International Forum of Sovereign Wealth Funds 1 Asset Allocation for the Short- and Long-Term Executive Summary This white paper explores challenges that International Forum of Sovereign Wealth Funds (IFSWF) members face as they balance short- and long-term investment objectives and proposes specific frameworks that may be useful in this endeavour. The investment landscape has evolved significantly in recent years and SWFs have contended with an ever-expanding array of investment opportunities in both public and private markets. In response, many are re-evaluating the methods they employ to construct portfolios and measure and manage portfolio risk. This paper also addresses the organizational challenges related to acquiring and maintaining the human talent that SWFs need to achieve their objectives. Our approach to this study was multifaceted and consisted of three distinct avenues of research. Specifically: I. We undertook an extensive review of the academic literature related to asset allocation challenges and solutions. II. We spoke to a leading researcher in the area of portfolio construction and risk management techniques. III. We surveyed a broad group of IFSWF members regarding the challenges that SWFs face and the ways that they address these challenges. Our goal throughout is to provide findings that are both descriptive, to enhance understanding of the issues involved, and prescriptive, to propose frameworks and solutions that may be helpful. The main body of this paper—which synthesizes inputs from parts I, II, and III as outlined above—presents our comprehensive findings. At the highest level, our key conclusions are as follows: In the years since the global financial crisis of 2008-2009, monetary policies across the globe have entered unfamiliar territory, interest rates have reached historic lows (some are even negative), return expectations have
74
Embed
Asset Allocation for the Short- and Long-Term · Section 4 presents portfolio construction approaches that take short- and long-term investment horizons into account. We consider
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
International Forum of Sovereign Wealth Funds 1
Asset Allocation for the Short- and Long-Term
Executive Summary
This white paper explores challenges that International Forum of Sovereign Wealth Funds (IFSWF) members
face as they balance short- and long-term investment objectives and proposes specific frameworks that may be
useful in this endeavour. The investment landscape has evolved significantly in recent years and SWFs have
contended with an ever-expanding array of investment opportunities in both public and private markets. In
response, many are re-evaluating the methods they employ to construct portfolios and measure and manage
portfolio risk. This paper also addresses the organizational challenges related to acquiring and maintaining the
human talent that SWFs need to achieve their objectives.
Our approach to this study was multifaceted and consisted of three distinct avenues of research. Specifically:
I. We undertook an extensive review of the academic literature related to asset allocation challenges
and solutions.
II. We spoke to a leading researcher in the area of portfolio construction and risk management
techniques.
III. We surveyed a broad group of IFSWF members regarding the challenges that SWFs face and the
ways that they address these challenges.
Our goal throughout is to provide findings that are both descriptive, to enhance understanding of the issues
involved, and prescriptive, to propose frameworks and solutions that may be helpful. The main body of this
paper—which synthesizes inputs from parts I, II, and III as outlined above—presents our comprehensive
findings. At the highest level, our key conclusions are as follows:
In the years since the global financial crisis of 2008-2009, monetary policies across the globe have entered
unfamiliar territory, interest rates have reached historic lows (some are even negative), return expectations have
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 2
declined, market volatility has increased, and a variety of new investment styles have emerged. These changes
have forced investors, SWFs among them, to adapt their thinking and reconsider traditional approaches to
allocating investments and managing risks. At the same time, SWFs have become an important and rapidly
growing investor class and now comprise one of the world’s largest institutional asset pools. As SWFs have risen
in prominence they have found themselves on the front lines of the portfolio management challenges of this new
era. As they formulate investment strategies, it is critical for SWFs to adhere to their core investment beliefs and
employ methods that make the best use of available information to meet their specific objectives. This paper is
divided into seven sections, each of which explores challenges and solutions related to a particular area of
portfolio management.
In section 1, we present a discussion with Mark Kritzman, Senior Lecturer in Finance at the MIT Sloan
School of Business. As both a practitioner and leading asset allocation and risk management researcher,
Mark has deep expertise in portfolio and risk management methods. In this discussion, he shares his
insights regarding the portfolio management challenges faced by SWFs. This section summarizes many
of the key issues that we cover in greater detail in later sections.
In section 2, we discuss challenges associated with defining the opportunity set. The first step in
determining an optimal asset allocation is to identify suitable investments that may provide risk and/or
return benefits to a portfolio. We consider three separate frameworks that could help an SWF determine
whether a particular asset class or investment should be considered for inclusion.
Section 3 explores different approaches to forming future beliefs about asset class return, risk, and
diversification properties. SWFs face many challenges in investing for the long term, including being
subject to short-term evaluations and managing risk in exploiting tactical opportunities. We discuss the
significance of the investment horizon in determining the properties of asset classes as well as its impact
on the evaluation of investment managers. We also explore the concept of risk regimes and how this
framework can improve estimates of risk exposure. Finally, we review complexities that SWFs should
consider when evaluating alternative asset classes, including accounting for performance fees,
appraisal-based valuations, and liquidity.
Section 4 presents portfolio construction approaches that take short- and long-term investment horizons
into account. We consider portfolio construction methods that include different aversions to short- and
long-term risk. We show how to incorporate risk regime information into both strategic and tactical
allocation decisions. We also consider the implications of different investor preferences—for example,
distinguishing between upside and downside risk—and describe frameworks that can help investors
account for these preferences. Finally, we introduce the notion of asset class stability and discuss how it
can be used to produce portfolios with more stable risk characteristics over time.
Section 5 looks at techniques for measuring, evaluating, and communicating portfolio risk. SWFs must
communicate their investment decisions and the risks associated with those decisions to a wide array of
stakeholders that include governance bodies as well as the public. Managing stakeholder expectations
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 3
can be critical to maintaining confidence in both investment decisions and expected outcomes through
good times and bad. Most risk metrics focus implicitly on risk at the end of a specific investment horizon.
However, in practice, stakeholders are also keenly interested in understanding the losses that might
occur along the way. We explore the distinction between end-of-horizon and within-horizon risk and
discuss its practical relevance.
Section 6 provides an overview of the Reference Portfolio approach to portfolio management. It then
addresses how the methods presented in this paper can be complementary to the use of reference
portfolios. It expands on this notion by proposing the use of active risk budgets as a method of
constraining active decisions and managing risk.
Section 7 presents the key findings from a survey of SWFs intended to provide insights into areas of
interest and trends in investment preferences. In this section, we also discuss how SWFs align their
organizational structures to meet their objectives.
The views and interpretations expressed herein are those of the authors and do not necessarily reflect the views
Table of Contents ..................................................................................................................................................... 4
Fund Organization ........................................................................................................................................ 64
IFSWF Member Survey ................................................................................................................................ 65
Q: So you are basically incorporating information about the volatility of volatility in the portfolio construction
process?
MK: Yes.
Q: Thank you for those important insights on portfolio construction and risk estimation. It is evident that you have
delved much deeper into the portfolio construction process than most of us and we have certainly benefitted from
those efforts today. In the time we have left, we did want to address some questions directly from members of
the IFSWF. The first question is as follows: It seems that various parts of the world, Europe at first, are going to
go through a long period of very low interest rates. How do you think this will change the way we look at these
types of investments?
MK: You can address that in several ways. One is to define interest rate regimes. You can then characterize
your estimates of future return and risk of portfolio components contingent on what regime you expect to be in.
When you conduct an optimization, what you are doing is maximizing expected return minus some coefficient of
risk aversion times portfolio risk. That portfolio risk is characterized as a covariance matrix. So, what you can do
is to collect a long history of returns. You have information about when interest rates were low in history and
when interest rates were high in history. Instead of basing the risk of the asset classes on the full sample of
historical returns, divide the historical returns into two samples. One sample would be returns when interest rates
were below some level and the other would be returns when interest rates were above some level. You would
then calculate separate covariance matrices and condition expected returns based on what prevailed in the low
interest rate regime versus the high interest regime. Then, when you optimize your portfolio, instead of
maximizing expected return minus risk aversion times one covariance matrix, you would maximize expected
return minus one risk aversion coefficient times covariances estimated from the low interest rate regime minus
another risk aversion coefficient times covariances estimated from the high interest rate regime. So you have two
interest rate regimes. Earlier I spoke about a fragile regime and a resilient regime. You can take the same
approach but have it be conditioned on these different rate environments. Then the risk aversion coefficient that
you assign to these two covariance matrices can either reflect the relative aversion you have toward risk during
periods of high or low interest rates or, instead, it can reflect your expectation for what the future will hold. You
may argue, and I would argue, that interest rates are more likely to be higher in the future than they have been in
the recent past and you might put a higher probability on that when you do your optimization. That is one
approach.
The other thing to keep in mind is that interest rates historically, at least in the United States, have gone through
very long cycles. We had a declining interest rate environment from 1979 through just about the present. Short
rates went from about 20 percent down to zero. It is unlikely that that trend can continue. It can’t continue without
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 17
going significantly negative. I think it’s more likely that we’ll have a long and gradual increasing interest rate
environment.
The other thing that this implies is that the risk from fixed income assets is much greater than you think it is. For
example, there is a strategy called risk parity. What that means is that you structure a portfolio such that each of
the major components of the portfolio contributes the same amount to total portfolio risk. So you should lever up
your exposure to bonds and cut your exposure to equities. People have written articles showing that this risk
parity strategy approach has outperformed a 60/40 stock/bond portfolio going back to the 1920s. It turns out, that
is not true. They based that performance on the Sharpe ratio which has as its denominator standard deviation.
They converted the standard deviation of monthly returns to the standard deviation of longer horizon returns
using that heuristic I described earlier. If you take into account the lagged correlations of the asset’s returns then
the 60/40 portfolio outperformed the risk parity portfolio by as much as they argued it underperformed.
Anyway, the short answer is…and I have trouble giving short answers…I would condition my expected returns
and risk estimates on the sub-samples of high and low interest rates and use that information to build my
portfolio.
Q: I like the discussion and identification of this richness of risks and I understand the statistical qualities of these
other risk measures. What it presents is added complications regarding optimization and determining what is a
best portfolio…especially if you have multiple objectives. Generally, my board is happy if we do well versus
public plans, if we don’t have a high risk of losing money, if we show actuarial progress, or the equivalent of
actuarial progress, towards some long-term goal. It sounds like what you are describing is that, in general, the
profession has made more advancements in risk measurement than on the optimization side. What is your view?
MK: Well, I think there is a lot of misunderstanding of optimization. Let’s talk about mean variance for a minute. It
turns out that mean variance is much more robust than people give it credit for. I am a big fan of Harry
Markowitz, and he and I have had many discussions about this. Mean variance optimization requires one of two
things. Either that returns are approximately normally distributed or that investors have preferences that can be
reasonably described by just mean and variance. You do not need both of those to be true. You just need one or
the other to be true. So, mean variance does a pretty good job.
Now, you can amplify mean variance to take into account multiple objectives like you’ve just described. For
example, you may care about performance relative to your peers and you also may care about your absolute
performance. So, just as I described about how you can come up with covariance matrices based on different
regimes, you can come up with covariance matrices based absolute returns and covariance matrices based on
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 18
relative returns. You can specify the objective function of mean variance optimization to be expected return
minus absolute risk aversion times the covariances of absolute returns minus a measure of aversion to relative
risk times the covariance matrix of relative returns. So, basically, you are jointly optimizing for both absolute
volatility and tracking error relative to some portfolio of peer investors or some benchmark. That is one thing you
can do that is trivial to implement.
In terms of pension liabilities or actuarial progress, as you describe it, that is a really interesting question. This is
research that we are actually doing right now and I’ll be giving a talk at Oxford University in a couple of months
on the topic. If you want to hedge the monthly volatility of your liabilities, for example, the best hedge would
probably be some kind of fixed income asset. This is because high frequency volatility of liabilities is typically a
function of changes in discount rates. To be clear, when I say high frequency I mean monthly versus say yearly
rather than milliseconds. Bonds would be the best hedge for that. But over the long term, the low frequency
volatility of liabilities is a function of wage inflation and productivity growth. Equities are a better hedge against
that. Again you can construct an optimization process that balances your aversion to large drawdowns along the
way versus your aversion to the gradual erosion of your pension assets relative to your liabilities. That is another
thing you can do in the optimization process.
To the extent that you or your committee or stakeholders have preferences that can’t be well described by mean
and variance, there is another thing you can do. A typical example of this would be thresholds. If there is some
threshold where if you breached that threshold conditions would be qualitatively worse than if you suffer a loss
above that threshold, this is what we call a “kinked” utility function. If you have a situation where your returns are
not normally distributed and you have preferences that are affected by these thresholds then you can’t use mean
variance optimization. What you would use is what is called full-scale optimization. Full-scale optimization is just
plain direct utility maximization through the use of sophisticated search algorithms.
So you write down your utility function. You have some sample of returns. You plug those returns into the formula
for your kinked utility function and then you plug in a portfolio with one set of asset weights and calculate the
utility. Then you plug in another portfolio with another set of asset weights and calculate the utility. You do this
over and over again until you find the portfolio that has the highest utility. Now, that is computationally very
challenging, especially if you have portfolios that have more than just a few assets in them. However, it turns out
that there are optimizers that run full-scale optimization that can sample as many as half a million portfolios in
about 30 seconds. So, this is what I would use instead of mean variance optimization in the case where you
believe returns not to be approximately normally distributed and you have thresholds.
– End of discussion –
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 19
2. Defining and Selecting Asset Classes
The starting point for portfolio construction is the identification of the asset classes to be included in the portfolio.
One of the key challenges for SWFs is assessing the increasing assortment of investment opportunities available
in public and private markets across world. This includes determining whether an investment represents an
avenue for capturing a market risk premium or an opportunity to exploit alpha, and whether it provides
diversification benefits or if it is even suitable for the objectives of the fund. These are all important
considerations for SWFs to contemplate. However, the imprecision with which many investors approach defining
asset classes often results in inefficient diversification. Incorrectly determining investments as being similar would
result in an SWF neglecting an opportunity to diversify. Alternatively, incorrectly determining investments to be
distinct would result in a SWF deploying assets towards redundant investments to little or no benefit. Part of the
challenge in addressing these issues is that asset class definitions are often ambiguous. Delineating investment
strategies and asset classes requires an understanding of both the qualitative and quantitative aspects of an
investment.
Defining Asset Classes
The first approach to defining an asset class is through their investment attributes. Asset classes are defined as
a group of assets with common characteristics that include:2
Sensitivity to the same major economic and/or investment factors.
Risk and return characteristics that are similar.
A common legal or regulatory structure.
When asset classes are defined in this manner, the relationship between the returns of two different asset
classes would be expected to exhibit low correlations. This approach is useful in defining asset classes in
general terms. However, some ambiguity remains in terms of the degree to which assets are influenced by
specific economic factors and in the extent to which risk and return characteristics are similar. Furthermore, it
does not consider the assets currently held by an investor in determining whether it should be considered for
inclusion in an investor’s portfolio.
Asset Class Criteria for Portfolio Inclusion
A second approach looks beyond specific investment attributes and allows any group of assets that is treated as
2 F. J. Fabozzi and H. M. Markowitz, “The Theory and Practice of Investment Management,” John Wiley & Sons, Inc., Hoboken NJ.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 20
an asset class by investment managers to be designated as an asset class as long as it meets the following four
criteria for asset class status:3
1. An asset class should be relatively independent of other asset classes in the investor’s portfolio.
2. An asset class should be expected to raise the utility of the investor’s portfolio without selection skill on
part of the investor.
3. An asset class should be comprised of homogeneous investments.
4. An asset class should have the capitalization capacity to absorb a meaningful fraction of the investor’s
portfolio.
Independence is necessary to avoid investing in assets that do not provide efficiency benefits to the portfolio by
considering the redundancy of an asset class candidate against all of the other asset classes held by an investor.
Independence can be tested by constructing a portfolio that minimizes tracking error to the proposed asset class
by using a combination of the asset classes already held by the investor. If this “mimicking” portfolio exhibits a
high tracking error to the candidate asset class then it is reasonable to assume relative independence. A low
tracking error would suggest that the asset class would not provide meaningful benefits to the portfolio.
The criterion for increasing the expected utility of the investor’s portfolio distinguishes between return and
diversification benefits. Because expected utility (Expected Return – Risk Aversion x Variance) is a function of
both return and risk, an asset class can prove beneficial through either its return or its ability to diversify a
portfolio. That is, an asset class can be determined to be beneficial even if the average return it provides is below
that of the current portfolio as long as it exhibits sufficient diversification properties. Requiring that utility be
increased without the need for skill in asset selection differentiates between benefits provided by an asset class
and those afforded by superior active management.
The homogeneity requirement is to ensure that opportunities for diversification are not neglected. If assets
designated as constituents of an asset class are, in fact, dissimilar then it is likely that greater efficiency can be
achieved by partitioning the dissimilar components into another asset class.
The fourth requirement addresses the capacity of an asset class. This is of particular importance to SWFs in that
they represent some of the largest asset pools in the world. If an asset class is not sufficiently large enough to
absorb a significant portion of a SWF’s portfolio then it is likely that illiquidity would reduce expected return and
increase risk to the point of eroding any expected benefit provided by the asset class.
The four criteria detailed for asset class status provide a rigorous framework for defining and evaluating an asset
class in the context of an investor’s existing portfolio. However, it may be useful to have a simplified approach for
assessing the benefits of an asset class or investment. Assuming that an asset class is constrained to a positive
3 M. Kritzman, “Towards Defining an Asset Class,” Journal of Alternative Investments 2, no. 1(1999): 79.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 21
weight, its impact on the portfolio’s Sharpe ratio can be used to evaluate whether it should be considered for
inclusion in a portfolio. Exhibit 1 presents an inequality that provides a framework for assessing the benefits of an
asset class.45,6
If the Sharpe ratio of the new asset class is greater than the Sharpe ratio of the portfolio multiplied by the
correlation between the new asset class and the portfolio then the new asset class provides an expected benefit
to the portfolio. Otherwise, the new asset class provides no benefit and may actually detract from portfolio
efficiency. This evaluation framework can be applied to both asset classes as well as individual investments and
will be particularly useful for evaluating investments for inclusion in portfolios, especially when considering
benefits over different horizons.
Exhibit 1: Sharpe Ratio Framework for the Evaluation of Asset Class Benefits
E(Rnew)-Rf
σnew
> (E(Rp)-Rf
σp
) ρRnew,Rp
Where:
E(Rnew) = expected return of the new asset class
σnew = standard deviation of the new asset class
Rf = risk-free rate
E(Rp) = expected return of the portfolio
σp = standard deviation of the portfolio
ρRnew,Rp
= correlation between the new asset class and the portfolio
4 2017 CFA Program Curriculum Level III, CFA Institute
5 M. Blume, “The Use of “Alphas” to Improve Performance,” Journal of Portfolio Management, no. 11 (1984): 86-92.
6 E. Elton, M. Gruber, and J. Rentzler, “Professionally Managed, Publicly Traded Commodity Funds,” Journal of Business, Volume 60, Issue
2 (1987): 175-199.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 22
3. Estimating Asset Class Risk, Returns, and Correlations
Markowitz’s (1952) seminal work on portfolio theory begins with three simple sentences:
“The process of selecting a portfolio may be divided into two stages. The first stage starts with observations and
experience and ends with beliefs about future performances of available securities. The second stage starts with
the relevant beliefs about future performances and ends with the choice of portfolio.”7
It is evident from the very introduction of portfolio theory that beliefs are at the core of the process. This section
focuses on stage one, where the identification of the working set of asset classes for portfolio construction
proceeds to establishing future beliefs for the expected returns, standard deviations and correlations of those
assets. These estimates are the raw materials from which efficient portfolios are developed. While history can
(and should) inform those estimates, judgment plays a central role in how those estimates are developed and for
what purpose.
SWFs are generally tasked with the achievement of specific long-term objectives and are often required to
balance those objectives while pursuing short-term opportunities. Furthermore, SWFs will inevitably be evaluated
over short-term horizons. The balancing of this tension between short-term and long-term risks requires an
understanding of asset class characteristics over different investment horizons. Unfortunately, the standard risk
models used by academia and practitioners can underestimate risk over longer horizons. Consequently, it is
critical that SWFs inform their portfolio construction decisions using methods that allow them to make the best
use of available information in matching their specific objectives.
Return
A reasonable starting point for estimating expected returns is to assume that markets are fairly priced. This
implies that the return provided by an asset class represents a fair compensation for the risk of the asset class
within a broadly diversified market. These returns are called equilibrium returns, and are estimated by first
calculating the beta of each asset class with respect to a broad market portfolio based on historical standard
deviations and correlations. Estimates for the expected return of the market portfolio and the risk-free rate are
then used to scale the returns of asset classes according to their betas. Therefore, the equilibrium return for each
asset class is calculated as the risk-free return plus its beta multiplied by the excess return of the market
portfolio.
7 H. Markowitz, “Portfolio Selection,” Journal of Finance, March 1952.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 23
While markets are rarely, if ever, in equilibrium, market forces tend to have a powerful and persistent pull towards
producing long-run asset returns that are consistent with asset risks. Moreover, the expected return of each
asset class can be adjusted easily to accord with views about departures from fair value. Consider this example:
Suppose an investor estimates the market’s expected return to equal 7.0% and the risk-free return to equal
3.0%. Exhibit 2 presents asset class betas with the selected market portfolio along with the respective equilibrium
returns. These estimates, together with estimates of beta, are based on monthly returns from December 2000
through September 2016.8 Real Estate refers to listed real estate assets. Direct real estate would be expected to
have lower volatility and expected returns.
Exhibit 2: Expected Returns (Illustrative)
Asset Class β Equilibrium Views Confidence Blend
Developed Market Equities 1.45 8.8% - - 8.8%
Emerging Market Equities 1.91 10.6% - - 10.6%
Real Estate 1.52 9.1% - - 9.1%
Global Credit 0.44 4.7% 4.2% 50.0% 4.5%
Global Treasuries 0.24 4.0% 3.5% 25.0% 3.8%
Source: State Street Global Exchange, Datastream
Some asset classes may be expected to produce returns that differ from those that would occur if markets were
in equilibrium and perfectly integrated, especially if they are not typically arbitraged against other asset classes.
Suppose Global Credit is expected to return 4.2% and Global Treasuries to return 3.5% and that an investor has
different degrees of confidence in these views. These views can be blended with equilibrium returns to derive
expected returns. The blend column in Exhibit 2 shows the expected returns for each of the asset classes in this
analysis given specific views and confidence in those views.
While a variety of alternative methods to forecasting returns can be used, equilibrium returns serve as a
reasonable baseline for comparison. For a thorough discussion of different approaches to estimating expected
returns for both traditional and alternative asset classes the reader is directed to Ilmanen (2011).9
8 The market and asset class proxies used were as follows:
Market Portfolio – 60% MSCI AC World Index + 40% Bloomberg Barclays Global Aggregate Bond Index Developed Market Equities – MSCI World Index Emerging Market Equities – MSCI Emerging Market Index Listed Real Estate – FTSE EPRA/NAREIT Developed Real Estate Index Global Credit – Bloomberg Barclays Global Aggregate Corporate Index Global Treasuries – Bloomberg Barclays Global Treasury Index 9 Ilmanen, A. “Expected Returns: An Investor’s Guide to Harvesting Market Rewards.” Chichester, West Sussex, U.K.: John Wiley & Sons,
2011.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 24
Risk, Co-movement, Risk Regimes, and the Investment Horizon
The goal of a portfolio analysis is to identify the mix of assets that is expected to provide the highest return for the
least amount of risk. While expected returns drive the composition of portfolios towards higher returns, estimates
for the risk and co-movement characteristics of assets are what inform risk reduction. The challenge for SWFs is
in specifying the risks to be mitigated and producing the relevant estimates for those risks.
The first step in estimating risks and asset relationships is to understand what risk assets have exhibited
historically. This is accomplished by calculating volatilities of asset classes and the correlations between each
pair of assets over a relevant historical period. Generally, these historical estimates are calculated using time
series of asset class returns at monthly intervals.
SWFs will also be concerned with risk over longer intervals,
such as 1-, 3-, 5-, or 10-years. Estimates for lower-
frequency, or rather, longer-horizon statistics are routinely
extrapolated from higher-frequency monthly statistics using
a conventional heuristic. That commonly applied heuristic is
the scaling of risk by multiplying by the square root of time.
For example, the standard deviation of monthly returns is
multiplied by the square root of 12 to estimate the standard
deviation of annual returns. A second heuristic is the
commonly held belief that correlations are invariant to the
time interval used to measure them. For example, the
correlation of monthly returns for an asset pair is assumed to
be the same as the correlation of annual returns. These
methods are regularly used by academics and practitioners and are even programmed into most portfolio
construction/management software applications. Unfortunately, they often misestimate the true risk presented by
investments and can lead to significantly sub-optimal results for investors with long horizons.
The reason for this misestimation is that these approaches assume that all returns are independently and
identically distributed (I.I.D.). Contrary to this assumption, financial time series often exhibit serial dependence,
mean reversion, trending, and/or risk clustering. This leads to asset values evolving through time in ways that are
highly inconsistent with expectations drawn from their high-frequency standard deviations and correlations. To
derive estimates that include information about the evolution of asset returns through time it is necessary to
account for autocorrelations and lagged cross-correlations.
Considering the unique characteristics of financial time series while estimating long-horizon risk can be
accomplished by directly using cumulative periodic returns (e.g. 12-month, 36-month, 60-month, or 120 month)
Counter to what financial theory suggests,
the risk and diversification properties of
assets differ depending on the time intervals
over which they are measured. Assets that
appear highly correlated over monthly
intervals may be uncorrelated over multi-year
intervals. To strike an effective balance
between short- and long-term objectives,
SWFs should account for this divergence
when evaluating investment opportunities
and forming portfolios.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 25
for all complete overlapping periods in the historical sample and then calculating the estimate statistics. This
approach may provide a reasonable estimate if the historical period is sufficiently long. Alternatively, the following
analytical approach can be used.10
Estimating long-horizon risk begins with calculating the discrete returns of an asset class X over the high
frequency interval. In this instance, a monthly interval is assumed and the percentage change in the price of X
from one interval t-1 to the next t is:
rt=Pt
Pt-1
-1 (1)
It follows that the cumulative multi-period return equals the cumulative product of the quantity, one plus the
single-period returns, minus one.
The continuously compounded return of X is the logarithm of the quantity, one plus the discrete return. For ease
of notation going forward, the continuously compounded return of asset X is denoted with a lower-case x:
xt = ln (Pt
Pt-1
) = ln (1+rt) (2)
The cumulative multi-period continuous return equals the cumulative sum of single-period continuous returns.
The standard deviation of the cumulative continuous returns of x over the horizon specified as q periods,
xt+…+xt+q-1, is given by:
σ(xt+⋯+xt+q-1 ) = σx√q+2 ∑ (q-k)ρxt,xt+k
q-1
k=1
(3)
where σx is the standard deviation of x measured over single-period intervals. Note that if the lagged auto-
correlations of x all equal zero, the standard deviation of x will scale with the square root of the horizon, q.
However, to the extent the lagged auto-correlations differ from zero, extrapolating the single-period standard
deviation this way to the longer-horizon standard deviation may provide a poor estimate of the actual longer-
horizon standard deviation.
To estimate long-horizon correlations, a second asset, Y, is introduced whose continuously compounded rate of
return over the period t-1 to t is denoted as yt. The correlation between the cumulative returns of x and the
cumulative returns of y over q periods, is given by:
10 It is assumed throughout this section that the instantaneous rates of return for all assets are normally distributed with stationary means and
variances.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 26
ρ(xt+⋯+xt+q-1 , yt+⋯+y
t+q-1) =
qρxt,yt
+ ∑ (q-k)(ρxt+k,yt
+ρxt,yt+k
)q-1
k=1
√q+2 ∑ (q-k)ρxt,xt+k
q-1
k=1 √q+2 ∑ (q-k)ρyt,yt+k
q-1
k=1
(4)
The numerator equals the covariance of the assets taking lagged correlations into account, whereas the
denominator equals the product of the assets’ standard deviations as described by Equation (3). This equation
allows for assumed values for the auto-correlations of x and y, as well as the lagged cross-correlations between
x and y, in order to compute the correlations and standard deviations that these parameters imply over longer
horizons. These values can be drawn from historical estimates or can be adjusted based on expectations. While
particular choices for auto-correlation do not uniquely determine cross-correlations, it is important to note that
choices for some of the lags do impose bounds on the possible values for other lags.
Long horizon estimates calculated using equations 2, 3, and 4 are expressed in units of continuously
compounded growth. The following formulas can be used to convert the mean and standard deviation of each
asset into discrete units, which is required for optimization:
μd = eμc+σc
2/2-1 (5)
σd=√e2μc+σc2(e
σc2
-1) (6)
where μd and σd are the mean and standard deviation of the cumulative discrete returns, and μ
c and σc are the
mean and standard deviation of cumulative continuous returns. Similarly, it is also necessary to compute the
correlation between the cumulative discrete returns in terms of the means, standard deviations, and correlation
ρc of cumulative continuous returns.
ρd=
eρcσx,cσy,c-1
√(eσx,c
2
-1)√(eσy,c
2
-1)
(7)
Estimates for long-horizon risk that account for autocorrelation and lagged cross-correlations are important
beyond their use in portfolio optimization. Because the standard deviation measure is used in various ways for
evaluating the performance of investment strategies and estimating active risk, understanding the implications of
long-horizon risks can have an impact on the evaluation and selection of investment strategies. Consider that the
Sharpe ratio, a common method of ranking investment strategies, uses standard deviation as the denominator.
Estimates of long-horizon tracking error, which is the standard deviation of relative returns, which also
incorporates information about asset correlations over longer horizons, can also be affected. Consequently,
Information ratios, which equal the average excess return divided by tracking error, would be impacted. Kinlaw,
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 27
Kritzman, and Turkington (2015) provide a detailed analysis of the impact of the misestimation of long-horizon
risk on performance measurement.11
Developing estimates that can provide for a more realistic understanding of
the long horizon risks presented by investment strategies is an important part of the portfolio implementation and
risk management process.
Full sample estimates of asset risks and correlations from a
historical sample of monthly returns provide a reasonable
starting point for a portfolio analysis. However, these
estimates provide information regarding what occurs on
average and mask the fact that risks and correlations are not
necessarily stable through time. With respect to managing
portfolio risk, it is useful to understand how these parameters
change during periods of stress in the financial markets. This
requires partitioning the historical sample of returns into
normal and abnormal periods and can be accomplished
through the use of a risk measure known as turbulence.
Financial turbulence is defined as a condition in which asset prices, given their historical patterns of behavior,
behave in an uncharacteristic fashion, including extreme price moves, decoupling of correlated assets, and
convergence of uncorrelated assets. Financial turbulence often coincides with excessive risk aversion, illiquidity,
and devaluation of risky assets.12
Key benefits of using turbulence versus other measures of risk include the fact
that turbulence can be estimated for any set of assets and that it considers the volatilities and correlations of a
group of assets simultaneously.
To calculate estimates of asset characteristics during turbulent periods it is first necessary to identify periods
considered to be turbulent within a historical sample. For any group of assets, financial turbulence can be
determined by identifying outliers using the following multivariate distance measure:
dt =(yt-μ)Σ
-1(yt-μ)' (8)
Where:
yt = vector of asset returns for period t
μ = sample average vector of historical returns
Σ = sample covariance matrix of return series yt
11 Kinlaw, W., Kritzman, M., and Turkington, D. “The Divergence of High- and Low-Frequency Estimation: Implications for Performance
Measurement.” The Journal of Portfolio Management, Vol. 41, No. 3. 12
Kritzman, M. and Li, Y. 2010. “Skulls, Financial Turbulence, and Risk Management.” Financial Analysts Journal, vol. 66, no. 5 (September/October).
Too often, investors rely on long-run
averages to characterize the properties of
asset classes. In practice, investments
behave very differently during quiet markets
than during turbulent markets, when risk and
correlation are often higher. SWFs can
benefit from estimating their risk exposure
based on inputs from turbulent periods, when
losses are most likely to occur.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 28
The return series yt is assumed to be normally distributed with a mean vector μ and a covariance matrix Σ. For
12 return series, for example, an individual observation of yt would be the set of the 12 asset returns for a
specific measurement interval. A tolerance “distance” is then chosen and the distance, dt, for each vector in the
series is examined. If the observed dt is greater than the tolerance distance, the vector is defined as an outlier.
For two uncorrelated return series, Equation 8 simplifies to the following equation, which is the equation of an
ellipse with horizontal and vertical axes:
dt =(y-μ
y)
2
σy2
+(x-μ
x)
2
σx2
(9)
If the variances of the return series are equal, Equation 9 simplifies to a circle. For the general n-return normal
series case, dt is distributed as a chi-square distribution with n degrees of freedom. Under this assumption, if an
outlier is defined as falling beyond the outer 25 percent of the distribution and we have 12 return series, our
tolerance boundary is a chi-square score of 14.84. Using Equation 8, we calculate the chi-square score for each
vector in our series. If the observed score is greater than 14.84, that vector is an outlier and considered to fall
within the turbulent regime. This process partitions the historical sample of returns into normal and turbulent
periods. Alternatively, a Markov switching model can be used to partition historical returns using turbulence or
other indicators.13,14,15
Estimates of asset risk and co-movement for turbulent periods are then calculated using
the turbulent historical asset returns.
This process can be used with portfolio assets to develop estimates based on intrinsic turbulence, which is
turbulence that is specific to portfolio assets. Alternatively, the process can be used with a broad set of global
assets, such as developed market equities, emerging market equities, global fixed income, and commodities, to
identify a measure of extrinsic turbulence which can then be used to partition portfolio asset historical samples.
Regardless of the type of turbulence indicator used, these types of estimates can be useful in constructing
portfolios that are more resistant to turbulent periods.
Exhibit 3 shows estimates of standard deviations and correlations using the full historical sample of returns,
considering a 5-year investment horizon, and using turbulent regime information. Estimates are based on the
historical sample of daily returns. A review of turbulent estimates shows higher volatilities and generally higher
correlations than those seen in full sample estimates.
13 Kritzman, M., Page, S., and Turkington, D. 2012. “Regime Shifts – Implications for Dynamic Strategies” Financial Analysts Journal, vol. 68,
no. 3 (May/June). 14
Daily returns are used to calculate turbulent estimates so as to include important asset characteristics that might be missed if monthly returns are used. 15
Alternatively, a turbulence index can be calculated and then used to identify periods where turbulence values exceed a particular percentile within a rolling historical sample window.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 29
Exhibit 3: Standard Deviation and Correlation Estimates for Portfolio Assets (Illustrative)
Asset Class
Full Sample
Long Horizon
Turbulent
1 Developed Market Equities 15.6% 28.2% 23.2%
2 Emerging Market Equities 22.6% 36.9% 27.6%
3 Real Estate 18.8% 36.8% 27.2%
4 Global Credit 6.3% 8.9% 5.7%
5 Global Treasuries 6.8% 11.1% 7.3%
Full Sample 1 2 3 4 5
1 Developed Market Equities 1.00
2 Emerging Market Equities 0.86 1.00
3 Real Estate 0.81 0.77 1.00
4 Global Credit 0.56 0.59 0.68 1.00
5 Global Treasuries 0.16 0.21 0.34 0.75 1.00
Long-Horizon 1 2 3 4 5
1 Developed Market Equities 1.00
2 Emerging Market Equities 0.55 1.00
3 Real Estate 0.80 0.72 1.00
4 Global Credit 0.08 0.47 0.45 1.00
5 Global Treasuries -0.40 0.33 -0.04 0.70 1.00
Turbulent 1 2 3 4 5
1 Developed Market Equities 1.00
2 Emerging Market Equities 0.77 1.00
3 Real Estate 0.86 0.72 1.00
4 Global Credit 0.30 0.37 0.29 1.00
5 Global Treasuries 0.03 0.01 0.04 0.78 1.00
Source: State Street Global Exchange, Datastream, FactSet
Considerations for Alternative Assets
Alternative asset classes such as private equity, real estate, and hedge funds may offer a range of benefits, but
they also present analytical challenges. These types of investments have characteristics that differentiate them
from traditional asset classes in three important ways: appraisal-based pricing, performance-based fees, and
illiquidity. Each of these features can lead to substantial analytical challenges, and the potential for misguided
conclusions. Quantitative methods need to be refreshed and adapted to apply to alternative asset classes.
The common practice of using lagged appraisal-based pricing for non-traded portfolio companies (for private
equity funds) or specific properties (for real estate funds) can lead to a dramatic underestimation of risk. De-
smoothing of the illiquid asset returns is required to offset the reduction in the observed standard deviation
introduced by appraisals and fair value pricing. This is accomplished by estimating a first order autoregressive
model using least squares.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 30
The model is specified as:
rt= A0+A1+rt-1+ε (8)
To de‐smooth the time series, returns are computed as:
rt'=
rt-A1×rt-1
1-A1
(9)
Where:
rt' = de‐smoothed return observation at time t
rt = return observation at time t
A0 = intercept
A1 = regression coefficient
ε = error term
The asymmetric nature of performance-based fees is another consideration for alternative assets. These fees
can – counterintuitively – make returns seem less risky, if they are not accounted for properly. Consider a fund
charging 2% and a 20% performance fee. Estimating the standard deviation of the fund using net returns would
underestimate risk due to the 20% performance fee for returns above the hurdle rate. It is necessary to correct
for the downward bias in the observed standard deviation of the illiquid asset arising from the effect of
performance fees.
For a single fund that accounts for performance fees on an annual basis, the returns of the illiquid asset net of
fees can be converted to returns gross of fees, as shown below.
rn= rg-b - (max 0,p×(rg-b)) (10)
rg= {
rn+ b for rn<0rn
1-p+ b for rn≥0
(11)
Where
rn = return net of fees
rg = return gross of fees
b = base fee
p = performance fee
Alternative asset classes such as real
estate and private equity present a range of
analytical challenges to SWFs. Several
features of these investments – including
the fees they charge, the way they are
priced, and their illiquidity – can lead
investors to significantly underestimate their
risk. It is essential for SWFs to adjust for
these considerations when forming
portfolios and evaluating risk.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 31
In practice, a simulation can be performed to estimate the volatility dampening effect of performance fees when
fee accrual accounting is used. On average, the true standard deviation is estimated to be approximately 1.09
times larger than the standard deviation estimated from monthly net returns with fee accrual.16
It has already been shown that performance fees cause the observed standard deviation of a fund to understate
its risk. Performance fees also reduce the expected return of a group of funds which charge performance fees
beyond the average reduction of the individual funds’ expected returns. Consider, for example, a fund that
charges a base fee of 2% and a performance fee of 20%. A fund that delivers a 10% return in excess of the
benchmark on a $100 million portfolio will collect a $2 million base fee (2% x 100,000,000) and a $1.6 million
performance fee (20% x (10,000,000 – 2,000,000)), for a total fee of $3.6 million. The investor’s return net of
fees, therefore, is 6.4%.
Now suppose an investor hires two funds that each charges a base fee of 2% and a performance fee of 20%.
Assume as well that these funds both have expected returns of 10.0% in excess of the benchmark. The expected
fee for each fund is 3.6% (2% + 20% x (10% ‐ 2%)). Therefore, the investor might expect an aggregate return net
of fees from these two funds equal to 6.4%. This expectation would be justified, however, only if both funds’
returns exceed the base fee. If, instead, one fund produces an excess return of 30.0% and the other a ‐10.0%
excess return, and an equal amount of capital is allocated to each fund, the investor would pay an average fee of
4.8% rather than 3.6%, and the average return to the investor would equal 5.2% rather than 6.4%, even though
the funds still have an average excess return of 10%.17
These results are summarized in Exhibit 4.
Exhibit 4: Multi-Fund Return Impact
Excess Return Manager Fee
Return to Investor
Fund 1 10.0% 3.6% 6.4%
Fund 2 10.0% 3.6% 6.4%
Average 10.0% 3.6% 6.4%
Fund 1 30.0% 7.6% 22.4%
Fund 2 -10.0% 2.0% -12.0%
Average 10.0% 4.8% 5.2%
Impact 1.2%
Source: State Street Global Exchange
16 This is based on a simulation of 1,000 years of monthly fund returns from a normal distribution with an annualized mean of 8% and an
annualized standard deviation of 8%. It is assumed that an annual base fee of 2% and an additional annual hurdle rate of 3.5% before performance fees are charged. We record a 20% annual performance fee on an accrual basis.
17 This difference is equivalent to the difference in value between a portfolio of options and an option on a portfolio.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 32
This result is specific to the assumptions of this example. Nevertheless, it is easy to determine the typical
reduction in expected return by applying Monte Carlo simulation. Consider investment in 10 funds, each of which
has an expected excess return of 70%, a standard deviation of 15.0%, and a correlation of 0% with the other
funds. Also assume that LIBOR equals 4.0%. With these assumptions, the reduction to the collective expected
return of the funds equals about 0.7%. If the funds’ correlations were higher, the reduction would be smaller and
vice versa. This reduction in the collective return is a hidden fee arising from the fact that investors pay for
outperformance but are not reimbursed for underperformance.
In principal, this effect could be somewhat muted because most performance fee arrangements include claw
back provisions which require funds to offset prior losses before collecting performance fees. In most cases,
though, underperforming funds are either terminated, or the performance fees are reset without reimbursement
for prior losses.
One of the most important characteristics of alternative investments that must be assessed is liquidity, or rather,
the fact that they are generally less liquid than traditional assets. There are two aspects to consider. First, how
much of an illiquidity premium should be demanded from an alternative asset? Second, how should investors
account for illiquidity in the portfolio selection process?
While there is little consensus amongst academics regarding how much of a liquidity premium markets actually
provide, it is possible to determine the premium an investor should demand based on an investor’s specific uses
of liquidity. Locking up capital imposes a cost. This is because the portion of the portfolio allocated to illiquid
investments is no longer available for passive rebalancing, market timing, raising cash to meet periodic
demands, exiting unproductive investments, or taking advantage of new opportunities that might arise in the
future. The size of this cost is different for every investor, but it can be quantified. To the extent an investor
foregoes some portion of the benefits provided by liquidity, an allocation to illiquid assets should be justified by
an offsetting expected return that is equal to or greater than the penalty imposed by illiquidity.
The framework for incorporating liquidity in the portfolio construction context treats liquidity as a “shadow
allocation” that either bestows benefits or imposes penalties. If an investor deploys liquidity to raise a portfolio’s
expected utility beyond its original expected utility, a shadow asset is attached to tradable assets. If instead an
investor deploys liquidity to prevent a portfolio’s expected utility from falling, a shadow liability is attached non-
tradable assets. Because the impact of illiquidity is path dependent, the value of these assets and liabilities must
be estimated using simulation methods. This allows for a great deal of flexibility in extending this type of analysis
to address real-world concerns. Changing risk regimes, different investment horizons, varying degrees of
liquidity, trading costs, capital calls, and future cash outflows, amongst other things, can all be accounted for
within this framework.
The first step for an SWF is to identify its specific uses of liquidity. The ability to trade can be used to play
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 33
“defense,” for example responding to capital calls and rebalancing the portfolio. It can also be used to play
“offense,” such as engaging in market timing or seizing new investment opportunities. Using realistic
assumptions, thousands of hypothetical future portfolio return paths can be generated that account for the fund’s
specific uses of liquidity. Two set of portfolio returns are simulated. The first, assumes all assets are liquid and
tradable. The second uses the liquidity characteristics of the assets being considered. This highlights an
important benefit of this approach; it allows investors to address absolute and partial illiquidity within a single
framework. The differences between the two simulations are measured using certainty equivalent returns to
account for the differences in return as well as the differences in risk.18
For an investor with log wealth (mean-
variance) utility, the certainty equivalent return is computed as:
rCE=μTw-λwTΣw (12)
Where:
rCE = certainty equivalent return
μ = expected returns
Σ = covariance matrix
λ = coefficient of risk aversion
w = portfolio weights
The analytical construct of this framework can be demonstrated using a two-asset example. First, a mean-
variance analysis is used to solve for optimal allocations to liquid equity and liquid bonds without considering the
effect of liquidity.19
Optimal weights are identified by maximizing expected utility:
E(U)= rewe-rbwb-λ(σe2we
2+σb2wb
2+2ρb,e
σeσbwewb) (13)
Where:
E(U) = expected utility
re = expected equity return
rb = expected bond return
σe = equity standard deviation
σb = bond standard deviation
we = equity weight
wb = bond weight
18 The certainty equivalent return for a given portfolio is the amount of risk free return required such that a given investor is indifferent
between receiving that return and holding a particular risky portfolio. 19
While mean-variance optimization is used in this illustration, it can be applied to any portfolio formation process, including full-scale optimization, multi-period models, and even heuristic approaches.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 34
λ = coefficient of risk aversion
ρb,e = correlation of equity and bonds
The weights that equate the marginal utilities of equities and bonds, as shown below, are those that are optimal.
∂U
∂we
= re-λ(2σe2we+2ρ
b,eσeσbwb) (14)
∂U
∂wb
= rb-λ(2σb2wb+2ρ
b,eσeσbwb) (15)
Next illiquid equity is substituted for liquid equity. However, two adjustments are required. First, the downward
bias in the illiquid asset’s observed standard deviation that results from the effect of performance fees is
corrected. Second, illiquid equity returns are de-smoothed to offset the reduced observed standard deviation
introduced by appraisals and fair-value pricing. A shadow asset is then attached to the bond portion of the
portfolio and the expected return, standard deviation, and correlation are re-stated to account for the presence of
the shadow asset, as shown in equations (16), (17), and (18).20
rbl= rb+rl (16)
σbl2 = σb
2+σl2 (17)
ρbl,e
= ρ
b,e×σb
σbl
(18)
Where:
rbl = expected return of bonds with shadow liquidity asset
rl = expected return of shadow liquidity asset
σbl = standard deviation of bonds with shadow liquidity asset
σb = standard deviation of shadow liquidity asset
ρbl,e = correlation of bonds (with shadow liquidity asset) and equity
Exhibit 5 presents a simple numerical illustration of the analytical framework. It shows how the required return for
equities changed as liquid equities are switched to illiquid equities and then, step by step, adjusted for the effects
of performance fees, smoothing, and the inclusion of the shadow asset.
20 It is assumed that the shadow asset is uncorrelated with both stocks and bonds; an assumption that can be relaxed.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 35
Exhibit 5: Required Return for Liquid and Illiquid Equities
This same approach can be used to balance other combinations of risks. For example, because investors are
often measured against some benchmark or peer group, it may be useful to balance absolute and relative risks.
Here an investor could incorporate their aversion to absolute risk and their aversion to relative risk (tracking
error) in the optimization process. This simultaneously addresses concerns about absolute performance and
relative performance. However, instead of producing an efficient frontier in two dimensions the optimization
process produces an efficient surface in three dimensions: expected return, standard deviation, and tracking
error.
The efficient surface is bounded on the upper left by the traditional mean-variance efficient frontier, which
comprises of efficient portfolios in dimensions of expected return and standard deviation. The left most portfolio
on the mean-variance efficient frontier is the minimum risk asset. The right boundary of the efficient surface is the
mean-tracking error efficient frontier. It comprises portfolios that offer the highest expected return for varying
levels of tracking error. The left most portfolio on the mean-tracking error efficient frontier is the benchmark
portfolio because it has no tracking error. The efficient surface is bounded on the bottom by combinations of the
minimum risk asset and the benchmark portfolio. All of the portfolios that lie on this surface are efficient in three
dimensions. However, it does not necessarily follow that a three-dimensional efficient portfolio is always efficient
in any two dimensions. Consider, for example, the minimum risk asset. Although it is on both the mean-variance
efficient frontier and on the efficient surface, if it were plotted in dimensions of just expected return and tracking
error, it would appear very inefficient if the benchmark included high expected return assets such as stocks and
long-term bonds. This asset has a low expected return compared with the benchmark and yet a high degree of
tracking error.
Risk Regimes and Conditioned Covariance
To identify optimal portfolios based on views and attitudes toward two risk regimes, the standard mean–variance
objective function is first augmented to include a normal covariance matrix ΣN and an event covariance matrix ΣE
that reflects returns that occur during specific events such as periods of high turbulence, which are then assigned
probabilities. The vector of returns has a mean µ and a covariance matrix Σ. We replace the full-sample
covariance matrix Σ with
pΣN+(p-1)ΣE (21)
where p is the probability of falling within the normal sample and 1 – p is the probability of falling within the event
sample.
Substituting these two covariance matrixes into the standard equation for the expected utility, EU, of a portfolio
with a weight vector w yields:
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 40
EU=w'μ-λ[pw'ΣNw+(1-p)w'ΣEw] (22)
where λ equals aversion to full-sample risk.
Equation (22) allows an investor to express views about the respective probabilities of the two risk regimes, but it
assumes that they are equally averse to both regimes. To differentiate aversions to the two regimes, values are
assigned to reflect the relative aversion to each of the regimes. Those values are then rescaled so that they sum
to 2. For example, suppose aversion to normal risk equals 2 and aversion to event risk equals 3. Aversions are
then rescaled to equal 0.80 for normal risk and 1.20 for event risk as follows:
λN*=
2λN
λN+λE
(23)
λE*=
2λE
λN+λE
(24)
The probability-weighted normal and event covariance matrixes are then multiplied by their respective rescaled
risk aversions:
EU=w'μ-λ[λN*pw'ΣNw+λE
* (1-p)w'ΣEw] (25)
Although Equation (25) has the virtue of transparency, it is somewhat cumbersome. It can be simplified by
defining a grand or conditioned covariance matrix to equal:
Σ*=λN
*pΣN+λE
* (1-p)ΣE (26)
This definition allows the recasting of the objective function to
EU=w'μ-λ(w'Σ*w) (27)
which is the original Markowitz objective function. The conditioned covariance, Σ*, can then be used as part of the
standard portfolio optimization framework. A probability-weighted covariance matrix as given by equation (21)
can also be used for portfolio construction or in calculating forward looking risk estimates that are aligned with a
SWF’s probability beliefs about the likelihood of normal and event periods in the future.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 41
Risk Regimes and Tactical Shifts
While the identification of risk regimes was presented as a backward looking exercise in partitioning a historical
sample to develop regime specific estimates of asset characteristics, the process can also be used to inform
tactical portfolio allocation decisions to respond to changing risk conditions. This can provide for improved
performance as well as for more stable portfolio risk characteristics through time.
Financial turbulence, presented in Section 3, has been shown to be an effective risk measure for informing
tactical shifts. The differential performance of risky strategies during turbulent and nonturbulent periods, together
with the persistence of turbulence, can make it particularly useful in conditioning exposure to risk.
Another measure of risk that has been shown to be useful in measuring and predicting systemic risk in the
financial markets and informing tactical shifts is the absorption ratio. The absorption ratio measures systemic risk
and is calculated by measuring the proportion of variation in asset returns that is explained or “absorbed” by a
fixed number of factors. Rather than attempt to select specific relevant factors, a well-known statistical procedure
called Principal Components Analysis is used to identify the factors (or eigenvectors) that are most important in
terms of their contribution to overall variation in asset returns. Equation (28) shows the formula for calculating
systemic risk.
AR=∑ σEi
2n
i=1
∑ σAj2
N
j=1
(28)
Where:
AR = absorption ratio
N = number of assets
n = number of eigenvectors in numerator of absorption ration
σEi2
= variance of the ith eigenvector
σAj2 = variance of the j
th asset
The absorption ratio measure captures the extent to which a set of assets is unified or tightly coupled. A high
absorption ratio indicates that assets are tightly coupled and are collectively fragile in the sense that negative
shocks can travel more quickly and broadly than when assets are loosely linked. In contrast, a low absorption
ratio implies that risk is distributed broadly across disparate sources; hence, the assets are less likely to exhibit a
unified response to bad news. In short, the absorption ratio can be used to distinguish fragile market conditions
from resilient market conditions. While the level of absorption does provide a measure of market fragility, the
level is not particularly useful as a tactical signal as market fragility may remain elevated for extended periods of
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 42
time. What is useful is to identify significant changes in the level of absorption. To do this, an investor can
calculate the standardized shift in the level of an absorption index which is equal to today’s value minus the
previous year’s average, divided by the standard deviation over the previous year. High risk or fragile periods can
then be identified as those when the absorption ratio exhibits a 1-sigma increase and more resilient periods can
be identified as those when the absorption ratio sees 1-sigma decrease. It should be noted that a high absorption
ratio does not necessarily lead to asset depreciation or financial turbulence. It is simply an indication of fragility.
Financial turbulence and systemic risk measures can generally be applied to any set of asset returns. They can
be used individually to provide specific information about risk or in concert to provide a more holistic view of the
risk environment. Furthermore, the risk measures can be used with portfolio asset returns in order to provide a
measure of intrinsic or portfolio specific risk or they can be calculated using a broad set market index returns to
measure extrinsic or broad market risk.
The identification of a risk regime can be accomplished through the calculation of a selected risk measure (or
combination of risk measures) and determining if a specified threshold has been breached, or through the use of
a Markov switching model to determine if the current period falls within a particular risk regime. Portfolio
allocations can then be adjusted in a variety of ways given the determination of being in one risk regime or
another. For example, exposure to risk can be scaled according to pre-determined levels of turbulence such that
an investor accepts increasing amounts of risk when exposure to risk is most likely to provide rewards and
decreases exposure to risk when exposure to risk has a high probability of resulting in a loss. Alternatively, an
investor can identify tilts away from assets that tend to underperform towards assets that tend to outperform
given a particular risk regime.
Investor Utility Preferences
Institutional investors typically employ mean-variance optimization to create portfolios, in part because it only
requires knowledge of the expected returns, standard deviations and correlations of the portfolio’s components.
The approach is aligned with the use of log wealth utility which is well documented and can be approximated by
a quadratic that is a function of mean (expected return) and variance. The significance of approximation is that if
log-wealth utility can be well approximated using mean and variance for a sufficiently wide range of returns, then
maximizing a function of mean and variance will approximately maximize expected utility. In the case of log
wealth utility, maximizing utility is the equivalent of maximizing the geometric mean (long run growth rate) of the
portfolio since the log of 1+geometric mean is the average of the logs of (1+return).
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 43
The table in Exhibit 8 shows log-wealth utility at various
return levels alongside utility calculated using a quadratic
approximation based on mean and variance. Returns within
the range of approximately -30% and +40% are virtually
identical to one another.23
Although differences appear at -
40% and +50%, the approximation still provides fairly
reasonable results. Outside of this range the approximation
deteriorates at an increasing rate. The chart in Exhibit 8
provides a graphical comparison of the log-wealth utility
function and the quadratic approximation.
In 1956, Markowitz developed the Critical Line Algorithm
(CLA) that provides an efficient approach to tracing out a
mean-variance efficient frontier. Significant advances in
computing power since then now allow for direct utility
maximization, also known as full-scale optimization, to be
used as an alternative. With this approach a sophisticated search algorithm is used to identify a single optimal
portfolio or to trace out an efficient frontier of portfolios based on any description of investor utility preferences.24
The process uses a sample of asset returns and calculates a portfolio’s utility for every period in the sample and
the sums the utilities, iteratively searching to identify the combination of asset weights that yields the highest
expected utility over the entire sample.25
When used with mean and variance, full-scale optimization provides
(virtually) identical results to the CLA approach. Therefore, full-scale optimization is most useful when
considering alternative utility preferences or risk measures.
23 Markowitz, H. “Portfolio Selection,” Malden, MA: Blackwell Publishing, 1959.
24 Adler, T. and Kritzman, M. , Mean-Variance versus Full-Scale Optimisation: In and Out of Sample, Journal of Asset Management, Vol. 7, 5,
302–311, (2007).
25 Full scale optimization generally incorporates a historical sample of returns rescaled to reflect forward-looking risk and return estimates.
The relationships between assets are retained in the return series used rather than being provided explicitly. Because actual returns are used, it incorporates all of the features of the asset returns in the empirical sample that are not captured by descriptive summary statistics (mean, variance, and correlations), including skewness and kurtosis. Therefore, the historical period used should be sufficiently long to be representative of asset characteristics and relationships.
For example, the original sample may comprise monthly returns, but the investment horizon may be five years. Therefore, the covariance matrix using five-year, overlapping returns would need to be estimated. Log returns are used to calculate covariance matrices in order to remove the effect of compounding. In particular, each return observation is transformed by taking the natural logarithm of one plus the return. The multi-period compounded returns of a normally distributed asset will be highly skewed due to compounding and therefore not normally distributed; however the logarithms of the long-period returns will be normally distributed.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 47
sample.30
5. Proceed in this fashion until errors in covariances from all overlapping sub-samples are
computed.31
6. For all sub-samples, add the errors to the covariances of a base-case sample, which, for
example, could be the median sub-sample.32,33
Then, assuming normality, generate simulated
return samples from each error-adjusted covariance matrix.
7. Combine these return samples into a new large sample, which can be thought of as a stability-
adjusted return sample.
There are several features of this process that should be noted. First, the composite errors incorporate all three
sources of error. They reflect small-sample error because the sub-samples are smaller than the original sample.
They reflect independent-sample error, because each sub-sample is distinct from the remaining observations in
the large sample. And they capture interval error, because the sub-sample covariances are estimated from
longer-interval returns than the returns used to estimate the large-sample covariances.
It should also be noted that the resultant return distribution will not be normal despite the distributions of the sub
samples as well as the Central Limit Theorem. The stability-adjusted return distribution should be expected to
have fatter tails than a normal distribution. The Central Limit Theorem states that the sum of independent random
variables, which themselves need not be individually normally distributed, will approach normality as the quantity
of random variables increases.34
But we are not summing random variables. We are combining distributions.
For example, suppose the daily returns of a particular asset for a given month are approximately normally
distributed around a mean of 0.5%. And suppose their returns in the following month are again approximately
normal, but this time around a mean of -0.5%. If these daily returns are summed for the first day of the two
months, the second day of the two months, and so on, the 20 summed observations will also be normally
distributed, but around a mean of 0.0%. However, the 40 daily returns for the two-month period will not be
normally distributed. They will have a bimodal distribution with some observations clustering around a peak of
0.5% and others clustering around a peak of -0.5%.
Constructing efficient stability-adjusted portfolios is a matter of identifying the desired utility function (e.g.: log-
30 Overlapping samples are used to mitigate the distortion that could be caused by choosing a particular start date with independent samples.
For example, it could be that a particular period has very high risk and the subsequent period has very low risk. If we were to choose a start date such that we combined half of the first period with half of the subsequent period, we would not capture these extreme episodes of risk. 31
Any strong directional bias from the distribution of errors is removed by subtracting the median error from each individual sub-sample error. 32
The full-sample covariance matrix should not be used as the base case because the full sample embeds the small-sample error of all the sub samples. 33
Some of the sub-sample covariance matrices may not be positive semi-definite. Therefore standard corrections are applied to render all covariance matrices invertible. 34
In addition to independence, the Central Limit Theorem also assumes finite variances.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 48
wealth or kinked) and then using full-scale optimization with stability-adjusted return samples.35
While mean
variance optimization can provide reasonable results, the use of full-scale optimization accounts for every feature
of the data, even beyond kurtosis and skewness and is thus suitable for use with stability-adjusted distributions
and for utility functions that cannot be described by mean and variance.
The stability-adjusted optimization approach yields portfolios with different asset weights than when ignoring
errors or using Bayesian shrinkage for a fixed level of expected return. Stability-adjusted portfolios also exhibit
more stable risk over time. Other applications of the approach where accounting for errors has proven beneficial
are index tracking, proxy hedging for expensive currencies, and tracking liabilities or inflation.
35 Although it could be prohibitively challenging to test every possible asset mix in small increments, there are search algorithms that yield a
reasonably reliable solution in a few seconds. A particular algorithm based on evolutionary biology initiates several searches simultaneously and iteratively terminates those searches that are sure to fail, thus transferring the search energy to the remaining feasible searches.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 49
5. Evaluating Portfolio Risk
Investors typically measure risk as the probability of a given loss or the amount that can be lost with a given
probability at the end of their investment horizon. This view of risk only considers the result at the end of the
investment horizon and ignores what might happen along the way. This section focuses on techniques for
measuring and communicating the risks associated with portfolios and considers various aspects of those risks
that can help inform strategic portfolio choice as well as active decisions.
Risk and the Investment Horizon
Exhibit 10 illustrates the distinction between risk based on ending outcomes and risk based on outcomes that
might occur along the way. Each line represents the path of a hypothetical investment through four periods. The
horizontal line represents a loss threshold, which in this example equals 10%. Exhibit 10 reveals that only one of
the five paths breaches the loss threshold at the end of the
horizon; hence we might conclude that the likelihood of a
10% loss equals 20%. However, four of the five paths at
some point during the investment horizon breach the loss
threshold, although three of the four paths subsequently
recover. If we also care about the investment’s performance
along the way to the end of the horizon, we would instead
conclude that the likelihood of a 10% loss equals 80%.
One might argue that calculation of daily value at risk
measures a strategy’s exposure to loss within an
investment horizon, but this is not true. Knowledge of the
value at risk on a daily basis does not reveal the extent to
which losses may accumulate over time. Moreover, even if
daily value at risk is adjusted to account for prior gains and losses, the investor still has no way to know at the
inception of the strategy, or at any other point, the cumulative value at risk to any future point throughout the
horizon, including interim losses that later recover.
We estimate probability of loss at the end of the horizon by: 1.) calculating the difference between the cumulative
percentage loss and the cumulative expected return, 2.) dividing this difference by the cumulative standard
deviation, and 3.) applying the normal distribution function to convert this standardized distance from the mean to
a probability estimate, as shown in Equation (29).
The approach to estimating risk exposure
that is espoused by most financial textbooks
contains a crucial flaw. This approach
typically measures the probability (or size) of
a loss that might occur at the END of an
investment horizon, whether that horizon is a
day, a month, or many years. In practice,
most investors are equally concerned with
how interim losses might accumulate along
the way. Indeed, to reach the long-term, an
investor much first survive the short term.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 50
Exhibit 10: Risk of Loss: Ending Wealth versus Interim Wealth (Illustrative)
Source: State Street Global Exchange
PE=N [ln(1+L)-μT
σ√T] (29)
Where,
PE = probability of loss at the end of the investment horizon
N[ ] = cumulative normal distribution function
ln = natural logarithm
L = cumulative percentage loss in periodic units
= annualized expected return in continuous units
T = number of years in horizon
= annualized standard deviation of continuous returns
The process of compounding causes periodic returns to be lognormally distributed. The continuous counterparts
of these periodic returns are distributed normally, which is why the inputs to the normal distribution function are in
continuous units.
$80
$90
$100
$110
$120
1 2 3 4 5
Wealt
h
Time
Path 1 Path 2 Path 3 Path 4 Path 5
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 51
To estimate value at risk, this calculation is turned around by specifying the probability and solving for the loss
amount, as shown in equation (30):
VaR = -(eμT-Zσ√T-1)W (30)
Where,
e = base of natural logarithm (2.718282)
Z = normal deviate associated with chosen probability (e.g. Z=1.645 for a 5% probability)
W = initial wealth
Both of these calculations pertain only to the distribution of values at the end of the horizon and therefore ignore
variability in value that occurs throughout the horizon. To capture this variability, a statistic called first passage
time probability is used.36
This statistic measures the probability (PW) of a first occurrence of an event within a
finite horizon. The following equation gives the probability of loss within the investment horizon, PW, which is the
probability that an investment will depreciate to a particular value over some horizon if it is monitored
continuously.
PW = N [ln(1+L)-μT
σ√T] +N [
ln(1+L)+μT
σ√T] (1+L)
2μ
σ2 (31)
Note that the first part of this equation is identical to the equation (29) for the end of period probability of loss. It is
augmented by another probability multiplied by a constant, and there are no circumstances in which this constant
equals zero or is negative. Therefore, the probability of loss throughout an investment horizon must always
exceed the probability of loss at the end of the horizon. Moreover, within horizon probability of loss rises as the
investment horizon expands in contrast to end of horizon probability of loss, which diminishes with time. This
effect supports the notion that time does not diversify all measures of risk and that the appropriate equity
allocation is not necessarily horizon dependent.
We can use the same equation to estimate continuous value at risk. Whereas value at risk measured
conventionally gives the worst outcome at a chosen probability at the end of an investment horizon, continuous
value at risk gives the worst outcome at a chosen probability from inception to any time during an investment
horizon. It is not possible to solve for continuous value at risk analytically. Numerical methods must be used.
Estimating continuous value at risk is accomplished by setting equation (31) equal to the chosen confidence level
and solving iteratively for L. Continuous value at risk equals L times initial wealth.
36 The first passage probability is described in Karlin, S. and Taylor, H., A First Course in Stochastic Processes, 2nd edition, Academic Press,
1975.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 52
Risk Regimes and Stress Testing
Most risk measures weight a sample’s observations equally in order to estimate risk parameters. Although this
procedure may produce reasonable estimates for the full investment horizon, it likely misrepresents a portfolio’s
risk attributes during periods of turbulence or financial crisis when asset and manager returns tend to become
more volatile and more highly correlated. Thus, the diversification that characterizes the sample, on average,
disappears when it is most needed.
The conventional approach for measuring VaR uses the full-sample covariance matrix to compute the portfolio’s
standard deviation and considers the probability distribution only at the end of the investment horizon. Exposure
to loss can be measured more reliably by estimating covariances from the turbulent subperiods, when losses are
more likely to occur, and by accounting for interim losses as well as losses that occur only at the conclusion of
the investment horizon.
Exhibit 11 shows three portfolios—conservative to aggressive—together with assumptions for their expected
returns and two estimates of standard deviation. One estimate of standard deviation, “Full-sample risk,” is based
on the full-sample covariance matrix of monthly returns beginning in January 1977 and ending in December
2006. The other estimate of standard deviation, labeled “Turbulent risk,” is based on the covariance matrix from
the turbulent subsample. Turbulence was calculated according to Equation 8, in which each return vector
consisted of returns of the five asset-level indices for a particular month, and average vector μ and covariance
matrix Σ were calculated from monthly returns during the entire 30-year history. The threshold for identifying
turbulent periods was set at 75 percent, which means that roughly 25 percent of the months fell within turbulent
subperiods.37
Note how risk increases for each portfolio when turbulence risk is used.
If the 2007–08 financial crisis is considered as a once-in-a-century event, Exhibit 12 shows that the conventional
approach to measuring exposure to loss badly underestimated the riskiness of these portfolios. The turbulence-
based approach, in contrast, anticipated the exposure to loss of these portfolios much more accurately. To be
clear, it should be noted that the turbulence-based approach does not offer a more reliable estimate of when an
extreme event will occur; rather, it gives a more reliable estimate of the consequences of such an event. Also
note that turbulence is a relative measure. If the world becomes more turbulent, for example, the threshold for
separating turbulent periods from nonturbulent periods will rise.
37 The results are not particularly sensitive to the 75th percentile threshold.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 53
Exhibit 11: Efficient Portfolios, Expected Returns, and Two Risk Estimates
Asset Class Conservative
Portfolio Moderate Portfolio
Aggressive Portfolio
U.S. Stocks 22.86% 35.23% 48.15%
Non-U.S. Stocks 16.59% 24.22% 32.19%
U.S. Bonds 49.95% 32.81% 14.89%
Real Estate 3.85% 2.59% 1.28%
Commodities 6.75% 5.16% 3.49%
Expected Return 7.60% 8.37% 9.17%
Full-sample Risk 7.77% 10.12% 12.86%
Turbulent Risk 10.68% 13.68% 17.33%
Note: “Full-sample” risk was estimated from the full-sample covariance matrix; “Turbulent-sample” risk was estimated from the covariance matrix of the turbulent sample.
Source: State Street Global Exchange
Exhibit 12: VaR and Realized Returns
Portfolio
VaR for Full Sample,
End of Horizon
VaR for Full Sample,
End of Horizon
Maximum Loss from Inception (Jan/07-Sep/09)
Maximum Drawdown
(Jan/07-Sep/09)
Conservative 2.10% 26.20% 19.60% 25.80%
Moderate 9.90% 35.10% 29.42% 35.50%
Aggressive 10.70% 45.00% 38.96% 45.30%
Note: The horizon is five years.
Source: State Street Global Exchange, Datastream
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 54
6. Reference Portfolios
While the methods detailed in this paper are presented in the context of a traditional portfolio construction
framework it is important to acknowledge the reference portfolio framework that has recently gained the attention
of SWFs as a method of increasing implementation flexibility and accommodating the inclusion of private market
investments, such as hedge funds and private equity, in the portfolio management process. To do this the
approach replaces the policy portfolio with a “reference” portfolio.
A reference portfolio is a notional diversified portfolio designed to achieve specific goals that is implemented with
only simple, passive, low cost, liquid investments. It provides a baseline for investment performance and is used
to determine the effectiveness of active portfolio management efforts. Although the reference portfolio and the
actual portfolio share the same goals, the composition of the two portfolios can differ substantially due to active
portfolio management decisions implemented to exploit opportunities to add value relative to the returns offered
by the traditional assets used in the reference portfolio. The use of a reference portfolio as a benchmark allows
for greater versatility in asset selection and portfolio composition relative to a bucketed policy benchmark
approach. In effect, investors increase flexibility in implementation at the expense of the implicit risk controls
imposed by a policy benchmark. Consequently, the management of active risk depends, to a great extent, on
investor judgement.
The methods and approaches discussed in this paper are not at all incongruent with the reference portfolio
approach and can only serve to better inform portfolio and risk management decisions. The distinction between a
portfolio of traditional assets and one of active and/or alternative assets does not preclude investors from seeking
to construct reference and active portfolios that are efficient in terms of expectations for the capital markets or
from seeking a more holistic understanding of the risks borne by exposure to financial markets and those that
result from active decisions. In fact, establishing pre-determined limits on active risk can be a useful method of
retaining important risk controls for those applying this approach.
While generally not part of the reference portfolio framework, the determination and use of a reasonable tracking
error budget relative to the reference portfolio imposes discipline in selecting and implementing active decisions.
Determining an active risk budget is not a trivial exercise as it demands a thoughtful and deliberate determination
of a suitable reference portfolio as well as a clear understanding of how a SWF expects to generate excess
returns. Furthermore, the active risk budget may consider both short- and long-term investment horizons. If not
chosen carefully, the reference portfolio and/or the active risk budget could impose unintended constraints on a
portfolio manager’s ability to add value. The identification of the reference portfolio and active risk budget can be
accomplished simultaneously using the following framework:
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 55
1. Specify the SWF’s specific investment goal
A key characteristic of a reference portfolio is that it should be appropriate for achieving specified
investment goals. Consequently, the identification of a reference portfolio is, first and foremost, a function
of the SWF’s goals. This may include quantifying the level of risk a SWF is willing to bear in seeking to
maximize return or provide for broader goals such as maintaining the fund’s reputation relative to peers.
2. Define the opportunity set of reference and active portfolio assets
The selection of investments used in the reference portfolio should be a deliberate process that includes
the development of criteria for the inclusion of assets in the reference portfolio. These criteria would then
be applied to the universe of assets available to a SWF in order to determine the opportunity set of
assets for use in the construction of the reference portfolio. The opportunity set of active portfolio assets,
which are assets that are expected to be used to add value in excess of reference portfolio assets,
should also be identified.
3. Develop estimates of return and covariance for reference and active portfolio assets
The results of a portfolio analysis are a function of the inputs used. This paper has detailed a number of
different approaches to forming beliefs about the future performances of investments. Special
consideration must be given to unlisted and/or illiquid assets as risks for these types of investments are
often under-estimated due to periodic valuation biases and the asymmetric nature of performance fees.
The estimates developed in this step will be used as part of the portfolio optimization process as well as
for determining ex-ante tracking error estimates of possible portfolio implementations.
4. Define portfolio constraints
Portfolio constraints are used to incorporate professional judgment in limiting risks that are not
adequately expressed as a function of volatility. Additional constraints can include limitations on
domestic or foreign assets as well as limitations on private/illiquid assets.
5. Conduct a portfolio analysis with reference assets and identify a suitable reference portfolio
A portfolio analysis is conducted to identify the set of efficient portfolios based on the estimates in step 3
and the constraints identified in step 4. The selection of a reference portfolio would begin with identifying
the subset of efficient portfolios that exhibit return and risk characteristics acceptable to the SWF.
Portfolios can be evaluated using various criteria including:
Multi-period returns: The outputs of a portfolio analysis are presented in terms of expected return and
standard deviation. The expected return for a portfolio is the weighted sum of the expected returns of
portfolio constituents. These expected returns are arithmetic means (expected values) and not geometric
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 56
means (compound returns). Approximations to geometric mean using portfolio arithmetic mean and
standard deviation can be used to estimate the long-run return provided by portfolios. Alternatively,
simulation could be used to assess portfolio outcomes. This provides an understanding of a portfolio’s
ability to achieve long-term goals.
Stress testing of portfolios assuming multiple risk regimes: Understanding portfolio characteristics in both
normal markets and turbulent markets is useful in determining an appropriate reference portfolio.
End-of-horizon and within-horizon exposure to loss: Investors typically measure risk as the probability of
a given loss at the end of their investment horizon. Exposure to loss within the investment horizon is
substantially greater than investors normally assume and is an important characteristic to consider in
selecting a portfolio.38
6. Evaluate the tracking error of possible active portfolio implementations and identify a suitable
active risk budget
Estimates of ex-ante tracking error are calculated using estimates developed in step 3 and possible
active portfolio weights. Possible active weights should consider the SWF’s methods and expertise in
pursuing excess returns. Considering some investor approaches, such as determining the tracking error
impact of opportunistically shifting active portfolio weights over time, may require the use of simulation
methods.
An important by-product of this exercise is that additional portfolio constraints may be identified that may help in
the management of active risk. A mean variance tracking error framework that considers both absolute risk and
relative risk can then be used to implement and update the active portfolio on an ongoing basis to keep
allocations within active risk budgets. Alternatively, the fund can be allocated using other methods with desired
allocations being informed by estimates of active risk.
38 Kritzman, M. and Rich, D. 2002. “The Mismeasurement of Risk.” Financial Analysts Journal, vol. 58, no. 3 (May/June).
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 57
7. Survey Results: The Experience of Sovereign Wealth Funds
As the financial markets have evolved, SWFs have had to balance the application of financial theory with the
complexities presented by real world circumstances. To gain insight as to the specific challenges faced by SWFs,
how their approaches, organizations, and fund allocations have adapted to changing markets, and how they are
positioning themselves for the future, the working group conducted a survey of IFSWF members. The survey
focused on current fund asset allocations, the evolution of those allocations, private market investments, and
how fund organizations have evolved to facilitate change. Ten funds participated in the survey. The funds were
broadly distributed across the world and represented a variety of fund types. Despite the wide range of different
investment objectives and disparate geographic locations of these funds, the results provide insight into how
SWFs have evolved. In the interest of protecting each fund’s anonymity, responses are not attributed to any
specific SWFs or their investment teams. The complete survey is provided in Appendix.
Current Fund Asset Allocations
This section of the survey focuses on identifying the asset classes currently held by SWFs and the distribution of
those asset classes across markets. Details regarding allocations across various asset segments were also
explored. While the unique nature of the SWFs surveyed limits drawing firm conclusions about specific
allocations, they can be indicative of general asset preferences. Exhibit 13 presents the current average
allocation across broad asset classes for diversified SWFs.
This indicates that the diversified SWFs in the survey group are primarily allocated to more traditional investment
categories and are more heavily weighted to fixed income than equity. Allocations to infrastructure/real estate
and hedge funds are comparatively much smaller. Additional detail regarding these allocations can be gained by
reviewing allocation the preferences across various asset segments presented in Exhibit 14.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 58
Exhibit 13: Current Average Allocation for Diversified Sovereign Wealth Funds
Source: IFSWF Survey Completed July 2016
This shows that SWF investments are focused primarily in foreign markets. This may be a function of
preferences or due to specific policies. Interestingly, only a small amount of assets currently devoted to emerging
market investments with the majority of assets being directed to the developed world. Actively managed
investments are favored over passive investments by only a small margin and listed investments outweigh non-
listed investments by a ratio of three-to-one.
The survey also delved into the distribution of allocations to specific asset classes across geographic regions.
Exhibit 15 shows the percentage of surveyed SWFs that currently have assets allocated to specific regions. Here
it can be seen that listed equity allocations received allocations from most funds across all of the global regions
(notable values are highlighted in orange). The region receiving the greatest percentage of fund allocations is,
unsurprisingly, North America followed by Europe and then Asia. This is confirmed by survey responses that list
the United States, the United Kingdom, and Japan as the top three countries for investments and also aligns
closely with the fact that these three countries represent the largest markets in the world as measured by market
capitalization.39
Only a small percentage of funds have allocations within the MENA region. It is notable that
European listed equities are the only assets common to all funds.
With regard to how SWFs access specific investments across different regions, the survey suggests that funds
are flexible in that they invest both directly and through funds with traditional assets (listed equities, government
and corporate bonds) being the top areas where SWFs had only direct investments.
39 “Bank of America/Merrill Lynch’s Transforming World Atlas: Investment Themes Illustrated by Maps” March 2016.
34%
53%
8% 5%
Equities Fixed income Infrastructure/Real Estate Hedge funds
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 59
Exhibit 14: Fund Allocation Across Various Asset Segments
Source: IFSWF Survey Completed July 2016
28.5%
71.5%
Domestic Markets Foreign Markets
73.2%
26.8%
Listed Non-Listed
46.0%
54.0%
Passive Active
92.5%
7.5%
Developed Markets Emerging Markets
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 60
Exhibit 15: Percentage of Funds Allocated to Specific Assets Across Regions
North America Asia Europe MENA Rest of World
Listed Equities 90% 90% 100% 70% 80%
Private Equity 40% 50% 50% 10% 40%
Venture Capital 30% 10% 20% 0% 10%
Government Bonds 80% 70% 70% 30% 50%
Corporate Debt 80% 60% 70% 30% 50%
Infrastructure 40% 30% 40% 20% 30%
Real Estate 30% 30% 50% 10% 20%
Hedge Funds 40% 30% 30% 20% 20%
Liquid Assets 60% 20% 20% 10% 0%
Other Assets 20% 0% 0% 0% 0%
Source: IFSWF Survey Completed July 2016
The Evolution of Fund Allocations
This section of the survey presents information on the evolution of SWF asset allocations beginning with how
SWFs have been altering their allocation over the recent past (3-5 years) and ending with how SWF anticipate
changing their portfolios in the future. As with the exploration of current asset allocations the survey first looks
into changes in broad asset categories and then probes further into various asset segments.
Exhibit 16 presents the percentage of funds with specific changes in allocations across broad asset classes in
both domestic and foreign markets over the recent past (notable values are highlighted in orange). Here we find
that, within domestic markets, there is no consensus in the percentage of funds making changes to any particular
asset class. The most significant area of agreement is in increasing Private Equity.
Across foreign markets there appears to have been more agreement as to how SWFs have actively changed
their allocations. Here we find that the areas where the highest percentages of funds have actively increased
their allocations were listed equity, private equity, and real estate asset classes. These increases appear to have
been funded with decreases in both government and corporate fixed income. This aligns with survey responses
where listed equities, private equities, and real assets were amongst the most commonly listed relevant asset
classes introduced to portfolios over the last three to five years.
Exhibit 17 presents the percentage of SWFs with specific allocation changes across various asset segments in
the recent past (3-5 Years) versus future expectations. Here we see that there has been some agreement as to
where changes have been made in the recent past with the highest percentage of funds agreeing in increasing
infrastructure/real estate along hedge funds and other assets. Emerging market assets saw the greatest
agreement in increases with half of the funds having increased allocations. Non-listed assets were also a focus
for funds. The greatest agreement in decreases came in the listed assets and developed market segments.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 61
Exhibit 16: Percentage of Funds with Specific Allocation Changes Across Broad Asset Categories in the Recent Past (3-5 Years)
Domestic Markets
Increase No Change Decrease N/A
Equities
Listed Equities 10% 20% 20% 50%
Private Equity 30% 20% 0% 50%
Venture Capital 20% 10% 0% 70%
Fixed Income Government Bonds 0% 10% 10% 80%
Corporate Debt 20% 10% 0% 70%
Infrastructure / Real Estate
Infrastructure 20% 20% 0% 60%
Real Estate 20% 0% 20% 60%
Hedge Funds and
Other Assets
Hedge Funds 10% 30% 0% 60%
Liquid Assets 0% 20% 0% 80%
Other Assets 20% 0% 0% 80%
Foreign Markets
Increase No Change Decrease N/A
Equities
Listed Equities 50% 10% 20% 20%
Private Equity 50% 0% 0% 50%
Venture Capital 20% 10% 0% 70%
Fixed Income Government Bonds 0% 20% 50% 30%
Corporate Debt 20% 20% 30% 30%
Infrastructure / Real Estate
Infrastructure 20% 20% 0% 60%
Real Estate 30% 10% 10% 50%
Hedge Funds and
Other Assets
Hedge Funds 10% 20% 10% 60%
Liquid Assets 10% 20% 10% 60%
Other Assets 10% 10% 0% 80%
Source: IFSWF Survey Completed July 2016
With regard to anticipated changes in the future, the survey suggests the majority of funds expect that they will
not adjust current allocations. A small percentage of funds will continue with existing trends that have seen
increases in allocations to infrastructure/real estate, non-listed assets, and emerging market assets. The limited
percentage of funds indicating future changes in the survey could be an indication of uncertainty regarding the
evolution of the capital markets in the future.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 62
Exhibit 17: Percentage of Funds with Specific Allocation Changes Across Various Asset Segments in the Recent Past (3-5 Years) versus Future Expectations
Recent Past
Increase No Change Decrease N/A
Asset Categories
Equities 20% 60% 20% 0%
Fixed Income 10% 60% 20% 10%
Infrastructure / Real Estate
30% 40% 10% 20%
Hedge Funds and Other Assets
30% 30% 10% 30%
Target Markets
Domestic Markets 10% 70% 0% 20%
Foreign Markets 0% 80% 10% 10%
Listed 10% 60% 30% 0%
Non-Listed 40% 40% 10% 10%
Actively Managed 20% 70% 10% 0%
Passively Managed 20% 50% 20% 10%
Geographies
Developed Markets 0% 60% 40% 0%
Emerging Markets 50% 30% 0% 20%
North America 20% 70% 0% 10%
Europe 10% 80% 10% 0%
Asia 20% 70% 0% 10%
MENA 10% 70% 0% 20%
Rest of the World 20% 50% 10% 20%
Future Expectations
Increase No Change Decrease N/A
Asset Categories
Equities 20% 70% 0% 10%
Fixed Income 0% 70% 10% 20%
Infrastructure / Real Estate
20% 50% 0% 30%
Hedge Funds and Other Assets
10% 40% 10% 40%
Target Markets
Domestic Markets 0% 70% 0% 30%
Foreign Markets 10% 70% 0% 20%
Listed 0% 80% 0% 10%
Non-Listed 20% 60% 10% 20%
Actively Managed 10% 80% 0% 10%
Passively Managed 0% 70% 10% 20%
Geographies
Developed Markets 0% 90% 0% 10%
Emerging Markets 20% 50% 0% 30%
North America 0% 70% 10% 20%
Europe 0% 90% 0% 10%
Asia 10% 70% 0% 20%
MENA 10% 60% 0% 30%
Rest of the World 10% 60% 0% 30%
Source: IFSWF Survey Completed July 2016
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 63
Private Markets
One of the greatest challenges faced by SWFs is in allocating to private market investments. These investments
have unique characteristics that must be understood as well as higher fees that must be considered as part of
the decision to invest in these assets. Generally, this requires intellectual resources with the professional aptitude
and skillset necessary to fully comprehend the legal and operational complexities of these types of investments.
This section of the survey focuses on gaining greater insight regarding what SWFs perceive to be the challenges
of investing in private markets and what they believe to be the keys to the successful navigation of private market
investments. For additional information on how SWFs are addressing the challenges of private market investing
please see the IFSWF’s whitepaper titled “Comparison of Member’s Experiences Investing in Public versus
Private Markets” which provides insights gathered from interviews with various SWFs, one of the world’s
foremost academic researchers of private markets, and an extensive review of academic literature.
Exhibit 18 ranks the biggest challenges to investing in private markets identified by SWFs along with what they
believe to be the keys to successfully investing those markets. Interestingly, fees and insufficient in-house
resources rank below concerns about lack of transparency and illiquidity. This is consistent with the ranking of
the keys to success where investment and operational due diligence along with institutional relationships and
manager alignment are ranked highest. These two items are the primary methods portfolio managers have for
decreasing the opacity often present in private markets and to increasing comprehension of risks, such as
illiquidity, that transcend those implied by measures of volatility.
Exhibit 18: Private Market Investments: Risks and Key for Success
Rank What are the greatest challenges to investing in private markets?
1 Lack of Transparency
2 Illiquidity
3 Lack of appropriate benchmark
4 Fees
5 Insufficient in-house resources
Rank What are the key for success in private market investing?
1 Investment and operational due diligence process
2 Institutional relationships & manager alignment
3 In-house resources & human resources policies
4 Governance structure & stakeholder communication
5 Speed of decision making
6 Sophistication of risk management systems
7 Size of assets under management
Source: IFSWF Survey Completed July 2016
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 64
Fund Organization
To this point SWF challenges have been primarily viewed as issues that emanate from sources outside of the
control of SWFs. This section of the survey turns the focus to SWFs as investment management organizations
and explores how they are adapting and organizing themselves to manage change and the expansion into new
asset classes.
The survey first explored the organizational solutions implemented to access the human and intellectual
resources needed to manage new asset classes. Respondents were asked to indicate whether they have
implemented the full in-house management of new asset classes, worked through partnership/cooperation to
manage assets, or outsourced the management of assets to an external manager. The results show that SWFs
are comfortable outsourcing asset management to external managers when new assets were traditional
investments such as listed equities and fixed income. When it came to private market investments, SWFs
demonstrated a preference towards fully managing in-house and, to a lesser extent, in partnering or cooperating
with an outside resource. This suggests that SWFs view the management of traditional assets as a commodity
and the management of private or alternative assets as an area that requires specialized resources and/or
competencies. Because the additional resources need for private assets are primarily intended to address issues
with transparency and risk management, having the organization play an active role in gathering and sharing
objective insights to inform asset management decisions is preferred. This coincides with interest from IFSWF
members in how to build out the human and intellectual resources within their organizations to manage private
assets.
The second area of inquiry for this section was to understand how SWFs have organized themselves to manage
the expansion into new asset classes. In this section respondents were asked to indicate whether they have
created a new business division to manage the new assets, created a new team within an existing business
division, added new resources to current teams, or made no changes to resources/organization. By a wide
margin, the approach taken by SWFs has been to add new resources to existing teams. Establishing new
business divisions was indicated twice for private assets and once for listed equities. Only a small number of
respondents indicated that no new resources were required.
The final area of focus for the survey was to understand SWF perceptions of the most relevant competencies
required in expanding to new asset classes. An overarching theme across the responses provided by SWFs,
outside of the obvious need for analytical aptitude and commercial acumen, was the importance of being able to
build and maintain relationships. Whether it is was with regard to building internal capacity through cooperation
with outside resources, through establishing relationships with consultants/advisors, or through establishing
communication channels with external managers, the key competency mentioned revolved around relationship
building. This has important implications for identifying external investment resources, for hiring investment
talent, and for establishing and maintaining a collaborative culture within investment teams.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 65
8. Appendix
IFSWF Member Survey
The working group prepared a list of survey questions intended to gain insight as to the specific challenges faced
by SWFs, how their approaches, organizations, and fund allocations have adapted to changing markets, and
how they are positioning themselves for the future.The survey questions are presented below for reference.
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 66
Section I: Current Asset Allocation Overview
Question 1: Assets Under Management: Please provide an indication of the amount of assets under
management: $______ Billions.
Question 2: Asset Categories: What is the relative allocation of your current investments between the following
asset categories?
Equities
(Both Listed and Private Equity) Fixed Income
Infrastructure / Real Estate
Hedge Funds and Other Assets
Portfolio %
Question 3: Target Market: What is the relative allocation of your current investments between the following
markets?
% of Portfolio
Domestic Markets
Foreign Markets
Listed Assets
Non-Listed Assets
Actively Managed
Passively Managed
Question 4: Geographic Distribution: What is the relative allocation of your current investments between the
following geographies?
% of Portfolio
Developed Markets
Developed Markets
North America
Europe40
Asia
MENA41
Rest of World
Please indicate the top 3 countries of your current investments
Top 3 Countries
1
2
3
40 Excluding Turkey and Russia as they are grouped with Asia.
41 MENA: Middle East North Africa region
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 67
Section II: Evolution of Asset Allocation
Question 5: Asset Evolution: How would you describe the recent past evolution (e.g. the last 3-5 years) of
previous categories? Which are the expectations for their evolution in the near future?
Please provide a qualitative indication using the following indicators:
: Increase
= : Stable
: Decrease
Recent Evolution Future Expectation
Asset Categories
Equities
Fixed Income
Infrastructure / Real Estate
Hedge Funds and Other Assets
Target Markets
Domestic Markets
Foreign Markets
Listed
Non-Listed
Actively Managed
Passively Managed
Geographies
Developed Markets
Emerging Markets
North America
Europe
Asia
MENA
Rest of the World
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 68
Section III: Details on Current Asset Allocation
Question 6: Asset Class Matrix: Which of the following asset classes are currently present within your
investments portfolio?
Please mark with an X the asset classes present in your portfolio:
Domestic Market Foreign Markets
Direct Through Funds Direct Through Funds
Equities
Listed Equities
Private Equity
Venture Capital
Fixed Income Government Bonds
Corporate Debt
Infrastructure / Real Estate
Infrastructure
Real Estate
Hedge Funds and
Other Assets
Hedge Funds
Liquid Assets (e.g. Mutual Funds, Money
Market)
Other Assets (Please Specify):
Question 7: Regional Asset Class Matrix: Which of the following asset classes are currently present within
your investments portfolio?
Please mark with an X the asset classes present in your portfolio:
North America Asia Europe MENA Rest of World
Equities
Listed Equities
Private Equity
Venture Capital
Fixed Income Government Bonds
Corporate Debt
Infrastructure / Real Estate
Infrastructure
Real Estate
Hedge Funds and
Other Assets
Hedge Funds
Liquid Assets (e.g. Mutual Funds, Money
Market)
Other Assets (Please Specify):
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 69
Section IV: Details on Evolution of Asset Allocation
Question 8: Portfolio Diversification: Which are the most relevant asset classes that have been introduced in
the recent past in your portfolio (e.g. the last 3-5 years)? What are the expectations for possible introductions in
the near future?
Please indicate the most relevant three asset classes:
Asset Classes Recently Introduced
Asset Classes Under Consideration
1 1
2 2
3 3
Question 9: Asset Matrix Evolution: How has the allocation of the various asset classes evolved in the recent
past in your portfolio (e.g. the last 3-5 years)?
Please provide a qualitative indication using the following indicators:
: Increase
= : Stable
: Decrease
Domestic Markets Foreign Markets
Equities
Listed Equities
Private Equity
Venture Capital
Fixed Income Government Bonds
Corporate Debt
Infrastructure / Real Estate
Infrastructure
Real Estate
Hedge Funds and
Other Assets
Hedge Funds
Liquid Assets (e.g. Mutual Funds, Money
Market)
Other Assets (Please Specify):
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 70
Section V: Private Markets
Question 10: Challenges: What are the biggest challenges you face investing in private markets?
Please rank the following from 1 to 6 (with 1 representing the biggest challenge):
Ranking
Lack of transparency
Lack of appropriate benchmark
Fees
Illiquidity
Insufficient resources in-house
Other (Please Specify)
Question 11: Success Factors: What are the key success factors for private markets?
Please rank the following from 1 to 7 (with 1 representing the greatest success factor):
Ranking
Investment and operational due diligence
In-house resources and human resource policies
Institutional relationships and manager alignment
Governance structure and stakeholder communication
Speed of decision making
Sophistication of risk management systems
Amount of assets under management
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 71
Section VI: Organization
Question 12: Management: How have you decided to manage the expansion towards new asset classes?
Please indicate the new asset class and mark with an X the implemented organizational solution:
New Asset Class Name
Indicate new asset class here →
Fully managed in-house
Through Partnership /Co-operation
External Manager
Question 13: Organizational Set-up: How have you organized to be able to manage the expansion towards
new asset classes?
Please indicate the new asset class and mark with an X the implemented organizational solution:
New Asset Class Name
Indicate new asset class here →
New business division
New team in current business division
New resources in current team
No new resources / organization
Question 14: Competencies: What additional competencies have you implemented to be able to manage the
expansion towards new asset classes?
Please indicate the new asset class and the three most relevant competencies:
New Asset Class Name
Indicate new asset class here →
1
2
3
Asset Allocation for the Short- and Long-Term
International Forum of Sovereign Wealth Funds 72
References
Adler, T. and Kritzman, M. (2007). Mean-Variance versus Full-Scale Optimisation: In and Out of Sample, Journal
of Asset Management, Vol. 7, no. 5: 302–311
Blay, K., and H. Markowitz. Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume One).
New York, NY: McGraw –Hill, 2013.
Blume, M. (1984) “The Use of “Alphas” to Improve Performance,” Journal of Portfolio Management, no. 11: 86-
92.
Chow, G.1995. “Portfolio Selection Based on Return, Risk, and Relative Performance.” Financial Analysts
Journal, vol. 51, no. 2 (March/April):54-60.
Chow, G., Jacquier, E., Kritzman, M., and Lowry, K. 1999. “Optimal Portfolios in Good Times and Bad.” Financial
Elton, E., Gruber, M., and Rentzler, J. (1987) “Professionally Managed, Publicly Traded Commodity Funds,” Journal of Business, Volume 60, Issue 2: 175-199. F. J. Fabozzi and H. M. Markowitz, “The Theory and Practice of Investment Management,” John Wiley & Sons,
Inc., Hoboken NJ.
Ilmanen, A. “Expected Returns: An Investor’s Guide to Harvesting Market Rewards.” Chichester, West Sussex,
U.K.: John Wiley & Sons, 2011.
Karlin, S. and Taylor, H., A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975.
Kinlaw, W., Kritzman, M., and Turkington, D. 2014. “The Divergence of High- and Low-Frequency Estimation:
Causes and Consequences.” The Journal of Portfolio Management, Vol. 40, No. 5 (40th Anniversary).
Kinlaw, W., Kritzman, M., and Turkington, D. 2015. “The Divergence of High- and Low-Frequency Estimation:
Implications for Performance Measurement.” The Journal of Portfolio Management, Vol. 41, No. 3.
Kinlaw, W., Kritzman, M., and Turkington, D. 2013. “Liquidity and Portfolio Choice: A Unified Approach.” The
Journal of Portfolio Management, vol. 39, no. 2 (Winter).
Kinlaw, W., Kritzman, M., and Turkington, D. 2012. “Toward Determining Systemic Importance.” Journal of