tracing frontier

8/14/2019 Tracing Frontier

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Second Investment CourseNovember 2005

Topic Four:

Portfolio Optimization: Analytical Techniques


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Overview of the Portfolio Optimization Process

The preceding analysis demonstrates that it is possible for investorsto reduce their risk exposure simply by holding in their portfolios asufficiently large number of assets (or asset classes). This is thenotion of nave diversification, but as we have seen there is a limit tohow much risk this process can remove.

Efficient diversificationis the process of selecting portfolio holdingsso as to: (i) minimize portfolio risk while (ii) achieving expectedreturn objectives and, possibly, satisfying other constraints (e.g., noshort sales allowed). Thus, efficient diversification is ultimately aconstrained optimizationproblem. We will return to this topic in thenext session.

Notice that simply minimizing portfolio risk without a specific returnobjective in mind (i.e., an unconstrained optimizationproblem) isseldom interesting to an investor. After all, in an efficient market,any riskless portfolio should just earn the risk-free rate, which theinvestor could obtain more cost-effectively with a T-bill purchase.


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The Portfolio Optimization Process As established by Nobel laureate Harry Markowitz in the 1950s, the

efficient diversification approach to establishing an optimal set of portfolio

investment weights(i.e., {wi}) can be seen as the solution to the followingnon-linear, constrained optimization problem:

Select {wi} so as to minimize:

subject to: (i) E(Rp) = R*

(ii) Swi= 1

The first constraint is the investors return goal (i.e., R*). The secondconstraint simply states that the total investment across all 'n' asset

classes must equal 100%. (Notice that this constraint allows any ofthe wito be negative; that is, short selling is permissible.)

Other constraints that are often added to this problem include: (i) All wi> 0(i.e., no short selling), or (ii) All w

i< P, where P is a fixed percentage

]w2w...w[2w]w...[w n,1nn1nn1-n2,121212

n

2

n

2

1

2

1

2

p


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Solving the Portfolio Optimization Problem

In general, there are two approaches to solving for theoptimal set of investment weights (i.e., {wi}) depending

on the inputs the user chooses to specify:

1. Underlying Risk and Return Parameters: Asset class expectedreturns, standard deviations, correlations)

a. Analytical(i.e., closed-form) solution: True solution but

sometimes difficult to implement and relatively inflexible at

handling multiple portfolio constraints

b. Optimal search: Flexible design and easiest to implement, but

does not always achieve true solution

2. Observed Portfolio Returns: Underlying asset class risk and

return parameters estimated implicitly


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The Analytical Solution to Efficient Portfolio OptimizationFor any particular collection of assets, the efficient frontierrefers to the set of portfolios that offers

the lowest level of risk for a pre-specified level of expected return. Given information about theexpected returns, standard deviations, and correlations amongst the securities, we have seen that

efficient portfolio weights can be determined analytically by solving the following problem:

Select {wi} so as to minimize:

subject to: (i) E(Rp) = R*

(ii) Swi= 1.

To make the solution somewhat more transparent, we can first rewrite the problem in matrixnotation. Assuming there are "n" securities available, define:

V = (n x n) covariance matrix (i.e., the diagonal elements are the n variances and

the off-diagonal elements are the correlation coefficients amongst thesecurities);

w = (n x 1) vector of portfolio weights;

R = (n x 1) vector of expected security returns;

i = (n x 1) "unit" vector (i.e., a vector of ones).

With this notation, the efficient frontier problem can be recast as:

Min [(0.5) w' V w]

wsubject to

R* = w' R

and 1 = w' i

]w2w...w[2w]w...[w n,1nn1nn1-n2,121212

n

2

n

2

1

2

1

2

p


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The Analytical Solution to Efficient Portfolio Optimization (cont.)

Note here that the variance of the portfolio (i.e., w' Vw) has been divided by two; this is merely an

algebraic convenience and does not change the values of the optimal weights.

One approach to solving this constrained optimizationproblem is the Lagrange-multiplier method.

That is, to convert the constrained problem into an unconstrained one, select the vector of portfolio

weights so as to:

Min L = Min [ (0.5)w'Vw -

1(R

*-w

'R) -

2(1 -w

'i ) ]

where 1 and 2 are the Lagrangean multipliers. Notice that this method incorporates the

constraints directly into the orginal function by creating two new variables to be solved.

Consequently, the solution can proceed by differentiating L with respect to w, 1and 2. The firstorder conditions of the minimum are:

L

w= V w - 1R - 2i = 0

(1)

L

1

= R*

- w'R = 0 = R

*- R

'w

(2)

L

2

= 1 - w'i = 0 = 1 - i

'w

(3)

Solving (1) for wyields:

= 1V-1R+ 2V-1i =V-1[Ri] (4)


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The Analytical Solution to Efficient Portfolio Optimization (cont.)

For a particular value of R*, solve for 1* and 2* by combining the constraint equations in (3a)

and (3b) with an expanded form of (4):

R

*

=R

'

w =

1

*

R

'

V

-1

R +

2

*

R

'

V

-1

(5a)

and

1 = i'w = 1

*

i'V

-1R + 2

*

i'V

-1

(5b)

Also, define the following efficient set constants:

A = R'V

-1i = i

'V

-1R (6a)

B = R

'V

-1R

(6b)

C = i'V

-1 (6c)

Substituting (6) into (5) yields:

B A

AC

1

2

M R

(7)

or:

M-1

R

(8)

Substituting (8) into (4) leaves:

*= V

-1[ R i ] M

-1 R*

1 (9)

where w* is the vector of weights for the minimum variance portfolio having an expected return ofR* and a variance of 2= w*'Vw*.


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Example of Mean-Variance Optimization: Analytical Solution

(Three Asset Classes, Short Sales Allowed)


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Example of Mean-Variance Optimization: Analytical Solution (cont.) (Three

Asset Classes, Short Sales Allowed)


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Example of Mean-Variance Optimization: Optimal Search Procedure

(Three Asset Classes, Short Sales Allowed)


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(Three Asset Classes, No Short Sales)


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Measuring the Cost of Constraint: Incremental Portfolio Risk

Main Idea: Any constraint on the optimization process imposes a cost to the

investor in terms of incremental portfolio volatility, but only if that constraint isbinding (i.e., keeps you from investing in an otherwise optimal manner).


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Mean-Variance Efficient Frontier With and Without Short-Selling


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Optimal Search Efficient Frontier Example: Five Asset Classes


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(Five Asset Classes, No Short Sales)


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Mean-Variance Optimization with Black-Litterman Inputs

One of the criticisms that is sometimes made about the mean-

variance optimization process that we have just seen is that the

inputs (e.g., asset class expected returns, standard deviations, and

correlations) must be estimated, which can effect the quality of the

resulting strategic allocations.

Typically, these inputs are estimated from historical return data.

However, it has been observed that inputs estimated with historical

datathe expected returns, in particularlead to extreme portfolio

allocations that do not appear to be realistic.

Black-Litterman expected returns are often preferred in practice for

the use in mean-variance optimizations because the equilibrium-

consistent forecasts lead to smoother, more realistic allocations.


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BL Mean-Variance Optimization Example (cont.)

These inputs can then be used in a standard mean-variance optimizer:


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This leads to the following optimal allocations (i.e., efficient frontier):


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Another advantage of the BL Optimization model is that it provides a

way for the user to incorporate his own views about asset classexpected returns into the estimation of the efficient frontier.

Said differently, if you do not agree with the implied returns, the BLmodel allows you to make tactical adjustmentsto the inputs and stillachieve well-diversified portfolios that reflect your view.

Two components of a tactical view:- Asset Class Performance

- Absolute(e.g., Asset Class #1 will have a return of X%)

- Relative(e.g., Asset Class #1 will outperform Asset Class #2 by Y%)

- User Confidence Level

- 0% to 100%, indicating certainty of return view

(See the article A Step-by-Step Guide to the Black-Litterman Modelby T. Idzorek of Zephyr Associates for more details on thecomputational process involved with incorporating user-specifiedtactical views)


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Suppose we adjust the inputs in the process to include two tactical views:

- US Equity will outperform Global Equity by 50 basis points (70% confidence)

- Emerging Market Equity will outperform US Equity by 150 basis points (50% confidence)


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The new optimal allocations reflect these tactical views (i.e., more Emerging MarketEquity and less Global Equity:


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This leads to the following new efficient frontier:


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Optimal Portfolio Formation With Historical Returns: Examples

Suppose we have monthly return data for the last threeyears on the following six asset classes:

- Chilean Stocks (IPSA Index)

- Chilean Bonds (LVAG & LVAC Indexes)

- Chilean Cash (LVAM Index)

- U.S. Stocks (S&P 500 Index)

- U.S. Bonds (SBBIG Index)

- Multi-Strategy Hedge Funds (CSFB/Tremont Index)

Assume also that the non-CLP denominated assetclasses can beperfectly and costlessly hedged in fullif

the investor so desires


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Optimal Portfolio Formation With Historical Returns: Examples (cont.)

Consider the formation of optimal strategic asset allocations under awide variety of conditions:

- With and without hedging non-CLP exposure

- With and Without Investment in Hedge Funds

- With and Without 30% Constraint on non-CLP Assets

- With different definitions of the optimization problem:

- Mean-Variance Optimization- Mean-Lower Partial Moment (i.e., downside risk) Optimization

- Alpha-Tracking Error Optimization

Each of these optimization examples will:

-Use the set of historical returns directly rather than the underlying set ofasset class risk and return parameters

- Be based on historical return data from the period October 2002September 2005

- Restrict against short selling (except those short sales embedded in thehedge fund asset class)


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1. Mean-Variance Optimization: Non-CLP Assets 100% Unhedged


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One Consequence of the Unhedged M-V Efficient Frontier

Notice that because of the strengthening CLP/USDexchange rate over the October 2002September 2005

period, the optimal allocationfor any expected return

goal did not include any exposure to non-CLP asset

classes

This unhedged foreign investmentefficient frontier is

equivalentto the efficient frontier that would have

resulted from a domestic investment only constraint.

The issue of foreign currency hedging will be considered

in a separate topic


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Mean-Variance Optimization: Non-CLP Assets 100% Hedged


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Unconstrained M-V Efficient Frontier: 100% Hedged

E(R) p Relative p Wcs Wcb Wcc Wuss Wusb Whf

5.00% 0.71% 1.000 2.20% 9.45% 68.45% 0.59% 0.00% 19.31%

6.00% 1.02% 1.000 3.36% 13.82% 53.57% 0.74% 0.00% 28.51%

7.00% 1.34% 1.000 4.51% 18.19% 38.68% 0.89% 0.00% 37.72%

8.00% 1.67% 1.000 5.67% 22.56% 23.80% 1.04% 0.00% 46.93%

9.00% 2.00% 1.000 6.83% 26.93% 8.92% 1.19% 0.00% 56.13%

10.00% 2.33% 1.000 8.58% 26.81% 0.00% 0.68% 0.00% 63.93%

11.00% 2.72% 1.000 11.18% 20.28% 0.00% 0.00% 0.00% 68.54%

12.00% 3.16% 1.000 13.75% 13.98% 0.00% 0.00% 0.00% 72.28%

13.00% 3.63% 1.000 16.32% 7.67% 0.00% 0.00% 0.00% 76.01%14.00% 4.11% 1.000 18.88% 1.37% 0.00% 0.00% 0.00% 79.75%


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Comparison of Unhedged (i.e. Domestic Only) and

Hedged (i.e., Unconstrained Foreign) Efficient Frontiers

Expected Return Unhedged M-V p Hedged M-V p Relative p5.00% 1.13% 0.71% 1.598

6.00% 1.65% 1.02% 1.620

7.00% 2.18% 1.34% 1.624

8.00% 2.71% 1.67% 1.6249.00% 3.24% 2.00% 1.622

10.00% 3.77% 2.33% 1.617

11.00% 4.30% 2.72% 1.581

12.00% 4.83% 3.16% 1.531

13.00% 5.36% 3.63% 1.480

14.00% 5.90% 4.11% 1.433


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A Related Question About Foreign Diversification

What allocation to foreign assets in a domestic investment portfolioleads to a reduction in the overall level of risk?

Van Harlow of Fidelity Investments performed the following analysis:

- Consider a benchmark portfolio containing a 100% allocation to U.S.

equities

- Diversify the benchmark portfolio by adding a foreign equity allocation in

successive 5% increments

- Calculate standard deviations for benchmark and diversified portfolios

using monthly return data over rolling three-year holding periods during

1970-2005- For each foreign allocation proportion, calculate the percentage of

rolling three-year holding periods that resulted in a risk level for the

diversified portfolio that was higher than the domestic benchmark


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Portfolio Risk Reduction and Diversifying Into Foreign Assets

United States, 1970-2005

0%

5%

10%

15%

20%

25%

30%

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65%

Foreign Stock Allocation

FrequencyofHigherRisk(vsDomesticOnly)

Rolling3YearPeriods1970-2005


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Foreign Diversification Potential (cont.)

Ennis Knupp Associates (EKA) have provided an alternative way of quantifying thediversification benefits of adding international stocks to a U.S. stock portfolio:

EKA concludes that international diversification adds an important elementof risk control within an investment program; the optimal allocation from astatistical standpoint is approximately 30%-40% of total equities, althoughthey generally favor a slightly lower allocation due to cost considerations.

Impact of Diversification on Volatility

1971 - 2001

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

0 10 20 30 40 50 60 70 80 90 100

Percentage in Foreign Stocks

Volatility


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Foreign Diversification Potential: One Caveat

During recent periods, it appears as though the correlations between U.S.and non-U.S. markets are increasing, reducing the diversification benefits ofnon-U.S. markets.

While this is true, the fact that these markets are less than perfectlycorrelated means that there is still a diversification benefit afforded toinvestors who allocate a portion of their assets overseas.

Rolling 3-Year CorrelationsU.S. and Non-U.S. Stocks1971-2001

0.00.10.20.30.40.50.60.70.80.91.0

1974

1975

1977

1978

1979

1980

1982

1983

1984

1985

1987

1988

1989

1990

1992

1993

1994

1995

1997

1998

1999

2000

2002


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More on Mean-Variance Optimization:

The Cost of Adding Additional Constraints

Start with the following base case:- Six asset classes: Three Chilean, Three Foreign (Including

Hedge Funds)

- No Short Sales

- 100% Hedged Foreign Investments- No Constraint on Total Foreign Investment

- No Constraint on Hedge Fund Investment

Consider the addition of two more constraints:- 30% Limit on Foreign Asset Classes

- No Hedge Funds


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Additional Constraints: 30% Foreign Investment


5.00% 0.71% 1.000 2.20% 9.45% 68.45% 0.59% 0.00% 19.31%

6.00% 1.02% 1.000 3.36% 13.82% 53.57% 0.74% 0.00% 28.51%

7.00% 1.40% 1.040 7.00% 16.77% 46.22% 0.63% 0.00% 29.37%

8.00% 1.85% 1.109 10.87% 19.60% 39.53% 0.50% 0.00% 29.50%

9.00% 2.34% 1.171 14.73% 22.43% 32.84% 0.37% 0.00% 29.63%

10.00% 2.84% 1.218 18.59% 25.25% 26.15% 0.24% 0.00% 29.76%

11.00% 3.35% 1.233 22.45% 27.97% 19.58% 0.10% 0.00% 29.90%

12.00% 3.87% 1.226 26.31% 30.93% 12.76% 0.00% 0.00% 30.00%

13.00% 4.39% 1.212 30.16% 33.90% 5.94% 0.00% 0.00% 30.00%14.00% 4.92% 1.195 33.99% 36.01% 0.00% 0.00% 0.00% 30.00%


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Additional Constraints: 30% Foreign Investment & No Hedge Funds


5.00% 1.03% 1.449 5.82% 11.38% 72.44% 4.63% 5.73% 0.00%

6.00% 1.51% 1.477 8.75% 16.68% 60.13% 6.66% 7.78% 0.00%

7.00% 1.99% 1.485 11.68% 21.98% 47.82% 8.69% 9.83% 0.00%

8.00% 2.48% 1.488 14.62% 27.28% 35.50% 10.72% 11.88% 0.00%

9.00% 2.97% 1.488 17.55% 32.57% 23.19% 12.76% 13.93% 0.00%

10.00% 3.46% 1.485 20.57% 37.75% 11.69% 14.64% 15.36% 0.00%

11.00% 3.96% 1.455 23.98% 42.31% 3.71% 15.85% 14.15% 0.00%

12.00% 4.46% 1.412 27.45% 42.55% 0.00% 16.67% 13.33% 0.00%

13.00% 4.98% 1.373 30.97% 39.03% 0.00% 17.13% 12.87% 0.00%14.00% 5.51% 1.339 34.57% 36.16% 0.00% 17.52% 11.74% 0.00%


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2. Mean-Downside Risk Optimization Scenario

Start with Same Base Case as Before:

- Six Asset Classes: Three Domestic, Three Foreign

- Fully Hedged Foreign Investments; No Short Sales

- No Constraint on Foreign Investments

- No Constraint on Hedge Funds

Downside Risk Conditions:

- Threshold Level = 2.93% (i.e., annualized return from Chileancash market)

- Power Factor for Downside Deviations = 2.0


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Mean-Downside Risk Optimization: Non-CLP Assets 100% Hedged


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E(R) LPMp Rel. LPMp Wcs Wcb Wcc Wuss Wusb Whf

5.00% 0.19% 1.000 3.36% 12.91% 67.61% 0.65% 0.00% 15.47%

6.00% 0.26% 1.000 5.13% 16.98% 54.49% 0.61% 0.00% 22.79%

7.00% 0.33% 1.002 7.12% 20.75% 42.13% 0.42% 0.00% 29.58%

8.00% 0.45% 1.098 11.04% 24.30% 34.66% 0.00% 0.00% 30.00%

9.00% 0.59% 1.233 14.90% 27.81% 27.29% 0.00% 0.00% 30.00%

10.00% 0.76% 1.361 18.79% 32.63% 18.58% 0.00% 0.00% 30.00%

11.00% 0.93% 1.432 22.68% 37.57% 9.75% 0.00% 0.00% 30.00%

12.00% 1.12% 1.422 26.46% 43.47% 0.07% 0.89% 0.00% 29.11%

13.00% 1.32% 1.401 30.29% 39.71% 0.00% 0.00% 0.00% 30.00%

14.00% 1.53% 1.384 33.99% 36.01% 0.00% 0.00% 0.00% 30.00%


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3. Alpha-Tracking Error Optimization Scenario

Start with Same Base Case as Before:

- Six Asset Classes: Three Domestic, Three Foreign

- Fully Hedged Foreign Investments; No Short Sales

- No Constraint on Foreign Investments or Hedge Funds

Optimization Process Defined Relative to Benchmark Portfolio:- Minimize Tracking Error Necessary to Achieve a Required Level of

Excess Return (i.e., Alpha) Relative to Benchmark Return

- Benchmark Composition: Chilean Stock: 35%; Chilean Bonds: 30%,

Chilean Cash: 5%; U.S. Stock: 15%; U.S. Bonds: 15%; Hedge Funds:

0%

Notice that Benchmark Portfolio Could Be Defined as Average Peer

Group Allocation


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Alpha-Tracking Error Optimization: Non-CLP Assets 100% Hedged


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Unconstrained a-TE Efficient Frontier: 100% Hedged

a TE Relative TE Wcs Wcb Wcc Wuss Wusb Whf

0.20% 0.07% 1.000 35.23% 30.87% 2.16% 15.01% 14.83% 1.90%

0.40% 0.13% 1.000 35.51% 31.35% 0.00% 14.94% 14.46% 3.74%

0.60% 0.22% 1.000 35.96% 30.57% 0.00% 14.60% 13.46% 5.41%0.80% 0.31% 1.000 36.41% 29.79% 0.00% 14.26% 12.45% 7.08%

1.00% 0.40% 1.000 36.85% 29.02% 0.00% 13.93% 11.45% 8.75%

1.20% 0.49% 1.000 37.30% 28.24% 0.00% 13.59% 10.44% 10.42%

1.40% 0.59% 1.000 37.75% 27.47% 0.00% 13.25% 9.44% 12.09%

1.60% 0.68% 1.000 38.19% 26.69% 0.00% 12.92% 8.43% 13.76%

1.80% 0.78% 1.000 38.64% 25.92% 0.00% 12.58% 7.43% 15.43%

2.00% 0.87% 1.000 39.09% 25.14% 0.00% 12.25% 6.42% 17.10%


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0.20% 0.09% 1.336 35.52% 30.68% 3.80% 14.86% 13.89% 1.25%

0.40% 0.18% 1.325 36.04% 31.37% 2.60% 14.72% 12.78% 2.50%

0.60% 0.27% 1.225 36.56% 32.05% 1.39% 14.58% 11.67% 3.75%

0.80% 0.35% 1.151 37.08% 32.76% 0.16% 14.45% 10.56% 5.00%

1.00% 0.44% 1.108 37.58% 32.42% 0.00% 14.15% 9.40% 6.45%

1.20% 0.54% 1.083 38.09% 31.91% 0.00% 13.83% 8.23% 7.94%

1.40% 0.63% 1.068 38.59% 31.41% 0.00% 13.52% 7.06% 9.42%

1.60% 0.72% 1.058 39.10% 30.90% 0.00% 13.20% 5.89% 10.91%

1.80% 0.82% 1.051 39.60% 30.40% 0.00% 12.88% 4.72% 12.40%

2.00% 0.91% 1.045 40.11% 29.89% 0.00% 12.56% 3.55% 13.89%


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Additional Constraints: 30% Foreign Investment & No Hedge Funds


0.20% 0.10% 1.558 35.68% 30.92% 3.40% 15.24% 14.76% 0.00%

0.40% 0.21% 1.546 36.36% 31.83% 1.81% 15.49% 14.51% 0.00%

0.60% 0.31% 1.430 37.05% 32.77% 0.19% 15.73% 14.27% 0.00%0.80% 0.42% 1.353 37.75% 32.25% 0.00% 15.84% 14.16% 0.00%

1.00% 0.53% 1.314 38.45% 31.55% 0.00% 15.94% 14.06% 0.00%

1.20% 0.64% 1.292 39.16% 30.84% 0.00% 16.03% 13.97% 0.00%

1.40% 0.75% 1.278 39.86% 30.14% 0.00% 16.12% 13.88% 0.00%

1.60% 0.87% 1.268 40.60% 29.70% 0.00% 16.19% 13.51% 0.00%

1.80% 0.98% 1.261 41.34% 29.29% 0.00% 16.25% 13.13% 0.00%

2.00% 1.10% 1.255 42.07% 28.88% 0.00% 16.31% 12.74% 0.00%


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The Portfolio Optimization Process: Some Summary Comments

The introduction of the portfolio optimization process was an important stepin the development of what is now considered to be modern finance theory.These techniques have been widely used in practice for more than fiftyyears.

Portfolio optimization is an effective tool for establishing the strategic assetallocation policyfor a investment portfolio. It is most likely to be usefullyemployed at the asset class level rather than at the individual security level.

There are two critical implementation decisionsthat the investor must make:- The nature of the risk-return problem:

- Mean-Variance, Mean-Downside Risk, Excess Return-Tracking Error

- Estimates of the required inputs:- Expected returns, asset class risk, correlations

Portfolio optimization routines can be adapted to include a variety ofrestrictions on the investment process (e.g., no short sales, limits on foreigninvesting).

- The cost of such investment constraints can be viewed in terms of the incrementalvolatility that the investor is required to bear to obtain the same expectedoutcome

tracing frontier

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