Non-normality of Market Returns
A framework for asset allocation decision-making
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Foreword
When market stress and disruptions hit portfolios
especially hard, it is natural for investors to ask
whether anything could have been done differently. We revisit our tools and strategies looking
for ways to improve them. And often, it is in such
times of reflection that we arrive at important
insights and begin to craft effective solutions.
The turmoil of 2007/2008 is just one among many such
financial crises over the past 30 years—rare and unpredictable
events that can, and should, challenge conventional ideas about
portfolio construction.
Of course, we can never know in advance when such events will
occur and how bad they will be. We do know, however, that we
empirically observe such events with much greater frequency
than current models allow for. So, the proper line of questioning
for investors becomes: Can current risk management frameworks be modified to better capture the long-term, downside
risk associated with these rare but dangerous anomalies—i.e.
frameworks that take into account more than just one-standarddeviation event, and better reflect their observed frequency?
The goal of this paper is to present such a revised framework.
In broad scope, we believe that two specific weaknesses in conventional risk assessment may be contributing to a quantifiable
underestimation of portfolio risk:
• Frameworks assuming “normality”1: Conventional asset
allocation frameworks typically make a range of assumptions
about the “normality” of asset returns, the most problematic
of which are that returns are independent from period to
1
The normal distribution—recognizable by its bell shaped probability density function—
is a statistical distribution commonly used to model asset class returns in traditional
Mean-Variance Optimization frameworks.
2
Though “independence” is not in strict statistical terms a form of “normality”, we include it
here because the assumption of “independence” is one of the central tenets of conventional
asset allocation frameworks built around the concept of “normality” of asset returns.
period2 and normally distributed. In reality, we can empirically
observe that in many cases returns are not independent, and
in all cases they are not normally distributed.
• Inadequate risk measures: If one adopts an asset allocation
framework that incorporates non-normality, then standard
deviation becomes ineffective as the primary quantifier of
portfolio risk.
Using the latest statistical methods, however, we believe that
these shortcomings can be addressed. And based on the results
presented here, we argue that a modified risk framework may
help investors improve portfolio efficiency and resiliency.
Of course, no single statistical framework—no matter how
robust—can render portfolios immune to such extreme events.
Even the most important concepts and discoveries in finance
continue to evolve through a process of dialogue and
refinement.
But our hope is that the work presented here will be an effective step toward confronting and clarifying some critically
unresolved issues in asset allocation—specifically, the habit of
sacrificing accuracy in favor of ease-of-use when addressing the
issue of non-normality of returns. This trade-off is potentially
very harmful as it understates portfolio risk, and given advancements in statistical techniques, it is also no longer necessary.
The challenge of asset allocation is one of both art and science,
and to this end we hope that this work will prompt both discussion and further research, all geared toward helping investors
better address the most pressing issues of the day. We appreciate all comments, questions, and contributions that this paper
might prompt from clients and colleagues alike.
Abdullah Z. Sheikh, FIA, FSA
Director of Research
Strategic Investment Advisory Group
J.P. Morgan Asset Management | 1
Table of Contents
1
Foreword
3
Overview
7
Empirical evidence of non-normality
in asset returns
14 Incorporating non-normality into an asset
allocation framework
24 Estimating downside portfolio risk in a
non-normal framework
27 Implications of non-normality on optimal
portfolio solutions 31 Conclusion: Incorporating non-normality
can lead to more efficient portfolios
32 Appendix
2 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Overview
Non-Normality and Its Impact on Portfolio Risk
As sudden market disruptions go, the events of the past
year—the sub-prime mortgage crisis and ensuing global financial
meltdown—are not without precedent. Just in the last three
decades, investors have been faced with a number of financial
crises:
•
•
•
•
•
•
•
Latin American debt crisis in the early 1980s
Stock market crash of 1987
U.S. Savings & Loan crisis in 1989–1991
Western European exchange rate mechanism crisis in 1992
East Asian financial crisis of 1997
Russian default crisis and the LTCM Hedge Fund crisis in 1998
Bursting of U.S. Technology bubble in 2000-2001
While these anomalous events are rare, we observe such
extreme “non-normality” in real-world markets more frequently
than current risk management approaches allow for. Said
another way, we believe that conventionally derived portfolios
carry a higher level of downside risk than many investors
believe, or current portfolio modeling techniques can identify.
The primary reason for this underestimation of risk lies in the
conventional approach to applying mean-variance theory, which
was pioneered by Harry Markowitz in 1952. Traditional meanvariance frameworks have become the bedrock of top-down
asset allocation decision making. As suggested above, a standard
assumption in the mean-variance framework, and indeed many
other holistic asset allocation frameworks, is that future asset
class returns will be independent from period to period and
normally distributed.
Despite being widely recognized as overly simplistic, such broad
assumptions of “normality” have appeal due to the ease with
3
In this context, the converse is also true, i.e. empirical observations of non-normality are
the result of extreme market scenarios.
4
Asset classes include U.S. Aggregate Bonds, U.S. Large Cap Equity, International Equity,
Emerging Markets Equity, Real Estate Investment Trusts, Hedge Fund of Funds and Private
Equity.
5
“Influenced” in this context refers to a statistically significant coefficient when one month’s
return is regressed against the previous month’s return.
which they can be implemented. To implement an asset allocation framework based on normal asset return distributions,
practitioners need only make two assumptions for each asset
class (mean and standard deviation) and one assumption for
each pair (co-variance). The latter applies because one implicitly
assumes the relationship between each pair of asset classes is
linear (another problematic issue discussed later in the paper).
But what if asset returns are not normally distributed?
In fact, in the real world we can observe empirically that returns
are not normally distributed. This leads us to ask: How would
incorporating non-normality affect the strategic asset allocation
process?
Empirically Identifying Non-Normality
Each of the market events mentioned above was due to an
incidence of non-normality of one sort or another3. And the
practical implication is that incorporating non-normality into
the asset allocation process would force recognition of greater
downside risk to the portfolio, precisely from such extreme,
unexpected negative events.
Our focus here is on capturing the impact of non-normality
on downside portfolio risk—as well as the asset allocation/
optimization process—rather than identifying the specific sources
of non-normality itself (although we do discuss economic or
behavioral factors where relevant).
Specifically, we test seven asset classes4 and empirically confirm
three primary categories of non-normality:
• Serial Correlation: A critical pillar of many traditional asset
allocation frameworks (i.e. frameworks built on a premise
of “normality”), is the assumption that asset returns from
period to period are independent and identically distributed.
However, if one month’s return is ‘influenced’5 by the previous
month’s return, then there may be a need to account for this
effect in future asset projections. Typically, traditional asset
allocation frameworks do not allow for serial correlation,
J.P. Morgan Asset Management | 3
but we identify serial correlation in four of the seven asset
classes we model.
Serial correlation, if not adjusted for in the underlying data,
masks true asset class volatility and biases risk estimates downwards, leading to underestimation of overall portfolio risk.
• “Fat” Left Tails (Negative Skewness and Leptokurtosis): Our
second form of non-normality relates to observing negative
returns in greater magnitude and with a higher probability
than implied by the normal distribution. This phenomenon
is commonly referred to as “fat” left tails.
Exhibit 1 illustrates this phenomenon for monthly dollar-hedged
International Equity returns over the ten years to October 2008.
The chart plots the empirical (i.e. observed) probability density
function of the data relative to a normal distribution.
One can see that the observed return series (blue line) is more
peaked, has a higher density at the extreme left, and leans
further to the right than the normal distribution (orange line).
The rightward lean is its “negative skewness”. The consequence
of this skewness is that the left slope of the blue line is longer
than the left slope of the orange line—i.e. it has a longer tail,
which indicates a greater magnitude of extreme negative events.
In addition, the blue line is taller at its apex and shows a higher
density at the extreme left end (i.e. leptokurtosis). In particular,
the higher density at the left tail indicates a higher probability
of extreme negative events.
An asset allocation framework based on the normal distribution would understate both the frequency and magnitude of
extreme negative events, as well as their potential effects on
portfolio returns and efficiency.
• Correlation Breakdown: The simple correlations often used
in traditional asset allocation models assume a linear relationship between asset classes—i.e. they assume that the
relationship between the variables at the extremes is similar
to their relationship at less extreme values. Using simple
linear correlations is the equivalent of assuming that the
‘joint’ distribution of asset returns is (multivariate) normal.
Joint distributions capture how asset classes behave together
rather than individually.
However, we find that in many cases correlations under extreme
conditions are quite different than under normal conditions. In
other words, the expected linear correlations breakdown and
asset classes exhibit quite different joint behavior. The relationships, in fact, are not linear and the assumption of linearity
(by using linear correlation matrices) underestimates the
probability of joint negative returns under extreme conditions.
Relying on linear correlation matrices tends to overestimate
the benefits of portfolio diversification during periods of high
market volatility. This leads to a systematic underestimation of
downside portfolio risk.
Statistical Approaches for Incorporating Non-Normality
Fortunately, we have at our disposal sophisticated statistical
tools that allow us to correct for these types of non-normality.
Exhibit 1: International Equities—“Fat” Left Tails in historical returns
12
Empirical
10
Normal
Density
8
6
4
“Fat” left tails
2
0
-20%
-15%
-10%
-5%
0%
5%
Monthly return
10%
Source: J.P. Morgan Asset Management. For illustrative purposes only.
15%
20%
• Unsmoothing Serial Correlation: Serial correlations can be
“unsmoothed”. That is, we can correct for the influence of
prior-period returns and restore independence to singleperiod returns. To arrive at our new adjusted return series
we apply a variation of Fisher-Geltner-Webb’s well-established
“unsmoothing” methodology6. Our ‘new’ adjusted return
series is better reflective of the risk characteristics of the
asset class. Notably, the new unsmoothed return series has
the same mean as our original return series, but shows a
higher volatility, and thus higher downside risk.
6
For a more detailed treatment, please refer to Fisher, J., D. Geltner, and B. Webb. 1994.
Value Indices of Commercial Real Estate: A Comparison of Index Construction Methods.
Also, Fisher, J. and D. Geltner. 2000. De-Lagging the NCREIF Index: Transaction Prices and
Reverse-Engineering.
4 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
• Modeling “Fat” Left Tails (Negative Skewness and Leptokurtosis): “Fat” left tails can be addressed using Extreme
Value theory—a body of work specifically designed to look at
the probability of high-risk, but low-probability, events such
as floods, earthquakes and large insurance losses. In other
words, its focus is estimating tail risk.
By applying Extreme Value theory we can create asset return
distribution models that are a closer “fit” to the return series we
observe empirically, much more similar than normal distributions.
Exhibit 2 shows a probability density function for dollar-hedged
International Equities, calculated using Extreme Value Theory
(orange line).
Exhibit 2: International Equities—Applying Extreme Value theory
for a better fit
12
Empirical
Extreme value
10
Density
8
6
4
Much better fit
for left tail
2
0
-20%
-15%
-10%
-5%
0%
Monthly return
5%
10%
15%
Source: J.P. Morgan Asset Management. For illustrative purposes only.
It is a much closer approximation to empirically observed
performance in terms of negative skewness (rightward tilt) and
leptokurtosis (“fat” left tail). This process can be applied to all
assets in the portfolio, the overall result being an increase in
the portfolio’s downside risk profile.
• Simulating Correlation Breakdown: Finally, we can address
the issue of correlation breakdown using ‘copula’ theory7—a
body of work that explicitly looks at the impact of contagion
or converging correlations at the total portfolio level. Copulas
7
Copulas have been applied extensively to the pricing of Collateralized Debt Obligations—in
addition to other areas in finance. For a detailed treatment, please refer to “An Introduction
to Copulas” by Nelson R. R. 1999.
8
A key tenet of behavioral finance is the idea of loss aversion, i.e. a tendency of investors to
prefer avoiding losses than making gains. This translates to risk preferences that are
asymmetric in nature.
9
Our model calculates real portfolio value by discounting the nominal portfolio value using
projected inflation. Inflation itself is projected stochastically in our framework.
are mathematical functions that allow us to model the joint
distribution of asset returns separately from the marginal
(i.e. individual asset class) distributions.
• By considering joint distributions, we turn our focus to how
asset classes behave together rather than individually. In
particular, copulas allow us to model an increased incidence
of joint negative returns (i.e. the “fatter” joint left tails) in
our simulation results, just as we observe empirically in
real-world market data. And again, using a more accurate
proxy for observed behavior results in recognizing higher
downside portfolio risk, specifically due to increased dependence of asset returns during periods of market stress.
A Better Risk Quantifier: Conditional Value at Risk
We believe that empirical evidence suggests an imperative to
incorporate various types of “non-normality” into the asset
allocation and portfolio modeling process, specifically to better
understand and model downside portfolio risk. Yet if we take
this step, we have to ask whether or not our conventional risk
measure (i.e. standard deviation) is up to the new task.
We would argue that, in a framework based on non-normality,
standard deviation may not be investors’ most appropriate
measure of portfolio risk because it rewards the desirable
upside movements as hard as it punishes the undesirable
downside movements. This is generally inconsistent with
investor risk preferences—primarily as observed in the field
of behavioral finance8.
Conditional Value at Risk (CVaR95) overcomes many of the drawbacks of standard deviation as a risk measure. Primarily, as it
only measures risk on the downside, it captures both the asymmetric risk preferences of investors and the incidence of “fat”
left tails induced by skewed and leptokurtic return distributions.
Further, given the widespread use by major institutional investors and regulators of its first cousin—Value at Risk—we judge it
to be the most appropriate risk measure to incorporate it into
our framework.
We define Conditional Value at Risk (CVaR95) as the average real9
portfolio loss (or gain) relative to the starting portfolio value in
the worst five percent of scenarios, based on our 10,000 Monte
Carlo simulations. It is simply the average real loss (or gain) in
the worst 500 (5% of 10,000) scenarios i.e. the left tail of the
portfolio distribution.
J.P. Morgan Asset Management | 5
Incorporating Non-Normality: A Potentially Better Way
to Assess and Manage Downside Risk
Through rigorous analysis, we examine the impact of incorporating non-normality into the asset allocation process and
determine that it reveals increased downside risk, compared
with portfolios optimized using conventional mean-variance
frameworks.
These results are compelling: ignoring empiric observations of
non-normality in equity (and equity type) return distributions
understates portfolio downside risk. Likewise, using standard
deviation, rather than more behaviorally attuned conditional
value-at-risk measures, may in fact inadvertently increase
rather than decrease downside risk. Such an underestimation
of downside risk can have severe consequences for investors,
even in extreme cases presenting a solvency risk.
The remainder of this paper is focused on statistical proofs of
non-normality and the steps we apply to allow for them in the
asset allocation and optimization process. The applied methodologies can be, statistically speaking, quite intricate; but they
are very logical in their flow, and following the sequence of
steps is very straightforward.
But again, what we would emphasize here is the goal and overall impact of this work: to better capture downside portfolio risk
that is observed in real-world markets, but missed by traditional asset allocation frameworks. The intended result is a more
robust portfolio modeling approach that may help investors
improve portfolio efficiency and resiliency, in light of a clearer
understanding of portfolio risk.
6 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Empirical evidence of non-normality in
asset returns
Addressing the problem of non-normality
entails employing quantitative techniques that
are rather more sophisticated compared to
traditional asset allocation techniques. However,
our end goal is conceptually very simple: to
produce an asset allocation framework that
balances future portfolio return expectations
with the potential for more robust downside
risk management.
First, however, we must test to see whether and what kinds of
non-normality may be present in empirically observed asset
returns. In this section, we present the results of such tests,
which indicate the presence of several types of non-normality
that are typically not allowed for in traditional asset allocation
frameworks—to the detriment of risk estimation.
We address three categories of non-normality10 in the
following order:
A. Serial correlation in asset returns—this occurs when one
period’s return is correlated to the previous period’s return,
inducing dependence over time.
B. “Fat” left tails (negative skewness and leptokurtosis)—this
occurs when extreme negative returns are observed, with a
magnitude and frequency greater than implied by the ‘normal’
distribution.
C. Correlation breakdown in joint asset class returns—this
occurs during periods of high market volatility and is typically
not captured by linear correlation matrices.
Throughout this section, we base our tests on ten years of
monthly return data to October 31, 200811. In later sections we
then incorporate these empirical results into a revised asset
allocation framework.
Exhibit 3: Testing for serial correlation in monthly asset class returns
Benchmark index
Test statistic
P-value
Reject null
hypothesis of no
serial correlation
Statistically
significant lag
Aggregate Bonds
Barclays Capital U.S.
Aggregate Index
4.46
0.62
No
None
U.S. Equity
S&P500 Total Return Index
3.59
0.73
No
None
International Equity
MSCI EAFE (hedged) Index
4.48
0.03
Yes
One
Emerging Markets
Equity
MSCI Emerging Markets
Index
5.73
0.02
Yes
One
Real Estate Investment
Trusts (REITs)
FTSE / S&P NAREIT Index
1.48
0.96
No
None
Hedge Fund of Funds
HFRI Fund of Funds
Diversified Index
16.18
0.00
Yes
One
Private Equity
Dow Jones Wilshire Microcap
Total Return Index
4.50
0.03
Yes
One
For those familiar with this
statistical procedure, we note that
the complete official moniker for
the “Q-statistic” is the Ljung-Box
Q-statistic.
It is computed as follows:
Where Tj is the j-th serial
correlation and T is the number
of observations.
Source: J.P. Morgan Asset Management. For illustrative purposes only.
10
There are other forms of non-normality one can observe in asset returns. For example,
heteroskedasticity or serial correlation in volatilities (also called volatility clustering) is
another such form.
11
Our sources for these return streams are Datastream, Bloomberg, Barclays Capital, MSCI,
HFRI, FTSE, and Dow Jones.
J.P. Morgan Asset Management | 7
A. Serial Correlation of Returns
For each asset class, we formally test for serial correlation by
calculating what is called the “Q-statistic.” In this test, we are
looking to either affirm or reject the “null hypothesis,” which
asserts that serial correlation does not exist in the data.
If the Q-statistic for a given asset class has significance at a 5%
confidence level (i.e. a p-value of less than 0.05), we can conclude that there is sufficient evidence to reject the null hypothesis of no serial correlation. In this case, we must allow for the
effect of serial correlation on future asset class returns. If the
p-value is higher than 0.05, the null hypothesis is not rejected
and we conclude that there is insufficient evidence to reject the
null hypothesis of no serial correlation.
In Exhibit 3 on the previous page, we show the results of testing
for serial correlation in the return series of our universe of seven
asset classes.
We find statistical evidence of first-order serial correlation12 in
the returns of several asset classes: International Equity, Emerging Markets Equity, Hedge Fund of Funds, and Private Equity.
We hypothesize that serial correlation is common in alternative
investment strategies—such as Hedge Fund of Funds and Private
Equity—because they often hold illiquid and hard-to-price assets. The difficulty in valuing the underlying assets at regular intervals requires managers or administrators to estimate prices
(for example, with reference to the closest marketable security
or based on certain economic indicators).
In the case of International Equity and Emerging Markets Equity,
the presence of serial correlation is a very recent phenomenon.
In fact, it is a direct consequence of the financial crises experienced globally by investors over 2008 and into 2009.
Public equity markets around the world have fallen precipitously over 2008 and into 2009, displaying month-upon-month
of negative returns. In Developed International Equity, nine of
the last twelve monthly returns to October 2008 were negative.
In the case of Emerging Markets Equity, eight of the last twelve
monthly returns were negative. This has had the effect of inducing statistically significant serial correlation in monthly public
equity market returns for International and Emerging Markets
Equity.
Unfortunately, the presence of serial correlation distorts the
true risk characteristics of an asset class. In particular, it is
incorrect to model future returns assuming independence, if
in fact returns are serially correlated (as our statistical tests
indicate). If it is not corrected, serial correlation can reduce risk
estimates from a time series by inappropriately smoothing
asset class volatility
B. Evidence of “Fat” Left Tails
Our second form of non-normality relates to observing negative returns in greater magnitude and with a higher probability
Exhibit 4: histogram of U.S. equity returns—ten years ending
october 2008
25
20
Frequency
If current asset prices are derived—for example—by updating
last month’s asset prices (after allowing for changes in the
economic environment since the last valuation), then serial
correlation reflects a gradual recognition of the true underlying
value of the asset. Such a valuation methodology would explain
our empirical observation of first order serial correlation in the
asset returns of our alternative asset classes. And a consequence of gradual (rather than instantaneous) recognition of
the true underlying value of an asset class is that conventional
risk estimates derived from a serially correlated return stream
will be underestimated.
Key Statistics
Mean 0.1%
Median 0.7%
Maximum 9.8%
Minimum -16.8%
15
10
5
0
-18% -15% -12% -9%
12
First order serial correlation implies that the return from one month is correlated with the
return from the previous month.
Std Dev. 4.4%
Skewness -0.67
Kurtosis 4.20
Jarque-Bera 16.0
-6%
-3%
0%
3%
6%
Monthly return
Source: J.P. Morgan Asset Management. For illustrative purposes only.
8 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
9%
12%
Exhibit 4 shows a simple histogram of monthly returns for U.S.
Equities over the last ten years. It illustrates that the distribution is clearly not symmetric. In fact it has a very noticeable
negative skew (i.e. leans to the right) as well as excess kurtosis
(i.e. is more peaked than a normal distribution; it has a kurtosis
value of 4.15, which is above the “normal” value of 3.0). We can
confirm this intuitive observation with a simple statistical test:
the Jarque Bera (J-B) test13 is a simple test of non-normality
and conclusively rejects the null hypothesis of normality in the
return distribution.
When we repeat the J-B test using our other asset classes, we
find that those results, too, reject the null hypothesis. Their
returns also exhibit non-normality. We can illustrate the understatement of downside risk by comparing two types of density
functions:
Exhibit 5: U.S. Bonds—“Fat” left tails in historical returns
50
40
Density
than implied by the normal distribution. As an example, let us
consider U.S. Equities.
Normal
0
-6%
The “fat” left tail phenomenon implies that using a normal
distribution to model returns underestimates the frequency
and magnitude of downside events.
-4%
-2%
0%
Monthly return
2%
4%
6%
Exhibit 6: U.S. Equities—“Fat” left tails in historical returns
12
10
Empirical
Density
8
In Exhibits 5 through 11, we show density function results for all
seven asset classes. Our focus, in particular, is on the left tails
of the density function and on whether the normal distribution
underestimates the probability of outcomes at these extremes.
These kernel (or empirical) density functions clearly indicate “fat”
left tails relative to those implied by the normal distribution. The
table in Exhibit 12 summarizes the results of our tests, which
identify non-normality in all seven asset classes and “fat” left tails
across all asset classes.
Empirical
20
10
1. The density function of the normal distribution14
2. The kernel (i.e. empirical) density function ‘implied’ by actual
historical returns
30
Normal
6
4
2
0
-30%
-20%
-10%
0%
Monthly return
10%
20%
Exhibit 7: International Equities—“Fat” left tails in historical
returns
12
10
Empirical
Density
8
Normal
6
4
2
13
The Jarque-Bera (J-B) test measures the departure from normality of a sample, based on its
skewness and kurtosis. The J-B test statistic is defined as N/6 * (S2 + (K-3)2 / 4) where N is the
sample size, S is the skewness, and K is the kurtosis.
14
The normal distribution density function is parameterized using the Method of Maximum
Likelihood based on our sample of underlying historical data.
0
-20%
-15%
-10%
-5%
0%
5%
Monthly return
10%
15%
20%
Source: J.P. Morgan Asset Management. For illustrative purposes only.
J.P. Morgan Asset Management | 9
Exhibit 8: Emerging Markets Equities—“Fat” left tails in historical
returns
Exhibit 9: U.S. REITS—“Fat” left tails in historical returns
9
7
8
6
7
6
4
Density
Density
5
Empirical
3
5
Empirical
4
Normal
Normal
3
2
2
1
1
0
-40%
-30%
-20%
-10%
0%
10%
Monthly return
20%
30%
Exhibit 10: Hedge Fund of funds—“Fat” left tails in historical returns
8
28
7
24
6
-10%
0%
10%
Monthly return
20%
30%
Density
Normal
40%
Empirical
4
12
3
8
2
4
1
Normal
0
0
-100%
-20%
5
Empirical
16
-30%
Exhibit 11: Private Equity—“Fat” left tails in historical returns
32
20
Density
0
-40%
40%
-75%
-50%
-25%
0%
25%
Monthly return
50%
75%
-30%
100%
-20%
-10%
0%
Monthly return
10%
20%
30%
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Exhibit 12: Empirical evidence of “fat” left tails in asset returns
Skewness
Kurtosis
J-B Test Statistic
Rejects Normality
“Fat” Left Tail Compared to Normal
Aggregate Bonds
-0.70
4.03
15.15
Yes
Yes
U.S. Equity
-0.67
4.20
15.99
Yes
Yes
International Equity
-0.91
4.01
21.66
Yes
Yes
Emerging Markets Equity
-0.82
4.44
23.93
Yes
Yes
Real Estate Investment Trusts (REITs)
-2.30
14.18
731.05
Yes
Yes
Hedge Fund of Funds
-0.36
7.26
93.44
Yes
Yes
0.0
4.17
6.81
Yes
Yes
Private Equity
Source: J.P. Morgan Asset Management. For illustrative purposes only.
10 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
C. Correlation Breakdown: Converging Correlations
During Periods of Market Stress
It is a common observation in financial literature that correlations
tend to be quite unstable over time. Depending on the historical
period, different asset classes can provide varying degrees of
diversification. Market events such as the 1987 stock market
crash and the 1997 Asian Financial crisis illustrate how contagion—the risk of one market negatively impacting and destabilizing another—can lead to cases of converging correlations.
In this section, we test the hypothesis that correlations and volatility are positively related. In particular, we investigate whether
correlations between asset classes tend to increase during
periods of high market volatility and stress, compared to periods
of relative calm.
Exhibit 13: CBOE VIX since inception
70
Expected volatility %
To test our hypothesis15, we analyze simple “Pearson” correlations16 over two distinct historical periods—a full ten-year period
versus a shorter, high-volatility period of market stress. Our
chosen volatility measure is the the Chicago Board Options
Exchange Volatility Index (CBOE VIX). Exhibit 13 plots CBOE VIX
values since its inception.
The CBOE VIX tracks expected equity volatility over a 30-day
period17. It is widely regarded as one of the most reliable measures of market uncertainty available, and it is traded in real
time on a daily basis. We identify “high-volatility” periods in
relation to the long-term VIX average.
More specifically, our two test periods are defined as follows:
60
1. The control period is a full ten-year history, from October
1997 to September 2007, during which the VIX averaged 20.9%.
50
40
30
2. The period of market stress used for comparison is from
August 1998 to September 1999, during which time the VIX
averaged 30.9%—considerably higher than its ten-year average.
This high-volatility period captured both the Asian and the Russian financial crises, which led to significant market uncertainty.
20
10
0
Jan90
If this proves to be the case, it means that the benefits of portfolio diversification may be overestimated by traditional frameworks that rely on linear correlation matrices. In other words,
diversification may not materialize precisely when an investor
needs it most—i.e. during periods of high market volatility.
Jan92
Jan94
Jan96
Jan98
Jan00
Jan02
Jan04
Jan06
Jan08
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Exhibit 14: SIMPLE CORRELATIONS OVER TEN YEARS TO SEPTEMBER 2007
U.S. Bonds
U.S. Bonds
U.S. Equity
International
Equity
Emerging
Real Estate
Markets Equity Investment Trusts
Hedge Fund of
Funds
Private Equity
1.00
U.S. Equity
-0.21
1.00
International Equity
-0.35
0.81
1.00
Emerging Markets Equity
-0.23
0.71
0.74
1.00
Real Estate Investment Trusts (REITs)
0.05
0.31
0.23
0.33
1.00
Hedge Fund of Funds
-0.14
0.46
0.61
0.70
0.19
1.00
Private Equity
-0.15
0.61
0.62
0.70
0.33
0.73
1.00
Source: J.P. Morgan Asset Management. For illustrative purposes only.
15
Note that although we do not test whether the changes in correlations are statistically significant, we do believe the changes illustrate our broad point of correlation convergence during
periods of market stress.
16
Pearson correlations are used to indicate the strength and direction of a linear correlation
between two random variables.
More precisely, the CBOE VIX represents the implied volatility of S&P 500 index options.
The VIX is quoted in terms of percentage points and translates, roughly, to the expected
movement in the S&P 500 index over the next 30-day period, on an annualized basis.
17
J.P. Morgan Asset Management | 11
“The results are striking. Nearly all the
correlation coefficients increased to some
degree (during this period of high market
volatility), with only four showing a decrease.
Our main diversifiers did not provide the
diversification benefits anticipated.”
12 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Exhibit 14 shows correlation coefficients (ρ) over the full
(monthly) ten-year period to September 2007 for all seven
asset classes.
anticipated. The correlation coefficients with U.S. Equities were
as follows:
Long-run CoefficientDuring Market Stress
Our correlation matrix indicates that investing in certain pairs
of assets should provide diversification benefits—i.e. they
should either be negatively or lowly correlated. For example,
let’s consider correlations of asset classes with U.S. Equities.
Based on historical correlations over the entire ten years, we
expect the following asset classes to provide some diversification
benefits relative to a U.S. Equity-dominated portfolio:
•
•
•
•
U.S. Bonds -0.21 -0.02
REITs 0.31 0.58
Hedge Fund of Funds
0.46 0.60
Private Equity 0.61
0.86
Our analysis reaffirms our intuition, suggesting ample evidence
of correlation convergence during periods of market stress.
U.S. Bonds (ρ= -0.21)
REITs (ρ = 0.31)
Hedge Funds of Funds (ρ = 0.46)
Private Equity18 (ρ = 0.61)
Again, conventional frameworks which assume normality—in
this case normal distribution of joint returns—prove inadequate
to account for what we observe in real-world markets. And
inappropriately assuming linearity of correlations can lead to
a significant underestimation of joint negative returns during a
market downturn.
In times of market stress, however, these correlations break
down. Exhibit 15 shows a simple correlation matrix for the
high-volatility period from August 1998 to September 1999.
We have highlighted in orange where correlations have
converged (i.e. reducing diversification) and in green where
correlations have diverged (i.e. increasing diversification).
The results are striking. Nearly all the correlation coefficients
increased to some degree, with only four showing a decrease.
Our main diversifiers did not provide the diversification benefits
Exhibit 15: Correlations during high volatility period (AugUST 1998—SeptEMBER 1999)
U.S. Bonds
U.S. Bonds
1.00
U.S. Equity
U.S. Equity
-0.02
1.00
International Equity
-0.42
0.75
International
Equity
Emerging
Markets Equity
Real Estate
Investment Trusts
Hedge Fund of
Funds
Private Equity
1.00
Emerging Markets Equity
-0.26
0.86
0.82
1.00
Real Estate Investment Trusts (REITs)
0.04
0.58
0.36
0.70
1.0
Hedge Fund of Funds
-0.42
0.60
0.73
0.69
0.48
1.0
Private Equity
-0.08
0.86
0.70
0.81
0.76
0.72
1.00
Source: J.P. Morgan Asset Management. For illustrative purposes only.
18
Private Equity may be seen as a return enhancer rather than a diversification play. We include
it in our diversification analysis—despite a relatively high correlation—for illustrative purposes.
J.P. Morgan Asset Management | 13
Incorporating non-normality into an asset
allocation framework
Based on empirical testing, we have demonstrated that conventional asset allocation frameworks understate downside risk because they do
not allow for non-normality. However, we believe
that achieving optimal portfolio efficiency—based
on a more precise estimation of portfolio risk—
requires investors to incorporate non-normal
return distributions into the asset allocation and
portfolio optimization process.
In this section, we offer a statistical methodology for incorporating the three categories of non-normality already discussed. We
address them in the same order as they were presented in the
previous section:
A. Serial correlation
characteristics of the underlying data generating process. Notably, our new series should have a higher volatility, demonstrate
no serial correlation and have a cross correlation structure with
other asset classes similar to our original data.
Our procedure for “unsmoothing” returns (or correcting for
serial correlation in the underlying data series) is a two-step
process:
STEP 1:
We determine the correlation coefficient at lag one20
(i.e. previous month’s return) for each return series, by
running the following regression:
Rt = a + bR(t-1)
where Rt represents the return at time t
The regressions indicate the following estimates (Exhibit 16) for
the “‘serial” correlation coefficients, based on monthly returns.
B. “Fat” left tails (negative skewness and leptokurtosis)
C. Correlation breakdown (converging correlations)
A. Incorporating the Impact of Serial Correlation
Our empirical work in the last section shows that certain asset
classes—International Equity, Emerging Markets Equity, Hedge
Fund of Funds and Private Equity—test positive for serial correlation. This result implies that, over time, single period returns
are influenced by, and not independent from, previous periods.
First-order serial correlation, if not corrected for in the underlying data, will mask true asset class volatility and bias risk
estimates downwards, leading to underestimation of asset
class risk and, hence, portfolio risk.
In order to ‘adjust’ for the impact of serial correlation on
asset returns, we apply a variation of Fisher-Geltner-Webb’s
“unsmoothing” methodology19, whereby we re-calculate an
“unsmoothed” return stream from our original data. The new
adjusted return series should be more reflective of the risk
Exhibit 16: Correlation coefficients at lag one
^b
Statistically significant*
0.21
Yes
Emerging Markets Equity
0.25
Yes
Hedge Fund of Funds
-0.42
Yes
Private Equity
0.21
Yes
International Equity
Source: J.P. Morgan Asset Management. Note the coefficient b reflects the strength of the autocorrelation. For illustrative purposes only.
* Statistically significant at the 5% level
For a more detailed treatment, please refer to Fisher, J., D. Geltner, and B. Webb. 1994.
Value Indices of Commercial Real Estate: A Comparison of Index Construction Methods.
Also, Fisher, J. and D. Geltner. 2000. De-Lagging the NCREIF Index: Transaction Prices and
Reverse-Engineering.
19
Note International Equity, Emerging Markets Equity, Hedge Fund of Funds and Private Equity
returns test positive for serial correlation at lag one.
20
14 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
STEP 2:
We then produce our “unsmoothed” return series as follows,
based on the “serial” correlation coefficient already derived:
^
^ ) / (1—b)
Rt (corrected) = (Rt—bR
t-1
After applying the methodology outlined above, we obtain
a new data series Rt (corrected) that should display no serial
correlation. Once again we use the Q statistic to test the
unsmoothed data for serial correlation. The results are shown
below (Exhibit 17).
Exhibit 17: Tests for serial correlation on corrected data for up to
six lags
Test statistic
p-value
Evidence of serial
correlation
0.03
0.86
No
Emerging Markets Equity
0.06
0.80
No
Hedge Fund of Funds
0.04
0.85
No
Private Equity
0.05
0.82
No
International Equity
Source: J.P. Morgan Asset Management. For illustrative purposes only.
The test indicates that our revised data is no longer serially correlated. More important, our transformations of the underlying
data series do not alter the mean return of the series over the
time period considered. However, removing the effects of serial
correlation does increase the standard deviation of returns, as
shown in Exhibit 18 below. As discussed, the higher volatility of
our “unsmoothed” return series is in line with our expectations,
as serial correlation masks volatility and understates risk.
Exhibit 18: Volatilities before and after “unsmoothing” for serial
correlation
Annualized volatility
before
“unsmoothing”
Annalized volatility
after
“unsmoothing”
International Equity
15.25%
18.93%
Emerging Markets Equity
24.21%
31.42%
Hedge Fund of Funds
6.51%
10.30%
Private Equity
23.51%
29.17%
Finally, we evaluate the impact on asset correlations to assess
the viability of our unsmoothed data series—as our unsmoothing procedure should not affect correlations. We compare our
original correlation matrix using index data (Exhibit 19) with
our new correlation matrix derived from unsmoothed data
(Exhibit 20).
We see that, as anticipated, the impact of unsmoothing on asset
correlations is negligible.
B. Incorporating the Impact of “Fat” Left Tails into our
Framework
All of our asset classes show “fat” left tails, compared to the
normal distribution. Our approach to dealing with such left tail
risk involves what is called a “semi-parametric approach”: that
is, instead of modeling returns using a single distribution (such
as is traditionally done with the normal distribution), we fit the
left and right tails separately from the interior of the distribution.
Hence, we segment each return distribution into three parts—
the left tail, right tail and interior (See Exhibit 21). We then
“fit” each segment of the distribution separately. This approach
enables us to more accurately capture the potential distributions associated with the empirical data in each segment of
the distribution.
To fit the left and right tails of the distribution we turn to
Extreme Value theory—a body of work specifically designed to
look at the probability of low-probability but high-risk events.
The central premise behind Extreme Value theory is that the
probability of observing extreme values (such as negative
financial market returns), can be modeled using any one of
several Extreme Value distributions.21
Our choice of Extreme Value distribution is the Generalized
Pareto distribution (GPD). We choose the GPD distribution,
rather than other Extreme Value distributions, due to the ease
with which it can be adapted to modeling financial market
returns.
Source: J.P. Morgan Asset Management. For illustrative purposes only.
21
An Extreme Value distribution is defined as the limiting distribution of extreme values sampled
from any independent randomly distributed process.
J.P. Morgan Asset Management | 15
Exhibit 19: Correlations based on smoothed (or raw) data over ten years to October 2008
U.S. Bonds
U.S. Equity
International
Equity
Emerging
Real Estate
Hedge Fund of
Markets Equity Investment Trusts
Funds
Private Equity
U.S. Bonds
1.00
U.S. Equity
-0.07
1.00
International Equity
-0.16
0.87
1.00
Emerging Markets Equity
-0.03
0.76
0.82
1.00
Real Estate Investment Trusts (REITs)
0.17
0.46
0.39
0.44
1.00
Hedge Fund of Funds
0.08
0.57
0.69
0.78
0.30
1.00
Private Equity
-0.03
0.65
0.68
0.71
0.42
0.76
1.00
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Exhibit 20: Correlations based on unsmoothed data over ten years to October 2008
U.S. Bonds
U.S. Equity
International
Equity
Emerging
Real Estate
Hedge Fund of
Markets Equity Investment Trusts
Funds
U.S. Bonds
1.00
U.S. Equity
-0.07
1.00
International Equity
-0.14
0.85
1.00
Emerging Markets Equity
-0.02
0.76
0.82
1.00
Real Estate Investment Trusts (REITs)
0.17
0.46
0.35
0.40
1.00
Hedge Fund of Funds
0.10
0.57
0.69
0.77
0.25
1.00
Private Equity
-0.02
0.64
0.67
0.69
0.40
0.74
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Exhibit 21: Hypothetical dissection of return distribution
Left tail
Interior
Right tail
Source: J.P. Morgan Asset Management. For illustrative purposes only.
16 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Private Equity
1.00
The “Fitting” Process is Done in Two Steps:
• First: Fitting the left (and right) tail of the Generalized Pareto
distribution (GPD) to our historical data. This is done by calibrating the tail of the GPD to extreme values in the sample
using the Method of Maximum Likelihood. Extreme values are
defined by a threshold. This method is commonly referred to
as the Peaks-Over-Threshold method within Extreme Value
theory.
• Second: Fitting the interior of a distribution is accomplished
using a non-parametric (kernel or empirical) approach.
• What we are looking for is a better fit to the actual distribution of empirically observed data (blue line)—as shown in the
kernel density function.
• What we see is that in each case (Exhibits 22 through 28), the
GPD distribution (orange line) is a much more accurate fit to
real-world data than the “normal” distribution.
• The improved fit is particularly noticeable in the left tail,
where the normal distribution understates risk.
Hence, in aggregate, we derive a semi-parametric probability
density function to describe the data generating process. We refer to this broad approach as a semi-parametric GPD approach.
We can test the results of the “fitting” process against both the
normal and kernel (or empirical) distributions used to establish
“fat” left tails in the previous section:
Generalized Pareto Distribution
For those interested in the Generalized Pareto distribution,
we provide supporting equations here.
The probability density function for the generalized Pareto
distribution with shape parameter k ≠ 0, scale parameter σ,
and threshold parameter θ, is given by:
Exhibit 22: u.s. bonds—fitting the semi-parametric gpd distribution
to historical data
50
Density
40
Empirical
for θ < x, when k > 0, or for θ < x <–σ/k when k < 0.
Semi-parametric GPD
In the limit for k = 0, the density is
for θ < x.
30
f(xI k,σ,θ)
20
10
0
-.60%
-40%
-20%
0%
20%
Monthly return
40%
1+k
-1
1
k
If k = 0 and θ = 0, the Generalized Pareto distribution is
equivalent to the exponential distribution. If k > 0 and θ =
σ, the generalized Pareto distribution is equivalent to the
Pareto distribution.
Source: J.P. Morgan Asset Management. For illustrative purposes only.
J.P. Morgan Asset Management | 17
Exhibit 23: U.S. Equities—Fitting the semi-parametric GPD
distribution to historical data
12
Exhibit 24: International Equities—Fitting semi-parametric GPD
distribution to historical data
12
Empirical
Semi-parametric GPD
10
Semi-parametric GPD
8
Density
Density
8
6
6
4
4
2
2
0
-40%
Empirical
10
-30%
-20%
-10%
0%
Monthly return
10%
Exhibit 25: Emerging Markets Equities—Fitting the semi-parametric GPD
distribution to historical data
7
0
-20%
20%
9
-5%
0%
Monthly return
5%
10%
15%
Empirical
8
Semi-parametric GPD
Semi-parametric GPD
7
5
6
4
Density
Density
-10%
Exhibit 26: U.S. REITs—Fitting the semi-parametric GPD distribution
to historical data
Empirical
6
-15%
3
5
4
3
2
2
1
0
-100%
1
-80%
-60%
-40%
-20%
Monthly return
0%
20%
7
Semi-parametric GPD
6
20
5
Density
Density
8
24
16
3
8
2
4
1
-10%
0%
10%
-40%
-20%
0%
20%
20%
30%
Empirical
Semi-parametric GPD
4
12
0
-20%
-60%
Exhibit 28: Private Equity—Fitting the semi-parametric GPD
distribution to historical data
Empirical
28
-80%
Monthly return
Exhibit 27: Hedge fund of Funds—Fitting the semi-parametric
GPD distribution to historical data
32
0
-100%
0
-40%
Monthly return
Source: J.P. Morgan Asset Management. For illustrative purposes only.
18 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
-30%
-20%
-10%
0%
10%
Monthly return
20%
30%
40%
To enhance our conviction in these results, we need to statistically test this “goodness of fit” both for the whole distribution as well as specifically for the tails. Below we apply the
Kolmogorov-Smirnov (KS)22 empirical distribution test under two
hypotheses: first, the null hypothesis that returns are normally
distributed, and second, under the hypothesis that they are
sampled from a semi-parametric GPD. Exhibit 29 below sets out
our results. We also show the probability that our null hypothesis stands statistical significance.
Given our focus is on ‘left’ tail risk, however, we can also gauge
fit by considering the Quantile-Quantile (QQ) plots of the
empirical data against those implied by the semi-parametric
GPD distribution23.
We see that in each case, the lowest quintiles are a much better fit with the semi-parametric GPD distribution than with the
normal distribution, indicating that the semi-parametric GPD
distribution more accurately captures the frequency of extreme
negative events.
Our empirical tests confirm our view that the semi-parametric
GPD distribution provides a better fit for the historical data—
when compared to the normal distribution—for all our asset
classes.
Exhibit 29: Testing goodness of fit of Normal and semi-parametric GPD distribution
Normal
Semi-parametric GPD
KS Statistic
p Value
Reject Semiparametric GPD
Yes
0.05
0.89
No
Yes
0.06
0.75
No
Yes
0.12
0.08
No
0.11
0.13
No
0.05
0.89
No
0.05
0.92
No
0.10
0.18
No
KS statistic
p Value
Reject Normality
U.S. Aggregate Bonds
0.50
0.00
U.S. Large Cap Equity
0.49
0.00
EAFE Unhedged Equity
0.46
0.00
Emerging Markets
0.46
0.00
Yes
Real Estate Investment Trusts (REITs)
0.45
0.00
Yes
Hedge Fund of Funds
0.47
0.00
Yes
Private Equity
0.47
0.00
Yes
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Exhibit 30: U.S. Bonds—QQ Plot (Normal Distribution)
Exhibit 31: U.S. Bonds—QQ Plot (Semi-parametric GPD distribution)
.03
Quantiles of semi-parametric GPD
.04
Quantiles of Normal
.03
.02
.01
.00
-.01
-.02
-.03
.02
.01
.00
-.01
-.02
-.03
-.04
-.04
-.02
.00
.02
.04
-.04
-.02
.00
.02
.04
Quantiles of data
Quantiles of data
Source: J.P. Morgan Asset Management. For illustrative purposes only.
QQ plots illustrate how different quantiles of a sample compare to those implied by the theoretical density function. All data points being on a straight 45 degree line indicate a perfect fit
of the data with the distribution.
23
22
The Kolmogorov-Smirnov test is a goodness of fit test to test whether a sample comes from a
particular distribution. Our null hypothesis is tested at 5% significance.
J.P. Morgan Asset Management | 19
Exhibit 32: U.S. Equities—QQ Plot (Normal Distribution)
Exhibit 33: U.S. Equities—QQ Plot (semi-parametric GPD distribution )
.15
Quantiles of semi-parametric GPD
.12
Quantiles of Normal
.08
.04
.00
-.04
-.08
-.12
.10
.05
.00
-.05
-.10
-.15
-.20
-.15
-.10
-.05
.00
.05
-.20
.10
-.15
-.10
Quantiles of data
Exhibit 34: International Equities—QQ Plot (Normal Distribution)
.05
.10
.15
Quantiles of semi-parametric GPD
.10
Quantiles of Normal
.00
Exhibit 35: International Equities—QQ Plot
(semi-parametric GPD distribution )
.15
.05
.00
-.05
-.10
.10
.05
.00
-.05
-.10
-.15
-.15
-.15
-.10
-.05
.00
Quantiles of data
.05
-.15
.10
-.10
-.05
.00
.05
.10
.1
.2
Quantiles of data
Exhibit 36: Emerging Markets Equities—QQ Plot (Normal Distribution)
Exhibit 37: Emerging Markets Equities—QQ Plot
(semi-parametric GPD distribution )
.2
.20
Quantiles of semi-parametric GPD
.15
Quantiles of Normal
-.05
Quantiles of data
.10
.05
.00
-.05
-.10
-.15
.1
.0
-.1
-.2
-.3
-.20
-.3
-.2
-.1
.0
.1
.2
Quantiles of data
Source: J.P. Morgan Asset Management. For illustrative purposes only.
20 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
-.3
-.2
-.1
.0
Quantiles of data
Exhibit 38: REITs—QQ Plot (Normal Distribution)
Exhibit 39: REITs—QQ Plot (semi-parametric GPD distribution )
.15
Quantiles of semi-parametric GPD
.1
Quantiles of Normal
.10
.05
.00
-.05
-.10
.0
-.1
-.2
-.3
-.15
-.4
-.3
-.2
-.1
.0
.1
-.4
-.3
-.2
Exhibit 40: Hedge fund of Funds—QQ Plot (Normal Distribution)
.1
.12
Quantiles of semi-parametric GPD
.04
Quantiles of Normal
.0
Exhibit 41: Hedge fund of Funds—QQ Plot
(semi-parametric GPD distribution)
.06
.02
.00
-.02
-.04
-.06
.08
.04
.00
-.04
-.08
-.08
-.04
.00
Quantiles of data
.04
-.08
.08
-.04
.00
.04
.08
Quantiles of data
Exhibit 42: Private Equity—QQ Plot (Normal Distribution)
Exhibit 43: Private Equity—QQ Plot (semi-parametric GPD distribution )
.3
.20
Quantiles of semi-parametric GPD
.15
Quantiles of Normal
-.1
Quantiles of data
Quantiles of data
.10
.05
.00
-.05
-.10
-.15
.2
.1
.0
-.1
-.2
-.20
-.3
-.2
-.1
.0
.1
Quantiles of data
.2
.3
-.3
-.2
-.1
.0
.1
.2
.3
Quantiles of data
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Note that the results shown are for illustrative purposes and are based on applying Extreme Value theory to the raw ‘smoothed’ data. In our framework—because we require independence of asset
returns—we remove the effect of serial correlation before fitting a semi-parametric GPD.
J.P. Morgan Asset Management | 21
Copulas are mathematical functions that allow us to model the
joint distribution of asset returns separately from the marginal
(or individual asset class) distributions. By considering joint
distributions, we turn our focus to how asset classes behave
together (rather than individually, as we did in the last section).
In a traditional framework, joint distributions are captured by
simple (Pearson) correlations. Unfortunately, simple correlations assume a linear relationship between individual asset
classes, and we have already shown empirically that linearity
breaks down at the extremes. Copulas, on the other hand,
allow us to differentiate the relationship between asset
classes in times of market stress, from more normal times.
In particular, we observe empirically that not only do individual
asset classes have “fatter” left tails than allowed in a normal
distribution, combinations of asset classes exhibit “fat” joint left
tails—i.e. a higher incidence of joint negative returns in times of
market stress. Copulas allow us to model both the “fat” marginal
left tails for individual asset classes (using the semi-parametric
GPD approach explained earlier), while at the same time modeling
more accurately the “fat” joint left tails (joint negative returns)
as empirically observed.
Just as the application of Extreme Value Theory involves choosing from among several types of Extreme Value Distributions,
applying copula theory requires us to choose a copula type.
Here we use the t-copula (based on the Student t distribution)24
because it enables us to better capture the effects of converging correlations—i.e. allows fatter tails than the normal
distribution.
24
The Student t distribution comes from the same family of distributions as the Normal. A Student t distribution is calibrated on only one parameter i.e. the degrees of freedom. The higher
the degrees of freedom, the closer it is to a normal. At 30 degrees of freedom, the Student t
distribution is virtually identical to the normal distribution. Hence, assuming a lower degree of
freedom parameter imposes significant increased dependence between asset classes returns.
In Exhibit 44, we illustrate the impact of modifying the normal
marginal distributions by adding a t-copula with a low degreeof-freedom parameter.
Exhibit 44: Modeling joint dependence in asset returns using copulas
Monthly simulated Hedge Fund of Funds returns
One of our empirical findings in the previous section was that
correlations converge during periods of high market volatility
and stress. Our approach to dealing with this phenomenon is
to apply ‘copula’ theory—a body of work that explicitly looks at
the impact of contagion or converging correlations at the total
portfolio level.
This approach better reflects the empirical observations of converging correlations during periods of high volatility, which tend
to be accompanied by negative returns. Specifically, we model
joint “fat” left tails by adjusting the copula’s “degree-of-freedom”
parameter: 30 degrees-of-freedom is virtually identical to a
normal distribution. By lowering the parameter, we can increase
asset class dependence to model observed asset class behavior
in times of market stress.
Illustration of traditional linear correlations
10%
8%
6%
4%
2%
0%
-2%
-4%
-6%
-8%
-10%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
Monthly simulated U.S. Large Cap Equity returns
Monthly simulated Hedge Fund of Funds returns
C. Incorporating the impact of converging correlations
into our framework
10%
Illustration of increased joint dependence through copulas
8%
6%
4%
2%
0%
-2%
-4%
-6%
-8%
-10%
-20%
-15%
-10%
-5%
0%
5%
10%
Monthly simulated U.S. Large Cap Equity returns
Source: J.P. Morgan Asset Management. For illustrative purposes only.
22 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
15%
20%
In order to show the effect of modeling increased joint dependence, we produce a scatter plot of 2,500 simulations of U.S.
Large Cap Equity returns and Hedge Fund of Funds returns
first, assuming normality with simple correlations; and second,
assuming normality with a Student t-copula added (with two
degree of freedom25).
Compared to the traditional multivariate normal framework,
our simulations show the impact of increased dependence,
through a higher joint incidence of negative returns between
U.S. Large Cap Equities and Hedge Fund of Funds (highlighted).
This illustrates the phenomenon of “fat” joint left tails we
alluded to earlier.
Our simulations also illustrate increased dependence of other
extreme joint occurrences. In other words, we can see an
increased incidence of negative U.S. Large Cap Equity returns
with positive Hedge Fund of Funds returns (the top left quadrant)
or positive U.S. Equity returns with positive Hedge Fund of
Funds returns (the top right quadrant). This is visible as the
“star” pattern in the scatter plot.
This occurs because our Student t distribution is symmetric
around the mean (like the Normal distribution) and, hence,
incorporating “fat” joint tails increases the probability of
observing all extreme joint outcomes (positive and negative)—
not just extreme joint negative outcomes. This can be seen as
a drawback of our choice of the Student t distribution, as—in
reality—correlations do exhibit asymmetric behavior, i.e. the
convergence properties are different in extreme positive
markets compared to extreme negative markets.
Exhibit 45: Incorporating non-normality—Summary of model
Step 1: Input historical data
Step 2: Remove serial
correlation from returns
Unsmoothed data
Step 3: Fit marginal distributions
(using Extreme Value theory) to allow for
“fat” left tails
Step 4: Calibrate Student t-copula
to allow for joint “fat” tails—i.e. increased
dependence during periods of
market stress
Source: J.P. Morgan Asset Management. For illustrative purposes only.
However, our choice of t-copula is a vast improvement when
compared to linear correlation matrices used in traditional
models, as the latter completely ignore the effect of correlation
convergence. Incorporating “fat” joint left tails using copula
theory allows us to more robustly model the empirical
phenomenon of correlation breakdown during periods of
market stress.
25
Note our model calibrates the degrees of freedom parameter for the Student t distribution
assuming the marginal distributions are semi-parametric GPD. However, to illustrate the effect
of adding the copula, we choose a low degree of freedom parameter of two and assume the
marginal distributions are normal (rather than semi-parametric GPD). This allows us to isolate
and illustrate the effect of fat ‘joint’ tails separately from fat ‘marginal’ tails.
J.P. Morgan Asset Management | 23
Estimating downside portfolio risk in a
non-normal framework
Now that we have a statistically solid framework
for incorporating non-normality into the asset
allocation process, we can apply this framework
to a hypothetical U.S. domiciled investor. We will
assume a well-diversified portfolio with an initial
value of $1 billion and allocations across our
seven major asset classes. Detailed allocations to
each asset class are shown in Exhibit 46 below.
We can now generate forward looking projections of our hypothetical portfolio’s real value26 in our newly revised Monte Carlo
simulation framework. For each of our ten forward years, we
generate 10,000 simulations27. We then measure downside portfolio risk as the Conditional Value at Risk at the 5th percentile of
the portfolio.
Exhibit 46: Hypothetical portfolio allocation
Asset class
Current Allocation
Total bonds
30%
U.S. Aggregate Bonds
30%
Total equity
55%
U.S. Large Cap Equity
40%
International Equity (hedged)
10%
Emerging Markets Equity
Total alternatives
5%
15%
REITs
5%
Hedge Fund of Funds
5%
Private Equity
5%
We define Conditional Value at Risk (CVaR95) as the average
(real) portfolio loss (relative to the starting value) in the worst
five percent of scenarios, based on our 10,000 Monte Carlo
simulations. It is simply the average real loss in the worst 500
(5% of 10,000) scenarios (i.e. the left tail of the portfolio
distribution).
A Better Risk Quantifier: Conditional Value at Risk
We believe that empirical evidence suggests an imperative to
incorporate various types of “non-normality” into the asset
allocation and portfolio modeling process, specifically to better
understand and model downside portfolio risk. Yet if we take
this step, we have to ask whether or not our conventional risk
measure (i.e. standard deviation) is up to the new task.
We would argue that, in a framework based on non-normality,
standard deviation may not be investors’ most appropriate
measure of portfolio risk because it punishes the desirable
upside movements as hard as it punishes the undesirable downside movements. This is generally inconsistent with investor risk
preferences—primarily as observed in the field of behavioral
finance28.
Conditional Value at Risk (CVaR95) overcomes many of the
drawbacks of standard deviation as a risk measure. Primarily,
as it only measures risk on the downside, it captures both the
asymmetric risk preferences of investors and the incidence of
“fat” left tails induced by skewed and leptokurtic return distributions. Further, given the widespread use by major institutional
investors and regulators of its first cousin—Value at Risk—we
judge it to be the most appropriate risk measure to incorporate
it into our framework.
Key statistics
Expected arithmetic return
Expected volatility
Expected compound return
Sharpe ratio
9.1%
10.0%
26
Our model calculates real portfolio value by discounting the nominal portfolio value using
projected inflation. Inflation itself is projected stochastically in our framework.
27
We believe 10,000 simulations should reduce simulation error sufficiently to allow us to
draw robust inferences from the results.
8.7%
0.51
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Sharpe ratio calculated based on expected risk free return of 4.0% per year as per
J.P. Morgan Asset Management Long Term Capital Market Assumptions (please see
Appendix for details).
A key tenet of behavioral finance is the idea of loss aversion i.e. a tendency of investors to
prefer avoiding losses than making gains. This translates to risk preferences that are
asymmetric in nature.
28
24 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
The calculation for CVaR95 is illustrated graphically for the current
allocation below. Exhibit 47 shows the projected frequency of
the portfolio’s real gains (or losses) at the end of ten years. We
calculate CVaR95 by taking the average real cumulative loss in
the worst 5% of simulations.
Exhibit 47: Histogram of projected cumulative portfolio gain (loss)
at the end of ten years assuming non-normality
800
The reason non-normality can impact asset allocation is that
the downside risk associated with different asset classes is very
different. Most obviously, equity and equity type asset classes
entail greater degrees of downside risk than, for example, fixed
income type investments. Hence, because the downside risk
characteristics of different asset classes are different—and
cannot be accounted for using traditional modeling techniques
or risk measures such as standard deviation—the efficient
allocations in our CVaR95 motivated non-normal framework
must also be different from a traditional framework.
700
600
Frequency
We should note that while the increased risk associated with
modeling non-normality is striking, incremental risk in itself is
not necessarily a reason for changing a plan’s asset allocation
(unless an absolute threshold value has been breached). More
specifically, if one assumes an arbitrarily (but equally) higher
downside risk for all asset classes, this in itself would not
impact the efficiency of the portfolio—i.e. its efficiency would
not change at all, albeit the CVaR95 would now be higher.
500
400
300
200
100
(489)
(289)
(90)
110
309
509
709
908
1,108
1,307
1,507
1,706
1,906
2,106
2,305
2,505
2,704
2,904
3,104
3,303
3,503
3,702
3,902
4,101
4,301
4,501
4,700
4,900
5,099
5,299
5,498
5,698
5,898
0
Expected portfolio gain (loss) at the end of ten years ($ mm)
CVaR95 is defined as the average real cumulative loss in the worst 5% or 500 simulations.
This is equal to $168 million for the current portfolio.
Source: J.P. Morgan Asset Management. For illustrative purposes only.
For this reason—not merely due to higher CVaR95 figures—we
believe investors need to quantitatively incorporate the impact
of non-normality into the asset allocation process.
In the next section, we consider the impact of non-normality
on optimized portfolio solutions for investors with a long-term
investment horizon.
The CVaR95 of the current allocation, based on our new methodology, is $168 million. In other words, the portfolio can expect
to lose (on average) $168 million in the worst five percent of
cases (based on our simulation results). This risk is significant. It
indicates a real return (i.e. after allowing for inflation) of -16.8%
on the portfolio over an extended time horizon—a result our
investor is unlikely to be very happy with.
For the same portfolio, however, risk calculations that assume
normality29 would result in a CVaR95 figure of $74 million. Incorporating non-normality more than doubles our prior estimate
of CVaR95. In absolute terms, the risk underestimation is $94
million or 9.4% of the portfolio’s initial value.
29
Our estimates of CVaR95 under a traditional framework were derived with an identical asset
allocation, except that asset returns were assumed to be individually and jointly normally
distributed. Risk and correlations were derived based on the same ten year historical period
from November 1998 to October 2008.
J.P. Morgan Asset Management | 25
“The reason non-normality can impact asset
allocation is that the downside risk associated
with different asset classes is very different. Most
obviously, equity and equity type asset classes
entail greater degrees of downside risk than, for
example, fixed income type investments.”
26 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Implications of non-normality on optimal
portfolio solutions
In this section we assess the impact of
non-normality on portfolio efficiency, using
CVaR95 as the risk measure of choice30.
Our optimization process involves minimizing CVaR95 for a given
expected return (or equivalently, maximizing return for a given
CVaR95). In any case, the portfolios we derive here are “efficient”
in that they offer investors the optimal tradeoff between
(downside) risk and reward. The optimization process is based
on linear programming techniques31 applied to 10,000 Monte
Carlo simulations (generated in the previous section).
We construct an efficient frontier comprising ten “efficient”
portfolios:
• Our first portfolio comprises our minimum CVaR95 portfolio—
such that there is no other asset allocation (derived from
our simulations) that minimizes the portfolio’s CVaR95 further
over ten years. It is our minimum risk portfolio.
• Our tenth portfolio comprises our highest return portfolio—
such that there is no other asset allocation (derived from
our simulations) that maximizes the return of the portfolio
further over ten years.
• The intermediate eight portfolios are equally distributed
between the first and tenth in terms of expected return, but
minimizing CVaR95 over ten years at each point along the
efficient frontier.
30
It is worth noting that when optimizing under the assumption of normality, the choice of
risk measure (whether standard deviation or CVaR95) is irrelevant. This is because a normal
distribution is symmetric about its mean—so choosing a different percentile (e.g. two standard
deviations)—does not impact the derived solution. Under non-normality, the choice of risk
measure is critical to the derived solution.
31
Linear programming is required because as we move away from traditional “normal”
assumptions, there are no shortcuts to computing total portfolio risk. Individual downside
risk measures—such as standard deviation—can no longer be aggregated to the total
portfolio level using analytical formulae.
32
Our traditional efficient frontier was derived under a Monte Carlo simulation framework
except that asset returns were assumed to be individually and jointly normally distributed.
Exhibit 48: Efficient frontier based on traditional and non-normal
framework
Expected compound return (% per year)
12%
11%
10%
9%
8%
7%
Normal framework
6%
5%
(200)
Non-normal Mean CVaR framework
0
200
400
600
Conditional
Value
at
Risk
95
($
millions)
Source: J.P. Morgan Asset Management. For illustrative purposes only.
800
1,000
In aggregate, a plot of the expected compound return (in % per
year) versus the CVaR95 (in $ millions) constitutes our efficient
frontier (Exhibit 48).
The efficient frontier created using a non-normal framework
differs markedly from the equivalent frontier produced from a
traditional model32 built on assumptions of normality. Notably all
points along the “new” efficient frontier demonstrate—for a given
expected return—significantly higher risk, as captured by CVaR95.
Detailed allocations underlying each efficient frontier are shown
in the Appendix.
These results confirm our intuition that capturing the nonnormality associated with equity and alternative return distributions should increase the magnitude of CVaR95 of any portfolio
with an allocation to these asset classes. Crucially, as traditional
frameworks fail to capture non-normality, they underestimate
the downside risk associated with such portfolios. Hence, the
efficient frontier derived from a traditional model is likely to
show a misleading, lower risk figure for portfolios with a similar
expected return.
J.P. Morgan Asset Management | 27
Optimal Portfolio Solution for our Hypothetical Investor
Optimization Constraints
In this section, we ask ourselves how incorporating non-normality
into our return distributions affects our optimal portfolio solution
for our hypothetical investor.
Optimizations in general—whether based on
modern or, as in our case, post-modern portfolio
theory—suffer from certain inherent drawbacks. A particular drawback associated with
traditional frameworks is the sheer number of
constraints applied to the optimization process.
Constraints reduce the credibility of the optimization process as, by applying constraints, the
practitioner is guiding the model towards a
particular solution—rather than allowing the
model to calculate one itself independently.
Exhibit 49 compares our current hypothetical investor with a
portfolio allocation using an identical target arithmetic return of
9.1%, but optimized using our revised non-normal framework33.
For illustrative purposes, we also show the optimized allocation
derived from a traditional-mean variance framework. We do not
impose any constraints on our optimization process.
33
All optimizations are based on J.P. Morgan Long Term Capital Market Return Assumptions.
In deriving our optimal portfolios, we do not
apply any constraints. Hence, we let the model
dictate the direction and magnitude of results.
Exhibit 49: Optimization results
Current allocation
Optimized, Unconstrained
Normal allocation
30.0%
0.0%
Total bonds
Optimized, Unconstrained
Non-normal allocation
34.5%
U.S. Aggregate Bonds
30.0%
0.0%
34.5%
Total equity
55.0%
18.3%
36.0%
U.S. Large Cap Equity
40.0%
8.9%
21.7%
International Equity
10.0%
7.6%
5.8%
5.0%
1.9%
8.5%
Emerging Markets Equity
Total alternatives
15.0%
81.7%
29.5%
REITs
5.0%
11.9%
11.1%
Hedge Fund of Funds
5.0%
64.1%
7.0%
Private Equity
5.0%
5.6%
11.5%
100.0%
100.0%
100.0%
9.1%
9.1%
9.1%
10.0%
8.6%
9.5%
8.7%
8.6%
8.7%
0.51
0.59
0.54
$168 mm
$206 mm
$148 mm
23%
12%
Total
Key statistics
Target expected arithmetic return
Expected volatility
Expected compound return
Sharpe ratio
CVaR95 ($ million) allowing for non-normality
CVaR95 vs. current allocation
Return per unit of CVaR95
0.30
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Sharpe ratio calculated assuming risk free return of 4.0%.
28 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
0.25
0.35
Our results indicate that the ‘optimal’ portfolio using a traditional mean-variance framework actually increases (rather than
decreases) risk by 22.6% relative to the current allocation—
as defined by CVaR95. As a traditional framework minimizes
standard deviation34—which we argue is an inadequate risk
measure—it inadvertently exposes our investor to even worse
scenarios on the downside than the current allocation.
However, the more significant issue by far—and the biggest
drawback of the traditional approach—is that it produces a highly
concentrated and impractical asset allocation. This is because
in the absence of formal constraints, it over allocates to asset
classes based on small differences in assumptions. This limits
our ability to draw useful insights into the portfolio construction
process, using such a framework.
On the other hand, our non-normal CVaR95 based framework
produces a diversified solution with allocations across the asset
class spectrum. No single asset class significantly dominates the
portfolio. Despite the fact that our investor already holds quite
a diversified portfolio, our framework suggests that there is still
scope for our investor to improve portfolio efficiency further.
The “optimal” portfolio identified by the non-normal framework improves both the expected Sharpe ratio and reduces the
CVaR95 relative to the current allocation. This signifies that our
optimal portfolio is more efficient (in Sharpe ratio and CVaR95
space) than the current hypothetical portfolio. Based on our
Long Term Capital Market Return assumptions, the allocations
to fixed income and alternatives increase, while the allocation to
equities decreases.
34
Minimizing standard deviation for a given expected return target is the equivalent of
maximizing Sharpe ratio.
J.P. Morgan Asset Management | 29
Comparing a Traditional Model with
a Non-normal Framework
So, how does each form of non-normality affect our optimal
portfolio solution relative to a traditional mean-variance
optimization framework?
Exhibit 50 shows the marginal and then combined impact of
each form of non-normality on the optimal portfolio solution,
when compared to a traditional mean-variance framework35.
(Note the analysis below is based on J.P. Morgan’s Long Term Expected Return
Assumptions. It is worth noting that the results below will vary depending on the
expected return assumptions i.e. relative attractiveness of the asset classes.)
Each arrow in the table indicates the broad directional impact
of the particular form of non-normality on the allocation to the
asset class—relative to the allocation implied by a mean-variance
approach. For example, an up arrow indicates the allocation to
the asset class increases in a non-normal framework, relative to
the allocation implied by a traditional framework. Each subsequent
column allows for the prior form (or forms) of non-normality—
hence, the attribution in each case is truly marginal.
Exhibit 50: Attribution analysis of non-normality
Phenomenon 1:
Serial correlation
Phenomenon 2:
“Fat” left tails
Phenomenon 3:
Converging
correlations
U.S. Bonds
U.S. Equity
International
Equity
Emerging
Markets Equity
REITs
Hedge Fund
of Funds
Private Equity
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Note the impact of the form of non-normality on the optimal portfolio solution varies
along the efficient frontier. The analysis in this section is broadly representative of a
portfolio with an expected (arithmetic) return of 9.0%—based on our Long Term
Capital Market Return Assumptions.
35
Serial correlation
The impact of serial correlation is negative for International
Equity, Emerging Markets Equity, Hedge Fund of Funds and
Private Equity as these asset classes test positive for first order
serial correlation. Recall that the presence of serial correlation
increases the risk associated with the particular asset class—
hence its negative marginal impact. As U.S. Bonds, U.S.
Equity and REITs do not test positive for serial correlation,
the marginal impact for these asset classes is positive
(relative to a traditional mean-variance framework).
“Fat” left tails
Our second form of non-normality—“fat” left tails—has a
negative marginal impact on optimal allocations to U.S. Equities, REITs and Hedge Fund of Funds (relative to a traditional
mean-variance framework). This is driven by the asset classes’
distributional properties—its negative skewness, excess kurtosis
and extreme negative values (all of which are bad for investors).
In particular, our model indicates higher excess kurtosis for
U.S. Equity, REITs and Hedge Fund of Funds compared to U.S.
Bonds, International Equity, Emerging Markets Equity and
Private Equity (after allowing for serial correlation). This
implies that these asset classes have “fatter” left tails than the
other asset classes, when compared to a normal distribution.
Hence, our model reduces allocations to U.S. Equity, REITs
and Hedge Fund of Funds (relative to a traditional framework)
due to the prevalence of this phenomenon.
Converging correlations
Our calibration results for the copula and optimization results
suggest correlation convergence during periods of market stress
is much more pronounced for International Equity and Hedge
Fund of Funds (after allowing for serial correlation and “fat”
left tails). Hence, the model reduces allocations to these asset
classes relative to traditional mean-variance frameworks.
In other words, the degree of non-linearity in correlations is
much more severe for International Equity and Hedge Fund
of Funds, compared to the other asset classes. This implies
that diversification benefits from these asset classes do
not materialize to the extent implied by linear correlation
matrices (during periods of market stress). This detracts from
relative allocations.
30 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Conclusion
Incorporating non-normality can lead to more efficient portfolios
Until recently, investors have been constrained
in their ability to incorporate non-normality into
the asset allocation process. But now, with the
availability of sophisticated statistical tools, we
can meet this challenge. Why should we change?
The most straightforward answer is that this is how the world
really works—i.e. we empirically observe non-normality with
much greater frequency that current mean-variance frameworks allow for. The more important answer is that ignoring
non-normality in equity (and equity type) return distributions
significantly understates downside portfolio risk—in the worst
of the worst-cases, potentially posing a solvency risk for the
investor.
We also believe that investors need to allow for downside risk in
a more robust fashion than standard deviation measures have
traditionally assumed. For this reason we recommend CVaR95.
This measure is a better fit for investors’ asymmetric risk preferences, as well as the “fat” left tails recognized by non-normal
asset allocation frameworks.
Finally, an asset allocation incorporating non-normality has the
benefit of reducing the need for external constraints. Investors
often impose such constraints in an effort to get normal frameworks to provide non-normal solutions—i.e. to better reflect the
non-normality we see in the real world. A framework that builds
in non-normality up front, however, provides much a more
direct, efficient, and elegant way of addressing the problem.
Ultimately, we believe the quantitative results illustrate the
point best, and speak for themselves: incorporating non-normality
may reduce the portfolio’s volatility, improve its efficiency
(Sharpe ratio), and reduce its risk relative to unpredictable,
extreme negative events.
So, we argue for a new asset allocation framework because
beyond its pure statistical merit, there lies a significant,
practical benefit for investors: the potential to improve portfolio
efficiency and resilience, in light of a clearer understanding of
portfolio risk.
Limitations of reliance
It should be noted that a quantitative framework is only one input into the asset allocation process and cannot replace the professional skill and judgment necessary to
arrive at an appropriate strategy. The importance of allowing for subjective—and often
qualitative—factors in decision making remains. Further, there is always an explicit
need to account for the investor’s specific circumstances, including liabilities, when
arriving at an appropriate portfolio allocation.
J.P. Morgan Asset Management | 31
Appendix
Exhibit 51: J.P. Morgan Asset Management Asset Allocation Framework—Graphical User Interface
32 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Detailed Asset Allocations and Assumptions Underlying Efficient Frontiers
Exhibit 52: Efficient frontier portfolio allocations (Traditional Mean-Variance based on J.P. Morgan Long Term Expected Returns)
Mean CVaR Allocations
Portfolio 1
Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Portfolio 6 Portfolio 7 Portfolio 8 Portfolio 9 Portfolio 10
U.S. Aggregate Bonds
80.2
52.0
24.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
U.S. Large Cap Equity
0.0
3.0
7.0
10.3
7.9
5.3
2.7
0.2
0.0
0.0
International Equity
2.5
2.4
0.1
2.2
8.8
11.8
15.0
18.1
4.4
0.0
Emerging Markets Equity
0.0
0.0
0.0
0.0
4.8
12.7
20.5
28.4
43.3
100.0
Real Estate Investment Trusts
0.0
1.8
6.0
10.4
12.6
14.4
16.2
18.1
15.2
0.0
Hedge Fund of Funds
17.3
40.7
62.9
77.1
57.7
40.7
23.6
6.5
0.0
0.0
Private Equity
0.0
0.0
0.0
0.0
8.2
15.1
22.0
28.8
37.1
0.0
Exhibit 53: Efficient frontier portfolio allocations (CVaR95 motivated non-normal framework based on J.P. Morgan Long Term Expected Returns)
Mean CVaR Allocations
Portfolio 1
Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Portfolio 6 Portfolio 7 Portfolio 8 Portfolio 9 Portfolio 10
U.S. Aggregate Bonds
76.3
57.3
43.4
36.8
29.5
21.5
10.7
0.0
0.0
0.0
U.S. Large Cap Equity
7.5
15.8
20.9
21.5
22.6
23.7
25.1
24.7
6.8
0.0
International Equity
2.5
4.4
6.8
6.1
5.2
5.0
2.7
2.8
0.0
0.0
Emerging Market Equity
0.0
0.0
0.0
6.5
12.5
17.7
22.8
28.2
48.1
100.0
Real Estate Investment Trusts
3.6
7.1
9.7
10.8
11.7
12.8
15.0
16.7
12.2
0.0
Hedge Fund of Funds
10.1
15.4
15.1
9.2
4.3
0.0
0.0
0.0
0.0
0.0
Private Equity
0.0
0.0
4.1
9.1
14.1
19.3
23.7
27.6
32.9
0.0
j.p. morgan asset management Long-term capital market return assumptions
Assets
Expected
arithmetic return
Historic
annualized
volatility
Historic
correlations
U.S. Inflation
2.76%
1.30%
1.00
U.S. Aggregate Bonds
5.59%
3.58%
-0.03
1.00
U.S. Large Cap Equity
10.36%
15.15%
0.00
-0.06
1.00
International Equity
10.61%
15.27%
0.01
-0.16
0.87
1.00
Emerging Market Equity
13.50%
24.04%
0.04
-0.03
0.76
0.82
1.00
Real Estate Investment Trusts
9.95%
18.17%
0.10
0.17
0.46
0.39
0.44
1.00
Hedge Fund of Funds
8.12%
6.49%
0.10
0.08
0.57
0.68
0.78
0.30
1.00
Private Equity
13.09%
23.36%
0.00
-0.03
0.65
0.68
0.72
0.42
0.75
1.00
Source: J.P. Morgan Asset Management. For illustrative purposes only.
Based on ten years of monthly returns from November 1998–October 2008.
J.P. Morgan Asset Management | 33
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34 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
Authors
Abdullah Z. Sheikh, FIA, FSA
Director of Research
Strategic Investment Advisory Group
[email protected]
Hongtao Qiao, FRM
Strategic Advisor
Strategic Investment Advisory Group
[email protected]
J.P. Morgan Asset Management | 35
36 | Non-normality of Market Returns—A Framework for Asset Allocation Decision-Making
IMPORTANT DISCLAIMER
This document is intended solely to report on various investment views held by J.P. Morgan Asset Management. All charts and graphs are shown for illustrative purposes
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change without notice. We believe the information provided here is reliable but should not be assumed to be accurate or complete. The views and strategies described
may not be suitable for all investors. References to specific securities, asset classes and financial markets are for illustrative purposes only and are not intended to be,
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