Non-normality of Market Returns. A framework for asset allocation decision-making

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Non-normality of Market Returns

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About J.P. Morgan Asset Management—Strategic Investment Advisory Group

The Strategic Investment Advisory Group (SIAG) partners with clients to develop objective, thoughtful solutions to the broad investment policy issues faced by corporate and public defined benefit pension plans, insurance companies, endowments and foundations. Our global team is one of J.P. Morgan’s primary centers for thought leadership and advisory services for institutional clients in the areas of asset allocation, pension finance and risk management.

The team’s expertise is supported by powerful analytical capabilities for conducting asset/liability, risk budgeting and optimal asset allocation analysis, in line with client-specific investment guidelines, risk tolerance and return requirements. In response to the changing needs of CFOs, treasurers and CIOs, our suite of tools has been expanded to include corporate finance-based risk management analytics for assessing and proactively managing the impact of the pension plan on the corporation as a whole.

SIAG brings a deep knowledge and understanding of capital markets behavior to all its advisory services, ensuring that results and recommendations have real-world consistency and can be tested under a variety of market scenarios. Strategic Investment Advisory Group services are offered as part of an overall asset management relationship to complement the many ways in which J.P. Morgan Asset Management provides clients with value-added insights.

About J.P. Morgan Asset Management

For more than a century, institutional investors have turned to J.P. Morgan Asset Management to skillfully manage their investment assets. This legacy of trusted partnership has been built on a promise to put client interests ahead of our own, to generate original insight, and to translate that insight into results.

Today, our advice, insight and intellectual capital drive a growing array of innovative strategies that span U.S., international and global opportunities in equity, fixed income, real estate, private equity, hedge funds, infrastructure and asset allocation.

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When market stress and disruptions hit portfolios

especially hard, it is natural for investors to ask

whether anything could have been done

differ-ently. 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 frame-works 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-standard-deviation 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 con-ventional 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

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 ef-fective 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 advance-ments 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 discus-sion and further research, all geared toward helping investors better address the most pressing issues of the day. We appreci-ate 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

Foreword

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.

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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

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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 mean-variance 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

which they can be implemented. To implement an asset alloca-tion framework based on normal asset return distribualloca-tions, 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,

Overview

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}

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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 down-wards, 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 distribu-tion 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 rela-tionship 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 relation-ships, 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.

Unsmoothing Serial Correlation:

• Serial correlations can be

“unsmoothed”. That is, we can correct for the influence of prior-period returns and restore independence to single- period 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.

Source: J.P. Morgan Asset Management. For illustrative purposes only. 0 2 4 6 8 10 12 -20% -15% -10% -5% 0% Monthly return Density 5% 10% 15% 20% Empirical Normal

“Fat” left tails

ExhIBIT 1: INTERNATIONAL EqUITIES—“FAT” LEFT TAILS IN hISTORICAL RETURNS

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.

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Modeling “Fat” Left Tails (Negative Skewness and Lepto-•

kurtosis): “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).

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

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 depen-dence 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 draw-backs of standard deviation as a risk measure. Primarily, as it only measures risk on the downside, it captures both the asym-metric 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 inves-tors and regulainves-tors 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.

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. Source: J.P. Morgan Asset Management. For illustrative purposes only.

Empirical Extreme value

Much better fit for left tail

0 2 4 6 8 10 12 -20% -15% -10% -5% 0% 5% 10% 15% Monthly return Density

ExhIbIt 2: IntErnAtIonAl EquItIES—ApplyIng ExtrEmE VAluE thEory For A bEttEr FIt

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Incorporating Non-Normality: A Potentially Better Way to Assess and Manage Downside Risk

Through rigorous analysis, we examine the impact of incor-porating 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 method-ologies 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 over-all impact of this work: to better capture downside portfolio risk that is observed in real-world markets, but missed by tradition-al asset tradition-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.

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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.

Empirical evidence of non-normality in

asset returns

Source: J.P. Morgan Asset Management. For illustrative purposes only.

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 T_j is the j-th serial correlation and T is the number of observations.

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

ExhIBIT 3: TESTINg FOR SERIAL CORRELATION IN MONThLy ASSET CLASS RETURNS

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,}

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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 con-clude that there is sufficient evidence to reject the null hypoth-esis 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, Emerg-ing 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 as-sets. The difficulty in valuing the underlying assets at regular in-tervals requires managers or administrators to estimate prices (for example, with reference to the closest marketable security or based on certain economic indicators).

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 conse-quence 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.

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 experi-enced globally by investors over 2008 and into 2009. Public equity markets around the world have fallen precipi-tously 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 induc-ing 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 nega-tive returns in greater magnitude and with a higher probability

Monthly return Frequency 0 5 10 15 20 25 -18% -15% -12% -9% -6% -3% 0% 3% 6% 9% 12%

ExhIBIT 4: hISTOgRAM OF U.S. EqUITy RETURNS—TEN yEARS ENDINg OCTOBER 2008 Key Statistics Mean 0.1% Median 0.7% Maximum 9.8% Minimum -16.8% Std Dev. 4.4% Skewness -0.67 Kurtosis 4.20 Jarque-Bera 16.0

12_{First order serial correlation implies that the return from one month is correlated with the}

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than implied by the normal distribution. As an example, let us consider U.S. Equities.

Exhibit 4 shows a simple histogram of monthly returns for U.S. Equities over the last ten years. It illustrates that the distribu-tion 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 under-statement of downside risk by comparing two types of density functions:

1. The density function of the normal distribution14

2. The kernel (i.e. empirical) density function ‘implied’ by actual historical returns

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.

The “fat” left tail phenomenon implies that using a normal distribution to model returns underestimates the frequency and magnitude of downside events.

0 10 20 30 40 50 -6% -4% -2% 0% 2% 4% 6% Monthly return Density

ExhIBIT 5: U.S. BONDS—“FAT” LEFT TAILS IN hISTORICAL RETURNS

0 2 4 6 8 10 12 -30% -20% -10% 0% 10% 20% Monthly return Density

ExhIBIT 6: U.S. EqUITIES—“FAT” LEFT TAILS IN hISTORICAL RETURNS

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 2 4 6 8 10 12 -20% -15% -10% -5% 0% 5% 10% 15% 20% Monthly return Density

ExhIBIT 7: INTERNATIONAL EqUITIES—“FAT” LEFT TAILS IN hISTORICAL RETURNS Empirical Normal Empirical Normal Empirical Normal

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Empirical Normal Empirical Normal Empirical Normal Empirical Normal 0 1 2 3 4 5 6 7 -40% -30% -20% -10% 0% 10% 20% 30% 40% Monthly return Density

ExhIBIT 8: EMERgINg MARkETS EqUITIES—“FAT” LEFT TAILS IN hISTORICAL RETURNS 0 1 2 3 4 5 6 7 8 9 -40% -30% -20% -10% 0% 10% 20% 30% 40% Monthly return Density

ExhIBIT 9: U.S. REITS—“FAT” LEFT TAILS IN hISTORICAL RETURNS

0 1 2 3 4 5 6 7 8 -30% -20% -10% 0% 10% 20% 30% Monthly return Density

ExhIBIT 11: PRIVATE EqUITy—“FAT” LEFT TAILS IN hISTORICAL RETURNS

0 4 8 12 16 20 24 28 32 -100% -75% -50% -25% 0% 25% 50% 75% 100% Monthly return Density

ExhIBIT 10: hEDgE FUND OF FUNDS—“FAT” LEFT TAILS IN hISTORICAL RETURNS

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

Private Equity 0.0 4.17 6.81 Yes Yes

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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 conta-gion—the risk of one market negatively impacting and destabi-lizing another—can lead to cases of converging correlations. In this section, we test the hypothesis that correlations and vola-tility 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.

If this proves to be the case, it means that the benefits of port-folio diversification may be overestimated by traditional frame-works 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. To test our hypothesis15_{, we analyze simple “Pearson”}

correla-tions16_{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}

mea-sures 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: 1. The control period is a full ten-year history, from October 1997 to September 2007, during which the VIX averaged 20.9%. 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 Rus-sian financial crises, which led to significant market uncertainty.

0 10 20 30 40 50 60 70

Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Jan-08

Expec

ted v

olatilit

y %

ExhIBIT 13: CBOE VIx SINCE INCEPTION

ExhIBIT 14: SIMPLE CORRELATIONS OVER TEN yEARS TO SEPTEMBER 2007

U.S. Bonds U.S. Equity International Equity Markets EquityEmerging Investment TrustsReal Estate hedge Fund of Funds Private Equity

U.S. Bonds 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

15_{Note that although we do not test whether the changes in correlations are statistically}

signifi-cant, 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.

17_{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.

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“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.”

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Exhibit 14 shows correlation coefficients (ρ) over the full (monthly) ten-year period to September 2007 for all seven asset classes.

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 (

• ρ= -0.21)

REITs (

• ρ = 0.31)

Hedge Funds of Funds (

• ρ = 0.46)

Private Equity

• 18₍_{ρ = 0.61)}

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

anticipated. The correlation coefficients with U.S. Equities were as follows:

Long-run Coefficient During Market Stress

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. 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.

ExhIBIT 15: CORRELATIONS DURINg hIgh VOLATILITy PERIOD (AUgUST 1998—SEPTEMBER 1999)

U.S. Bonds 1.00

U.S. Equity -0.02 1.00

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

18_{Private Equity may be seen as a return enhancer rather than a diversification play. We include}

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Incorporating non-normality into an asset

allocation framework

Based on empirical testing, we have

demon-strated that conventional asset allocation

frame-works 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 incorporat-ing 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

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 cor-relation. 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 un-derlying 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

characteristics of the underlying data generating process. Nota-bly, 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:

R_t = a + bR_(t-1)

where R_t represents the return at time t

The regressions indicate the following estimates (Exhibit 16) for the “‘serial” correlation coefficients, based on monthly returns.

19_{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.

20_{Note International Equity, Emerging Markets Equity, Hedge Fund of Funds and Private Equity}

returns test positive for serial correlation at lag one.

b Statistically significant*

International Equity 0.21 Yes

Emerging Markets Equity 0.25 Yes

Hedge Fund of Funds -0.42 Yes

Private Equity 0.21 Yes

Source: J.P. Morgan Asset Management. Note the coefficient b reflects the strength of the auto-correlation. For illustrative purposes only.

* Statistically significant at the 5% level

ExhIBIT 16: CORRELATION COEFFICIENTS AT LAg ONE ^

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StEp 2:

We then produce our “unsmoothed” return series as follows, based on the “serial” correlation coefficient already derived: R_t (corrected) = (R_t—bR_t-1) / (1—b)

After applying the methodology outlined above, we obtain a new data series R_t (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).

The test indicates that our revised data is no longer serially cor-related. 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.

Finally, we evaluate the impact on asset correlations to assess the viability of our unsmoothed data series—as our unsmooth-ing 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 distribu-tions 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.

21_{An Extreme Value distribution is defined as the limiting distribution of extreme values sampled}

from any independent randomly distributed process.

Test statistic p-value Evidence of serial correlation

International Equity 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

ExhIBIT 17: TESTS FOR SERIAL CORRELATION ON CORRECTED DATA FOR UP TO SIx LAgS 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%

ExhIBIT 18: VOLATILITIES BEFORE AND AFTER “UNSMOOThINg” FOR SERIAL CORRELATION

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ExhIBIT 19: CORRELATIONS BASED ON SMOOThED (OR RAW) DATA OVER TEN yEARS TO OCTOBER 2008

U.S. Bonds 1.00

U.S. Equity -0.07 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

ExhIBIT 20: CORRELATIONS BASED ON UNSMOOThED DATA OVER TEN yEARS TO OCTOBER 2008

U.S. Bonds 1.00

U.S. Equity -0.07 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 1.00

Source: J.P. Morgan Asset Management. For illustrative purposes only. Left tail Interior Right tail

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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 cali-brating 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. Hence, in aggregate, we derive a semi-parametric probability density function to describe the data generating process. We re-fer 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:

What we are looking for is a better fit to the actual distribu-•

tion 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.

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:

for θ < x, when k > 0, or for θ < x <–σ/k when k < 0. In the limit for k = 0, the density is

for θ < x.

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.

f(x

_I

k,

σ

,

θ)

1+k

k

1 -1

Source: J.P. Morgan Asset Management. For illustrative purposes only. Empirical Semi-parametric GPD 0 10 20 30 40 50 -.60% -40% -20% 0% 20% 40% Monthly return Density

ExhIBIT 22: U.S. BONDS—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

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ExhIBIT 28: PRIVATE EqUITy—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

Semi-parametric GPD Empirical 0 1 2 3 4 5 6 7 8 -40% -30% -20% -10% 0% 10% 20% 30% 40% Monthly return Density Semi-parametric GPD Empirical 0 4 8 12 16 20 24 28 32 -20% -10% 0% 10% 20% 30% Monthly return Density

ExhIBIT 27: hEDgE FUND OF FUNDS—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

Semi-parametric GPD Empirical 0 1 2 3 4 5 6 7 -100% -80% -60% -40% -20% 0% 20% Monthly return Density

ExhIBIT 25: EMERgINg MARkETS EqUITIES—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

Semi-parametric GPD Empirical 0 1 2 3 4 5 6 7 8 9 -100% -80% -60% -40% -20% 0% 20% Monthly return Density

ExhIBIT 26: U.S. REITS—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA Semi-parametric GPD Empirical 0 2 4 6 8 10 12 -40% -30% -20% -10% 0% 10% 20% Monthly return Density

ExhIBIT 23: U.S. EqUITIES—FITTINg ThE SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

Semi-parametric GPD Empirical 0 2 4 6 8 10 12 -20% -15% -10% -5% 0% 5% 10% 15% Monthly return Density

ExhIBIT 24: INTERNATIONAL EqUITIES—FITTINg SEMI-PARAMETRIC gPD DISTRIBUTION TO hISTORICAL DATA

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To enhance our conviction in these results, we need to statisti-cally test this “goodness of fit” both for the whole distribu-tion 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 hypoth-esis stands statistical significance.

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.

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 bet-ter 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.

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.

23_{QQ plots illustrate how different quantiles of a sample compare to those implied by the}

theo-retical density function. All data points being on a straight 45 degree line indicate a perfect fit of the data with the distribution.

Quantiles of data Quantiles of Normal -.03 -.02 -.01 .00 .01 .02 .03 .04 -.04 -.02 .00 .02 .04

ExhIBIT 30: U.S. BONDS—qq PLOT (NORMAL DISTRIBUTION)

Quantiles of data Quantiles of semi-parametric GPD -.04 -.03 -.02 -.01 .00 .01 .02 .03 -.04 -.02 .00 .02 .04

ExhIBIT 31: U.S. BONDS—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION)

ExhIBIT 29: TESTINg gOODNESS OF FIT OF NORMAL AND SEMI-PARAMETRIC gPD DISTRIBUTION

normal Semi-parametric gpD

kS statistic p Value Reject Normality kS Statistic p Value parametric gPDReject Semi-

U.S. Aggregate Bonds 0.50 0.00 Yes 0.05 0.89 No

U.S. Large Cap Equity 0.49 0.00 Yes 0.06 0.75 No

EAFE Unhedged Equity 0.46 0.00 Yes 0.12 0.08 No

Emerging Markets 0.46 0.00 Yes 0.11 0.13 No

Real Estate Investment Trusts (REITs) 0.45 0.00 Yes 0.05 0.89 No

Hedge Fund of Funds 0.47 0.00 Yes 0.05 0.92 No

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Quantiles of data Quantiles of Normal -.12 -.08 -.04 .00 .04 .08 .12 -.20 -.15 -.10 -.05 .00 .05 .10

ExhIBIT 32: U.S. EqUITIES—qq PLOT (NORMAL DISTRIBUTION)

Quantiles of data Quantiles of semi-parametric GPD -.15 -.10 -.05 .00 .05 .10 .15 -.20 -.15 -.10 -.05 .00 .05 .10

ExhIBIT 33: U.S. EqUITIES—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION )

Quantiles of data Quantiles of semi-parametric GPD -.3 -.2 -.1 .0 .1 .2 -.3 -.2 -.1 .0 .1 .2

ExhIBIT 37: EMERgINg MARkETS EqUITIES—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION ) Quantiles of data Quantiles of Normal -.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 -.3 -.2 -.1 .0 .1 .2

ExhIBIT 36: EMERgINg MARkETS EqUITIES—qq PLOT (NORMAL DISTRIBUTION) Quantiles of data Quantiles of Normal -.15 -.10 -.05 .00 .05 .10 .15 -.15 -.10 -.05 .00 .05 .10

ExhIBIT 34: INTERNATIONAL EqUITIES—qq PLOT (NORMAL DISTRIBUTION)

Quantiles of data Quantiles of semi-parametric GPD -.15 -.10 -.05 .00 .05 .10 .15 -.15 -.10 -.05 .00 .05 .10

ExhIBIT 35: INTERNATIONAL EqUITIES—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION )

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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.

Quantiles of data Quantiles of semi-parametric GPD -.2 -.1 .0 .1 .2 .3 -.3 -.2 -.1 .0 .1 .2 .3

ExhIBIT 43: PRIVATE EqUITy—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION )

Quantiles of data Quantiles of Normal -.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 -.3 -.2 -.1 .0 .1 .2 .3

ExhIBIT 42: PRIVATE EqUITy—qq PLOT (NORMAL DISTRIBUTION)

Quantiles of data Quantiles of semi-parametric GPD -.08 -.04 .00 .04 .08 .12 -.08 -.04 .00 .04 .08

ExhIBIT 41: hEDgE FUND OF FUNDS—qq PLOT (SEMI-PARAMETRIC gPD DISTRIBUTION) Quantiles of data Quantiles of Normal -.06 -.04 -.02 .00 .02 .04 .06 -.08 -.04 .00 .04 .08

ExhIBIT 40: hEDgE FUND OF FUNDS—qq PLOT (NORMAL DISTRIBUTION) Quantiles of data Quantiles of Normal -.15 -.10 -.05 .00 .05 .10 .15 -.4 -.3 -.2 -.1 .0 .1

ExhIBIT 38: REITs—qq PLOT (NORMAL DISTRIBUTION)

Quantiles of data Quantiles of semi-parametric GPD -.3 -.2 -.1 .0 .1 -.4 -.3 -.2 -.1 .0 .1

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C. Incorporating the impact of converging correlations into our framework

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.

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 correla-tions assume a linear relacorrela-tionship 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 choos-ing 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 converg-ing correlations—i.e. allows fatter tails than the normal distribution.

This approach better reflects the empirical observations of con-verging 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.

In Exhibit 44, we illustrate the impact of modifying the normal marginal distributions by adding a t-copula with a low degree-of-freedom parameter.

24_{The Student t distribution comes from the same family of distributions as the Normal. A}

Stu-dent 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. Source: J.P. Morgan Asset Management. For illustrative purposes only.

ExhIBIT 44: MODELINg JOINT DEPENDENCE IN ASSET RETURNS USINg COPULAS

Illustration of increased joint dependence through copulas 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% Mon thly simulated h edge F und of F unds r eturns

Monthly simulated U.S. Large Cap Equity returns -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

Mon thly simulated h edge F und of F unds r eturns

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In order to show the effect of modeling increased joint depen-dence, 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.

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.

ExhIBIT 45: INCORPORATINg NON-NORMALITy—SUMMARy OF MODEL

Source: J.P. Morgan Asset Management. For illustrative purposes only. Step 1: Input historical data

Unsmoothed data Step 2: Remove serial correlation from returns

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

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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 hypo-thetical 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}

port-folio risk as the Conditional Value at Risk at the 5th_{percentile of}

the portfolio.

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 down-side 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 distri-butions. 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.

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.

28_{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.

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 5%

Total alternatives 15%

REITs 5%

Hedge Fund of Funds 5%

Private Equity 5%

key statistics

Expected arithmetic return 9.1%

Expected volatility 10.0%

Expected compound return 8.7%

Sharpe ratio 0.51

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).

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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.

The CVaR₉₅ of the current allocation, based on our new method-ology, 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 CVaR}

95 figure of $74 million.

Incor-porating 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.

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. 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.

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.

29_{Our estimates of CVaR}

95 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.

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. 0 100 200 300 400 500 600 700 800 (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} Fr equenc y

ExhIBIT 47: hISTOgRAM OF PROJECTED CUMULATIVE PORTFOLIO gAIN (LOSS) AT ThE END OF TEN yEARS ASSUMINg NON-NORMALITy