1 1.4
0.8-1.0 -1.2
1.91.1 -0.2
-0.3-0.7 -1.6
-0.45-1.10 -3.32
4.7-0.7-1.9
2.7 -4.4 -3.4
0.5 -1.5 -3.7
-2.4-2..8 -2.2
-4.4 -4.6 -2.1
7.8 2.7 1.1
-4.6 -5.2 -4.1
3.8 1.8-5.2
-1.5 -0.1 -0.2
-6.6 -6.6 -6.6
-5.2 -5.2 -5.2
-0.3 -0.4 -2.0
-0.1 -1.3 -2.2
-1.6 -1.9 -1.6
-1.3 -0.4 -4.4
-7.9 -7.9 0.6
5.1 4.0-1.2
5.1 3.4 2.0
-2.5 -2.8 -1.2
1.1 1.1-0.9
-0.4 -1.6 -2.2
-0.40 -2.19 -4.44
y = 16
5yr 10yr 20yr
-2.1-4.1 -3.1
2.9-4.4 -4.4
0.8-2.2 -3.1
-4.3-3.3 -2.5
-8.2-7.2 -2.6
8.33.3 1.3
-2.6-3.0 -2.0
-5.2-5.2 -5.2
-3.2-0.3 -0.3
-6.6-6.6 -6.6
-5.2-5.2 -5.2
-3.3 -1.6 -2.1
1.2-0.7 -2.3
1.8-0.6 -1.1
-4.6 -4.6 -4.6
-7.9 -7.9 -0.9
-7.16.2 2.1
5.33.2 2.1
-6.3-6.3 -6.3
2.31.1 -1.2
-2.2-2.5 -2.3
-2.12-2.89 -3.94
In this example we have to reduce the return of equities. In other scenarios we have to increase the return of equities, so the alpha can be positive or negative, but is negative in the vast majority of cases.
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Page 20 of 27
In Table 6 we find ERPs that are small and mostly negative, especially at higher levels of risk
aversion, although generally lacking statistical significance in many cases. For example, in order to
eliminate the historical existence of time diversification, the average ERP (versus bonds) for
a 10-year investment period for an investor with a risk aversion level of 4 would have had to have
been -.7% (as noted in Table 6). In other words, stocks would have had to underperform bonds,
on average, by -.7% per year over the historical period considered for there to be no potential benefit
from time diversification.
These findings suggest that while future parameter uncertainty should be a concern for investors,
the historical change in the return of equities required to eliminate the benefit of time diversification
would be substantial. While the ERP varies by period and level of risk aversion, it is generally
negative suggesting that for time diversification to provide no benefit to an investor the ERP must not
only disappear in the future, it must become negative.
8
8
No benefit is not the same thing as a cost, since the values in Table 6 are merely those where time diversification neither helps nor hurts the investor.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 21 of 27
Conclusions
Many investors use higher portfolio equity allocations for long-run goals despite an active
scientific debate over the existence of time diversification. Much of the evidence supporting time
diversification comes from 20th century U.S. asset return data that may be a historical anomaly.
In order to test whether time diversification is robust across a larger sample, we use historical real
returns from 20 countries each with 113 years of real return data resulting in a total of in
2,260 return years. We then determine optimal portfolio allocations based on varying levels of
risk aversion and investing periods.
Using a Constant Relative Risk Aversion utility function with varying risk levels and investment
time periods, we find that the optimal allocation to equities tends to increase over longer investment
periods. This adds to the empirical evidence supporting the existence of time diversification.
The average optimal equity allocation across the 20 countries increases by 1.3% per year for low risk
aversion (γ=2) portfolios, 2.1% per year for moderate risk aversion (γ=4) portfolios, 2.5% per
year for high risk aversion (γ=8) portfolios, and 2.7% per year for extremely high risk aversion (γ=16)
portfolios, based on our overlapping period analysis, with each being statistically significant
at the .01% level.
We find the average historical equity allocation over varying investment periods and levels of risk
aversion for U.S. equities is roughly consistent with the average equity allocation across all
20 countries, although certain countries such as Australia and Switzerland are considerably different
than the all-country average. We are further able to replicate time diversification using an
autoregressive model in a Monte Carlo setting and demonstrate how not considering this within
a modeling setting can have important implications for the results of a given simulation.
Next, we test to see how time diversification has changed over the 20th century and find that the
optimal equity allocations for shorter-term investors have been decreasing, but the optimal
allocations for longer-term investors have been increasing (i.e., the benefits time diversification have
increased over time). Finally, we determine that the ERP would need to have been zero or
slightly negative to eliminate the historical benefit of time diversification.
These findings have important practical implications for individual and institutional investors
with long investment periods, such as investors in target-date mutual funds and pension funds. While
conventional economic theory suggests that time diversification should not exist with certainty,
our analysis based on empirical historical data from 20 countries adds weight to the empirical evidence
from previous studies that time diversification does exist, or at least has existed historically.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 22 of 27
References
Barberis, Nicholas, Shleifer, Andrei and Robert Vishny. 1998. A Model of Investor Sentiment, Journal of Financial
Economics, 49 (3): 307 – 343.
Barberis, Nicholas. 2000. Investing for the Long Run when Returns are Predictable, Journal of Finance, 55: 225 – 264.
Bodie, Zvi. 1995. On the Risk of Stocks in the Long Run, Financial Analysts Journal, 51, 18 –22.
Campbell, John y. and Luis M. Viceira. 2003. Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford
University Press, New York, NY.
Dolvin, Steven D., Templeton, William K. and William J. Rieber. 2010. Asset Allocation for Retirement: Simple Heuristics
and Target-Date Funds, Journal of Financial Planning 23, 60 –71.
Estrada, Javier. 2013. Stocks, Bonds, Risk, and the Holding Period: An International Perspective, Journal of Wealth
Management, vol. 16, no. 2: 25 – 44.
Fabozzi, Frank J., Sergio M. Focardi, and Petter N. Kolm. 2006. “A Simple Framework for Time Diversification.”
Journal of Investing, vol. 15, no. 3: 8–17.
Hanna, Sean and Peng Chen. 1997. Subjective and objective risk tolerance: Implications for optimal portfolios, Financial
Counseling and Planning, vol. 8, no. 2 , 17 – 26.
Jegadeesh, Narasimhan, and Sheridan Titman. Profitability of momentum strategies: An evaluation of alternative explanations, No. w7159. National Bureau of Economic Research, 1999.
Pástor, L'uboš, and Robert F. Stambaugh. 2012. Are stocks really less volatile in the long run?, The Journal of Finance,
vol. 67, no. 2: 431– 478.
Samuelson, Paul. 1963. Risk and Uncertainty: A Fallacy of Large Numbers, Scientia, vol. 57, no. 6 (April/May): 1–6.
Siegel, Jeremy J. 2008. Stocks for the Long Run, 4th ed. McGraw Hill, New York, NY.
Xiong, James X. and Thomas M. Idzorek. 2011. The Impact of Skewness and Fat Tails on the Asset Allocation Decision,
Financial Analysts Journal, vol. 67, no. 2, March/April, 23–35.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 23 of 27
Appendix I: Regression Equity Allocation Results by Country Where γ = 4
1900–1940
1910–1950
1920–1960
1930–1970
1940–1980
1950–1990
1960–2000
1970–2010
Average
Inter Slope Inter Slope Inter Slope Inter Slope Inter Slope Inter Slope Inter Slope
Inter Slope Inter Slope
(%)(%) (%)(%) (%)(%) (%)(%)(%)(%) (%)(%) (%)(%) (%)(%) (%)(%)
Austria
100.0
0.0100.0
0.098.00.1 92.20.649.93.030.93.117.7
3.4 21.33.563.81.7
Australia
97.9-0.5 96.9
-0.8 76.10.4 69.01.4 72.60.4 27.1-1.0 9.1
-0.4
Belgium
13.3
4.8 60.0
1.266.00.2 75.61.163.9
-0.924.4
-1.328.9
-0.7 -5.24.740.91.1
Canada
30.6
4.2 57.9
1.754.92.0 73.91.889.40.759.32.542.3
3.0 45.72.256.72.3
Denmark
46.7
2.6 62.6
1.781.41.3 95.10.349.83.243.42.827.7
2.6 25.21.554.02.0
Finland
99.3
0.0 98.6
0.163.02.5 65.72.354.13.021.24.821.1
4.4 15.73.554.92.6
France
64.82.2 84.41.1 87.80.9 83.91.1 86.90.5 20.5-1.2 3.8
-0.3
Germany
84.0
0.6100.6
-1.096.1
-0.2 92.70.393.70.240.2
-0.320.8
0.7 13.72.967.70.4
Ireland
1.6
-0.1 59.1
2.384.71.1 89.60.739.93.631.93.137.0
2.5 23.62.745.92.0
Italy
69.9
2.1 78.9
1.577.91.6 78.81.585.8
-0.617.20.0 4.7
0.8 -5.23.551.01.3
Japan
42.5
1.4 42.7
3.943.23.9 43.23.943.23.967.32.115.5
3.3 17.90.339.42.8
Netherlands
31.9
-0.5 42.8
-0.534.32.9 59.92.558.62.563.21.956.2
2.1 13.64.245.01.9
New Zealand
100.00.0 100.00.0100.00.0 100.00.0 96.20.3 44.7 3.6 24.44.3 28.2 2.5 74.21.3
Norway
21.3
0.5 35.7
-0.265.32.4 62.62.229.33.026.12.425.7
1.8 22.12.436.01.8
South Africa
50.4
2.3 61.6
2.571.72.0 90.40.787.30.975.51.659.6
2.6 61.02.669.71.9
Spain
6.8
4.347.0
-0.541.6
-0.149.1
0.37.3
3.25.4
1.68.4
0.6 -7.02.1
19.81.4
Sweden
0.0
0.0 9.7
0.355.22.6 69.12.193.20.584.51.166.4
2.3 46.83.353.11.5
Switzerland
0.0
0.0 6.4
1.262.61.4 58.72.837.52.235.81.134.6
0.7 17.63.331.61.6
United Kingdom
51.3
3.3 65.8
2.379.01.4 90.40.739.93.137.82.731.3
3.1 30.23.853.22.5
United States
28.8
2.6 47.2
1.953.21.9 73.41.864.71.751.32.148.9
2.3 36.03.550.52.2
Average
47.1
1.5 62.9
0.969.61.4 75.71.462.21.740.41.629.2
1.9 20.12.850.9
1.7
4.3 1.7 56.60.2
-3.1 2.5 53.60.8
Appendix 2: Optimal Equity Allocations for ERP Test
y = 2
y = 4
y = 8
y = 16
1yr 5yr 10yr 20yr
1yr5yr 10yr20yr
1yr 5yr10yr20yr
1yr 5yr 10yr20yr
(%) (%)(%) (%)
(%)(%)(%)(%)
(%)(%)(%)(%)
(%)(%)(%)(%)
Austria
100100100100
66 73100100
41 4284 100
20 34 65 100
Australia
71 84 93 82
73 95 100 84
74 100100 85
73100100
Belgium
60 76100100
40 80100 79
37100100 71
54100100 67
Canada
100100100100
55 78100100
38 76100 100
29 80100 100
Denmark
61 82100100
38 65100100
27 56100 100
20 51 88 100
Finland
75 91100100
63 89100100
93 73100 100
100 60100 100
France
65100100100
44100100100
40100100 100
60100100 100
Germany
100 100100100
100100100 90
100100100 72
100100100 62
Ireland
6276100100
35 60 83100
17 6089100
46798 100
Italy
53 85100100
30100100 84
2610095 75
25100 93 72
Japan
61 92100100
7 92100100
0 8286 100
0 74 74 100
Netherlands
6873 84100
43 53 67 76
29 4860 70
235160 69
New Zealand
81 88100100
50 67100100
36 5993 100
31 55 95 100
Norway
5266 83100
34 49 71 82
38 5576 80
616278 80
South Africa
100100100100
50100100100
15 75100 100
0 69100 100
Spain
57445380
35111161
26 00 55
26 0 0 46
Sweden
66707474
41434456
281221 47
32 0 2 39
Switzerland
6062 74100
3221 2944
16 0 0 12
7 0 0 0
United Kingdom
100100100100
45 78100100
10 52100 100
0 38100 100
United States
7990100100
47 64 82100
28 5375100
154580 100
Average
74849397
46718488
366279 83
345977 81
88
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Page 24 of 27
About the Author
David Blanchett
David Blanchett, CFA is head of retirement research for the Morningstar Investment Management
division, which provides investment consulting, retirement advice, and investment management
operations around the world. In this role, he works closely with the division’s business leaders to provide research support for the group’s consulting activities and conducts client-specific research
primarily in the areas of financial planning, tax planning, and annuities. He is responsible for developing
new methodologies related to strategic and dynamic asset allocation, simulations based on
wealth forecasting, and other investment and financial planning areas for the investment consulting
group, and he also serves as chairman of the advice methodologies investment subcommittee.
Prior to joining Morningstar in 2011, Blanchett was director of consulting and investment research
for the retirement plan consulting group at Unified Trust Company in Lexington, Ky.
Blanchett has authored more than 35 articles that have been published in InvestmentNews, Journal
of Financial Planning, Journal of Index Investing, Journal of Indexes, Journal of Investing,
Journal of Performance Measurement, Journal of Pension Benefits, and Retirement Management
Journal. He won Journal of Financial Planning’s 2007 “Financial Frontiers Award” for his
research paper, “Dynamic Allocation Strategies for Distribution Portfolios: Determining the Optimal
Distribution Glide Path.”
Blanchett holds a bachelor’s degree in finance and economics from the University of Kentucky,
where he graduated magna cum laude, and a master’s degree in financial services from American
College. He also holds a master’s degree in business administration, with a concentration
in analytic finance, from the University of Chicago Booth School of Business, and he is currently pursuing
a doctorate degree in personal financial planning from Texas Tech University. Blanchett holds
the Chartered Financial Analyst (CFA), Certified Financial Planner (CFP), Chartered Life Underwriter
(CLU), Chartered Financial Consultant (ChFC), Accredited Investment Fiduciary Analyst (AIFA),
and Qualified Pension Administrator (QKA) designations.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
Morningstar and includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and
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Page 25 of 27
About Our Research
The research team within the Morningstar Investment Management division pioneers new
investment theories, establishes best practices in investing, and develops new methodologies to
enhance a suite of investment services. Published in some of the most respected peer-reviewed
academic journals, the team’s award-winning and patented research is used throughout the industry and
is the foundation of each client solution. Its commitment to ongoing research helps maintain its
core competencies in asset allocation, manager research, and portfolio construction. Rooted in a mission
to help individual investors reach their financial goals, its services contribute to solutions made
available to approximately 24.3 million plan participants through 201,000 plans and 25 plan providers.
The Morningstar Investment Management division creates custom investment solutions that combine
its award-winning research and global resources together with the proprietary data of its parent
company. This division of Morningstar includes Morningstar Associates, Ibbotson Associates, and Morningstar Investment Services, which are registered investment advisors and wholly owned subsidiaries
of Morningstar, Inc. With approximately $186 billion in assets under advisement and management,
the division provides comprehensive retirement, investment advisory, portfolio management, and index
services for financial institutions, plan sponsors, and advisors around the world.
Our research has practical applications. Each of these five components discussed in this paper is
either currently being used in, or is in development to be used in, Morningstar Retirement Manager
or Ibbotson’s Wealth Forecasting Engine.
®
SM
For more information, please visit http: //global.morningstar.com / mim
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Page 26 of 27
Important Disclosures
The above commentary is for informational purposes only and should not be viewed as an offer to buy or sell a particular
security. The data and/or information noted are from what we believe to be reliable sources, however Morningstar Investment
Management has no control over the means or methods used to collect the data/information and therefore cannot
guarantee their accuracy or completeness. The opinions and estimates noted herein are accurate as of a certain date and
are subject to change. The indexes referenced are unmanaged and cannot be invested in directly. Past performance
is no guarantee of future results. The charts and graphs within are for illustrative purposes only.
Monte Carlo is an analytical method used to simulate random returns of uncertain variables to obtain a range of possible
outcomes. Such probabilistic simulation does not analyze specific security holdings, but instead analyzes the identified
asset classes. The simulation generated is not a guarantee or projection of future results, but rather, a tool to identify a range
of potential outcomes that could potentially be realized. The Monte Carlo simulation is hypothetical in nature and for
illustrative purposes only. Results noted may vary with each use and over time.
The results from the simulations described within are hypothetical in nature and not actual investment results or
guarantees of future results.
This should not be considered tax or financial planning advice. Please consult a tax and/or financial professional for advice
specific to your individual circumstances.
© 2013 Morningstar. All Rights Reserved. These materials are for information and/or illustrative purposes only. The Morningstar
Investment Management division is a division of Morningstar and includes Morningstar Associates, Ibbotson Associates,
and Morningstar Investment Services, which are registered investment advisors and wholly owned subsidiaries of Morningstar,
Inc. All investment advisory services described herein are provided by one or more of the U.S. registered investment advisor
subsidiaries. The Morningstar name and logo are registered marks of Morningstar. This presentation includes proprietary materials of Morningstar. Reproduction, transcription or other use, by any means, in whole or in part, without the prior, written
consent of Morningstar is prohibited.
“The information, data, analyses, and opinions presented herein do not constitute investment advice; are provided as of
the date written and solely for informational purposes only and therefore are not an offer to buy or sell a security; and are not
warranted to be correct, complete or accurate. Past performance is not indicative and not a guarantee of future results.”
Monte Carlo simulation is an analytical method used to simulate random returns of uncertain variables to obtain a range
of possible outcomes. Such probabilistic simulations do not analyze specific security holdings, but instead analyzes the identified
asset classes or indexes. The simulation generated is not a guarantee or projection of future results, but rather, a tool to
identify a range of potential outcomes that could potentially be realized. The Monte Carlo simulation is hypothetical in nature
and for illustrative purposes only. Results noted may vary with each use and over time.
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in whole or in part, without the prior written consent of Morningstar is prohibited. The Morningstar Investment Management division is a division of
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Page 27 of 27