MARKOWITZ AND THE EFFICIENT FRONTIER

FNCE 455Class Session #3Lloyd KurtzSanta Clara University

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Topics

• Conceptual Basis

• Portfolios with Two Risky Assets

• The Efficient Frontier• The Cloud• The Frontier• The Separation Property (Tobin’s Line)

• Implications

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Conceptual Basis

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Why diversify?

“An investor who knew future returns with certainty would invest in only one security, namely the one with the highest future return. In no case would the investor actually prefer a diversified portfolio. But diversification is a common and reasonable investment practice. Why? To reduce uncertainty!”

- Harry Markowitz

Source: Nobelprize.org

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Rule #1 of Diversification

More is Better• All else equal, more securities increase diversification

and reduce volatility.

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Conventional Wisdom - 1987

Merrill Lynch, “Investment Performance Analysis”, Reference Manual and Glossary, cited in Cohen, Zinbarg, and Zeikel, Investment Analysis and Portfolio Management, 1987.

“In a typical, diversified investment portfolio of 30 or more common stocks, diversification eliminates so much of the specific [firm-specific] risk that roughly 85 to 95 percent of all the risk (in the portfolio) is market risk and only 5 to 15 percent is specific risk.”

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More Names Less Volatility

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30

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60

1 2 4 6 8 10 20 30 40 50 100 200 300 400 500 1000

Number of Stocks in Portfolio

Ave

rag

e P

ort

foli

o S

td D

evia

tio

n %

Based on data in M. Statman, "How Many Stocks Make a Diversif ied Portfolio?" Journal of Finance and Quantitative Analysis 22 , September 1987.

Firm-Specific Risk

Market

Risk

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Two Kinds of Risk

• Firm-Specific Risk• Can be diversified away

• Market Risk• Cannot be diversified away

Why would anyone accept firm-specific risk if they didn’t need to?

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There’s more to diversification than quantity

# of Holdings

Std. Dev., Five Years Ended 4/12

Largest

3 Holdings

(and %)

iShares Goldman Sachs Technology Index Fund

(IGM)233 23.1%

AAPL – 9.0%

IBM – 8.3%

MSFT – 8.0%

iShares S&P Global 100 Index Fund

(IOO)104 20.1%

XOM – 5.4%

IBM – 3.2%

MSFT – 3.2%

S&P 500 500 19.1%AAPL – 4.7%

XOM – 3.2%

IBM – 1.9%

Sources: iShares website and Morningstar, April 2012.

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Framing the Question

“It seemed obvious that investors that investors are concerned with risk and return, and that these should be measured for the portfolio as a whole.”

- Harry Markowitz

Source: Nobelprize.org

Portfolios with Two Risky Assets

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Rule #2 of Diversification

Portfolio risk depends on the correlation between the returns of the assets in the portfolio.

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Three Rules of 2RA Portfolios

• The rate of return on the portfolio is a weighted average of the returns on the component securities, with the investment proportions as weights.

• The expected rate of return on the portfolio is a weighted average of the expected returns on the component securities, with the investment proportions as weights.

• The variance of the rate of return on the two-risky-asset portfolio is...a sum of the contribution of the component security variances plus a term that involves the correlation coefficient (and hence, covariance) between the returns on the component securities.

See spreadsheet based on BK&M 6.2 on Angel

The Efficient Frontier

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The optimization problem

“How much return do you want to sacrifice in order to increase the probability that you will get what you planned for?”

- Milton Friedman

Source: The Myth of the Rational Market (p. 46)

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The cloud becomes a frontier

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17Asset Allocation Analysis Zephyr AllocationADVISOR: Nelson Capital Management

Efficient FrontierCase: Allocation Case Return vs. Risk (Standard Deviation)

0 2 4 6 8 10 12 14 165

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Risk (Standard Deviation)

Ret

urn

Vanguard 500 IndexVanguard Total BondVanguard Short-Term

Asset Allocations

Vanguard 500 Index

Vanguard Total Bond

Vanguard Short-Term

A portfolio of 20% stocks, 37% short-term bonds, and 43% longer-term bonds would have had the same volatility but better returns than a bond portfolio alone. Historical returns from January 1992 through February 2007.

Diversification benefit.

18Asset Allocation Analysis Zephyr AllocationADVISOR: Nelson Capital Management

Efficient FrontierCase: Allocation Case Return vs. Risk (Standard Deviation)

0 2 4 6 8 10 12 14 165

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9

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Risk (Standard Deviation)

Ret

urn

Vanguard 500 IndexVanguard Total BondVanguard Short-Term

Asset Allocations

Vanguard 500 Index

Vanguard Total Bond

Vanguard Short-Term

A capital allocation line (CAL) can be drawn from the risk-free rate to the most northwest point on the efficient frontier (highest Sharpe ratio). Points on this line are superior investment opportunities to all other points along the efficient frontier. (See BK&M p. 159)

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Markowitz-efficient portfolio with bond-equivalent risk.

Optimal risky portfolio.

The separation property

The property that implies portfolio choice can be separated into two independent tasks:

(1) Determination of the optimal risky portfolio, which is purely a technical problem, and

(2) The personal choice of the best mix of the risky portfolio and the risk-free asset.

- BK&M p. 163

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Implications

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Some MPT insights

• Rational investors should form portfolios to maximize return per unit of risk (volatility). • All that is needed to do this is a forecast, for each investment under

consideration of its return, risk, and correlation with other securities.

• Less-correlated assets are attractive!• Portfolio risk may be reduced by adding a risky asset, if it has a low

correlation with other assets in the portfolio.• This is true even for uncorrelated assets with relatively low expected

returns.

• Once the optimal risky portfolio has been identified, the best way to scale risk is by blending it with the risk-free.• Separation property

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Implementation challenges

• Markowitz’s Procedure Needs Three Inputs:• Expected Returns for each asset• Expected Volatility for each asset• Expected Correlations for all pairs of assets

• This is hard, especially when many assets are involved...• Past Returns ≠ Expected Returns• Past Volatility may not = Expected Volatility• Past Correlations may not = Expected Correlations

• Forecasts of returns, volatility and correlations may not be accurate.

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Significance (#1)

“With almost a disarming sleight-of-hand, Markowitz showed us that all the information needed to choose the best portfolio for any given level of risk is contained in three simple statistics: mean, standard deviation, and correlation.

“It suddenly appeared that you didn’t even need any fundamental information about the firm! The model requires no information about dividend policy, earnings, market share, strategy, quality of management – nothing about the myriad of things with which Wall Street analysts concern themselves!

“Virtually every major portfolio manager today consults an optimization program. They may not follow its recommendations exactly, but they use it to evaluate basic risk and return trade-offs.”

- William Goetzmann, An Introduction to Investment Theory

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Significance (#2)

“When I defended my dissertation as a student in the Economics Department of the University of Chicago, Professor Milton Friedman argued that portfolio theory was not Economics, and that they could not award me a Ph.D. degree in Economics for a dissertation which was not in economics. I assumed he was only half serious, since they did award me the degree without long debate. As to the merits of his arguments I am quite willing to concede: at the time I defended my dissertation, portfolio theory was not part of economics. But now it is.”

- Harry MarkowitzSource: Nobelprize.org