SAPM Module 7

8/8/2019 SAPM Module 7

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Originally proposed by HarryMarkowitz in 1950·s

First formal attempt to quantify therisk

Diversification reduces the risk

Round 20 stocks to the portfolio,

unsystematic risk can be reduced tozero



Wp %

35

20

0

Number of securities in portfolio

10 20 30 40 ...... 100+

Total risk

Systematic Risk

DiversifiableRisk



Markowitz, the father of modern portfolio theory,

developed the basic principle of portfoliodiversification in a formal way, in quantified

form, that shows why and how portfolio

diversification works to reduce the risk of a

portfolio to an investor.Modern Portfolio theory hypothesizes how

investors should behave.

As the no. of securities in the portfolio increases,

contribution of individual security·s riskdecreases due to offsetting effect of strong

performing and poor performing securities in the

portfolio and the importance of covariance

relationships among securities increases.



Holding two stocks is less risky than holdingone stock

Assumptions

Investors decisions are on twoparameters-expected return and variance

Investors are risk averse

Investors seek to achieve the highest

expected return at a given level of risk Investors have identical expectations

Common one period investment horizon



Portfolio Return

Actual Portfolio Return

RP = W1R1+ W2R2 + «WnRn

Expected Return of a Portfolio Return

E(RP) = W1 E(R1) + W2 E(R2) + «Wn E(Rn)

Portfolio Risk



States that among all investments with a

given return, the one with the least risk is

desirable; or given the same level of risk, the

one with the highest return is mostdesirable.



Security E(Ri) W

ATW 7% 3%

GAC 7% 4%

YTC 15% 15%FTR 3% 3%

HTC 8% 12%

ATW dominates GAC

ATW dominates FTR



1. Normal Diversification

This occurs when the investor combines more

than one (1) asset in a portfolio

1 5 10 20 30

Risk

Systematic Risk

Unsystematic

Risk

# of Assets

75% of Co.

Total Risk

25% of Co.

Total Risk



Unsystematic Risk

... is that portion of an asset·s total risk which

can be eliminated through diversification

Systematic Risk ... is that risk which cannot be eliminated

Inherent in the marketplace



Superfluous or Naive Diversification

Occurs when the investor diversifies in more

than 20-30 assets. Diversification for

diversification·s sake. a. Results in difficulty in managing such a

large portfolio

b. Increased costs

Search and transaction



This type of diversification considers the

correlation between individual securities. It is

the combination of assets in a portfolio that are

less then perfectly positively correlated. a. The two asset case:

Stk. A Stk. B

E(R) 5% 15%

W10

%20

%



Assume that the investor invests 50% of

capital stock in stock A and 50% in B

1. Calculate E(R)

E(Rp) = 7xi E(Ri) E(Rp) =.5(.05) + .5(.15)

E(Rp) = .025 + .075

E(Rp) = .10 or 10%

i=1

n



2. Graphically

A

B

PortfolioAB

E(R p)

15%

10%

5%

5 10 15 20 25 W



Portfolio Return of AB will always be on line AB

depending on the relative fractions invested in

assets A and B.

3. Calculating the risk of the portfolio Consider 3 possible relationships between A

and B

Perfect Positive Correlation

Zero Correlation Perfect Negative Correlation



A and B returns vary in identical pattern.

Hence, there is a linear risk-return

relationship between the two assets.



A

B

AB

E(R p)

15%

5%

10 15 20 W p



Therefore, the risk of portfolio AB is simply

the weighted value of the two assets· W.

In this case:

Wp = xA2 WA

2 + xB2 WB

2 + 2 xAxBWAWB VAB

Wp = .25(.10)2+.25(.20)2+2(.5)(.5)(.10)(.20)

Wp = .15 or 15%



A·s return is completely unrelated to B·s

return. With zero correlation, a substantial

amount of risk reduction can be obtained

through diversification.



A

B

AB

E(R p)

15%

10%

5%

10 11.2 20 W p

W p = .25(.10)2+.25(.20)2

W p } 11.2%



A·s and B·s returns vary perfectly inversely.

The portfolio variance is always at the lowest

risk level regardless of proportions in each

asset.



A

B

AB

E(R p)

15%

5%

5 10 20 W p

10%

W p = .25(.10)+.25(.20)+2(.5)(.10)(.20)(-1)

W p = .05 or 5%



Although there are no securities with

perfectly negative correlation, almost all

assets are less than perfectly correlated.

Therefore, you can reduce total risk (Wp) through diversification. If we consider many

assets at various weights, we can generate

the efficient frontier.



E(R p)

W p

M

Efficient

Frontier



The Efficient Frontier represents all the

dominant portfolios in risk/return space.

There is one portfolio (M) which can be

considered 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 weights of

each asset in the market.



Assume you have 20 assets. With the help of

the computer, you can calculate all possible

portfolio combinations. The Efficient

Frontier will consist of those portfolios withthe highest return given the same level of

risk or minimum risk given the same return

(Dominance Rule)



4. Borrowing and lending investment funds

at R 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



The efficient frontier is

convex as a result of risk

and return characteristics

of the portfolio, which

changes in a non-linearfashion as its components·

weighting are changed.











The capital asset pricing model defines therelationship between risk and return

E(RA) = Rf + FA(E(RM) ² Rf)

If we know an asset·s systematic risk, we canuse the CAPM to determine its expectedreturn

This is true whether we are talking aboutfinancial assets or physical assets



Pure time value of money ² measured by the

risk-free rate

Reward for bearing systematic risk ²

measured by the market risk premiumAmount of systematic risk ² measured by

beta



Consider the betas for each of the assets givenearlier. If the risk-free rate is 2.13% and the marketrisk premium is 8.6%, what is the expected return

for each?

Securit Beta Ex ected etur

DCLK 2.685 2.13 + 2.685(8.6) = 25.22%

KO 0.195 2.13 + 0.195(8.6) = 3.81%

INTC 2.161 2.13 + 2.161(8.6) = 20.71%

KEI 2.434 2.13 + 2.434(8.6) = 23.06%



SAPM Module 7

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