Portfolio Analysis

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Fundamentals of Investment ManagementFundamentals of Investment Management Hirt • BlockHirt • Block

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Portfolio Management and Capital Market Theory- Learning

Objectives

1. Understand the basic statistical techniques for measuring risk and return

2. Explain how the portfolio effect works to reduce the risk of an individual security.

3. Discuss the concept of an efficient portfolio4. Explain the importance of the capital asset

pricing model.5. Understand the concept of the beta

coefficient

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Table 21-1 Return and Probabilities for investments I and j

Return

Ki

Pi

(probability of

Ki occuring)

Possible state of the economy Return Kj

Pi

(probability of

Kj occuring)

5% 0.2 Recession 20% 0.27 0.3 Slow growth 8 0.3

13 0.3Moderate growth 8 0.3

15 0.2Strong economy 6 0.2

Investment jInvestment I

Ki Pi Ki Pi

5% 0.2 1.0%7 0.3 2.1%13 0.3 3.9%15 0.2 3.0%

10%

Kj Pj Kj Pj

20% 0.2 4.0%8 0.3 2.4%8 0.3 2.4%6 0.2 1.2%

10%

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Standard Deviation=

ki ki Pi (ki - k ) (ki - k )2 (ki - k )2 Pi

5% 10% 0.2 -5% 25% 5.0%7% 10% 0.3 -3% 9% 2.7%13% 10% 0.3 3% 9% 2.7%

15% 10% 0.2 5% 25% 5.0% 2 = (ki - k )2 Pi 15.4%

3.9%

Standard Deviation=

ki ki Pi (ki - k ) (ki - k )2 (ki - k )2 Pi

20% 10% 0.2 10% 100.00% 20.0%8% 10% 0.3 -2% 4.00% 1.2%8% 10% 0.3 -2% 4.00% 1.2%6% 10% 0.2 -4% 16.00% 3.2%

2 =(ki - k )2 Pi 25.6%

= 5.1%

Investment j

Investment i Note thatthe averageis the samefor each investmentbut that thestandarddeviation is different.Also note thatthis modelassumes nocorrelation between i and j.

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Portfolio Return k

Assume stocks x1 and x2 with parameters:

x1 = .5 K1 = 10% 1 = 3.9

x2 = .5 K2 = 10% 2 = 5.1

Definition of portfolio expected return according to equation 21-3.

Kp = x1 K1 + x2 K2

= .5(10 %) + .5(10 %) = 10%

Portfolio Effect ( 2 stocks, equal weight)

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Standard Deviation of a Two-Stock Portfolio ( 2 stocks, equal weight)

jiijjij2

j2

i22

p rxx2xx i

rij p

+1.0 4.5 p+ .5 3.9 0.0 3.2 - .5 2.3 - .7 1.8 -1.0 0.0

Calculated standard deviation with differingcorrelation coefficients.

= .52(3.9)2+.52(5.1)2+2(.5)(.5) rij (3.9)(5.1)

= 3.85 +6.4 + .5 rij 19.9

CorrelationCoefficient

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Developing and Efficient Portfolio

Many possible portfolios (i.e., combinations of investments)

The investor determines his personal risk-return criteria

An investor should select from the most efficient portfolios (i.e., those with the maximum return for a given risk).

Portfolios do not exist above the "efficient frontier"

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(Figure 21-3)

1 2 3 4 5 6 7 80

11

14

13

15

12

10

9

AB

C DE

FG

H

Expected return Kp

Portfolio standard deviation (p) (risk)

Efficientfrontier

Diagram of Risk-Return Trade-Offs

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1 2 3 4 5 6 7 80

11

14

13

15

12

10

9

AB

C DE

FG

H

Expected return Kp

Portfolio standard deviation (p) (risk)

Efficientfrontier

Inefficient portfolios

Diagram of Risk-Return Trade-offs

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Capital Asset Pricing Model

The CAPM introduces the risk-free

asset where RF = 0.

Under the CAPM, investors combine the risk-free asset with risky portfolios on the efficient frontier.

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(Fig21-8)Expected return K

p

RF

M

Portfolio standard deviation (p)

Z

Efficientfrontier

Initial: risk free point

Satisfies efficient frontier

Maximum attainable risk-

return

Risk Return line

The CAPM and Indifference Curves

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•The RFMZ line represents investment opportunities that are superior to the existing efficient frontier.

• RFMZ line is called capital market line.

•How do investors reach points on the RFMZ line?


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To attain line RFM Buy a combination of RFF and M

portfolio

To attain M Z Buy M portfolio and borrow additional

funds at the risk-free rate.

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Portfolio M is an optimum “market basket of investments.”

M portfolio can be represented by NYSE,or S&P 500.

Broadly based index is better than narrowly based index.

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Security Market Line

Refers to an individual stock Trade-off between risk & return Analogous to Capital Market Line for

market portfolios Formula is:

Ki = RF + bi (KM - RF)

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(Figure 21-12)

Risk (Beta)

RF

Market standard deviation

O

KM

1.0

Security MarketLine (CML)

return

Exp

ecte

d re

turn

Kp

Illustration of the Capital Market Line

2.0

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Measures excess return per unit of total risk.Also known as "excess return to variability" ratio.

Higher values indicate superior performance

Sharpe Approach

Sharpe measure

Total portfolio return - Risk-free rate Portfolio standard deviation=

Market data: KF = 5%Portfolio Data: kp = .12 p = 1.2 p = .14

= = 0.50.12 - .05 .14

SharpeMeasure

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Measures excess return per unit of systematic risk.

Also known as "excess return to volatility" ratio.

Higher values indicate superior performance

Treynor Approach

Treynor measure

Total portfolio return - Risk-free rate Portfolio Beta

=

Market data: KF = 6%Portfolio Data: kp = 0.10 p = 0.9

= = 0.044.10 - .06 0.9

TreynorMeasure

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18©The McGraw-Hill Companies, Inc.,1999

Jensen Approach

Alpha (average differential) return indicates the difference between a) the return on the fund and b) a point on the market line that corresponds to a beta equal that of the fund.

Alpha = the actual rate of return minus the rate of return predicted by the CAPM.

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19Portfolio Beta

Excess returns (%)

O

Market line

O .5 1.O 1.5

6

4

21

-1

3

-2-3

5Market

MZ

Y

Figure 22-2 Risk-Adjusted Portfolio Returns ML = (EMR)EMR is "excess market return"

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Jensen Approach

Jensen computed the alpha value of 115 mutual funds.

The average alpha was a negative 1.1% and only 39 out of 115 funds had a positive alpha.

Portfolio Analysis

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