14-Efficient Diversification II

BKM: Chapter 6

Portfolio Allocation

What about the case of many risky assets? The efficient frontier has the same shape Intuition is the same

Exp

ecte

d R

etur

n

Standard Deviation

Individual Assets

Tangency Portfolio

risk free rate Best possible CAL

Multiple AssetsGiven time series of returns for n stock, you can find the portfolio with maximum Sharpe ratio as follows:

Pick arbitrary weights. For the last stock, specify weight as 1-sum of others

Find the portfolio return using these arbitrary weights for each date (point to cells with weights)

Estimate the expected return and standard deviation

Calculate the Sharpe ratio of the portfolio

Use solver: maximize Sharpe ratio of portfolio by changing cells containing weights. Change all weights except last, which equals 1-sum of others

Diversification & Many Risky Assets

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Number of Assets in Portfolio

Stan

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n of

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rtfo

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Diversification

In any portfolio, what fluctuations are “cancelled out”?

Simple two-stock example:

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p

DELL

MSFT

rwrwr

rrr

portfolioon return Dellon return Microsofton return

Portfolio Variance

wMSFT

wDELL

Components of MSFT

Components of Dell

Remaining Components

wMSFT

wDELL

Portfolio Variance

wMSFT

wDELL

Vanishing Components

wMSFT

wDELL

Diversification

Portfolio Rule #5

The component of the stock return that is independent of the portfolio return is always “off-set” or “canceled out” in a portfolio.

Diversification

tMSFTtpMSFTMSFTtMSFT erbar ,,,

Remaining Component Vanishing Component

tDELLtpDELLDELLtDELL erbar ,,,

Remaining Component Vanishing Component

Market Portfolio

Assume the portfolio we hold is the market portfolio.

When we combine stocks into this portfolio, again there is a “remaining” and “vanishing” component to each stock.

We find this component for each stock by estimating the regression:

titMiMSFTti erbar ,,,

Market portfolio When we estimate this regression using a stock return

as the “y” variable and the market return as the “x” variable, we call the slope coefficient beta ()

We call the variance of the remaining component “systematic risk”

We call the variance of the vanishing component “unsystematic” or idiosyncratic, or “firm specific” risk.

Diversification and Many Risky Assets

R-square in this regression is

R-square in this regression is therefore the fraction of the total stock variance that “remains” or that is systematic.

)()(2

2

i

mi

rVarrVarR

Risk Systematic risk refers to fluctuations in asset prices

caused by macroeconomic factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic risk factors include the business cycle, inflation, monetary policy and technological changes.

Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-specific risk factors include litigation, patents, management, and financial leverage.