1 - 1

Investment Course - 2005

Day One:

Global Asset Allocation and Portfolio Formation

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Two Important Concepts Involving Expected Investment Returns

1. Investors perform two functions for capital markets:- Commit Financial Capital- Assume Risk

so,

E(R) = (Risk-Free Rate) + (Risk Premium)

2. The expected return (i.e., E(R)) of an investment has a number of alternative names: e.g., discount rate, cost of capital, cost of equity, yield to maturity. It can also be expressed as:

k = (Nominal RF) + (Risk Premium)= [(Real RF) + E(Inflation)] + (Risk Premium)

where:

Risk Premium = f(business risk, liquidity risk, political risk, financial risk)

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Historical Real Returns, 1954-2003: The Global Experience

Historical Real Returns, 1954-2003

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

Ann

ually

Com

poun

ded

Rea

l Ret

urn,

%

Equities

Bonds

Chile: Returns 1/54 – 6/03Chile*: Returns 1/54 – 12/71; 1/76 – 6/03Source: Global Financial Data

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Global Historical Volatility Measures, 1954-2003

Historical Risk, 1954-2003

0%

5%

10%

15%

20%

25%

30%

35%

40%

Ann

ually

Com

pou

nded

Rea

l Ret

urn,

%

Equities

Bonds

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Global Historical Risk Premia, 1954-2003

Risk Premia of Stocks and Bonds to Cash, 1954-2003

-2%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

Ann

ually

Com

poun

ded

Rea

l Ret

urn,

%

Equities-CashBonds-Cash

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Historical Returns and Risk for Various U.S. Asset Classes

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Historical Global Stock Market Volatility

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More on Historical Asset Class Returns: U.S. Experience

Stocks:

Bonds:

T-Bills:

Inflation:

1926-2004: Avg. Return 12.39% 6.19% 3.76% 3.13% Std. Deviation 20.31% 8.56% 3.14% 4.32% 1980-2004: Avg. Return 14.73 11.05 6.09 3.75 Std. Deviation 16.33 11.51 3.26 2.43 1995-2004: Avg. Return 14.00 9.45 3.92 2.49 Std. Deviation 21.09 9.32

1.90 0.81

2000-2004: Avg. Return -0.70 9.92 2.72 2.60 Std. Deviation 20.32 5.04 2.10 0.97 Source: Ibbotson Associates

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Historical Risk Premia vs. T-bills: U.S. Experience

Stocks: Bonds:

Stock - Bond

Difference:

1926-2004: 8.63% 2.43% 6.20%

1980-2004: 8.64 4.96 3.68

1995-2004: 10.08 5.53 4.55

2000-2004: -3.42 7.20 -10.62

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Performance of U.S.-Oriented Investment Strategies: 1975-2004

Growth

of $1

Avg.

Ann. Ret. Std. Dev.

Sharpe

Ratio

100% Stock $47.52 14.90% 16.13% 0.540

100% Bond $16.06 10.23 11.26 0.359

100% Cash $6.19 6.19 3.09 nm

“60-30-10”

Mix

$30.70 12.63 11.12 0.579

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Portfolio Management Strategy: Broad View

Passive Management Attempt to generate “normal” returns over time commensurate

with investor risk tolerance Typically achieved through diversified asset class selection and

asset-specific portfolio formation

Active Management Attempt to generate above-normal returns over time relative to

acceptable risk level Typically achieved either through periodic asset class or security-

specific portfolio adjustments

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Two Ways to Increase Returns (i.e., “Add Alpha”):

Tactical Allocation Decisions

- Global Market Timing

- Asset Class Timing

- Style/Sector Timing

Security Selection Decisions

- Stock or Bond Picking

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Allure of Tactical Market Timing

Suppose that on January 1st each year from 1975-2004, you put 100% of your money in what turned out to be the best asset class (stocks, bonds, or cash) at the end of the year.

This is equivalent to owning a perfect lookback option that entitles you to receive the return for the best performing asset class each year.

What difference would that type of tactical portfolio rebalancing make to your investment performance?

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Allure of Tactical Market Timing (cont.)

Growth

of $1

Avg.

Ann. Ret. Std. Dev.

Sharpe

Ratio

100% Stock $47.52 14.90% 16.13% 0.540

100% Bond $16.06 10.23 11.26 0.359

100% Cash $6.19 6.19 3.09 nm

“60-30-10”

Mix

$30.70 12.63 11.12 0.579

“Perfect Foresight” $237.68 20.48 10.92 1.309

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Danger of “Missing the Boat” (i.e., Not Being Invested):

Investment Period

S&P 500 Annualized

Return: 1980-1989

S&P 500 Annualized

Return: 1990-1999

Entire Decade

(2,528 Days) 12.6% 15.3%

Less: 10 Best Days 7.7 11.1

Less: 20 Best Days 4.7 8.1

Less: 30 Best Days 2.1 5.6

Less: 40 Best Days -0.3 3.4

Less: 10 Worst Days 21.0 20.1

Less: 20 Worst Days 24.7 23.4

Less: 30 Worst Days 27.5 26.4

Less: 40 Worst Days 30.5 28.9

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The Asset Allocation Decision

A basic decision that every investor must make is how to distribute his or her investable funds amongst the various asset classes available in the marketplace:

Stocks (e.g., Domestic, Global, Large Cap, Small Cap, Value, Growth) Fixed-Income (e.g., Government, Investment Grade, High Yield) Cash Equivalents (e.g., T-bills, CDs, Commercial Paper) Alternative Assets (e.g., Private Equity, Hedge Funds) Real Estate (e.g., Residential, Commercial) Collectibles (e.g., Art, Antiques)

The Strategic (or Benchmark) allocation is the proportion of wealth the investor decides to place in each of these asset classes. It is sometimes also referred to as the investor’s long-term normal allocation because it is presumed to be the “baseline” allocation that will remain in place until the investor’s life circumstances change appreciably (e.g., retirement)

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The Importance of the Asset Allocation Decision

In an influential article published in Financial Analysts Journal in July/August 1986, Gary Brinson, Randolph Hood, and Gilbert Beebower examined the issue of how important the initial strategic allocation decision was to an investor

They looked at quarterly return data for 91 pension funds over a ten-year period and decomposed the average returns as follows: Actual Overall Return (IV) Return due to Strategic Allocation (I) Return due to Strategic Allocation and Market Timing (II) Return due to Strategic Allocation and Security Selection (III)

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The Importance of the Asset Allocation Decision (cont.)

Graphically:

In terms of return performance, they found that:

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The Importance of the Asset Allocation Decision (cont.) In terms of return variation:

Ibbotson and Kaplan support this conclusion, but argue that the importance of the strategic allocation decision does depend on how you look at return variation (i.e., 40%, 90%, or 100%).

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Examples of Strategic Asset Allocations

Public Endowments:

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Examples of Strategic Asset Allocations (cont.)

Public Retirement Fund:

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Examples of Strategic Asset Allocations (cont.)

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Asset Allocation and Building an Investment Portfolio

I. Global Market Analysis

- Asset Class Allocation

- Country Allocation Within Asset Classes

II. Industry/Sector Analysis

- Sector Analysis Within Asset Classes

III. Security Analysis

- Security Analysis Within Asset Classes

and Sectors

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Asset Allocation Strategies

Strategic Asset Allocation: The investor’s “baseline” asset allocation, taking into account his or her return requirements, risk tolerance, and investment constraints.

Tactical Asset Allocation: Adjustments to the investor’s strategic allocation caused by perceived relative mis-valuations amongst the available asset classes. Ordinarily, tactical strategies overweight the undervalued asset class. Also known as market timing strategies.

Insured Asset Allocation: Adjustments to the investor’s strategic allocation caused by perceived changes in the investor’s risk tolerance. Usually, the asset class that experiences the largest relative decline is underweighted. Portfolio insurance is a well-known application of this approach.

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Sharpe’s Integrated Asset Allocation Model

C1 Capital Market Conditions

C2 Prediction Procedure

C3 Expected Returns, Risk

and Correlations

I1 Investor Assets, Liabilities

and Net Worth

I2 Investor's Risk Tolerance

Function

I3 Investor's Risk Tolerance

M1 Optimizer

M2 Investor's Asset Mix

M3 Returns

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Sharpe’s Integrated Asset Allocation Model (cont.)

Notice that the feedback loops after the performance assessment box (M3) make the portfolio management process dynamic in nature.

The strategic asset allocation process can be viewed as going through the model once and then removing boxes (C2) and (I2), thus removing any temporary adjustments to the baseline allocation.

Tactical asset allocation effectively removes box (I2), but allows for allocation adjustments due to perceived changes in capital market conditions (C2).

Insured asset allocation effectively removes box (C2), but allows for allocation adjustments due to perceived changes in investor risk tolerance conditions (I2).

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Measuring Gains from Tactical Asset Allocation

Example: Consider the following return and allocation characteristics for a portfolio consisting of stocks and bonds only.

Stock BondAllocation: Strategic 60% 40%

Actual 50 50

Returns: Benchmark 10% 7% Actual 9 8

The returns to active management (i.e., tactical and security selection) are:

Policy Performance: (.6)(.10) + (.4)(.07) = 8.80% Actual Performance: (.5)(.09) + (.5)(.08) = 8.50%

Active Return = - 30 bp

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Measuring Gains from Tactical Asset Allocation (cont.)

Also: (Policy & Timing): (.5)(.10) + (.5)(.07) = 8.50%

(Policy & Selection): (.6)(.09) + (.4)(.08) = 8.60%

so: Timing Effect: 8.50 – 8.80 = -0.30%

Selection Effect: 8.60 – 8.80 = -0.20%

Other: 8.50 – 8.60 – 8.50 + 8.80 = +0.20%

Total Active = -0.30%

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Example of Tactical Asset Allocation: Fidelity Investments

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Example of Tactical Asset Allocation: Texas TRS

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Example of Tactical Asset Allocation: Texas TRS

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Overview of Equity Style Investing

The top-down approach to portfolio formation involves prudent decision-making at three different levels:

Asset class allocation decisions Sector allocation decisions within asset classes Security selection decisions within asset class sectors

The equity style decision (e.g., large cap vs. small cap, value vs. growth) is essentially a sector allocation decision

There is tremendous variation in the returns produced by the myriad style class-specific portfolios, so investors must pay attention to this aspect of the portfolio management process

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Defining Equity Investment Style

The investment style of an equity portfolio is typically defined by two dimensions or characteristics:

- Market Capitalization (i.e., Shares Outstanding x Price)

- Relative Market Valuation (i.e., “Value” versus “Growth”)

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Equity Style Classification: Specific Terminology

Market Capitalization

- Large (> $10 billion)- Mid ($1 - $10 billion)- Small (< $1 billion)

Relative Valuation

- Value (Low P/E, Low P/B, High Dividend Yield, Low EPS Growth)

- Blend - Growth (High P/E, High P/B, Low Dividend Yield, High

EPS Growth)

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Equity Style Grid

Large-Cap

Value (LV)

Large-Cap

Blend (LB)

Large-Cap

Growth (LG)

Mid-Cap

Value (MV)

Mid-Cap

Blend (MB)

Mid-Cap

Growth (MG)

Small-Cap

Value (SV)

Small-Cap

Blend (SB)

Small-Cap

Growth (SG)

Value

Growth

Large

Small

1 - 36

Style Indexes & Representative Stock Positions: January 2005

- Russell 1000 Value

- ExxonMobil

Citigroup

- Russell 1000

- General Electric

Pfizer

- Russell 1000 Growth

- Microsoft

Wal-Mart

- Russell Mid Value

- Archer Daniels Midlan

Norfolk Southern

- Russell Mid

- Monsanto

Kroger

- Russell Mid Growth

- Apple Computer

Adobe Systems

- Russell 2000 Value

- Goodyear Tire & Rubber

Energen

- Russell 2000

- First Bancorp

Crown Holdings

- Russell 2000 Growth

- Allegheny Technologies

Aeropostale

Value

Growth

Large

Small

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Comparative Classification Ratios: January 2005(Source: Morningstar)

Fwd P/E: 15.3

P/B: 2.1

Div Yld: 2.2%

Fwd P/E: 17.8

P/B: 2.8

Div Yld: 1.6%

Fwd P/E: 21.6

P/B: 3.7

Div Yld: 0.9%

Fwd P/E: 16.1

P/B: 2.1

Div Yld: 1.8%

Fwd P/E: 18.4

P/B: 2.5

Div Yld: 1.2%

Fwd P/E: 22.9

P/B: 3.6

Div Yld: 0.4%

Fwd P/E: 13.7

P/B: 1.7

Div Yld: 1.7%

Fwd P/E: 13.1

P/B: 2.1

Div Yld: 1.1%

Fwd P/E: 12.3

P/B: 3.0

Div Yld: 0.3%

Value

Growth

Large

Small

1 - 38

Historical Equity Style Performance: 1991-2004 (Source: Frank Russell)

Style Class

Avg

Ann Ret Std Deviation

Sharpe

Ratio

LV 13.75% 13.36% 0.731

LB 12.67 14.44 0.602

LG 11.38 17.78 0.416

MV 16.01 13.42 0.896

MB 15.25 15.02 0.751

MG 14.03 21.76 0.462

SV 16.98 14.51 0.896

SB 14.58 18.40 0.576

SG 12.34 23.78 0.352

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Equity Style Rotation: 1991-2004

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

2004

SG SV SV LG LV LG LV LG MG SV SV MV SG MV

48.62% 26.52% 21.84% 3.12% 33.04% 21.72% 31.35% 35.84% 44.26% 21.54% 14.54% -8.56% 41.54% 21.93%

MG MV SB LB LB LB MV LB SG MV SB SV SB SV

41.10% 20.07% 17.86% 0.85% 32.60% 20.95% 30.55% 26.53% 39.03% 19.20% 5.15% -9.81% 40.49% 21.04%

SB SB LV SV LG LV LB MG LG MB MV LV SV MB

39.94% 18.02% 16.97% -1.11% 32.18% 20.19% 29.85% 20.53% 30.33% 9.78% 3.36% -14.84% 39.54% 19.07%

SV MB MV SB MV SV SV LV SB LV MB MB MG SB

36.59% 15.57% 14.84% -1.31% 30.54% 19.95% 28.50% 16.80% 21.01% 8.32% -3.62% -15.82% 36.84% 17.93%

MB LV MB LV MB MV LG MB LB SB SG SB MB LV

36.44% 13.28% 13.70% -1.53% 30.20% 19.00% 28.37% 12.35% 19.92% 0.52% -4.14% -20.21% 34.89% 15.60%

LG SG SG MG MG MB MB MV MB MG LV LB MV MG

36.44% 9.04% 13.34% -1.59% 29.96% 18.13% 26.60% 7.04% 17.83% -4.73% -4.95% -22.26% 33.45% 15.14%

MV LB MG MB SG MG MG SG LV LB LB MG LV SG

33.62% 8.94% 11.19% -1.60% 28.01% 17.21% 21.76% 6.79% 7.92% -6.66% -11.32% -29.08% 27.30% 14.74%

LB MG LB MV SB SB SB SB MV SG MG LG LB LB

29.99% 8.89% 9.91% -1.69% 25.73% 16.41% 21.51% 1.14% 0.70% -17.29% -15.09% -30.16% 27.00% 11.12%

LV LG LG SG SV SG SG SV SV LG LG SG LG LG

23.06% 5.19% 3.17% -1.72% 23.43% 12.78% 14.44% -4.35% -0.54% -22.07% -18.09% -32.39% 26.76% 8.96%

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Relative Return Performance:Value vs. Growth

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

40.00

50.00

1991

12

1992

06

1992

12

1993

06

1993

12

1994

06

1994

12

1995

06

1995

12

1996

06

1996

12

1997

06

1997

12

1998

06

1998

12

1999

06

1999

12

2000

06

2000

12

2001

06

2001

12

2002

06

2002

12

2003

06

2003

12

2004

06

2004

12

Ann

ualiz

ed R

etur

n D

iffer

ence

(%)

LV Outperforms

LG Outperforms

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Relative Risk Performance: Value vs. Growth

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

1991

12

1992

06

1992

12

1993

06

1993

12

1994

06

1994

12

1995

06

1995

12

1996

06

1996

12

1997

06

1997

12

1998

06

1998

12

1999

06

1999

12

2000

06

2000

12

2001

06

2001

12

2002

06

2002

12

2003

06

2003

12

2004

06

2004

12

Ann

ualiz

ed R

isk

Diff

eren

ce (%

) LV Riskier

LG Riskier

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Relative Return Performance: Large Cap vs. Small Cap

-50.00

-40.00-30.00

-20.00-10.00

0.00

10.0020.00

30.0040.00

50.00

1991

1219

9206

1992

1219

9306

1993

1219

9406

1994

1219

9506

1995

1219

9606

1996

1219

9706

1997

1219

9806

1998

1219

9906

1999

1220

0006

2000

1220

0106

2001

1220

0206

2002

1220

0306

2003

1220

0406

2004

12

Ann

ual

ized

Ret

urn

Diff

eren

ce (%

)

LB Outperforms

SB Outperforms

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Relative Risk Performance:Large Cap vs. Small Cap

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

1991

1219

9206

1992

1219

9306

1993

1219

9406

1994

1219

9506

1995

1219

9606

1996

1219

9706

1997

1219

9806

1998

1219

9906

1999

1220

0006

2000

1220

0106

2001

1220

0206

2002

1220

0306

2003

1220

0406

2004

12

Ann

ualiz

ed R

isk

Diff

eren

ce (%

) LB Riskier

SB Riskier

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Value vs. Growth: Global Evidence (Source: Chan and Lakonishok, Financial Analysts Journal, 2004)

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Equity Style Investing: Instruments and Strategies

Passive Style Alternatives- Index Mutual Funds- Exchange-Traded Funds (ETFs)

Active Style Alternatives- Investor Portfolio Formation- Open-Ended Mutual Funds

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Methods of Indexed Investing

Open-End Index Mutual Funds: There is a long-standing and active market for mutual funds that hold broad collections of securities that mimic various sectors of the stock market. Examples include the Vanguard 500 Index Fund, which recreates the holdings and weightings of the Standard & Poor’s 500, and the various Fidelity Select Funds, which reproduce the profiles of different industry sectors.

Exchange-Traded Funds (ETF): A more recent development in the

world of indexed investment products has been the development of exchange-tradable index funds. Essentially, ETFs are depository receipts that give investors a pro-rata claim on the capital gains and cash flows of the securities held in deposit.

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Index Fund Example: VFINX

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Index Fund Example (cont.)

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Top ETFs in the Large Blend Style Category

Name Category Style

Box YTD

Return % 1 mo

Return % 3 mo

Return % 1 yr

Return % 3 yr

Return % Trading Volume

Consumer Staples Select Sector SPDR (XLP) Large Blend 0.91 0.91 7.18 9.53 -0.79 1,183,000

I ndustrial Select Sector SPDR (XLI ) Large Value -4.06 -4.06 4.98 11.68 5.81 452,700

iShares Dow J ones US Cons Goods (I YK) Large Blend -0.36 -0.36 10.52 10.83 8.48 61,500

iShares Dow J ones US I ndustrial (I YJ ) Large Blend -3.98 -3.98 5.06 10.06 4.85 101,300

iShares Dow J ones US Total Market I nd (I YY) Large Blend -3.40 -3.40 5.11 6.01 3.70 22,100

iShares Morningstar Large Core I ndex (J KD) Large Blend -3.01 -3.01 5.54 --- --- 6,400

iShares NYSE Composite I ndex (NYC) Large Blend -3.29 -3.29 5.81 --- --- 600

iShares Russell 1000 I ndex (IWB) Large Blend -2.99 -2.99 4.77 5.84 3.42 136,900

iShares Russell 3000 I ndex (IWV) Large Blend -3.56 -3.56 4.58 5.45 3.78 166,000

iShares S&P 1500 Index (I SI ) Large Blend -3.57 -3.57 4.55 6.31 --- 33,400

iShares S&P 500 I ndex (IVV) Large Blend -3.11 -3.11 4.21 5.25 2.77 731,700

Rydex S&P Equal Weight (RSP) Large Blend -4.16 -4.16 5.71 9.22 --- 43,400

SPDRs (SPY) Large Blend -2.85 -2.85 4.52 5.57 2.76 60,817,300

streetTRACKS DJ Global Titans (DGT) Large Blend -2.67 -2.67 4.27 3.01 0.87 1,600

streetTRACKS Fortune 500 I ndex (FFF) Large Blend -3.12 -3.12 4.98 5.63 2.55 8,700

Vanguard Consumer Staples VI PERs (VDC) Large Blend 0.97 0.97 9.29 11.77 --- 58,600

Vanguard Large Cap VI PERs (VV) Large Blend -3.84 -3.84 4.66 5.66 --- 2,400

Vanguard Total Stock Market VIPERs (VTI ) Large Blend -3.48 -3.48 4.87 6.12 4.46 85,700

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ETF Example: SPY

1 - 51

ETF Example (cont.)

1 - 52

Growth of U.S. Equity Mutual Funds

Morningstar Mutual Fund Style Category:

Year LV LB LG MV MB MG SV SB SG Total

1991 133 158 117 60 46 79 25 29 42 689 1992 138 166 119 60 48 78 28 30 44 711 1993 154 178 124 65 53 78 31 30 49 762 1994 166 197 137 67 54 82 38 37 59 837 1995 211 238 174 69 62 106 47 52 77 1036 1996 269 303 228 87 69 150 62 71 112 1351 1997 346 371 293 102 95 183 79 97 152 1718 1998 406 436 351 126 101 221 96 123 206 2066 1999 500 571 421 167 121 288 120 147 262 2597 2000 615 778 651 212 140 415 193 201 383 3588 2001 750 1036 808 259 209 559 230 228 472 4551 2002 813 1198 1061 269 268 670 259 299 570 5407 2003 946 1407 1245 301 349 784 270 384 672 6358

Source: Morningstar, Frank Russell

1 - 53

Mutual Fund Performance Characteristics:1991-2003

Style Group

Avg. Annual

Fund Return (%)

Avg. Fund

Std. Dev. (%)

Sharpe Ratio

Large Value 11.81 13.03 0.587 Large Blend 11.43 13.98 0.520 Large Growth 12.44 17.57 0.471 Mid Value 13.85 13.09 0.740 Mid Blend 13.88 14.91 0.652 Mid Growth 14.57 21.35 0.488 Small Value 16.18 14.33 0.839 Small Blend 14.89 16.00 0.671 Small Growth 15.86 22.81 0.513

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Mutual Fund Performance Characteristics:1991-2003 (cont.)

Style Group

Avg. Fund Firm

Size ($MM)

Avg. Fund Expense Ratio

(%)

Avg. Fund

Turnover (%)

Median Tracking Error

(%) Large Value 24,966 1.41 69.24

4.72

Large Blend 38,137 1.32 73.32

4.24

Large Growth 37,596 1.54 100.41

6.28

Mid Value 6,672 1.56 87.06

6.60

Mid Blend 7,848 1.48 90.58

6.78

Mid Growth 5,924 1.64 135.76

7.88

Small Value 1,710 1.56 67.50

7.12

Small Blend 2,596 1.66 88.21

8.32

Small Growth 1,119 1.70 120.02

8.07

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Notion of Tracking Error

When managing an active investment portfolio against a well-defined benchmark (such as the Standard & Poor’s 500 or the IPSA index), the goal of the manager should be to generate a return that exceeds that of the benchmark while minimizing the portfolio’s return volatility relative to the benchmark. Said differently, the manager should try to maximize alpha while minimizing tracking error. Tracking error can be defined as the extent to which return fluctuations in the managed portfolio are not correlated with return fluctuations in the benchmark. The concept is analogous to the statistic (1 – R2) in a regression context. A flexible and straightforward way of measuring tracking error can be developed as follows: Let: wi = investment weight of asset i in the managed portfolio Rit = return to asset i in period t Rbt = return to the benchmark portfolio in period t. With these definitions, we can define the period t return to managed portfolio as:

N

iitR

1ipt w R

where: N = number of assets in the managed portfolio and:

1 w1

i

N

i

(i.e., the managed portfolio is fully invested).

1 - 56

Notion of Tracking Error (cont.)

We can then specify the period t return differential between the managed portfolio and the benchmark as:

.R - R R - w btptbt1

it

N

iitR

Notice two things about the return differential . First, given the returns to the N assets in the managed portfolio and the benchmark, it is a function of the investment weights that the manager selects (i.e., = f({wi}/{Ri}, Rb)). Second, can be interpreted as the return to a hedge portfolio where wb = -1. With these definitions and a sample of T return observations, calculate the variance of as follows:

.1) - (T

) - (

T

1t

2t

2

Then, the standard deviation of the return differential is:

2 = periodic tracking error,

so that annualized tracking error (TE) can be calculated as:

TE = P

where P is the number of return periods in a year (e.g., P = 12 for monthly returns, P = 252 for daily returns).

1 - 57

Notion of Tracking Error (cont.)

Generally speaking, portfolios can be separated into the following categories by the level of their annualized tracking errors:

Passive (i.e., Indexed): TE < 1.0% (Note: TE < 0.5% is normal)

Structured: 1.0% < TE < 3%

Active: TE > 3% (Note: TE > 5% is normal for active managers)

1 - 58

“Large Blend” Active Manager: DGAGX

1 - 59

Tracking Errors for VFINX, SPY, DGAGX

1 - 60

Risk and Expected Return Within a Portfolio

Portfolio Theory begins with the recognition that the total risk and expected return of a portfolio are simple extensions of a few basic statistical concepts.

The important insight that emerges is that the risk characteristics of a portfolio become distinct from those of the portfolio’s underlying assets because of diversification. Consequently, investors can only expect compensation for risk that they cannot diversify away by holding a broad-based portfolio of securities (i.e., the systematic risk)

Expected Return of a Portfolio:

where wi is the percentage investment in the i-th asset

Risk of a Portfolio:

Total Risk = (Unsystematic Risk) + (Systematic Risk)

E(R ) = w E(R )p ii = 1

n

i *

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

2n

21

21

2p

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Example of Portfolio Diversification: Two-Asset Portfolio

Consider the risk and return characteristics of two stock positions:

Risk and Return of a 50%-50% Portfolio:

E(Rp) = (0.5)(5) + (0.5)(6) = 5.50%and:

p = [(.25)(64) + (.25)(100) + 2(.5)(.5)(8)(10)(.4)]1/2 = 7.55%

Note that the risk of the portfolio is lower than that of either of the individual securities

E(R1) = 5% 1 = 8% 1,2 = 0.4

E(R2) = 6% 2 = 10%

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Another Two-Asset Class Example:

Suppose that a portfolio is divided into two different subportfolios consisting of stocks and bonds, respectively. Further assume that the subportfolios have the following risk and expected return characteristics: E(Rstock) = 12.0% stock = 21.2% = 0.18 E(Rbond) = 5.1% bond = 8.3% Then, an overall portfolio consisting of a 60%-40% mix of stocks and bonds would have the following characteristics:

E(Rp) = (0.6)(0.120) + (0.4)(0.051) = 0.0924 or 9.24% and

p = [(0.6)2(0.212)2 + (0.4)2(0.083)2] + [2(0.6)(0.4)(0.212)(0.083)(0.18)] = 0.0188

or p = (0.0188)1/2 = 0.1371 or 13.71%

For different asset mixes and different levels of correlation between stocks and bonds, the portfolio variance is given as: ( = 0.18) ( = 1) ( = -1) Portfolio wstock wbond E(Rp) p p p 1 0.00 1.00 5.10% 8.30% 8.30% 8.30% 2 0.25 0.75 6.83 8.87 11.53 0.93 3 0.28 0.72 7.04 9.16 11.93 0.00 4 0.40 0.60 7.86 10.58 13.46 3.50 5 0.50 0.50 8.55 12.06 14.75 6.45 6 0.75 0.25 10.28 16.40 17.98 13.83 7 1.00 0.00 12.00 21.20 21.20 21.20

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Example of a Three-Asset Portfolio:

Suppose that a portfolio is divided into three different subportfolios consisting of stocks, bonds, and cash equivalents, respectively. Further assume that the subportfolios have the following risk and expected return characteristics: E(Rstock) = 12.0% stock = 21.2% stock,bond = 0.18 E(Rbond) = 5.1% bond = 8.3% cash,stock = -0.07 E(Rcash) = 3.6% cash = 3.3% cash,bond = 0.22 Then, an overall portfolio consisting of a 60%-30%-10% mix of stocks, bonds, and cash equivalents would have the following characteristics:

E(Rp) = (0.6)(0.120) + (0.3)(0.051) + (0.1)(0.036) = 0.0909 or 9.09% and: p = [(0.6)2(0.212)2 + (0.3)2(0.083)2 + (0.1)2(0.033)2] +

{[2(0.6)(0.3)(0.212)(0.083)(0.18)] + [2(0.6)(0.1)(0.212)(0.033)(-0.07)]

+ [2(0.3)(0.1)(0.083)(0.033)(0.22)]} = 0.01793

or p = (0.01793)1/2 = 0.1339 or 13.39%

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Diversification and Portfolio Size: Graphical Interpretation

Total Risk

Portfolio Size1 20 40

0.20

0.40

Systematic Risk

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Advanced Portfolio Risk CalculationsTotal Portfolio Risk Suppose you have formed a portfolio consisting of N asset classes. Suppose also the portfolio

weight in the j-th asset class is denoted as wj while jk represents the covariance between assets j

and k (where jk equals the variance, j2, when j = k). With this notation, the return variance, p

2,

of the portfolio is given by:

N

1j

N

1kjkkj

2p σww σ (1)

or, equivalently:

N

1j

N

kj1k

jkkj

N

1j

2j

2j

2p σww σw σ (2)

The standard deviation of the portfolio is:

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2jp σww σw σ

(3)

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Advanced Portfolio Risk Calculations (cont.)Marginal Asset Risk

In order to compute the contribution of asset k’s risk to the overall risk of the portfolio, we can take

the derivative of equation (3) with respect to asset k’s weight in the portfolio:

N

kj1j

jkj2kk

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2j

k

p σw2 σw2 σww σw 2

1

w

σ

=

N

kj1j

jkj2kk

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2j σw σw σww σw

=

N

kj1j

jkj2kk

1/2-2p σw σw σ

which can be simplified to:

N

1jjkkjj

p

N

1jjkj

pk

p ρσσw σ

1 σw

σ

1

w

σ (4)

where j and k are the standard deviations of asset classes j and k, respectively, and jk is the

correlation coefficient between them.

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Advanced Portfolio Risk Calculations (cont.)

Equation (4) shows that the marginal volatility of asset k in a portfolio is the weighted sum of the k-

th row (or, equivalently, the k-th column) of the return covariance matrix divided by the standard

deviation of the portfolio. Notice that the magnitude of this marginal risk contribution is determined

by three factors: (i) the volatility of the asset itself, (ii) the asset’s weight in the portfolio, and (iii) the

asset’s covariance with all of the other portfolio holdings and their investment weights.

A convenient property of marginal volatilities is that the weighted sum over all assets is, in fact, the

overall volatility of the portfolio. By contrast, recall that the standard deviation of a portfolio is not

simply a weighted average of the standard deviations of the underlying assets whenever the

correlations between the asset classes are less that +1.0. However, by redefining the risk of asset k

within the portfolio taking those correlations into account—which is what equation (4) does—it is

possible to view overall portfolio risk as an additive statistic. To see this notice that:

N

1jjkj

p

N

1kk

k

pN

1kk σw

σ

1 w

w

σw

= pp

2p

N

1j

N

1kjkkj

p

σ σ

σ σww

σ

1

(5)

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Advanced Portfolio Risk Calculations (cont.)This feature provides a mechanism for the portfolio manager to “roll-up” marginal volatilities to a

higher level (e.g., the sector or country level) without having to recompute the derivatives. In other

words, assume that the marginal volatility of each of the N assets has been calculated. Assume also

that we are interested in knowing the aggregate marginal volatility of a collection of M assets where k

= 1 to M < N (i.e., the M assets comprise a subset of the total portfolio). The marginal volatility of

this sub-portfolio is given by:

M

1kk

M

1k k

pk

M

p

w

w

σw

w

σ (6)

This additive property also allows the portfolio manager to interpret the weighted marginal volatilities

directly as the asset’s contribution to overall portfolio risk or as the contribution to tracking error if

asset class returns are defined in excess of the returns to a benchmark. That is:

Asset k’s Marginal Risk: k

p

w

σ

(7)

Asset k’s Total Contribution to Risk: k

pk w

σw

(8)

Once again, equations (7) and (8) highlight two facts: (i) Asset k’s marginal volatility within the

portfolio depends not only on its own inherent riskiness (i.e. k) but also how it interacts with every

other asset held in the portfolio (i.e., jk), and (ii) Asset k’s total contribution to the risk of the overall

portfolio also depends on how much the manager invests in that asset class (wk).

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Example of Marginal Risk Contribution Calculations

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Fidelity Investment’s PRISM Risk-Tracking System: Chilean Pension System – March 2004

PRISM (CR) / Return, Volatility and Tracking Error for 200401 March 17, 2004 Mean Mean AFP Tracking Portfolio Excess Obs AFP Id Assets Volatility Error Return Return 1 LLLL 1 733 0.081731 0.003646 0.23543 0.004531 2 MMMM 2 738 0.081423 0.003514 0.23476 0.003862 3 NNNN 3 635 0.080780 0.004187 0.22947 -0.001425 4 PPPP 4 257 0.077193 0.004713 0.21804 -0.012857 5 QQQQ 5 469 0.079808 0.003791 0.22660 -0.004296 6 RRRR 6 54 0.073797 0.008308 0.22981 -0.001082 7 SSSS 7 31 0.061490 0.020927 0.20014 -0.030757 8 SISTEMA 8 2918 0.080525 0.000000 0.23089 0.000000

PRISM (CY) / AFP Value at Risk for 200401 March 17, 2004 50bp 200bp Tracking Shortfall Shortfall Error Probability Probability Obs AFP Assets (%) (%) (%) 1 LLLL 733 0.3646 8.5130 0.0000 2 MMMM 738 0.3514 7.7410 0.0000 3 NNNN 635 0.4187 11.6213 0.0001 4 PPPP 257 0.4713 14.4383 0.0011 5 QQQQ 469 0.3791 9.3576 0.0000 6 RRRR 54 0.8308 27.3640 0.8034 7 SSSS 31 2.0927 40.5582 16.9613 8 SISTEMA 2918 0.0000 . .

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Chilean Sistema Risk Tracking Example (cont.)PRISM (CX) / Risk Diagnostics March 17, 2004 1 Sistema-Relative Tracking Error for 200401 Contribution Active Worst Case to Implied Weight Contribution Tracking Error View Obs AFP Asset Class (%) (%) (%) (%) 1 LLLL PRD_BCD -0.2144 -0.0194 0.0023 -1.0527 2 LLLL PDBC_PRBC -1.1800 -0.0331 -0.0097 0.8206 3 LLLL BCP 0.1830 0.0111 -0.0003 -0.1709 4 LLLL RECOGN_BOND -1.2731 -0.1965 0.1040 -8.1706 5 LLLL CERO -0.2823 -0.0026 0.0002 -0.0617 6 LLLL PRC_BCU -1.1605 -0.0822 0.0193 -1.6626 7 LLLL BCE 0.0385 0.0011 0.0003 0.8206 8 LLLL PCD_PTF -0.0002 -0.0000 -0.0000 0.8206 9 LLLL ZERO -0.0184 -0.0017 0.0002 -1.0363 10 LLLL DEPOSITS_OFFNO 0.4018 0.0041 0.0001 0.0347 11 LLLL DEPOSITS_OFFUF 3.7074 0.1039 0.0304 0.8206 12 LLLL LHF -0.9636 -0.0363 0.0074 -0.7657 13 LLLL BEF -0.0080 -0.0011 0.0002 -2.4299 14 LLLL BSF -0.0395 -0.0029 0.0005 -1.3547 15 LLLL CC2 -0.2398 0.0000 0.0000 0.0000 16 LLLL CFI -0.3106 -0.0212 0.0004 -0.1428 17 LLLL CORPORATE_BOND -0.1932 -0.0084 0.0006 -0.3317 18 LLLL CTC_A -0.3847 -0.0826 0.0148 -3.8358 19 LLLL ENDESA -0.0505 -0.0095 0.0001 -0.2352 20 LLLL COPEC 0.1789 0.0424 0.0031 1.7448 21 LLLL ENTEL -0.0672 -0.0179 0.0015 -2.1698 22 LLLL CMPC 0.2043 0.0436 0.0104 5.1077 23 LLLL SQM-B -0.1966 -0.0356 -0.0010 0.5129 24 LLLL CERVEZAS 0.0069 0.0018 -0.0001 -1.2372 25 LLLL COLBUN 0.0184 0.0032 -0.0001 -0.4791 26 LLLL D&S -0.4237 -0.1334 0.0127 -2.9882 27 LLLL ENERSIS 0.3731 0.1107 0.0286 7.6550 28 LLLL Chile_SMALL_CAP 0.6830 0.0677 0.0039 0.5649 29 LLLL US 0.8959 0.1332 0.0607 6.7785 30 LLLL EUROPE 0.0364 0.0063 0.0022 6.0088 31 LLLL UK -0.0954 -0.0158 -0.0064 6.7521 32 LLLL JAPAN 0.0710 0.0149 0.0029 4.0322 33 LLLL GLOBAL 0.8999 0.1219 0.0518 5.7571 34 LLLL ASIA_EX_JAPAN -2.1238 -0.3475 -0.0165 0.7754 35 LLLL LATIN_AMERICA -0.5469 -0.0898 -0.0272 4.9767 36 LLLL EASTERN_EUROPE -0.1620 -0.0299 -0.0061 3.7790 37 LLLL EMERGING_MARKETS 1.8244 0.2729 0.0638 3.4975 38 LLLL G7_BOND 0.1123 0.0062 0.0008 0.7247 39 LLLL HY 0.0364 0.0041 0.0011 3.0627 40 LLLL EMB 0.2535 0.0304 0.0069 2.7101 41 LLLL MM 0.0094 0.0008 0.0002 2.5109 ========== ============ ============== 0.0000 -0.1870 0.3640

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Chilean Sistema Risk Tracking Example (cont.)PRISM (CX) / Risk Diagnostics March 17, 2004 4 Sistema-Relative Tracking Error for 200401 Contribution Active Worst Case to Implied Weight Contribution Tracking Error View Obs AFP Asset Class (%) (%) (%) (%) 1 PPPP PRD_BCD 0.5674 0.0515 0.0224 3.9475 2 PPPP PDBC_PRBC -1.1800 -0.0331 0.0101 -0.8560 3 PPPP BCP -0.5650 -0.0343 0.0010 -0.1737 4 PPPP RECOGN_BOND -0.2379 -0.0367 0.0009 -0.3819 5 PPPP CERO -0.2119 -0.0020 0.0001 -0.0523 6 PPPP PRC_BCU 0.8570 0.0607 0.0138 1.6098 7 PPPP BCE -0.0128 -0.0004 0.0001 -0.8560 8 PPPP PCD_PTF -0.0002 -0.0000 0.0000 -0.8560 9 PPPP ZERO -0.0184 -0.0017 -0.0006 3.3578 10 PPPP DEPOSITS_OFFNO 6.9872 0.0717 0.0131 0.1876 11 PPPP DEPOSITS_OFFUF -3.3930 -0.0951 0.0290 -0.856 12 PPPP LHF 1.2495 0.0471 0.0084 0.6692 13 PPPP BEF -0.0080 -0.0011 -0.0002 1.9925 14 PPPP BSF -0.0790 -0.0057 -0.0010 1.2991 15 PPPP CC2 -0.2401 0.0000 0.0000 0.0000 16 PPPP CFI -1.2552 -0.0858 0.0207 -1.6488 17 PPPP CORPORATE_BOND -0.6841 -0.0298 -0.0005 0.0663 18 PPPP CTC_A -0.4095 -0.0880 0.0246 -6.0021 19 PPPP ENDESA -0.1138 -0.0213 0.0045 -3.9277 20 PPPP COPEC -0.0081 -0.0019 0.0001 -0.8357 21 PPPP ENTEL 0.0744 0.0198 -0.0017 -2.2350 22 PPPP CMPC -0.0372 -0.0079 0.0007 -2.0070 23 PPPP SQM-B 0.0650 0.0118 -0.0013 -2.0004 24 PPPP CERVEZAS 0.0145 0.0038 0.0001 0.4622 25 PPPP COLBUN 0.1316 0.0232 -0.0009 -0.6567 26 PPPP D&S -0.4237 -0.1334 0.0434 -10.2519 27 PPPP ENERSIS 0.2388 0.0708 -0.0028 -1.1930 28 PPPP Chile_SMALL_CAP 0.8843 0.0876 -0.0165 -1.8696 29 PPPP US -0.1981 -0.0295 0.0147 -7.4319 30 PPPP EUROPE -0.2826 -0.0492 0.0297 -10.5277 31 PPPP UK -0.0419 -0.0069 0.0038 -8.9775 32 PPPP JAPAN 0.0953 0.0200 -0.0090 -9.4234 33 PPPP GLOBAL 0.1448 0.0196 -0.0122 -8.3883 34 PPPP ASIA_EX_JAPAN 0.0479 0.0078 -0.0051 -10.5855 35 PPPP LATIN_AMERICA -0.5673 -0.0931 0.0635 -11.1915 36 PPPP EASTERN_EUROPE -0.3052 -0.0563 0.0318 -10.4358 37 PPPP EMERGING_MARKETS -1.7698 -0.2647 0.2148 -12.135 38 PPPP G7_BOND -0.1264 -0.0070 0.0005 -0.3814 39 PPPP HY 0.3431 0.0386 -0.0167 -4.8702 40 PPPP EMB -0.0563 -0.0068 0.0023 -4.1709 41 PPPP MM 0.5247 0.0460 -0.0150 -2.863 ========== ============ ============== -0.0000 -0.5113 0.4708

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Notion of Downside Risk Measures: As we have seen, the variance statistic is a symmetric measure of

risk in that it treats a given deviation from the expected outcome the same regardless of whether that deviation is positive of negative.

We know, however, that risk-averse investors have asymmetric profiles; they consider only the possibility of achieving outcomes that deliver less than was originally expected as being truly risky. Thus, using variance (or, equivalently, standard deviation) to portray investor risk attitudes may lead to incorrect portfolio analysis whenever the underlying return distribution is not symmetric.

Asymmetric return distributions commonly occur when portfolios contain either explicit or implicit derivative positions (e.g., using a put option to provide portfolio insurance).

Consequently, a more appropriate way of capturing statistically the subtleties of this dimension must look beyond the variance measure.

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Notion of Downside Risk Measures (cont.):

We will consider two alternative risk measures: (i) Semi-Variance, and (ii) Lower Partial Moments

Semi-Variance: The semi-variance is calculated in the same manner as the variance statistic, but only the potential returns falling below the expected return are used:

Lower Partial Moment: The lower partial moment is the sum of the weighted deviations of each potential outcome from a pre-specified threshold level (), where each deviation is then raised to some exponential power (n). Like the semi-variance, lower partial moments are asymmetric risk measures in that they consider information for only a portion of the return distribution. The formula for this calculation is given by:

- = R

n p pn

p

)R - ( p = LPM

E(R)

- = R

2 p p

p

E(R)) -(R p = Variance-Semi

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Example of Downside Risk Measures:

To see how these alternative risk statistics compare to the variance consider the following probability distributions for two investment portfolios: Potential Prob. of Return for Prob. of Return for Return Portfolio #1 Portfolio #2 -15% 5% 0% -10 8 0 -5 12 25 0 16 35 5 18 10 10 16 7 15 12 9 20 8 5 25 5 3 30 0 3 35 0 3 Notice that the expected return for both of these portfolios is 5%: E(R)1 = (.05)(-0.15) + (.08)(-0.10) + ...+ (.05)(0.25) = 0.05

and

E(R)2 = (.25)(-0.05) + (.35)(0.00) + ...+ (.03)(0.35) = 0.05

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Example of Downside Risk Measures (cont.):

Clearly, however, these portfolios would be viewed differently by different investors. These nuances are best captured by measures of return dispersion (i.e., risk).

1. Variance As seen earlier, this is the traditional measure of risk, calculated the sum of the weighted squared differences of the potential returns from the expected outcome of a probability distribution. For these two portfolios the calculations are: (Var)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + ... + (.05)[0.25 - 0.05]2 = 0.0108 and (Var)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 + ... + (.03)[0.35 - 0.05]2 = 0.0114 Taking the square roots of these values leaves:

SD1 = 10.39% and SD2 = 10.65%

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Example of Downside Risk Measures (cont.):

2. Semi-Variance The semi-variance adjusts the variance by considering only those potential outcomes that fall below the expected returns. For our two portfolios we have: (SemiVar)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + (.12)[-0.05 - 0.05]2 +

(.16)[0.00 - 0.05]2 = 0.0054

and

(SemiVar)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 = 0.0034 Also, the semi-standard deviations can be derived as the square roots of these values:

(SemiSD)1 = 7.35% and (SemiSD)2 = 5.81% Notice here that although Portfolio #2 has a higher standard deviation than Portfolio #1, it's semi-standard deviation is smaller.

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Example of Downside Risk Measures (cont.):

3. Lower Partial Moments For these two portfolios, we will consider two cases (n = 1 and n = 2), both having a threshold level of 0% (i.e., = 0):

(i) LPM1 (LPM1)1 = (.05)[0.00 - (-0.15)] + (.08)[0.00 - (-0.10)] + (.12)[0.00 - (-0.05)] = 0.0215 and

(LPM1)2 = (.25)[0.00 - (-0.05)] = 0.0125

(ii) LPM2 (LPM2)1 = (.05)[0.00 - (-0.15)]2 + (.08)[0.00 - (-0.10)]2 + (.12)[0.00 - (-0.05)]2

= 0.0022 and

(LPM2)2 = (.25)[0.00 - (-0.05)]2 = 0.0006

For comparative purposes, it is also useful to take the square root of the LPM2 values. These are:

(SqRt LPM2)1 = 4.72% and (SqRt LPM2)1 = 2.50%

Notice again that Portfolio #2 is seen as being less risky when the lower partial moment risk measures are used.

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Overview of the Portfolio Optimization Process

The preceding analysis demonstrates that it is possible for investors to reduce their risk exposure simply by holding in their portfolios a sufficiently large number of assets (or asset classes). This is the notion of naïve diversification, but as we have seen there is a limit to how much risk this process can remove.

Efficient diversification is the process of selecting portfolio holdings so as to: (i) minimize portfolio risk while (ii) achieving expected return objectives and, possibly, satisfying other constraints (e.g., no short sales allowed). Thus, efficient diversification is ultimately a constrained optimization problem. We will return to this topic in the next session.

Notice that simply minimizing portfolio risk without a specific return objective in mind (i.e., an unconstrained optimization problem) is seldom interesting to an investor. After all, in an efficient market, any riskless portfolio should just earn the risk-free rate, which the investor 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 following non-linear, constrained optimization problem:

Select {wi} so as to minimize:

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

(ii) wi = 1

The first constraint is the investor’s return goal (i.e., R*). The second constraint simply states that the total investment across all 'n' asset classes must equal 100%. (Notice that this constraint allows any of the wi to 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 wi < P, where P is a fixed percentage

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

2n

21

21

2p

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Example of Mean-Variance Optimization: (Three Asset Classes, Short Sales Allowed)

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Example of Mean-Variance Optimization: (Three Asset Classes, No Short Sales)

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Mean-Variance Efficient Frontier With and Without Short-Selling

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Efficient Frontier Example: Five Asset Classes

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Example of Mean-Variance Optimization: (Five Asset Classes, No Short Sales)

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Efficient Frontier Example: 2003 Texas Teachers’ Retirement System

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Efficient Frontier Example: Texas Teachers’ Retirement System (cont.)

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Efficient Frontier Example: Texas Teachers’ Retirement System (cont.)

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Efficient Frontier Example: Chilean Pension System (Source: Fidelity Investments)

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond US Stock US Bond

Risk Premium 7.19% 2.65% 0.00% 4.89% 1.66% 5.83% 1.41%

Real Cash Return 0.60% 0.60% 0.60% 0.60% 0.60% 0.60% 0.60%

Expected Real Return 7.79% 3.25% 0.60% 5.49% 1.66% 6.43% 2.01%

Volatility 25.02% 6.75% 1.50% 12.57% 3.33% 14.75% 5.05%

Base Case Assumptions:

-Expected real returns based on 1954 – 2003 risk premiums

-Real returns for developed market stocks and bonds areGDP-weighted excluding US (equally-weighted returns for stocks and bonds are 5.73% and 1.39%, respectively)

- Chilean risk-premium volatility estimates exclude the period 1/72 – 12/75

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Efficient Frontier Example: Chilean Pension System (cont.)

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond US Stock US Bond

Domestic Stocks 100.00% 21.85% 13.51% 35.28% -20.91% 38.78% -23.13%Domestic Bonds 100.00% 31.04% 2.71% -0.98% -1.39% 2.37%Domestic Cash 100.00% 1.79% 10.31% 6.27% 3.77%DM Stocks 100.00% 26.01% 71.23% 11.06%DM Bonds 100.00% 7.54% 73.19%US Stocks 100.00% 16.56%US Bonds 100.00%

- Correlation matrix is based on real returns from the period 1/93 – 6/03 using Chilean inflation and based in Chilean pesos

- Real returns for developed market stocks and bonds areGDP-weighted excluding US

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Efficient Frontier Example: Chilean Pension System (cont.)

Unconstrained Frontier:

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Efficient Frontier Example: Chilean Pension System (cont.)

Fund A Fund B Fund C Fund D Fund EChile Stock 60% 50% 30% 15% 0%Chile Bond 40% 40% 50% 70% 80%Chile Cash 40% 40% 50% 70% 80%

All Foreign Investments 30% 30% 30% 30% 30%Min Total Equity 40% 25% 15% 5% 0%Max Total Equity 80% 60% 40% 20% 0%

Constraint Set:

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Efficient Frontier Example: Chilean Pension System (cont.)

Constrained Frontier for Fund A:

Point on EF

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond US Stock US Bond

Expected Return Volatility

1 10.0% 20.0% 40.0% 24.2% 0.0% 5.8% 0.0% 3.37 5.65 2 10.6% 21.0% 38.4% 22.3% 0.2% 7.2% 0.3% 3.43 5.73 3 11.3% 22.4% 36.3% 20.7% 0.5% 8.2% 0.6% 3.51 5.85 4 12.1% 23.8% 34.1% 19.3% 0.7% 9.0% 0.9% 3.59 6.00 5 13.2% 24.9% 32.1% 18.0% 0.9% 9.6% 1.3% 3.68 6.18 6 14.4% 25.7% 30.2% 16.9% 1.1% 10.0% 1.7% 3.76 6.38 7 15.7% 26.3% 28.3% 15.9% 1.4% 10.3% 2.0% 3.85 6.62 8 17.2% 26.9% 26.4% 15.0% 1.6% 10.6% 2.3% 3.95 6.89 9 18.7% 27.2% 24.5% 14.2% 1.9% 10.9% 2.6% 4.05 7.19

10 20.3% 27.5% 22.7% 13.5% 2.1% 11.0% 2.8% 4.16 7.51 11 22.1% 27.7% 20.9% 13.0% 2.3% 11.1% 3.0% 4.27 7.86 12 23.9% 27.8% 19.3% 12.4% 2.4% 11.1% 3.2% 4.38 8.23 13 25.8% 27.8% 17.8% 11.9% 2.3% 11.2% 3.3% 4.49 8.63 14 27.8% 27.6% 16.4% 11.4% 2.3% 11.3% 3.2% 4.61 9.06 15 29.9% 27.2% 15.3% 10.9% 2.1% 11.4% 3.2% 4.74 9.53 16 32.3% 26.7% 14.1% 10.5% 2.0% 11.5% 3.1% 4.87 10.04 17 34.8% 25.9% 12.9% 10.1% 1.8% 11.5% 2.9% 5.01 10.61 18 37.6% 25.1% 11.6% 9.6% 1.7% 11.6% 2.8% 5.17 11.23 19 40.9% 24.2% 10.3% 8.8% 1.4% 11.6% 2.7% 5.34 11.97 20 45.6% 23.0% 8.0% 7.9% 1.2% 12.2% 2.2% 5.63 13.08

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Efficient Frontier Example: Chilean Pension System (cont.)

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond

US Stock

US Bond

Expected Return Volatility

Unconstrained 42.8% 11.2% 0.0% 15.2% 1.6% 26.0% 3.2% 6.3% 13.9%Fund A 45.6% 23.0% 8.0% 7.9% 1.2% 12.2% 2.2% 5.6% 13.1%Fund B 35.5% 32.2% 10.1% 6.1% 1.7% 11.1% 3.3% 5.0% 10.6%Fund C 18.6% 40.3% 15.7% 6.0% 2.5% 10.7% 6.2% 4.0% 6.9%Fund D 6.5% 54.9% 14.5% 4.9% 5.4% 6.7% 7.1% 3.3% 4.9%Fund E 0.0% 63.8% 15.3% 0.0% 7.4% 0.0% 13.5% 2.6% 4.5%

Asset Allocations of Various Funds Using Point 20 on Unconstrained Frontier:

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Example of Mean-Lower Partial Moment Portfolio Optimization:(Five Asset Classes, No Short Sales)

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Estimating the Expected Returns and Measuring Superior Investment Performance

We can use the concept of “alpha” to measure superior investment performance:

= (Actual Return) – (Expected Return) = “Alpha”

In an efficient market, alpha should be zero for all investments. That is, securities should, on average, be priced so that the actual returns they produce equal what you expect them to given their risk levels.

Superior managers are defined as those investors who can deliver consistently positive alphas after accounting for investment costs

The challenge in measuring alpha is that we have to have a model describing the expected return to an investment.

Researchers typically use one of two models for estimating expected returns: Capital Asset Pricing Model Multi-Factor Models (e.g., Fama-French Three-Factor Model)

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Developing the Capital Asset Pricing Model

Recall that one of the most fundamental notions in all of finance is that an investor’s expected return can be expressed in terms of these activities: E(R) = (Risk-Free Rate) + (Risk Premium) Clearly, the practical challenge in measuring expected returns comes from assessing the risk premium component properly. The Capital Market Line (CML) offers one tractable definition of the risk premium by writing this relationship as follows:

m

mpp

RFR )E(R RFR )E(R

While this is a reasonable first step, the CML expresses the risk-expected return tradeoff that investors should expect in an efficient capital market if they are purchasing entire portfolios of securities. To be fully useful, financial theory must address the following question: What is the appropriate risk-expected return relationship for individual securities? The problem posed by individual securities (compared to fully diversified portfolio holdings) is that the risk of those securities contains both systematic and unsystematic elements. Simply put, investors cannot expect to be compensated for risk that they could have diversified away themselves (i.e., unsystematic risk).

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Developing the Capital Asset Pricing Model (cont.)

One way to handle this problem in the context of the CML is to adjust the number of “total risk” units that the investor assumes for security i (i.e., i) to account for just the systematic portion of that risk. This can be done by multiplying i by the security’s correlation with the market portfolio (i.e., rim):

m

mimii

RFR )E(R)r( RFR )E(R

Rearranging this expression leaves:

RFR] )[E(R r

RFR )E(R mm

imii

or: RFR] )[E(R RFR )E(R mii This is the celebrated Capital Asset Pricing Model (CAPM). Notice that the CAPM redefines risk in terms of a security’s “beta” (i.e., i), which captures that stock’s riskiness relative to the market as a whole. The graphical representation of the CAPM is called the Security Market Line (SML).

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Using the SML in Performance Measurement: An Example

Two investment advisors are comparing performance. Over the last year, one averaged a 19 percent rate of return and the other a 16 percent rate of return. However, the beta of the first investor was 1.5, whereas that of the second was 1.0.

a. Can you tell which investor was a better predictor of individual stocks (aside from the issue of general movements in the market)?

b. If the T-bill rate were 6 percent and the market return during the period were 14 percent, which investor should be viewed as the superior stock selector?

c. If the T-bill rate had been 3 percent and the market return were 15 percent, would this change your conclusion about the investors?

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Using the SML in Performance Measurement (cont.)

a. To tell which investor was a better predictor of individual stocks we look at their alphas. Alpha is the difference between their actual return an that predicted by the SML, given the risk of their individual portfolios. Without information about the parameters of this equation (risk-free rate and the market rate of return) we cannot tell which one is more accurate.

b. If RF = 0.06 and Rm = 0.14, then

Alpha1 = .19 – [.06+1.5(.14-.06)] = .19 - .18 = 0.01 Alpha2 = .16 – [.06+1(.14-.06)] = .16 - .14 = 0.02

Here, the second investor has the larger alpha and thus appears to be a more accurate predictor. By making better predictions the second investor appears to have tilted his portfolio toward undervalued stocks. c. If RF = 0.03 and Rm = 0.15, then

Alpha1 = .19 – [.03+1.5(.15 -.03)] = .19 - .21 = -0.02 Alpha2 = .16 – [.03+1(.15 -.03)] = .16 - .15 = 0.01

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Using CAPM to Estimate Expected Return: Empresa Nacional de Telecom

1. Expected Return/Cost of Equity (Assumes RF = 4.73%)

(i) RPm = 4.2%: E(R) = k = 4.73% + 0.79(4.2%) = 8.05%

(ii) RPm = 7.2%: E(R) = k = 4.73% + 0.79(7.2%) = 10.42%

2. Expected Price Change (Recall that E(R) = E(Capital Gain) + E(Cash Yield)):

(i) RPm = 4.2%: E(P1) = (4590)[1 + (.0805 - .0196)] = CLP 4869.53

(ii) RPm = 7.2%: E(P1) = (4590)[1 + (.1042 - .0196)] = CLP 4978.31

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Estimating Mutual Fund Betas: FMAGX vs. GABAX

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Estimating Mutual Fund Betas: FMAGX vs. GABAX (cont.)

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Estimating Mutual Fund Betas: FMAGX vs. GABAX (cont.)

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The Fama-French Three-Factor Model

The most popular multi-factor model currently used in practice was suggested by economists Eugene Fama and Ken French. Their model starts with the single market portfolio-based risk factor of the CAPM and supplements it with two additional risk influences known to affect security prices: A firm size factor A book-to-market factor

Specifically, the Fama-French three-factor model for estimating expected excess returns takes the following form:

(Rit – RFRt) = i + bi1(Rmt – RFRt) + bi2SMBt + bi3HMLt + eit where, in addition to the excess return on a stock market portfolio, two other risk factors are defined: SMB (i.e., “Small Minus Big”) is the return to a portfolio of small capitalization stocks less

the return to a portfolio of large capitalization stocks HML (i.e., “High Minus Low”) is the return to a portfolio of stocks with high ratios of

book-to-market values (i.e., “value” stocks) less the return to a portfolio of low book-to-market value (i.e., “growth”) stocks

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Estimating the Fama-French Three-Factor Return Model: FMAGX vs. GABAX

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Fama-French Three-Factor Return Model: FMAGX vs. GABAX (cont.)

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Fama-French Three-Factor Return Model: FMAGX vs. GABAX (cont.)

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Style Classification Implied by the Factor Model

FMAGX *

* GABAX

Small

Large

Value Growth

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Fund Style Classification by Morningstar

FMAGX GABAX

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Does Investment Style Consistency Matter?

Consider the style classification of two funds (A &B) over time:

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Does Investment Style Consistency Matter? (cont.)

Study conducted using several thousand mutual funds from all nine style classes over the period 1991-2003 (see K. Brown and V. Harlow, “Staying the Course: Performance Persistence and the Role of Investment Style Consistency in Professional Asset Management”)

Calculates style consistency measure for each fund using two different methods (i.e., R-squared from three-factor model, tracking error from style benchmark) and correlated these statistics with several portfolio characteristics, including returns

Estimated regressions of future fund returns on past performance, style consistency, and other controls (e.g., fund expenses, turnover, assets under management)

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Correlation of Style Consistency (i.e., R-Squared) With Other Fund Characteristics

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Regression of Future Predicted Returns on: (i) Past Performance (i.e., Alpha), (ii) Style Consistency (i.e., RSQ), and

(iii) Portfolio Control Variables in both Up and Down Markets

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Investment Style Consistency: Conclusions

In general, the findings strongly suggest that fund style consistency does matter in evaluating future fund performance

Overall, there is a positive relationship between fund style consistency and subsequent investment performance

However, the nature of how style consistency matters is somewhat complicated: In “up” markets, style-consistent funds outperform style- inconsistent

funds, everything else held equal The reverse is true in “down” markets: style-inconsistent funds

outperform style-consistent funds, everything else held equal “Up” and “down” markets are predictable in advance

Being able to maintain a style-consistent portfolio is a valuable skill for a manager to have

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Using Derivatives in Portfolio Management

Most “long only” portfolio managers (i.e., non-hedge fund managers) do not use derivative securities as direct investments.

Instead, derivative positions are typically used in conjunction with the underlying stock or bond holdings to accomplish two main tasks: “Repackage” the cash flows of the original portfolio to create a more

desirable risk-return tradeoff given the manager’s view of future market activity.

Transfer some or all of the unwanted risk in the underlying portfolio, either permanently or temporarily.

In this context, it is appropriate to think of the derivatives market as an insurance market in which portfolio managers can transfer certain risks (e.g., yield curve exposure, downside equity exposure) to a counterparty in a cost-effective way.

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The Cost of “Synthetic” Restructuring With Derivatives

Consider the relative costs of rebalancing a stock portfolio in two ways:

(i) physical rebalancing by trading the stocks themselves; or

(ii) synthetic rebalancing using future contracts

United States Japan United Kingdom France Germany Hong Kong Cost Factor (S&P 500) (Nikkei 225) (FT-SE 100) (CAC 40) (DAX) (Hang Seng) A. Stocks Commissions 0.12% 0.20% 0.20% 0.25% 0.25% 0.50% Market Impact 0.30 0.70 0.70 0.50 0.50 0.50 Taxes 0.00 0.21 0.50 0.00 0.00 0.34 Total 0.42% 1.11% 1.40% 0.75% 0.75% 1.34% B. Futures Commissions 0.01% 0.05% 0.02% 0.03% 0.02% 0.05% Market Impact 0.05 0.10 0.10 0.10 0.10 0.10 Taxes 0.00 0.00 0.00 0.00 0.00 0.00 Total 0.06% 0.15% 0.12% 0.13% 0.12% 0.15% Source: Joanne M. Hill, “Derivatives in Equity Portfolios,” in Derivatives in Portfolio Management, edited by T. Burns, Charlottesville, VA: Association for Investment Management and Research, 1998.

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The Hedging Principle

Suppose a portfolio manager holds a $100 million position in U.S. equity securities and she is

concerned with the possibility that the stock market will decline over the next three months. How

can she hedge the risk that her portfolio will experience significant declines in value?

1) Hedging With Stock Index Futures:

Actual Desired Economic Event Stock Exposure Futures Exposure

Stock Prices Fall Loss Gain

Stock Prices Rise Gain Loss

2) Hedging With Stock Index Options:

Actual Desired Economic Event Stock Exposure Hedge Exposure

Stock Prices Fall Loss Gain

Stock Prices Rise Gain No Loss

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The Hedging Principle (cont.)

Consider three alternative methods for hedging the downside risk of holding a long position in a $100 million stock portfolio over the next three months:

1) Short a stock index futures contract expiring in three months. Assume the current contract delivery price (i.e., F0,T) is $101 and that there is no front-expense to enter into the futures agreement. This combination creates a synthetic T-bill position.

2) Buy a stock index put option contract expiring in three months with an exercise price (i.e., X) of $100. Assume the current market price of the put option is $1.324. This is known as a protective put position.

3) (i) Buy a stock index put option with an exercise price of $97 and (ii) sell a stock index call option with an exercise price of $108. Assume that both options expire in three months and have a current price of $0.560. This is known as an equity collar position.

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1. Hedging Downside Risk With Futures

Expiration Date Value of a Futures-Hedged Stock Position: Potential Value of Short Cost of Net Futures Portfolio Value Futures Position Futures Contract Hedge Position 60 (101-60) = 41 0 (60+41) = 101

70 (101-70) = 31 0 (70+31) = 101

80 (101-80) = 21 0 (80+21) = 101

90 (101-90) = 11 0 (90+11) = 101

100 0 0 (100+0) = 101

110 (101-110) = -9 0 (110-9) = 101

120 (101-120) = -19 0 (120-19) = 101

130 (101-130) = -29 0 (130-29) = 101

140 (101-140) = -39 0 (140-39) = 101 Notice that this net position can be viewed as a synthetic Treasury Bill (i.e., risk-free) holding with a face value of $101.

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1. Hedging Downside Risk With Futures (cont.)

Graphically, restructuring the long stock position using a short position in the futures contract creates the following synthetic restructuring:

Now Three Months

Long Stock

Short Futures

Net Position:

Long T-Bill Long Stock (p = 0) (p = 1)

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2. Hedging Downside Risk With Put Options

Expiration Date Value of a Protective Put Position: Potential Value of Cost of Net Protective Portfolio Value Put Option Put Option Put Position 60 (100-60) = 40 -1.324 (60+40)-1.324 = 98.676

70 (100-70) = 30 -1.324 (70+30)-1.324 = 98.676

80 (100-80) = 20 -1.324 (80+20)-1.324 = 98.676

90 (100-90) = 10 -1.324 (90+10)-1.324 = 98.676

100 0 -1.324 (100+0)-1.324 = 98.676

110 0 -1.324 (110+0)-1.324 = 108.676

120 0 -1.324 (120+0)-1.324 = 118.676

130 0 -1.324 (130+0)-1.324 = 128.676

140 0 -1.324 (140+0)-1.324 = 138.676

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2. Hedging Downside Risk With Put Options (cont.)

Long Stock Plus Long Put: Equals:

100

Long Stock

Long Put

Expiration Date Stock Value

Terminal Position Value

98.676

-1.324

100Expiration Date Stock Value

Terminal Position Value

98.676

-1.324

Put-Protected Stock Portfolio

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3. Hedging Downside Risk With An Equity Collar

Expiration Date Value of an Equity Collar-Protected Position:

Potential Net Option Value of Value of Net Collar- Portfolio Value Expense Put Option Call Option Protected Position

60 (0.56-0.56)=0 (97-60)=37 0 60 + 37 = 97

70 (0.56-0.56)=0 (97-70)=27 0 70 + 27 = 97

80 (0.56-0.56)=0 (97-80)=17 0 80 + 17 = 97

90 (0.56-0.56)=0 (97-90)= 7 0 90 + 7 = 97

97 (0.56-0.56)=0 0 0 97 + 0 = 97

100 (0.56-0.56)=0 0 0 100 + 0 = 100

108 (0.56-0.56)=0 0 0 108 - 0 = 108

110 (0.56-0.56)=0 0 (108-110)= -2 110 - 2 = 108

120 (0.56-0.56)=0 0 (108-120)= -12 120 - 12 = 108

130 (0.56-0.56)=0 0 (108-130)= -22 130 - 22 = 108

140 (0.56-0.56)=0 0 (108-140)= -32 140 - 32 = 108

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3. Hedging Downside Risk With An Equity Collar (cont.)

97 108

Terminal Position

Value

97

108

Collar-Protected

Stock Portfolio

Terminal Stock Price

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Zero-Cost Collar Example: IPSA Index Options

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Zero-Cost Collar Example: IPSA Index Options (cont.)

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Another Portfolio Restructuring

Suppose now that upon further consideration, the portfolio manager holding $100 million in U.S. stocks is no longer concerned about her equity holdings declining appreciably over the next three months. However, her revised view is that they also will not increase in value much, if at all.

As a means of increasing her return given this view, suppose she does the following: Sell a stock index call option contract expiring in three months with an

exercise price (i.e., X) of $100. Assume the current market price of the at-the-money call option is $2.813.

The combination of a long stock holding and a short call option position is known as a covered call position. It is also often referred to as a yield enhancement strategy because the premium received on the sale of the call option can be interpreted as an enhancement to the cash dividends paid by the stocks in the portfolio.

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Restructuring With A Covered Call Position

Expiration Date Value of a Covered Call Position: Potential Value of Proceeds from Net Covered Portfolio Value Call Option Call Option Call Position 60 0 2.813 (60+0)+2.813 = 62.813

70 0 2.813 (70+0)+2.813 = 72.813

80 0 2.813 (80+0)+2.813 = 82.813

90 0 2.813 (90+0)+2.813 = 92.813

100 0 2.813 (100+0)+2.813 = 102.813

110 -(110-100) = -10 2.813 (110-10)+2.813 = 102.813

120 -(120-100) = -20 2.813 (120-20)+2.813 = 102.813

130 -(130-100) = -30 2.813 (130-30)+2.813 = 102.813

140 -(140-100) = -40 2.813 (140-40)+2.813 = 102.813

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Restructuring With A Covered Call Position (cont.)

Long Stock Plus Short Call: Equals:

100

Long Stock

Expiration Date Stock Value

Terminal Position Value

Short Call

2.813

100

Terminal Position Value

2.813

Covered Call Portfolio

102.813

Expiration Date Stock Value

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Some Thoughts on Currency Hedging and Portfolio Management

00 01 02 03 04

500

550

600

650

700

750

Exchange RateChilean Peso per U.S. Dollar High: 749Monthly: Feb 29, 2000 - Feb 28, 2005 Low: 502

Last: 573

Weakening CLP Strengthening CLP

Question: How much FX exposure should a portfolio manager hedge?

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Conceptual Thinking on Currency Hedging in Portfolio Management

There are at least three diverse schools of thought on the optimal amount of currency exposure that a portfolio manager should hedge (see A. Golowenko, “How Much to Hedge in a Volatile World,” State Street Global Advisors, 2003):

Completely Unhedged: Froot (1993) argues that over the long term, real exchange rates will revert to their means according to the Purchasing Power Parity Theorem, suggesting currency exposure is a zero-sum game. Further, over shorter time frames—when exchange rates can deviate from long-term equilibrium levels—transaction costs make involved with hedging greatly outweigh the potential benefits. Thus, the manager should maintain an unhedged foreign currency position.

Fully Hedged: Perold and Schulman (1988) believe that currency exposure does not produce a commensurate level of return for the size of the risk; in fact, they argue that it has a long-term expected return of zero. Thus, since the investor cannot, on average, expect to be adequately rewarded for bearing currency risk, it should be fully hedged out of the portfolio.

Partially Hedged: An “optimal” hedge ratio exists, subject to the usual caveats regarding parameter estimation. Black (1989) demonstrates that this ratio can vary between 30% and 77% depending on various factors. Gardner and Wuilloud (1995) use the concept of investor regret to argue that a position which is 50% currency hedged is an appropriate benchmark.

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Hedging the FX Risk in a Global Portfolio: Some Evidence

Consider a managed portfolio consisting of five different asset classes: Chilean Stocks (IPSA), Bonds (LVAC Govt), Cash (LVAC MMkt) US Stocks (SPX), Bonds (SBBIG)

Monthly returns over two different time periods: February 2000 – February 2005 February 2002 – February 2005

Five different FX hedging strategies (assuming zero hedging transaction costs):#1: Hedge US positions with selected hedge ratio, monthly rebalancing#2: Leave US positions completely unhedged#3: Fully hedge US positions, monthly rebalancing#4: Make monthly hedging decision (i.e., either fully hedged or completely

unhedged) on a monthly basis assuming perfect foresight about future FX movements

#5: Make monthly hedging decision (i.e., either fully hedged or completely unhedged) on a monthly basis assuming always wrong about future FX movements

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Investment Performance for Various Portfolio Strategies: February 2000-February 2005

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Investment Performance for Various Portfolio Strategies: February 2002-February 2005

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Sharpe Ratio Sensitivities for Various Managed Portfolio Hedge Ratios

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Currency Hedging and Global Portfolio Management: Final Thoughts

Foreign currency fluctuations are a major source of risk that the global portfolio manager must consider.

The decision of how much of the portfolio’s FX exposure to hedge is not clear-cut and much has been written on all sides of the issue. It can depend of many factors, including the period over which the investment is held.

It is also clear that tactical FX hedging decisions have potential to be a major source of alpha generation for the portfolio manager.

Recent evidence (Jorion, 1994) suggests that the FX hedging decision should be optimized jointly with the manager’s basic asset allocation decision. However, this is not always possible or practical.

Currency overlay (i.e., the decision of how much to hedge made outside of the portfolio allocation process) is rapidly developing specialty area in global portfolio management.