Portfoilo Beta

8/4/2019 Portfoilo Beta

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Portfolio Beta


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Expected Returns: The Risk-BasedApproach

This approach is more theoretically soundand used in practice by most corporatefinance professionals.

Two-step process:

1.Measure the risk of the asset.

2.Translate that risk measure into anexpected return estimate.


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How can we capture the systematic risk component

of a stocks volatility?

1. Measure the risk of the asset.

Systematic risks simultaneously affect many differentsecurities.

The market pays investors for bearing systematic risk -the risk that cannot be eliminated through diversification.

Standard deviation measures an assets total risk, whichis equal to the sum of its systematic and unsystematiccomponents.



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Collect data on a stocks returns and returns on a marketindex

Plot the points on a scatter plot graph

Y-axis: stocks return

X-axis: markets return

Plot a line (using linear regression) through the points

Slope of line equals beta, the sensitivity of a stocks returns

relative to changes in the overall market return.

Beta is a measure of systematic risk for a particularsecurity.



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Scatter Plot of Weekly Returns on The SharperImage vs. The S&P 500 Stock Index


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High Beta

By definition, the beta of the average stockequals 1.0.

The return on a high-beta stock like The Sharper

Image experiences dramatic up-and-downswings when the market return moves.

Because The Sharper Images beta equals 1.44,we can say that the return on The SharperImages shares moves, on average, 1.44 timesas much as does the market return.


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Scatter Plot of Weekly Returns on ConAgravs. The S&P 500 Stock Index


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Low Beta

Because the ConAgrasbeta equals 0.11, wecan say that the return on The ConAgras sharesmoves, on average, 0.11 times as much as does

the market return. A stock that can gain or lose 12 percent in a

week is volatile, but ConAgras volatility is onlyweakly related to fluctuations in the overall

market. Most of ConAgras risk is unsystematic and can

be eliminated through diversification.


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Beta measures systematic risk and links the risk

and expected return of an asset.

2. Translate that risk measure into an expectedreturn estimate.

Plot beta against expected return for two assets:

- A risk-free asset that pays 4% with certainty, withzero systematic risk and

- An average stock, with beta equal to 1, with anexpected return of 10%.

Draw a straight line connecting the two points.

Investors holding a stock with beta of 0.5 or 1.5, forexample, can find the expected return on the line.



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Expected returns

10%

1

Risk-free asset

0.2 0.4 0.6 0.8 21.2 1.4 1.6 1.8

Beta

4%

18%

14%

average

stock

What is the expected return for stock with beta = 1.5?

= 1.5

An investor willing toaccept an averagelevel of systematic risk,by holding a stock witha beta of 1.0, expects areturn of 10%. Byholding only the risk-free asset, an investorcan earn 4%, without

having to accept anysystematic risk at all.

Beta and Expected Returns


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The portfolio expected return equals the weighted average

of the portfolio assets expected returns.

E(Rp) = w1E(R1)+ w2E(R2)++wnE(Rn) w1, w2 , , wn : portfolio weights

E(R1), E(R2), , E(RN): expected returns of securities

Expected return of a portfolio with N securities

How does the expected return of a portfolio relate to theexpected returns of the securities in the portfolio?

Portfolio Expected Return


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Portfolio E(R) $ Invested Weights

IBM 10% $2,500 0.125

GE 12% $5,000 0.25

Sears 8% $2,500 0.125Pfizer 14% $10,000 0.5

E(Rp) = (0.125)(10%) + (0.25)(12%) +

(0.125)(8%) + (0.5)(14%)

= 12.25%

E(Rp) = w1E(R1)+ w2E(R2)++wnE(Rn)

Portfolio Expected Return


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Portfolio risk is the weighted average of systematicrisk (beta) of the portfolio constituent securities.

Portfolio Beta $ Invested Weights

IBM 1.00 $2,500 0.125

GE 1.33 $5,000 0.25

Sears 0.67 $2,500 0.125

Pfizer 1.67 $10,000 0.5

P = (0.125)(1.00) + (0.25)(1.33) + (0.125)(0.67) + (0.50)(1.67)= 1.38

But portfolio volatility is not the same as the weighted

average of all portfolio security volatilities.

Portfolio Risk


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The Dominance Principle

States that among all investments with agiven return, the one with the least risk isdesirable; or given the same level of risk,the one with the highest return is mostdesirable.


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Dominance Principle Example

Security E(Ri) ATW 7% 3%GAC 7% 4%YTC 15% 15%FTR 3% 3%HTC 8% 12%

ATW dominates GAC ATW dominates FTR


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Efficient Frontier Graph

E(Rp)

p

M

Efficient

Frontier


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Efficient Frontier

The Efficient Frontier represents all thedominant portfolios in risk/return space.

There is one portfolio (M) which can beconsidered the market portfolio if weanalyze all assets in the market. Hence,M would be a portfolio made up of assets

that correspond to the real relative weightsof each asset in the market.


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Efficient Frontier (continued)

Assume you have 20 assets. With thehelp of the computer, you can calculate allpossible portfolio combinations. The

Efficient Frontier will consist of thoseportfolios with the highest return given thesame level of risk or minimum risk given

the same return (Dominance Rule)


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Borrowing and lending investment funds atR to expand the Efficient Frontier.

a. We keep part of our funds in a saving

account Lending, OR

b. We can borrow funds for a greater

investment in the market portfolio


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E(Rp)

p

M

Efficient

Frontier

B

A

Lending

BorrowingCML

RF

Portfolio A: 80% of funds in RF, 20% of funds in M

Portfolio B: 80% of funds borrowed to buy more of M,

100% or own funds to buy M


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By using RF, the Efficient Frontier is nowdominated by the capital market line(CML). Each portfolio on the capitalmarket line dominates all portfolios on theEfficient Frontier at every point except M.


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The Portfolio Investment

E(Rp)

p

M

Efficient

Frontier

CML

RFMutual Fund Portfolios

with a cash position

Investors indifference curves are based on their degree

of risk aversion and investment objectives and goals.

Portfoilo Beta

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