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Quantitative Investing
Ibbotson Asset Allocation Conference
Robert Litterman
March, 2008
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Clearly Articulated Investment Beliefs Should
Drive Investment Strategy
Our Investment Beliefs (an example)
1) There is only one basic source of long run wealth creation, the growth of
the economy.
2) In equilibrium the investment portfolios of all individuals should reflect
that source of wealth. The portfolios would all have weights proportional to
market capitalizations which would reflect the economys expected future
productive capacity.
3) The world is clearly (given, among other indications, the diversity ofportfolios) not in equilibrium.
4) The market as a whole has shown itself to be subject to extended
periods of overreaction. Nonetheless, capital markets are competitive, and
though not entirely efficient, are becoming more efficient over time.
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Clearly Articulated Investment Beliefs Should
Drive Investment Strategy (continued)
Our Investment Beliefs (an example)
5) Deviations from equilibrium provide opportunities for some disciplined
investors with superior skills and information and typically a longer than
average investment horizon to outperform the market return on a risk
adjusted basis.
6 ) The key to superior investment performance is superior fundamental
research through which we can identify disequilibrium phenomena.
7) Systematically capturing returns from such phenomena requires rigorousobjective research, state of art risk measurement and portfolio construction,
efficient trading, and patience.
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Risk Is a Scarce Resource
Risk provides the energy that creates returns
However, risk also creates the opportunity for losses
Losses should be limited in a bad scenario
Thus, risk appetite is limited
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How much pain is too much?
Imagine a very bad economic scenario:
Equity markets decline globally by 50%
This decline reflects extensive defaults and depressed economic
activity for several years
In this environment long term investors should be increasing their
allocation to equity.
Will the investor be able to?
How much can an individual afford to lose? (10%, 20%, ?)
The answer to this question is the most important determinant of long
term asset allocation.
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An Optimal Portfolio
Maximizes return for a given level of risk
For illustrative purposes only.
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The Mathematics of Risk & Return
The Utility Function (Expected Return) is Linear
The Constraint Function (Risk) is Nonlinear
(And Depends on the Correlation of Returns)
A
C
B
A = Old Portfolio B = New Investment C = New Portfolio
A
C
B
correlation = .6
correlation = .2
correlation = -.2
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Different Levels of Portfolio Aggregation
Can Highlight Different Dimensions of Risk
In understanding the sources of risk and return at times it may be
useful to focus on:
Each cash flow
Individual securities
However, the most important determinants are:
Asset class allocations
Factors driving overall valuations In particular, beta, the exposure to global market returns
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When Is a Portfolio Optimal?
A portfolio is optimal when, at the margin the following ratio is
identical for each asset or other investment activity:
Change in Expected Excess Return
Change in Portfolio Risk
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Why Should This Be True?
If not, the fund can be improved:
Take funds from the lowest (per unit of contribution to portfolio
risk) returning activity
Move funds into a higher returning activity
Adjust cash to keep portfolio risk constant
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Focus on the Risk / Return Frontier
PortfolioExpec
tedR
eturn
Portfolio RiskFor illustrative purposes only.
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The Bottom Line
Ultimately there is a risk budget
Every decision depends on:
Expected excess return
And the marginal impact on portfolio risk
Efficient allocations require this ratio to be the same at every
margin
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Quantitative Models Measure Marginal
Contribution to Risk
The marginal contribution to risk of each asset
Depends on:
The covariance of that asset with every other asset
The amounts invested in each asset
Can be calculated given:
Portfolio holdings
Volatilities and correlations
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The Marginal Risk Contribution Determines
Implied Views
If a portfolio is optimal, the implied expected excess returns must
be proportional to the marginal contribution to portfolio risk
We refer to these expected excess returns for which the portfolio is
optimal as the implied views of the portfolio
We call this a risk based approach to asset allocation because
risk is measurable and risk measurement implies a set of views for
which any portfolio is optimal.
Are those implied views reasonable?
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Implied Views Guide Behavior
If these implied views conform with current expectations
Portfolio structure is appropriate
Otherwise
Adjustments should be made to increase expected portfolioreturn
Consider increasing investments in assets with expected
returns above the implied views; decreasing those below
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Equilibrium Theory providesa neutral starting point for Expected Returns
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Modern Investment ManagementAn Equilibrium Approach
Institutional investors are adjusting to an environment of low interest rates and reduced expected returns from
equities. What does the equilibrium theory suggest?
A book by Bob Litterman and 22 Goldman Sachs
Asset Management investment professionals
A re-examination of investment strategy with a focus on alpha vs. beta
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Sources of Total Return in a $1 Million Portfolio:
There are Three Fundamental
Sources Of Portfolio Return
1 Real Risk Free Rate
Risk free rate = 4.5%
$ 45,000
2 Market Risk Premium
20% Equity Allocation
0.82% ER
3.6% Vol
70% Equity Allocation
2.58% ER
9.6% Vol
3 Active Manager Return
100% Indexed
0.0% ER
1.5% Tracking Error
at IR = 0.5
Beta
Alpha
$ 8189 $ 25,779
$ 0 $ 7500
For illustrative purposes only.
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And Three Sources of Portfolio Risk
Interest rate risk, usually from liabilities:
Uncompensated risk
Can be hedged via derivatives or bonds
Market risk:
Basically available for free (no fees)
Has relatively low expected return per unit of risk
Active risk:
Uncorrelated risk implies low impact on portfolio risk
Skill-based
Opportunities require deviations from equilibrium
Active management fees
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Which Risks Should Be Compensated?
An answer was provided by the capital asset pricing model: an
equilibrium model
When all investors maximize expected return subject to a risk
constraint and markets are efficient
Expected excess return (the equilibrium risk premium) is
proportional to the beta of an asset
Why?
Beta measures the marginal impact of increasing asset weight
on the risk of the market portfolio
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Why Focus on Equilibrium?
The world is not in equilibrium
The academic theory is nonetheless relevant for investors
Deviations from equilibrium provide opportunities
But investors taking advantage of these opportunities push thecapital markets back toward equilibrium and greater efficiency
So the equilibrium framework helps investors to identify
opportunitiesit provides the hurdle rate, the required expected
return for taking additional risk
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Equilibrium Expected Excess Returns
There are several versions of Global CAPM Equilibrium
We focus on a particularly simple one: Fischer Blacks Universal Hedging
An assumption on risk aversion determines:
A constant degree of currency hedging A risk premium or excess return on all assets
Fischers model is calibrated to long run market returns
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Forward Looking Equilibrium Risk Premia Lead to
Better Behaved Optimal Asset Allocations
Equilibrium risk premia justify market capitalization portfolio weights
Asset Class MarketCapitalization
Equilibrium RiskPremium
US Equity 23.7% 3.53%
Non-US Developed Equity 26.3% 3.51%
Non-China Emerging Equity 4.8% 4.08%
China Equity 0.9% 4.51%
Global Fixed Income 42.5% 0.03%
High Yield 1.7% 1.27%
Private Equity 0% 4.69%
Real Estate 0% 2.63%
Hedge Funds 0% 0.48%
Market Cap above as of Sept 30th, 2007.All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-term assumptions aresubject to high levels of uncertainty regarding future economic and market factors that may affect future performance. They are
hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.
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Current Estimates of Global CAPM
Equilibrium Risk Premia
All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-term assumptions aresubject to high levels of uncertainty regarding future economic and market factors that may affect future performance. They are
hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.
4.50%
5.00%
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Strategic Asset Allocation providesa long term neutral anchor for
fund investment policy
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Steps toward a Strategic Asset Allocation
Specify the liability structure or fund objective: e.g. maximize real
wealth creation
Determine a risk tolerance
Start with global equilibrium risk premia
Tilt in the direction of long-term views
Optimize
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The Equity Allocation Drives the Overall Level of Risk
Market Capitalization Weights Risk Decomposition
1 The risk decomposition is the contribution of each asset class to the total portfolio variance.All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-term assumptionsare subject to high levels of uncertainty regarding future economic and market factors that may affect future performance. They are
hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.For illustrative purpose only
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Black-Litterman Model combinesMarket Equilibrium with Investor Views
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Incorporating Views In Portfolios
Step 1. Define What a View Is
A simple view:
UST yields will decline 50 bps in six months
Equivalently the Expected Return UST
= 3.3%
If the view is uncertain:
UST
= 3.3% + UST
where
UST
= Expected Return
UST
= Uncertainty in the Expected Return
UST N( 0, ) Measures Uncertainty
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A more complicated view:
Globally bonds will outperform stocks by about 3%
Equivalently: FI - MSCI = 3.0% +
Or: (1, -1) * ( FI , MSCI ) = 3.0% +
In general a view is represented as:
pv * = q +
where the weights pv define the view portfolio.
Incorporating Views In Portfolios
Step 2. Create a General Representation
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Views Can Reflect Long-Run or Short-Run
Opportunities
In forming a Strategic Benchmark, views should reflect long-term
deviations from equilibrium
Examples:
Emerging markets will outperform developed markets
Infrastructure returns will exceed their beta
Excess returns to commodities will be positive
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Tactical Views
In forming a Tactical Asset Allocation, views should reflect short-
term deviations from equilibrium
Examples:
Global stock markets will outperform global bond markets by
only 2 percent this year (a relatively bearish view on stocks)
Real estate will underperform US equities by 5 percent this year
Chinese equities will outperform other emerging markets by 5
percent this year
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Incorporating Views in Portfolios
Step 3. The Black-Litterman Model
Start with neutral expected returns derived from the global CAPM
equilibrium
Add a set of views:
p1 * = q1 + 1 1 N( 0, 1 )
p2 * = q2 + 2 2 N( 0, 2 )
p3 * = q3 + 3 3 N( 0, 3 )
Black-Litterman combines the views with equilibrium and provides
as output
BL = Black-Litterman Expected Returns
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An Example: Using the Black-Litterman model
Asset Class Symbol Volatility Equilibrium RiskPremium
China Equity C 35.7% 4.51%
US Equity US 14.5% 3.53%
Non-US Developed Equity DE 14.3% 3.51%
Non-China Emerging Equity EE 22.0% 4.08%
Global Fixed Income GFI 3.0% 0.03%
High Yield HY 8.4% 1.27%Private Equity PE 20.9% 4.69%
Real Estate RE 16.0% 2.63%
Hedge Fund Portfolio HF 3.6% 0.48%
In this example there are nine assets.
We start with the equilibrium risk premia -- a set of expected excess returns that are
for each asset proportional to the beta of that asset with the global market portfolio:
All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-term assumptions aresubject to high levels of uncertainty regarding future economic and market factors that may affect future performance. They are
hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.
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Equilibrium Risk Premia Provide a Reasonable
Starting Point for Asset Allocation Exercises
Equilibrium Expected
Excess Returns
Optimizer maximizes expected
return for a given level of risk
Implies Market
Capitalization
Weights
5%
6%
7%
45%Market Cap above as of Sept 30th, 2007.All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-term assumptions aresubject to high levels of uncertainty regarding future economic and market factors that may affect future performance. They are
hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.
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Suppose you have the view described below.
How would you adjust your expected returns?
1. Global stock markets will outperform global bond markets by
only 2 percent this year
One approach is to make adjustments directly to theexpected excess returns that drive an asset allocation
exercise.
Another approach is to use the Black-Litterman Global Asset
Allocation Model.
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The direct adjustment of expected excess returns
is a complex and often frustrating exercise
4.5%
5.0%
Equilibrium expected returns above reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategiclong-term assumptions are subject to high levels of uncertainty regarding future economic and market factors that may affect futureperformance. They are hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.
Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.For illustrative purposes only.
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The optimal portfolio does something strange:
It allocates 10 percent to real estate
70%
?Market Cap above as of Sept 30th, 2007. For illustrative purposes only.
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The Black-Litterman Model converts the views into a
set of consistent expected excess returns
4.5%
5.0%
One not entirely
obvious implication of
stocks doing poorly is
that real estate is
likely to do less well
Equilibrium expected returns above reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategiclong-term assumptions are subject to high levels of uncertainty regarding future economic and market factors that may affect futureperformance. They are hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.
Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.For illustrative purposes only.
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One can also specify higher or lower degrees of
confidence in a view
4.5%
5.0%
Equilibrium expected returns above reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategiclong-term assumptions are subject to high levels of uncertainty regarding future economic and market factors that may affect futureperformance. They are hypothetical indications of a broad range of possible returns. Please see p68 for a summary of the assumptions.Expected returns are estimates of hypothetical average returns of economic asset classes derived from statistical models. There can beno assurance that these returns can be achieved. Actual returns are likely to vary. Please see additional disclosures.For illustrative purposes only.
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The Black-Litterman expected excess returns
lead to a well behaved optimal portfolio
70%
Market Cap above as of Sept 30th, 2007. For illustrative purposes only.
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In fact, in the simplest context the deviations from
the market cap portfolio are the view portfolio
1400%
The bearish equity view is expressed in the Black-Litterman model as the following equation:
1.7%*C + 42.6%*US + 47.2% * DE + 8.6% * EE = GFI + 2.0
For illustrative purposes only.
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More generally, the Black-Litterman model
allocates risk to a combination of view portfolios
In the absence of:
A benchmark
Constraints
Transactions costsThe optimal portfolio is a linear combination of the market and
the view portfolios
PBL
= 0
x M + 1x p
1+
2x p
2+
3x p
3
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Where is the Value Added?
In this simplest context the optimal portfolio is intuitive and obvious:
Tilt away from the market portfolio
Tilt toward an optimal combination of the view portfolios which we call the OTP,
or Optimal Tilt Portfolio
Black-Litterman determines these optimal weights
The value added also shows up in more complex environments
When transactions costs matter
Or when there are constraints
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In more complicated contexts the optimal portfolio
is not so obvious
Here we show the optimal portfolio, based on three views, and withand without a constraint on the allocation to Private Equity
60%When private equity
is constrained the
allocations to public
equity increase
Market Cap above as of Sept 30th, 2007. For illustrative purposes only.
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The Role of Active Management
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Adding Active Risk Can Dramatically Shift the
Portfolio Frontier Upward
Note: Simulated performance results do not reflect actual trading and have certain inherent limitations. Please see appendix for further disclosures.
Adding exposure to active risk can boost long-run expected
returns without meaningfully increasing fund volatility.
Sharpe
Ratio
improves
with the
addition of
market
independen
t return
ExcessReturn (%)
Volatility(%)
SharpeRatio
Original Portfolio (50% equity) 1.7 8.1
Additional Market Risk (25% equity) 0.8 3.9
New Portfolio 2.5 11.8 0.21
Original Portfolio (50% equity) 1.7 8.1
1.5% Active Risk (assumed IR = 0.5) .7 1.5
New Portfolio 2.4 8.2 0.29
Benefit from Diversification (0.2)
Benefit from Diversification (1.4)
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Active risk typically makes a very small
contribution to overall portfolio volatility
For illustrative purposes only.Simulated performance results do not reflect actual trading and have inherent limitations. Please see additional disclosures.
12%
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The Active Risk Puzzle:
Why do funds have such modest expectations?
Optimal Risk Allocations Reveal Modest IR Expectations
Volatility = 6.0%
Volatility = 9.0%
Source: Goldman Sachs Asset Management.
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
0 0.1 0.2 0.3 0.4 0.5 0.6
Aggregate Active Risk Information Ratio
OptimalAllocationtoA
ctiveRisk
Allocations to
active risk of
typical funds
range between
50 and 200
basis points
Possible Explanations:
Funds may be unsure of their ability to select skilled managers
Career risk
Governance restrictions
Active risk and strategic asset allocation have historically been linked
For illustrative purposes only.Simulated performance resultsdo not reflect actual tradingand have inherent limitations.Please see additionaldisclosures.
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Alternative investments encompass a diverse range of
strategies and are a good source of active risk
Private Equity
Real Estate
Hedge Funds
Commodities
Overlays such as GTAA (Global Tactical Asset Allocation) and
Active Currency Management
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What Makes Alternative Investments Attractive?
Attributes of Alternative Investments include:
Historically attractive absolute returns versus traditional asset
classes
Lower correlations that can provide protection in bear markets
Therefore may provide excess returns above the equilibrium
hurdle rate
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Appendix
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Strategic Long-Term AssumptionsRisk and Return characteristics
All tracking error assumptions reflect GSAM Global Investment Strategies estimates for above-average active managers and are basedon a historical study of the results of active management [see Active Risk Budgeting in Action: Evaluating Historical Characteristics ofTraditional Managers by Yoel Lax, Tarun Tyagi, and Kurt Winkelmann (GSAM Strategic Research, October 2003)], which is availableupon request. Expected returns are estimates of hypothetical average returns of economic asset classes derived from statisticalmodels. There can be no assurance that these returns can be achieved. Actual returns are likely to vary. Please see additionaldisclosures. All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-termassumptions are subject to high levels of uncertainty regarding future economic and market factors that may affect future performance.They are hypothetical indications of a broad range of possible returns. Please see additional disclosures.
Asset Class S i T A i
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All numbers reflect GSAM Global Investment Strategies strategic assumptions as of a certain date. Strategic long-termassumptions are subject to high levels of uncertainty regarding future economic and market factors that may affect futureperformance. They are hypothetical indications of a broad range of possible returns. Please see additional disclosures.
Strategic Long-Term AssumptionsCorrelations
A di
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Appendix
The currency market affords investors a substantial degree of leverage. This leverage presents the potential forsubstantial profits but also entails a high degree of risk including the risk that losses may be similarly substantial. Suchtransactions are considered suitable only for investors who are experienced in transactions of that kind. Currencyfluctuations will also affect the value of an investment.
Emerging markets securities may be less liquid and more volatile and are subject to a number of additional risks,including but not limited to currency fluctuations and political instability.
High-yield, lower-rated securities involve greater price volatility and present greater credit risks than higher-rated fixedincome securities.
An investment in real estate securities is subject to greater price volatility and the special risks associated with directownership of real estate.
The portfolio risk management process includes an effort to monitor and manage risk, but does not imply low risk.
Indices are unmanaged. The figures for the index reflect the reinvestment of dividends but do not reflect the deduction of
any fees or expenses which would reduce returns. Investors cannot invest directly in indices.
References to indices, benchmarks or other measures of relative market performance over a specified period of time areprovided for your information only and do not imply that the portfolio will achieve similar results. The index compositionmay not reflect the manner in which a portfolio is constructed. While an adviser seeks to design a portfolio which reflectsappropriate risk and return features, portfolio characteristics may deviate from those of the benchmark.
A di
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Appendix
There may be conflicts of interest relating to the Alternative Investment and its service providers, including Goldman Sachsand its affiliates, who are engaged in businesses and have interests other than that of managing, distributing and otherwiseproviding services to the Alternative Investment. These activities and interests include potential multiple advisory,transactional and financial and other interests in securities and instruments that may be purchased or sold by theAlternative Investment, or in other investment vehicles that may purchase or sell such securities and instruments. Theseare considerations of which investors in the Alternative Investment should be aware. Additional information relating tothese conflicts is set forth in the offering materials for the Alternative Investment.
Past performance is not indicative of future results, which may vary. The value of investments and the income derived frominvestments can go down as well as up. Future returns are not guaranteed, and a loss of principal may occur.
Effect of Fees
The following table provides a simplified example of the effect of management fees on portfolio returns. Assume a portfoliohas a steady investment return, gross of fees, of 0.5% per month and total management fees of 0.05% per month of themarket value of the portfolio on the last day of the month. Management fees are deducted from the market value of the
portfolio on that day. There are no cash flows during the period. The table shows that, assuming all other factors remainconstant, the difference increases due to the compounding effect over time. Of course, the magnitude of the differencebetween gross-of-fee and net-of-fee returns will depend on a variety of factors, and this example is purposely simplified.
Gross NetPeriod Return Return Differential1 year 6.17% 5.54% 0.63%2 years 12.72 11.38 1.3410 years 81.94 71.39 10.55
A di
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Appendix
Alternative Investments such as hedge funds are subject to less regulation than other types of pooled investmentvehicles such as mutual funds, may make speculative investments, may be illiquid and can involve a significant useof leverage, making them substantially riskier than the other investments. An Alternative Investment Fund may incurhigh fees and expenses which would offset trading profits. Alternative Investment Funds are not required to provideperiodic pricing or valuation information to investors. The Manager of an Alternative Investment Fund has totalinvestment discretion over the investments of the Fund and the use of a single advisor applying generally similartrading programs could mean a lack of diversification, and consequentially, higher risk. Investors may have limitedrights with respect to their investments, including limited voting rights and participation in the management of the
Fund.
Alternative Investments by their nature, involve a substantial degree of risk, including the risk of total loss of aninvestor's capital. Fund performance can be volatile. There may be conflicts of interest between the AlternativeInvestment Fund and other service providers, including the investment manager and sponsor of the AlternativeInvestment. Similarly, interests in an Alternative Investment are highly illiquid and generally are not transferablewithout the consent of the sponsor, and applicable securities and tax laws will limit transfers.
Strategic Long Term Assumptions
The data regarding strategic assumptions has been generated by GSAM for informational purposes. As such data isestimated and based on a number of assumptions; it is subject to significant revision and may change materially withchanges in the underlying assumptions. GSAM has no obligation to provide updates or changes. The strategic long-term assumptions shown are largely based on proprietary models and do not provide any assurance as to futurereturns. They are not representative of how we will manage any portfolios or allocate funds to the asset classes.
A di
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Appendix
These examples are for illustrative purposes only and are not actual results. If any assumptions used do not prove tobe true, results may vary substantially.
This material is provided for educational purposes only and should not be construed as investment advice or an offeror solicitation to buy or sell securities.
Simulated Performance
Simulated performance is hypothetical and may not take into account material economic and market factors thatwould impact the advisers decision-making. Simulated results are achieved by retroactively applying a model withthe benefit of hindsight. The results reflect the reinvestment of dividends and other earnings, but do not reflect fees,transaction costs, and other expenses, which would reduce returns. Actual results will vary.
Views and opinions expressed are for informational purposes only and do not constitute a recommendation by GSAM
to buy, sell, or hold any security. Views and opinions are current as of the date of this presentation and may be
subject to change, they should not be construed as investment advice.
Opinions expressed are current opinions as of the date appearing in this material only. No part of this material may,without GSAMs prior written consent, be (i) copied, photocopied or duplicated in any form, by any means, or (ii)distributed to any person that is not an employee, officer, director, or authorized agent of the recipient.
Copyright 2007, Goldman, Sachs & Co. All rights reserved.
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