Portflio Analysis

7/28/2019 Portflio Analysis

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Fundamentals of Investment Management Hirt Block

11

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 portfolio

4. Explain the importance of the capital assetpricing 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% 0.2

7 0.3 Slow growth 8 0.3

13 0.3Moderate

growth 8 0.3

15 0.2Strong

economy 6 0.2

InvestmentInvestment 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 )

2Pi

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%

s2 = (ki - k )2 Pi 15.4%

s = 3.9%

Standard Deviation=

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

2Pi

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%

s2 =

(ki - k )2 Pi 25.6%

s 5.1%

Investment j

Investment i Note thatthe average

is the same

for each

investment

but that the

standard

deviation

is different.

Also notethat

this model

assumes no

correlation

betweeni and j.


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Portfolio Return kAssume stocks x1 and x2 with parameters:

x1 = .5 K1 = 10% s1 = 3.9x2 = .5 K2 = 10% s2 = 5.1

Definition of portfolio expected returnaccording 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 si

rij sp+1.0 4.5 p

+ .5 3.90.0 3.2- .5 2.3- .7 1.8-1.0 0.0

Calculated

standard deviationwith differing

correlation

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 mostefficient portfolios (i.e., those with the

maximum return for a given risk).

Portfolios do not exist above the "efficientfrontier"


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(Figure 21-3)

1 2 3 4 5 6 7 80

11

14

13

15

12

10

9

A

B

C DE

FG

H

Expected return Kp

Portfolio standard deviation (sp) (risk)

Efficient

frontier

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

A

B

C DE

FG

H

Expected return Kp

Portfolio standard deviation (sp) (risk)

Efficientfrontier

Inefficientportfolios

Diagram of Risk-Return Trade-offs


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Capital Asset Pricing Model

The CAPM introduces the risk-free

asset where sRF = 0.Under the CAPM, investors combine

the risk-free asset with riskyportfolios on the efficient frontier.


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(Fig21-8)Expe

cted

ret

urn

Kp

RF

M

Portfolio standard deviation (sp)

Z

Efficient

frontier

Initial: risk

free point

Satisfies

efficient frontier

Maximumattainable risk-

return

Risk Return line

The CAPM and Indifference Curves

F d t l f I t t M t




1111

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 theRFMZ line?


F d t l f I t t M t




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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.

F d t l f I t t M t




1313


Portfolio M is an optimum

market basket of investments.

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

Broadly based index is betterthan narrowly based index.

F d t l f I t t M t Hi Bl k




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Security Market Line

Refers to an individualstock

Trade-off between risk & return Analogous to Capital Market Line for market

portfolios

Formula is:

Ki = RF + bi (KM - RF)

F d t l f I t t M t Hi t Bl k




1515

(Figure 21-12)

Risk (Beta)

RF

Market standard deviation

O

KM

1.0

Security Market

Line (CML)

return

Expected

return

Kp

Illustration of the Capital Market Line

2.0

Fundamentals of Investment Management Hi t Bl k




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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 bp = 1.2 sp = .14

= = 0.50.12 - .05

.14

SharpeMeasure

Fundamentals of Investment Management Hi t Bl k




1717

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 bp = 0.9= = 0.044

.10 - .06

0.9

TreynorMeasure





1818The 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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Portfolio Beta

Excess returns (%)

O

Market line

O .5 1.O 1.5

6

4

2

1

-1

3

-2

-3

5

MarketM

Z

Y

Figure 22-2 Risk-Adjusted Portfolio Returns

ML = a b (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.

Portflio Analysis

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