Long-Term
Capital Market
Assumptions 2016
Elena V. Black
Yubo Qiu
Retirement Actuarial Services
The Principal Financial Group®
October 2016
Long-Term Capital Market Assumptions 2016
Elena V. Black, PhD, CFA, FSA, EA, MAAA, FCA
Yubo Qiu, CFA, FSA, EA, MAAA
Retirement Actuarial Services
The Principal Financial Group®
October 2016
©2016 Principal Financial Services, Inc.
May not be redistributed without express written consent.
Table of Contents
Session
Introduction
Methodology overview
Economic environment
Page
4
6
8
Inflation
8
Interest rate environment
11
Broad asset class expected returns
16
Fixed Income
16
Equity
18
Real Estate
23
Returns, risks and correlations
Conclusion
References
Appendices
25
26
27
29
A: Comparison of CMA 2014 and CMA 2016
29
B: Return, risk and correlation summary table
30
C: Benchmarks for asset classes
31
D: Statistical analysis of returns
32
Introduction
This paper documents the analysis and describes the models employed to develop the Principal
Retirement Actuarial Services (Principal RAS) capital market assumptions (CMA) 2016. The purpose is
to assist Principal RAS actuaries and clients in setting economic assumptions for U.S. pension actuarial
valuations and pension plan modeling.
In developing CMA 2016, we focus on forward-looking models and market indicators. The long-term
nature of pension plan obligations indicates a forecasting period of 20-30 years as our intended
investment horizon. Table 1 below summarizes the Principal RAS CMA 2016.
Table 1 - Principal RAS CMA 2016 summary
Expected return
Risk
Geometric
Arithmetic
Standard deviation
U.S. Equity - Large Cap
6.50%
7.85%
17.20%
U.S. Equity - Mid Cap
U.S. Equity - Small Cap
Non-U.S. Equity
Real Estate
6.50%
6.50%
6.50%
8.10%
8.55%
8.10%
18.80%
21.50%
18.90%
REITs
Real Estate (direct property)
Fixed Income
6.10%
5.50%
7.95%
5.80%
20.20%
7.90%
Cash
TIPS
Core Bond
Aggregate Credit Bond
Long Credit Bond
Long Gov't/Credit Bond
Long Gov't Bond
Ultra-Long Gov't Bond
High Yield
1.55%
2.85%
3.60%
4.05%
4.75%
3.90%
2.65%
1.85%
6.30%
1.55%
3.05%
3.75%
4.25%
5.15%
4.30%
3.30%
4.10%
6.70%
0.80%
6.40%
5.10%
6.40%
9.20%
9.40%
11.80%
21.90%
9.30%
Equity
For purposes of setting long-term economic assumptions for pension valuations, we conduct a CMA
review study every 3-5 years which is a standard practice within the actuarial profession. However, due
to the current economic and market conditions, we believe that our study is warranted sooner. The
previous Principal RAS CMA report was issued in December 2014 [1].
The 2016 study generally reflects market information as of June 30, 2016, unless specifically noted
otherwise. Expected geometric returns are lower for CMA 2016 compared to CMA 2014, reflecting
current economic and market conditions, as well as revised forward expectations. Appendix A shows
the comparison of CMA 2016 to CMA 2014. Appendix B presents a complete matrix of returns, risks,
and correlations for the CMA 2016 set.
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The remainder of this report is structured as follows. Our methodology review is followed by a section
discussing the key economic fundamental variables. A building block approach ensures consistency
between economic assumptions. Our starting point is inflation which is the key building block linking
all economic assumptions. The second key assumption is future economic growth, estimated by an
outlook for U.S. Gross Domestic Product (GDP) growth. The third key economic assumption is the
future interest rate environment. Accordingly, the construction of the future Treasury and
corporate/credit yield curves concludes the economic environment section.
The next section develops expected geometric returns for a variety of broad asset classes, starting
with fixed income, followed by equity and real estate. In the last part of the report we address risk
measures and correlations, as well as compute the expected arithmetic returns. A list of references
follows the conclusion of this report.
Appendices contain additional and supplemental information and are referenced throughout the
paper, as appropriate. The content of the supplemental information is as follows.
•
Appendix A compares Principal RAS CMA 2016 to CMA 2014.
•
Appendix B presents the complete matrix of returns, risk measures, and correlations for all
broad asset classes in our universe.
•
Appendix C provides a list of market benchmark indices for each of the broad asset classes.
Whenever historical information is utilized in the analysis, the relevant data for the appropriate
market index was obtained from either FACTSET or Wilshire Compass.
•
Appendix D contains a primer for the arithmetic and geometric means, as well as the
lognormal distribution.
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Methodology overview
Our philosophy in setting the capital market assumptions is to consider, first and foremost, forwardlooking market indicators and valuation models. As mentioned in the introduction, our focus is on the
long-term horizon, which we define as 20-30 years. For each assumption, we reflect both historical and
current market data and employ forward-looking valuation models, as well as analysis of historical
data and trends. We review future economic and capital markets’ forecasts developed by other
credible studies and refer to the outlook of recognized economists and organizations. We carefully
balance results produced by the mathematical forward-looking models, the historical analysis and
expert opinion to arrive at our CMA recommendations.
Financial professionals and academic researchers utilize different approaches to forecast expected
returns for broad capital markets asset classes. Here are the methodologies we employ in analyzing
each topic of interest:
1. Analysis of the current economic environment and implications for the future.
2. Forward-looking valuation models, reflecting estimates of long-term expectations.
3. Historical analysis of economic and market data and trends.
4. Forecasts and economic outlook from investment firms and other experts.
As mentioned in the introduction, the three key economic fundamental assumptions for our models
are inflation, economic growth, and interest rate environment. Our study starts with setting long-term
inflation expectations based on a variety of sources. We consider market expectations inherent in yield
differences between Treasury Inflation Protected Securities (TIPS) and nominal Treasury securities.
Other sources include Federal Reserve (Fed) inflation target, Federal Banks outlook, Congressional
Budget Office (CBO) forecasts, consensus among investment firms, and historical analysis. For our
assumption of future economic growth we look to CBO forecasts of U.S. Gross Domestic Product
(GDP) growth.
To forecast the future interest rate environment, we consider current yield curves as the starting point.
For a long-term view, we set our equilibrium assumption for the Treasury curve based on 1-, 10-, and
30-year Treasury rates, keeping the shape of the curve consistent with a historically “normal” shape.
The 10-year Treasury long-term rate serves as the anchor and is based on our equilibrium expectation
which is compared to the CBO outlook and the range indicated by other investment firms’
assumptions. We transition the current Treasury curve to the long-term expected curve over the next
5 years.
The long-term corporate/credit curve is built from the Treasury curve by adding on credit spreads.
Transition from current corporate/credit rates to the long-term view is also achieved over 5 years. We
carefully reconcile other relationships between corporate/credit and Treasury yields at different
maturities, as well as between the short and the long-term ends of the corporate/credit curve. Once
the future interest rate environment is established, we calculate expected returns for the fixed income
classes based on the appropriate yields and year-to-year yield movements, durations, and convexities.
6 of 33
Our expected geometric return is estimated as the geometric mean of forecasted returns over the next
25 years.
The forward-looking valuation models for high yield, equity, and real estate share the concept of
internal rate of return (IRR). The idea is that the fair current market value is equal to the future income
cash flows or payouts, discounted by the rate of future return. The “constant growth” equity discount
dividend model (DDM) is a simple implementation of this concept. In its most simple form, the DDM
translates to a very intuitive equation: the return is equal to the sum of the dividend yield and the
nominal growth rate. For this study, however, we employ a slightly more complex multi-stage DDM,
because we feel that the expectations of the upward interest rates’ trend warrant this approach.
For the real estate class, our model is similar to the equity model, estimating future expected return as
a sum of the real estate dividend yield and the estimate of future dividend growth. The Real Estate
Investment Trusts (REITs) model is again based on the same concept, although with a slightly different
approach to estimating the long-term assumption for REITs dividend yield and the future growth.
For each class, when utilizing more than one approach, we give most weight to results from the
forward-looking models. As a final step we review appropriate forecasts by other investment or
actuarial firms. Our recommendations take in consideration all of this information.
For measures of risk and volatility, as well as relationships between different asset classes’ returns, we
mostly look to historical information. This is the approach adopted by many investment experts
because these measures tend to be historically stable. We anticipate that historical averages of risk
measures and correlations are likely to persist into the future. For this purpose we use the monthly or
the annual return data series, as appropriate, going as far back as available for each of our benchmark
indices, listed in Appendix C.
We develop expected arithmetic returns under the lognormal distribution of returns assumption. For
educational purposes, we provide additional information on arithmetic and geometric means in
Appendix D.
7 of 33
Economic environment
Inflation
Our analysis starts with inflation, which is a fundamental building block that links all economic
assumptions. Inflation represents the overall price change of goods and is measured in a variety of
ways. In this analysis, inflation refers to the Consumer Price Index for all urban consumers (CPI-U).
Our long-term inflation assumption represents an expectation of inflation over the investment horizon
of 20 to 30 years.
Market-based expected inflation
Many participants in financial markets and investment professionals compare yields of TIPS with
nominal Treasury securities of the same maturity to develop a market expectation of inflation. This
difference indicates inflation expectation over the maturity period. Although some Federal Reserve
Banks adjust for illiquidity and inflation risks, this adjustment does not materially impact resulting
averages. We summarize market-based expected inflation over the forecast horizon of 10-, 20- or 30year periods, based on the appropriate maturity TIPS and nominal Treasury securities. The data period
includes monthly data through June 2016. The graph in Figure 1 illustrates that during the last 5 years
the market breakeven rates of inflation fluctuate in a range approximately from 1.50% to 2.50%, with
a markedly downward trend starting in 2013.
Figure 1
Source: data from Federal Reserve and calculations by the Principal RAS
Historical averages of market-based inflation expectations indicated by yield differences of TIPS and
nominal Treasuries are shown in Table 2. With the focus on more recent information we determine the
range of market-indicated inflation as 1.75% to 2.10%.
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Table 2- Market-based average expected inflation
Forecast horizon
Full period
Last 5 years
Last 2 years
Last year
Current market-based range
10-Year
2.12
2.02
1.72
1.54
20-Year
2.29
2.11
1.78
1.57
1.75% to 2.10%
30-Year
2.22
2.16
1.87
1.72
Source: data from Federal Reserve and calculations by the Principal RAS
Federal Reserve target
The Federal Reserve, as the U.S. central bank, sets the inflation target and other monetary policy. In
the last two decades, inflation has been more stable than in the past primarily due to the Fed’s
inflation target policy. Federal Open Market Committee (FOMC) meeting minutes from June, 2016 [2]
state the Fed’s long term Personal Consumption Expenditure (PCE) inflation target continues at 2%.
The Bureau of Economic Analysis (BEA) published monthly reconciliation of percent change CPI with
percent change in the PCE price index and historically inflation (CPI) is slightly higher than PCE. While
we acknowledge this slight difference, we believe the 2% PCE inflation target represents the
consistent inflation policy from the Fed [3, 4].
Federal Reserve Bank of Cleveland forecast
As of June 2016, the Federal Reserve Bank of Cleveland [5] anticipated 1.87% inflation for a 20-year
outlook and 2.04% inflation for a 30-year outlook. The graph in Figure 2 illustrates the monthly
outlook of 20-year and 30-year inflation since January 2011, as estimated by the Federal Reserve Bank
of Cleveland. Based on this information, we place expected inflation from this source in the range of
1.85% to 2.05%.
Figure 2
Source: Federal Reserve Bank of Cleveland
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Congressional Budget Office forecast
The CBO 11-year economic outlook [6] forecasts CPI inflation gradually increasing from a current
1.35% (2015) to an ultimate 1.96% rate. After 2020, according to the CBO forecast, inflation remains
approximately at the same level [7]. Over the entire 11-year horizon the CBO expected inflation
averages to 1.93%. Assuming future inflation remains at the ultimate rate of 1.96%, the last year of
CBO forecast, for 9 more years, the average over the 20-year period is approximately 1.94%.
Accordingly, we set CBO inflation forecast range from 1.90% to 2.00%.
Consensus forecast from investment firms
Many investments firms regularly publish CMA reports and we reviewed the most recent CMA 2016
from several organizations. The inflation expectation is set at 2.25% by J. P. Morgan [8], 1.55% by
Wilshire Associates [9], and at 2.20% BNY Mellon [10]. The range of inflation expectation from other
firms [11, 12, 13, 14] is from 1.55% to 2.30%, averaging to 1.95%.
Historical analysis
We reviewed CPI-U historical inflation rates. Over time the historical rates show significant variability.
Because inflation is controlled by the Fed’s policy, historical inflation prior to the 1990’s is less relevant
for our purposes of forecasting future inflation. Over the last 20 years, average inflation is 2.19% with
1.05% standard deviation. The historical averages are 1.86% over last 10 years and 1.54% over the last
5 years. Accordingly, we set the inflation range indicated by historical analysis as 1.55% to 2.15%.
Conclusion: inflation recommendation
We recommend a long-term inflation assumption of 2.00%. This is based on the information discussed
in this section and summarized in Table 3 below:
Table 3
Market-based expected inflation (TIPS)
Federal Reserve target
Federal Reserve Bank of Cleveland forecast
Congressional Budget Office (CBO) forecast
Consensus forecast from investment firms
Historical analysis
Our recommendation
Inflation
1.75%
1.85%
1.90%
1.55%
1.55%
2.00%
2.00%
2.10%
2.05%
2.00%
2.30%
2.15%
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Interest rate environment
As discussed in the Introduction and Methodology overview sections, the interest rate environment is
another of the key fundamental economic assumptions. Our approach is to develop expected longterm equilibrium yield curves and transition from current to long-term curves over a 5-year period. To
ensure consistency, we develop the U.S. Treasury curve first, anchored to the 10-year Treasury rate.
For the corporate/credit yield curve we add on the credit spreads while reconciling relationships
between long- and short-maturities’ credit spreads, as well as the long- and short-end of the
corporate/credit curve.
U.S. Treasury yield curves
By definition, the yield is the indicator of the market expectation of future returns. In the current
economic environment when interest rates are hovering at their historic lows, most economists
anticipate rising interest rates in the next few years. Accordingly, the first step in our analysis is to
develop a long-term equilibrium Treasury yield curve. We use Constant Maturity Treasury (CMT) rates
for our analysis. Figure 3 depicts current (June 30, 2016) and our long-term yield curve for the U.S.
Treasuries. We assume that the current Treasury curve will rise gradually to the long-term curve over
the next 5 years.
Figure 3
Source: current Treasury yields from http://www.treasury.gov and Principal RAS long-term estimates
In developing the long-term Treasury curve, we start with the 10-year rate. The 10-year CMT rate was
1.49% as of June 30, 2016. CBO forecasts the 10-year Treasury rate as rising from 1.49% in 2016 to
4.10% in 2019 and remaining at this level thereafter. Our historical data analysis based on long-term
rates (most recent 20-year period) results in an estimate of the equilibrium of 4.39%. Investment
firms, for example, J. P. Morgan [8], BNY Mellon [10], and Voya [11] forecast their 10-year Treasury
11 of 33
rate in the range of 4.00% to 4.40%. Balancing this information, we recommend the long-term
equilibrium rate for 10-year Treasuries to be 4.10%.
Anchoring the Treasury curve to the 10-year rate, we develop the long-term1-year and 30-year rates
according to the historical yield curve slope and curvature. The slope is defined as the difference
between 1- and 10-year rates and is estimated at 150 basis points (bps). The curvature, defined as the
1-year rate plus the 30-year rate minus twice the 10-year rate, is estimated at negative 100 bps. Both
numbers are historical averages of slope and curvature over the last 20 years of monthly data. This
results in the long-term 1-year and 30-year rates of 2.60% and 4.60%, respectively. We note that
historical analysis confirms that the Long Government Bond asset class has similar yields as the 30year Treasury, so we estimate the equilibrium yield for this class as 4.60%.
In our last step related to the Treasury rates, we extend the long-term Treasury yield curve to duration
around 25 years (long-term Treasury STRIPS). The acronym STRIPS stands for ‘separate trading of
registered interest and principal securities’. One of our LDI broad asset classes is benchmarked to 20+
Treasury STRIPS. We estimate the equilibrium rate for the 20+ STRIPS at 4.60%, same as the 30-year
Treasury rate, reflecting zero additional spread under assumption of flat yield curve structure at ultralong durations.
Long-term assumption for corporate/credit yield curve and other yields
Similar to the long-term equilibrium Treasury curve, next we develop our long-term expectation for
the corporate/credit yield curve. The corporate/credit yields are generally higher to compensate for
additional risk when investing in credit securities compared to U.S. Treasuries. The difference is the risk
premium or credit spread which generally tends to revert over time to the long-term historical
average. Accordingly, we construct our long term corporate/credit yield curve based on the
equilibrium Treasury curve and assumed long-term credit spreads.
For the long end of the credit curve, we add a credit spread of 125 bps (determined by the historical
average) to the equilibrium 30-year Treasury rate. The resulting long term rate of 5.85% represents the
Long Credit Bond class with duration of about 13 to14 years. This asset class is benchmarked to the
Bloomberg Barclays U.S. Long Credit Index. For the intermediate duration (approximately 7 years)
credit bond, benchmarked to the Bloomberg Barclays U.S. Aggregate Credit Index, we determine the
long-term rate of 4.95% by referring to the historical “slope” of 90 bps between intermediate and long
ends of the credit curve. For the short end of the credit yield curve, we consider average difference
between the yields of Bloomberg Barclays U.S. Aggregate Credit Index and Bloomberg Barclays U.S.
Aggregated Credit (1-5Y) Index of 40 bps. Thus, the short end yield of the equilibrium credit curve is
assumed to be 4.55%.
Sometimes, a pension actuary may need to set a specific assumption based on the high quality
corporate bond yields. Due to the quality difference and credit versus corporate distinction, we note
that the historical average yield difference between the two Long and two Intermediate (duration
about 4.5 years) benchmark indices is about 40 bps. Accordingly assuming a parallel adjustment down
12 of 33
by 40 bps of equilibrium credit curve, we obtain long-term equilibrium high quality corporate yields at
4.15%, 4.55% and 5.45% for short, intermediate, and long durations, respectively.
Figure 4 depicts the long-term yield curves discussed in this section.
Figure 4
Next, we estimate yields for the “mixed” fixed income classes. Figure 5 depicts the long-term yields for
broad asset classes included in our CMA 2016. The “mixed” fixed income classes we include are the
Core Bond, benchmarked to the Bloomberg Barclays U.S. Aggregate Index, and the Long Term Bond
(Long Gov’t/Credit Bond), benchmarked to the Bloomberg Barclays U.S. Long Gov’t/Credit Index.
The Long Term Bond make-up is approximately 40% government and 60% credit. Accordingly, we
calculate 5.35% for the long-term yield for this asset class as appropriately weighted yields of the Long
Government and the Long Credit yields. Finally, for the Core Bond yield, we use the average historical
spread of 35 bps over the 10-year Treasury rate. The result of 4.45% reconciles well to the historical
difference of 85 bps from the Long Term Bond yield. All yield and yield differentials’ averages reflect
most recent 30 years of monthly data.
13 of 33
Figure 5
14 of 33
Summary of interest rates and yields
Table 4 summarizes the current yields (June 30, 2016) and the long-term yields, as well as a brief
rationale for each fixed income broad asset class included in our CMA 2016 set.
Table 4 - Inflation, Treasury and fixed income broad asset classes’ yields/rates
Brief rationale for long-term yield
Current
yield
Long-term
yield
Based on differential between yields on inflationprotected and nominal Treasuries and other credible
forecasts’ consensus
Based on CBO forecast, historical analysis and expert
consensus
0.73%
2.00%
1.49%
4.10%
Based on historical yield slope (slope refers to the
difference in yield between 10-year and1-year Treasury)
of 150 bps, and the 10-year Treasury expectations
Based on historical yield curvature of negative 100 bps,
and 1-year and 10-year Treasury expectations
0.45%
2.60%
2.30%
4.60%
Building blocks
Inflation
10-year Treasury
1-year Treasury
30-year Treasury
Fixed income broad asset classes
Long Government
Bond
Same yield as 30-year Treasury, based on historical
analysis
2.17%
4.60%
Long Credit Bond
Based on 125 bps credit spread over Long Government
bond
4.16%
5.85%
Aggregate Credit
Bond
Based on long credit yield minus slope of 90 bps (slope
refers to the difference in yield between Long Credit
and Aggregate Credit benchmarks)
2.78%
4.95%
Core Bond
Based on historic spread over 10-Year Treasury during
last 30 years of approximately 35 bps
1.91%
4.45%
Long Term Bond
Based on 40% Long Gov't and 60% Long Credit
3.36%
5.35%
Ultra-Long Gov’t
Bond
Assumed zero spread in the ultra-long end of Treasury
curve
2.34%
4.60%
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Broad asset classes expected return
Fixed Income
Fixed Income other than High Yield
For Fixed Income classes, the future returns are calculated based on current yields, long-term
equilibrium yields developed in the previous section, and the transition period of 5 years. The exception
is the High Yield asset class which is addressed in a separate sub-section below. For each asset class we
use the appropriate duration and convexity to calculate returns over future 20-, 25- and 30-year
periods. Our ‘best estimate’ of expected geometric return for each class is the 25-year geometric mean
of the forecasted returns. The 20- to 30-year periods’ calculations serve as the reasonable assumption
range.
Fixed income investments’ returns can be decomposed into income or “coupon” return and the “price
change” due to the movement in the yields. The price change can be closely estimated using the
concepts of duration (D) and convexity (C). If the yields change by ∆y during the year, the price change
will be approximately equal to the negative D multiplied by ∆y. When yields are increasing, as we
assume in this study, the price change is negative and applies downward pressure on returns.
Convexity measures the change in the duration itself due to the yield changes. At higher yields
duration is lower approximately by C multiplied by ∆y. We use appropriate convexity to calculate
durations at the equilibrium level in 5 years and adjust duration every year linearly to converge to the
equilibrium duration.
For example, as of June 30, 2016, the Long Credit Bond yield was 4.16%, duration was 14.0 years and
convexity was 141. At the equilibrium rate of 5.85%, duration is reduced to 12.0 years. The forecasted
return for the first year, when yield increases to 4.50%, is equal to 0.60%. This is calculated as 4.16%
minus 14.0 multiplied by the rate increase during the year. The duration at the beginning of the second
year, at the higher rate of 4.50%, is calculated to be 13.6 years applying the linear convergence from
the current (14.0) to the equilibrium (12.0).
High Yield
We use a different model for the High Yield (HY) asset class because the HY return expectations
should also reflect expected defaults and recovery experience. This model is detailed in the Wilshire
High Yield Update [15] paper and is a version of the IRR model discussed in the Methodology overview
section. We forecast income cash flows and future asset levels over 10 years of varying yields and
default rates. After 10 years, the stable growth model applies and we utilize a general formula to
calculate the present value of the future cash flows starting at this point.
To determine future cash flows, we need to project HY yields and asset levels for each future year. In
order to do this, we start with determining the long-term equilibrium HY credit spread of 5.27% over
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the 10-year Treasury rate. The model transitions the current spread of 5.78% to the long-term spread
over a 10-year period. The long-term spread is determined as the historical average over the last 20
years. The future spreads are added to the forecasted 10-year Treasury rates to develop HY yields.
In order to forecast the asset level of the HY asset class, we project the loss rate, which is determined
as defaults less recovery. The default rates are forecasted to transition from the current level of 3.47%
to the long-term default rate of 4.43% over a 5-year period. The long-term default rate of 4.43% is
determined based on historical averages for the period from 1986 to 2015 [16]. We assume recovery
of 35% in all years. The recovery rate is based on analysis from historical data: 2015 at 38% and the
average of last 30 years at 43%. Future cash flows are calculated by applying the forecasted yields to
the asset level at the beginning of each year, and the asset level is reduced by the loss rate forecasted
for that year. Finally, the internal rate of return is calculated to be 6.30%.
Fixed Income summary
Table 5 below summarizes information used to calculate returns and the resulting expected geometric
returns for all Fixed Income asset classes.
Table 5
Fixed Income
Asset Classes
Cash
10-year Treasury Bond
30-year Treasury Bond
TIPS
Core Bond
Aggregate Credit Bond
Long Credit Bond
Long Government/Credit Bond
Long Government Bond
Ultra-Long Government Bond
High Yield
Yield
Duration
Current
Long-term
0.36%
1.49%
2.30%
0.76%
1.91%
2.78%
4.16%
3.36%
2.17%
2.34%
1.70%
4.10%
4.60%
2.10%
4.45%
4.95%
5.85%
5.35%
4.60%
4.60%
Current
7.27%
Yield
Long-term
9.37%
*Geometric return calculated over 25 years and is rounded to 5 bps
0
0
9.2
8.1
21.8
16.1
Return the same as 10-yr T
5.5
4.6
7.3
6.4
14.0
12.0
15.6
12.8
18.0
14.0
27.1
24.0
Geometric
return*
1.55%
2.85%
2.45%
2.85%
3.60%
4.05%
4.75%
3.90%
2.65%
1.85%
Default Rate
Current
Long-term
3.47%
4.43%
Geometric
Return*
6.30%
Current
Long-term
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Equity
Introduction
Financial and investment professionals have developed a large arsenal of models to estimate future
stock returns. The methodologies include observations and lessons from market history, analysis of
historical data, and forward-looking valuation models, relying on economic fundamentals. Equity
returns are extremely volatile and exhibit cyclical patterns. The cyclic nature of stock returns means
that historical analysis can lead to vastly different conclusions regarding market return expectations,
depending on the period analyzed. We conclude that historical analysis alone is not sufficient for the
task, and focus on discount cash flow models (or IRR models) to evaluate market expectations of
future equity returns.
The anchor for our equity analysis is the U.S. Large Cap Equity class which is benchmarked to the S&P
500 Index. Once we select a return expectation for this class, we analyze other equity classes in
reference to the U.S. Large Cap Equity class.
Methodology for U.S. Large Cap Equity
As discussed in the Methodology overview section, the underlying concept for the forward-looking
valuation models is the IRR calculation. The model forecasts future income cash flows and determines
the rate of return discounting by which results in the current asset value. For equity, the future cash
flows are determined by future dividend streams. The first two models that we use are both versions of
the dividend discount model (DDM).
1. Multi-stage Dividend Discount Model
Many versions of the multi-stage DDM use different methodologies to forecast dividend
streams that vary during different future periods (see, for example, [17]). In our version, for
the first 2 years we use projected S&P 500 earnings from FACTSET representing investment
bank consensus. For the second 9-year stage, we project earnings with the growth rate based
on the real GDP growth from the CBO forecasts and our assumption on the future inflation.
The last and final stage assumes stable earnings’ growth, with the real rate of growth
estimated as the 2020-2023 average of the GDP real growth. Dividends are calculated from
earnings using the payout ratio of 0.40 which is the historical average for the S&P 500 Index in
the last thirty years. Details on the S&P 500 Index data and assumptions for this model are
given in the next section. The internal rate of return is calculated at 6.48%.
2. Stable Growth Dividend Discount Model
The stable growth DDM, also known as the Gordon Model (see, for example, [18], [19]), is a
one-stage simple variant of the DDM. Under the assumption of constant growth, the
calculation simplifies to an intuitive equation: The equity expected return is equal to dividend
yield plus future dividend growth. In this model, the variation comes from the way the dividend
yield and the future dividend growth are estimated. For example, dividend yield can be
assumed to be the S&P 500 current dividend yield or defined by historical averages. It can also
be calculated by using historical earnings averages multiplied by the estimated payout ratio.
The dividend growth is closely tied to economic growth and is typically estimated by the sum
18 of 33
of the expected inflation and the real U.S. GDP growth. Different approaches to component
estimates result in a range of 6.23% to 6.54%.
3. Siegel Model
The “Siegel Model” (see, for example [19]) forecasts compound real stock returns as
normalized earnings’ yields, calculated as normalized earnings divided by the price. Earnings
are paid as dividends or reinvested. This model is based on the observation that the S&P 500
earnings’ yields in the past track closely with 20-25 year rolling averages of future S&P 500
returns. Earnings’ yields can be estimated as reciprocals of the price-to-earnings or P/E ratios
which are included in the standard valuation information calculated for stocks or equity
indices. In particular, for the S&P 500 Index we obtained the valuation data from Bloomberg
and Standard & Poor’s starting in 1960 for annual earnings and S&P 500 levels. The predictive
power of the earnings observed historically is not perfect, even in the past. We consider this
model for completeness but give it less weight than the equity DDM. The expected equity
return range obtained using this model is a bit higher than other models’ results at 7.20% to
7.33%.
4. Historical risk premium over 10-year Treasury
Although we believe historical analysis is not as good a forecasting tool as the forward-looking
models, we calculated historical equity premiums over 10-year Treasury returns on a
compound basis. We estimated expected equity return as our forecast for 10-year Treasury
return of 2.85% plus estimated risk premium based on historical averages over the most recent
20 to 30 years. The resulting range is 5.65% to 6.40%.
5. Consensus among major investment/actuarial firms
As the last step, we review the most recent CMA reports published by other investment and
actuarial firms. Although the investment periods vary for these studies, we study these CMA
reports for the current peer consensus. We review and evaluate their rationale, methodologies
and investment horizons, as well as changes from prior estimates. For the task at hand, the
CMA 2016 reports pointed to a fairly broad range of 6.50% to 7.20% for the future expected
equity returns.
U.S. Large Cap Equity data and summary results from all sources
The multi-stage DDM was based on the following data/assumptions:
•
S&P 500 Index level of 2,044 as of December 31, 2015
•
Base earnings of $106.3 per share for 2015. Consensus forecasts as of mid-2015 for 2015,
2016 and 2017 of $118.3, $120.8, and $128.3, respectively
•
Dividend payout ratio of 0.40 (average from 1986 to 2015)
•
Expected inflation 2.00% (our 2016 forward-looking inflation assumption)
•
The U. S. GDP real growth rate from the CBO forecast. The CBO forecast is for years from
2016 to 2023. For years after that, we assumed the ultimate rate of real GDP growth rate to
equal the average of 2020-2023 at 2.03%.
The stable growth DDM was based on the following estimates:
•
Trailing dividend yields range from 2.08% to 2.34% (based on S&P 2015 earnings yield and
range of payout ratios from 0.40 to 0.45)
19 of 33
•
Real dividend growth range of 2.06% to 2.11% based on the average real GDP growth
forecasted by CBO and averaged over the CBO forecast horizon, as well as for the next 20
years (2.03% used for years after 2024)
•
Expected inflation of 2.00%
The Siegel model used the current (2015) earnings yield of 5.20% as a low end estimate. The average
of the S&P 500 earnings yields over the last 20-years serve as the high estimate (5.33%). Historical
averages of the S&P 500 return risk premium over 10-year Treasury returns were ranging from 2.80%
to 3.55% over the last 20 to 30 years.
Table 6 - Summary of the U.S. Large Cap Equity models
Model
Multi-stage DDM
Stable growth DDM
Siegel Model (earnings yield)
Historical risk premium over 10-year Treasury
Consensus among investment/actuarial firms
Our recommendation
Expected
geometric return
6.48%
6.23% to 6.54%
7.20% to 7.33%
5.65% to 6.40%
6.50% to 7.20%
6.50%
Considering the current market and economic outlook we feel that the multi-stage DDM model
provides the best estimate. All ranges except for the Siegel Model estimates are consistent. As we
mention earlier this result is presented here for completeness. Balancing information from the sources
discussed with emphasis on multi-stage DDM result, we recommend a long-term expected geometric
return for U.S. Large Cap Equity at 6.50%.
U.S. Small Cap Equity
We believe that the historical record, although mixed, is supportive of the view that there is no
expectation of U.S. Small Cap Equity outperforming U.S. Large Cap Equity in the long term. Over the
last 30 years the geometric mean of the small-cap premium (over large-cap) is actually negative 1.1%.
Over the last 10 years or over the last 20 years, this premium is negative 0.5% and negative 0.2%,
respectively. The graph in Figure 5 illustrates a theory that the small-cap premium is inversely
proportional to the large-cap stock’s performance on a real return basis. Consider the periods ending
between 1975 and 1987, for example, versus the more recent period ending between 1997 and 2009.
The most recent history lends support to our no-premium view.
20 of 33
Figure 6
Source: Wilshire Compass database and the Principal RAS calculations
Based on the no-premium view, we recommend using the same long-term expected compound annual
return for U.S. Small Cap Equity as U.S. Large Cap Equity of 6.50%.
Non-U.S. Equity
The view of a no-premium expectation for non-U.S. Equity (developed countries) versus U.S. Equity is
gaining acceptance among financial and investment professionals. Just as in the case with the Small
Cap premium, this view is supported by historical market data. For the full period of availability for the
non-U.S. stock data (going back to 1970), the compound real return on non-U.S. Equity is 5.2% (in USD
returns). Compare this number to the same period compound real return on U.S. Equity of 5.9%. Over
the same period, we calculated a non-U.S. Equity-premium over U.S. Equity (on the real compound
basis) by dividing annual real wealth accumulation for non-U.S. Equity by the U.S. Equity annual real
wealth accumulation factor. The geometric average of this measure is 0.993.
Based on this historical data, just as in the case of the U.S. Small Cap Equity, our recommendation for a
non-U.S. Equity expected return assumption is 6.50%, the same as U.S. Equity.
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Historical comparison of non-U.S. and U.S. Small Cap Equity to U.S. Large Cap Equity
Table 7 summarizes empirical support for a no-premium expectation comparing U.S. Large Cap Equity
to U.S. Small Cap Equity and to non-U.S. Equity (developed countries).
Table 7
Compound Real Rates
Over the last 10 years
Over the last 20 years
Over the last 30 years
Premium over U.S. Large Cap (10 years)
Premium over U.S. Large Cap (20 years)
Premium over U.S. Large Cap (30 years)
U.S. Large
Cap Equity
5.3%
5.9%
7.6%
U.S. Small
Cap Equity
4.9%
5.7%
6.3%
-0.5%
-0.2%
-1.1%
Non-U.S.
Equity
1.6%
2.6%
5.0%
-3.5%
-3.1%
-2.3%
22 of 33
Real Estate
Real Estate Investment Trust securities (REITs)
A Real Estate Investment Trust (REIT) is a company that owns and, in most cases, operates incomeproducing real estate. Investing in U.S. Real Estate Investment Trusts securities (or REITs) is a stock
investment which is similar to any other stock that represents ownership in an operating business. As
mentioned in the Methodology overview, we use a model similar to the equity multi-stage DDM.
Our REITs valuation model is a cash flow discount model (or IRR model) with the following inputs and
assumptions:
•
Our 10-year forecast for the 10-year Treasury rate, transitioning from current 1.49% to longterm equilibrium of 4.10% over the next 5 years.
•
REITs’ dividend yield premium (versus 10-year Treasury rate) transition from the current 2.10%
to the long-term equilibrium of 1.08%. This long-term premium is based on the historical
average over the last 20 years.
•
For the stable growth stage of the model, the growth rate is assumed to be three quarters of
the long-term inflation. According to Wilshire Associates [9], who examined dividend growth
historically, the REITs are able to pass through about three quarters of the long-term inflation
through rent and dividend increases.
The internal rate of return resulting from this model and inputs is 6.11% and we set our
recommendation for REITs expected return at 6.10%.
Real Estate Direct Property
For the Real Estate (RE) Direct Property asset class, we used market data from the NCREIF National
Property Index. It consists of U.S. commercial real estate properties that have been acquired, at least
in part, on behalf of tax-exempt institutions and are held in a fiduciary environment. Since the
benchmark index assumes no leverage, our RE assumption is for RE Direct Property (unlevered).
As is the case with high yield, equity, and REITs, for Real Estate we primarily rely on the forwardlooking cash flow discount model. As before in the simple version, the expected return is estimated by
Real Estate dividend yield plus the estimated growth in Net Operation Income (NOI). Again, the
variation comes from different approaches in estimating the dividend yield and the growth rate [20,
21]. For this purpose, we used the NCREIF data obtained from NCREIF database. Using a payout ratio
of 68% based on the 25 average, we obtained a range of dividend yields of 3.5% to 4.0%. The U.S.
dividend yield is the Cap Rate multiplied by payouts, and we estimated the NOI growth to fall in a
range of 1.50% to 1.80%. This model resulted in the expected return range of 5.00% to 5.80%
As a secondary source of information, we also considered historical analysis of Real Estate returns over
inflation. This historical analysis points to a range of 5.20% to 5.80%, based on the historical average of
real returns over the last 10 years and 30 years.
23 of 33
Finally, 2016 RE class return expectations by other experts, such as Wilshire and JP Morgan, fall in a
range of 5.50% to 5.80%.
Accordingly, our recommended assumption for the RE asset class expected geometric return is 5.50%.
24 of 33
Returns, risks, and correlations
As discussed in the Methodology overview section, the historical data for risk and correlation measures
are relatively stable and it is a common practice among investment and financial professionals to
estimate these measures directly from the historical data. For this purpose, for each broad asset class
we use monthly and annual return data from the Wilshire Compass and FACTSET databases for the
appropriate market benchmarks indicated in Appendix C. The data for most indices are available since
the 1970s except for the Treasury STRIPS where historical data go back to the early 1990s.
Annual standard deviations are developed using 30 years of data (or the full period of availability, if
less) for all classes. For Real Estate returns, data are available on a quarterly basis, and the annual
standard deviation is calculated directly based on the annual returns compounded from quarterly.
When monthly data are used, a series of monthly returns produce a monthly standard deviation. To
calculate annualized standard deviation from the monthly data series we use the “mathematically
correct” formula [22], rather than a common approximation of multiplying by the square root of 12.
We agree with the author that this is the appropriate method of annualizing monthly standard
deviation because annual return is not a sum of monthly returns but rather a compound measure.
We note that the exact formula treats an annual return as the compound return of twelve
independent monthly returns. We analyzed standard deviation obtained by using monthly versus
annual data for fixed income classes due to more pronounced serial correlation than for other broad
asset classes. We believe that annual calculation is more appropriate for these classes.
Our CMA development focus is on the geometric expected returns as the logical outcome of the
forward-looking IRR models and our historical analysis was also conducted on a compound, i.e.,
geometric, basis. For certain applications, however, the arithmetic expected returns are the
appropriate measures. For each asset class we calculate the arithmetic expected return from the
geometric expected return and the standard deviation under an assumption of log-normal distribution
of returns. Please see Appendix D for detailed background on the geometric and the arithmetic means,
and the lognormal distribution. Conversion formulas are well known and can be found in multiple
sources (see, for example, [23, 24]).
Finally, the correlation matrix is developed directly from the monthly or annual data, as appropriate,
for the full period of availability. The correlation matrix completes the Principal RAS CMA 2016 set
and it is required in order to calculate investment characteristics of a portfolio. Appendix B presents
the arithmetic return, the geometric return, the standard deviation for each class and the correlation
matrix.
25 of 33
Conclusion
In conclusion, after significant research, analysis and review of the consensus between various firms’
and agencies’ expectations, we believe that our economic outlook and market expectations are
reasonable and appropriate for the purpose of setting economic assumptions for pension plans over
the long term horizon of 20 to 30 years. Our capital market assumptions should be considered as a set
including future economic environment, expected returns, risks, and correlations. Actuaries should
exercise professional judgment in setting economic and other actuarial assumptions, and other
relevant considerations, such as professional guidance, requirements, and best practices, should be
taken into account.
26 of 33
References
The list of relevant literature is organized in order of appearance in the body of this report.
[1] E. V. Black, Y. Qiu, “Long-Term Capital Market Assumptions”, Principal Financial Group,
Retirement Actuarial Services, 2014
[2] Federal Reserve, Minutes of the Federal Open Market Committee, June 14-15, 2016
[3] J. G. Haubrich, S. Millington, “PCE and CPI Inflation: What’s the Difference”, Federal Reserve
Bank of Cleveland, April 2014,
http://www.clevelandfed.org/research/trends/2014/0514/01infpri.cfm
[4] D. Short, “Two Measures of Inflation and Fed Policy”, September 30, 2016,
http://www.advisorperspectives.com/dshort/updates/2016/09/30/two-measures-of-inflation-andfed-policy
[5] Information from the Federal Bank of Cleveland can be found (http://www.clevelandfed.org/)
[6] Congressional Budget Office, “The Budget and Economic Outlook: Fiscal Years for 2016 to
2026”, January 2016
[7] Excel supplemental materials, specifically Figure 2, posted on
(https://www.cbo.gov/publication/51129)
[8] J.P. Morgan Asset Management, “2016 Long-Term Capital Market Assumptions”, October 2015
[9] M. Rush, R. Walker, “2016 Asset Allocation Return and Risk Assumptions”, Wilshire Consulting,
January 26, 2016
[10] M. Rausch, “10-year Capital Market Return Assumptions – Calendar Year 2016”, BNY Mellon
Fiduciary Solutions, 2016
[11] Voya Investment Management Multi-Asset Strategies and Solutions Team, “2016 Long-Term
Capital Market Forecasts”, January 2016
[12] SEI Portfolio Strategies Group, “Capital Market Assumptions Update”, April 2016
[13] AonHewitt, “Capital Market Assumptions, as of 30 September 2015”, 2015
[14] E. Freedman, “It could Be Different This Time”, Captrust Consulting, January 27, 2016
[15] Benjamin J. Yang, “High Yield Market Update”, Wilshire Investment Research, January 14,
2005
[16] Moody’s Investors Service, “Annual Default Study: Corporate Default and Recovery Rates,
1920-2015
[17] Roger Ibbotson, Peng Chen, “Stock Market Returns in the Long Run: Participating in Real
Economy”, Financial Analyst Journal, 2002
[18] John Campbell, “Forecasting U.S. Equity Returns in the 21st Century”, Harvard University, July
2001
[19] J.R. Ritter, “Economic Growth and Equity Returns”, Pacific-Basin Finance Journal 13, 2005
[20] Blackrock, “Long-Term Income Opportunities in U.S. Real Estate”, June 2013
[21] National Council of Real Estate Investment Fiduciaries (NCREIF), http://www.ncreif.org
[22] Paul D. Kaplan, “What’s Wrong with Multiplying by the Square Root of Twelve”, Morningstar,
January 2013
[23] D. Pachamanova, F. Fabozzi “Simulation and Optimization in Finance: modeling with MATLAB,
@RISK, or VBA” (2010), John Wiley & Sons (2010)
27 of 33
[24] American Academy of Actuaries, the Pension Committee, “Exposure Draft, Selecting
Investment Return Assumptions Based on Anticipated Future Experience”, April 2016
28 of 33
Appendix A: Comparison of CMA 2014 and CMA 2016
Geometric Expected Return
Asset Class
Equity
U.S. Equity - Large Cap
U.S. Equity - Mid Cap
U.S. Equity - Small Cap
Non-U.S. Equity
Real Estate
REITs
Real Estate (direct property)
Fixed Income
Cash
TIPS
Core Bond
Aggregate Credit Bond
Long Credit Bond
Long Gov't/Credit Bond
Long Gov't Bond
Ultra-Long Gov't Bond
High Yield
Arithmetic Expected Return
CMA 2014
CMA 2016
Change
CMA 2014
CMA 2016
Change
7.45%
7.45%
7.45%
7.45%
6.50%
6.50%
6.50%
6.50%
-0.95%
-0.95%
-0.95%
-0.95%
8.80%
9.10%
9.55%
9.20%
7.85%
8.10%
8.55%
8.10%
-0.95%
-1.00%
-1.00%
-1.10%
6.55%
5.95%
6.10%
5.50%
-0.45%
-0.45%
8.35%
6.30%
7.95%
5.80%
-0.40%
-0.50%
1.80%
3.90%
4.15%
4.65%
5.65%
5.10%
4.20%
4.00%
5.90%
1.55%
2.85%
3.60%
4.05%
4.75%
3.90%
2.65%
1.85%
6.30%
-0.25%
-1.05%
-0.55%
-0.60%
-0.90%
-1.20%
-1.55%
-2.15%
0.40%
1.80%
4.10%
4.25%
4.80%
6.05%
5.55%
4.80%
6.20%
6.30%
1.55%
3.05%
3.75%
4.25%
5.15%
4.30%
3.30%
4.10%
6.70%
-0.25%
-1.05%
-0.50%
-0.55%
-0.90%
-1.25%
-1.50%
-2.10%
0.40%
29 of 33
U.S. Equity Small Cap
Non-U.S. Equity
REITs
Real Estate
Cash
TIPS
Core Bond
Agg. Credit Bond
Long Credit Bond
Long Gov't/Credit
Bond
Long Gov't Bond
Ultra-Long Gov't bond
High Yield
Agg. Credit Bond
Long Credit Bond
Long Gov't/Credit
Long Gov't Bond
Ultra-Long Gov't
High Yield
U.S. Equity Mid Cap
Exp. arithmetic return
Exp. geometric return
Standard deviation
U.S. Equity Large Cap
U.S. Equity Mid Cap
U.S. Equity Small Cap
Non-U.S. Equity
REITs
Real Estate (direct )
Cash
TIPS
Core Bond
U.S. Equity Large Cap
Fixed Income
RE
Equity
Appendix B: Return, risk and correlation summary table
7.85
6.50
17.2
1.00
0.94
0.83
0.64
0.58
0.21
0.03
0.02
0.21
0.34
0.33
0.22
0.09
-0.16
0.58
8.10
6.50
18.8
0.94
1.00
0.94
0.66
0.65
0.09
0.01
0.07
0.20
0.32
0.30
0.18
0.06
-0.19
0.65
8.55
6.50
21.5
0.83
0.94
1.00
0.60
0.65
0.08
0.01
-0.01
0.11
0.21
0.21
0.09
-0.02
-0.22
0.62
8.10
6.50
18.9
0.64
0.66
0.60
1.00
0.47
0.15
0.02
0.10
0.15
0.26
0.26
0.15
0.03
-0.19
0.52
7.95
6.10
20.2
0.58
0.65
0.65
0.47
1.00
0.11
-0.01
0.24
0.18
0.30
0.33
0.22
0.09
-0.02
0.59
5.80
5.50
7.9
0.21
0.09
0.08
0.15
0.11
1.00
0.00
-0.09
-0.06
-0.23
-0.11
0.11
0.20
0.19
-0.36
1.55
1.55
0.8
0.03
0.01
0.01
0.02
-0.01
0.00
1.00
0.04
0.12
0.06
0.02
0.02
0.03
0.01
-0.03
3.05
2.85
6.4
0.02
0.07
-0.01
0.10
0.24
-0.09
0.04
1.00
0.77
0.73
0.68
0.70
0.61
0.51
0.28
3.75
3.60
5.1
0.21
0.20
0.11
0.15
0.18
-0.06
0.12
0.77
1.00
0.94
0.90
0.94
0.90
0.77
0.25
4.25
4.05
6.4
0.34
0.32
0.21
0.26
0.30
-0.23
0.06
0.73
0.94
1.00
0.97
0.91
0.79
0.64
0.49
5.15
4.75
9.2
0.33
0.30
0.21
0.26
0.33
-0.11
0.02
0.68
0.90
0.97
1.00
0.94
0.84
0.71
0.48
4.30
3.90
9.4
0.22
0.18
0.09
0.15
0.22
0.11
0.02
0.70
0.94
0.91
0.94
1.00
0.96
0.87
0.23
3.30
2.65
11.8
0.09
0.06
-0.02
0.03
0.09
0.20
0.03
0.61
0.90
0.79
0.84
0.96
1.00
0.95
0.02
4.10
1.85
21.9
-0.16
-0.19
-0.22
-0.19
-0.02
0.19
0.01
0.51
0.77
0.64
0.71
0.87
0.95
1.00
-0.12
6.70
6.30
9.3
0.58
0.65
0.62
0.52
0.59
-0.36
-0.03
0.28
0.25
0.49
0.48
0.23
0.02
-0.12
1.00
30 of 33
Appendix C: Benchmarks for asset classes
Asset class
Benchmark utilized
U.S. Equity - Large Cap
U.S. Equity - Mid Cap
U.S. Equity - Small Cap
Non-U.S. Equity
REITs
Real Estate (direct property)
Cash
TIPS
Core Bond
Aggregate Credit Bond
Long Credit Bond
Long Gov't/Credit Bond
Long Gov't Bond
Ultra-Long Gov't Bond
High Yield
S&P - 500 Index
Russell - Midcap Index
Russell - 2000 Index
MSCI - EAFE Index ($Gross)
FTSE NAREIT - All Equity REITs Index
NCREIF - National Property Index
Citi - 3 Month T-Bill Index
Bloomberg Barclays - U.S. TIPS Index
Bloomberg Barclays - U.S. Aggregate Index
Bloomberg Barclays - U.S. Aggregate Credit Index
Bloomberg Barclays - U.S. Credit Long Index
Bloomberg Barclays - U.S. Gov't/Credit Long Index
Bloomberg Barclays - U.S. Government Long Index
Bloomberg Barclays U.S. Treasury Strips (20+ Y)
Bloomberg Barclays - U.S. High Yield Index
Bloomberg Barclays U.S. Aggregate Corporate Long (AA or >)
Bloomberg Barclays U.S. Aggregate Corporate Intermediate (AA)
Long Corporate High Quality Bond
Intermediate Corporate High Quality
Bond
Source: whenever historical information is utilized in the analysis, the relevant data for the
appropriate market index was obtained from either FACTSET or Wilshire Compass.
31 of 33
Appendix D: Statistical analysis of returns
Geometric mean and arithmetic mean and lognormal distribution
First, some general theoretical comments are in order with regard to statistical analysis of asset
returns and the distinction between geometric and arithmetic means. When estimating expected
returns, the geometric mean of the historical returns represents the backward-looking measure as it
quantifies annualized compound returns measuring annual wealth accumulation over multi-year
periods. When forecasting asset returns for future single annual periods, the arithmetic mean is a more
appropriate measure. However, when forecasting annualized compound return over multi-year
periods, the geometric mean should be used.
The arithmetic mean is a simple average of the elements in the asset return series. For a historical
return series, the arithmetic mean is an average of annual returns. The geometric mean of a return
series is a compound annual rate of return over the multi-year period. A simple example illustrates the
difference between an arithmetic mean and a geometric mean. Let us assume that we invested $100
for a two-year period into a stock that over the first year returned 20% and over the second year lost
20%. At the end of the first year, we have $120 and at the end of the second year we have $96. The
arithmetic mean is 0% but the geometric mean is the square root of $96 divided by $100 minus 1, or
negative 2%. It is clear from this example that the geometric mean is the appropriate measure to
analyze a historical return series. It is also fitting for the forward-looking forecasting of the annual
compound returns in multi-year periods. Note that in this example, the arithmetic mean overstates the
actual return over a multi-year period. This is because our wealth over the two-year period is actually
reduced from $100 to $96, while a 0% annual return implies that we experienced neither a gain nor a
loss. An example of a forward-looking compound annual return measure is the yield on a fixed income
investment. Since our models in this study are either based on analysis of historical return series or on
the forward-looking yield-and-growth type models, our conclusions are generally in terms of
geometric means.
Because the arithmetic mean better represents typical asset performance for a future single annual
period, it is more appropriate for forecasting studies (such as ALM studies) or accounting expense
calculations for corporate pension plans. In general, the arithmetic mean is greater than the geometric
mean and the two are equal only for a series consisting of constant terms. The difference between the
arithmetic mean and the geometric mean is positively related to the variance of the series. In order to
calculate the arithmetic mean given the geometric mean and variance, it is a standard practice among
financial and investment professionals to assume that the annual wealth factor (one plus annual
return) follows a lognormal distribution. The back-of-the-envelope “rule of thumb” is that the
difference between the arithmetic and the geometric means is approximately equal to half of the
variance. However, this study utilizes the exact formula relating the geometric and the arithmetic
means of the log-normally distributed random variable (see, for example, [22] for additional details on
modeling assets and lognormal distribution).
32 of 33
Standard deviation
The formula relating the arithmetic and geometric means of a lognormal random variable also
requires the standard deviation of the distribution. The standard deviation, or equivalently the
variance, of the asset class returns is typically estimated from the historical return series of an
appropriate market benchmark index. Although the result depends on the length of the period
selected, it is usually sufficiently stable from period to period. The methodology of the calculation has
material impact on the resulting standard deviation. Monthly returns are often used for this purpose,
allowing for a longer time series. A common rule of thumb is to estimate the annual standard deviation
by multiplying the monthly standard deviation by the square root of twelve. While we feel this is a
good approximation for a “back-of-the-envelope” estimate, we also believe that it is not appropriate
for quantitative investment analysis for the following reason. The approximate formula treats the
annual return as a sum of independent monthly returns. But the annual return is a compound measure
of monthly returns, so this study utilizes the exact formula that reflects this fact.
Equivalent arithmetic returns assuming lognormal distribution
The last step in determining the expected returns for broad asset classes, appropriate for forecasting
annual returns, is to calculate the equivalent arithmetic expected returns given compound return
expectations and standard deviations for each broad asset class. As mentioned earlier, we use the
formula relating geometric and arithmetic means of the lognormal distribution. The table in Appendix
A show results of these calculations and compares both geometric and arithmetic expected returns for
our CMA sets.
Please note that when forecasting or modeling annual returns, it is appropriate to use the arithmetic
mean as it is the expected value of the annual return distribution (see for example [22]). For a portfolio
or a linear combination of the asset classes, annual expected return is calculated as a weighted
average of asset classes’ expected annual returns according to the asset allocation of the portfolio.
When forecasting compound annual returns for multi-year periods for a portfolio, or the estimated
expected value of this measure, one can either utilize simulation models or use an approach allowing
for closed formulas relating the arithmetic and geometric means as we used for broad asset classes.
For the latter approach, it is not appropriate to weight average geometric means of the broad asset
classes according to portfolio asset allocation because it does not reflect the relationship between the
asset classes and the diversification effect. Instead, the portfolio’s expected annual return and variance
reflecting correlations between asset classes should be calculated. Assuming a lognormal distribution
of portfolio returns, the geometric mean can be calculated and used as an estimate of the portfolio’s
compound annualized return expectation [23, 24].
Whether one uses simulation models or a “closed-formula” approach, it is appropriate to estimate
expected annual or compound returns for a portfolio by selecting a point-estimate within reasonable
confidence intervals that reflect sources of uncertainly around expected investment return. These
sources may include rebalancing tolerance levels around the portfolio’s target asset allocation,
assumptions on return probability distribution and the appropriate length of the compounding period
as well as the very nature of stochastic forecasting.
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