1

EXAMPLE: PORTFOLIO RISK & RETURN

Suppose:Georgia ThermoPacific Electron

Expected return 15% 21%Variance 784 1764Standard deviation 28 42

Note: Expected return = average of possible returnsVariance = average of possible squared deviations from expected returnStandard deviation = square root of variance

)(r)( 2

)(

2

PORTFOLIO RISK

Portfolio variance = sum of boxes

2

1

2

1X

1221

211221

XXXX

1221

211221

XXXX

2

2

2

2X

returns of ncorrelatio

returns of covariance12

3

PORTFOLIO RISK: EXAMPLE

Georgia Pacific Thermo Electron

GeorgiaPacific .62 *282 .6*.4*.4*28*42

=282 =113

Thermo .6*.4*.4*28*42 .42 *422

Electron =113 =282

Variance = 282 +282 + (2 X 113) = 790Std dev. = (790)1/2 = 28.1%

Note: assumes correlation is .4

4

Return and Risk for Portfolios

BA

N

i

N

jijji

N

iiiP

N

iii

XXX

RX E

) Corr(j and isecurity of returns of Covariance

isecurity return of Variance

portfolio in securities ofNumber N

isecurity on turnExpectedre)E(

isecurity in invested funds of Proportion

where

variancePortfolio

1 subject to

)() E(portfolioa of return Expected

RR

RX

X

R

BAij

2

i

i

i

1 11

222

N

1ii

1p

5

SUPPOSE EXPECTED RETURNS ARE AS FOLLOWS:

USA 13.7 PercentCANADA 18.3BELGIUM 12.5FRANCE 14.4GERMANY 12.5ITALY 19.7NETHERLANDS 17.4SWITZERLAND 13.0JAPAN 11.3UK 17.2

6

EXAMPLE:INTERNATIONAL PORTFOLIO SELECTION

1 VARIABILITY OF DIFFERENT MARKETS 1980-85

USA 14.9 PercentCANADA 20.8BELGIUM 16.3FRANCE 18.4GERMANY 13.5ITALY * 28.4NETHERLANDS 18.4SWITZERLAND 11.8JAPAN 11.4UK 17.0

7

CORRELATIONS BETWEEN RETURNS ON DIFFERENT MARKETS

USA CAN BEL FRA GER ITA NET SWI JAP UKUSA 1.00CAN 0.78 1.00BEL 0.11 0.09 1.00FRA 0.27 0.19 0.41 1.00GER 0.37 0.33 0.30 0.28 1.00ITA 0.12 0.29 0.08 0.19 0.04 1.00NET 0.28 0.35 0.38 0.31 0.52 0.43 1.00SWI 0.50 0.56 0.36 0.38 0.59 0.20 0.49 1.00JAP 0.34 0.32 0.30 0.23 0.36 0.21 0.45 0.37 1.00UK 0.46 0.51 0.34 0.31 0.47 0.40 0.45 0.58 0.55 1.00

8

EFFICIENT PORTFOLIOS

Exp StdPort ret dev USA CAN BEL FRA GER ITA NET SWI JAP UK1 19.7 28.4 - - - - - 100.0 - - - -2 19.0 19.9 - 50.0 - - - 50.0 - - - -3 18.7 18.2 - 47.0 - - - 39.0 14.0 - - -4 18.0 15.2 - 30.0 - - - 16.0 28.0 - - 26.05 17.4 13.9 - 24.0 - 13.0 - 10.0 26.0 - - 27.06 17.1 13.4 - 23.0 5.0 14.0 - 9.0 23.0 - - 25.07 16.8 13.1 - 22.0 7.0 14.0 4.0 9.0 21.0 - - 23.08 16.6 12.8 2.0 20.0 8.0 14.0 5.0 9.0 20.0 - - 22.09 16.2 12.3 5.0 17.0 9.0 13.0 7.0 10.0 17.0 3.0 - 19.010 13.6 9.5 13.0 2.0 11.0 7.0 13.0 10.0 2.0 18.0 21.0 -11 13.4 9.4 17.0 - 11.0 7.0 14.0 10.0 0.0 19.0 23.0 -12 13.4 9.3 16.0 - 11.0 6.0 14.0 10.0 - 19.0 23.0 -13 12.6 9.1 12.0 - 11.0 4.0 13.0 3.0 - 20.0 37.0 -

PERCENT INVESTED IN EACH COUNTRY

9

Security Market Line (SML)Risk and return relationship for individual security

Under CAPM, investor is concerned with the risk of themarket portfolio, M

2

Risk of individual security should be measured by itscontribution to the total risk of the market portfolio; therefore,the relevant risk measure of individual security is iM; not i

Equation of SML Expected return on asset

RM M

RF

0 1

rRrR

rRrR

fMifi

iMi

fMiM

fi

EE

MSet

EM

E

)()(

)()(

2

2

10

Understanding beta,

Two major components of risk:I. Company-unique riskII. Systematic risk

Systematic risk is a stock's responsiveness to movements inthe "general market," which cannot be eliminated throughdiversification.

Systematic risk is measured by the slope of the "characteristic line," as shown below:

The slope of the characteristic line is called "beta" infinance parlence.

The dispersion about the characteristic line (regression line)represents the company unique risk.

What will happen to the systematic risk and the company-unique risk as we add other stocks to our portfolio?

Stock Returns 10

5

5 10 15 MarketReturns

11

Capital Market Line (CML)• Equilibrium relationship between E(Rp) and σp for efficient

portfolios• Linear efficient set of CAPM by combining Market portfolio

with risk free (rf) borrowing and lending• CML only permits to well-diversified portfolios; portfolios not

employing M, the market portfolio, will plot below the CML• Equation of CML:

E(Rp)=rf + [(E(RM)-Rf )/ σM] σ(Rp)• Slope of CML: price of risk

{E(RM) – Rf }/ σM

• Price of time: Rf

12

Capital Asset Pricing Model (CAPM)

• Developed by Sharpe, Treynor, Lintner and Mossin• An equilibrium theory of how to price and measure risk of

portfolios as well as individual security• Concerning decomposition of risk into two components:

systematic (market, non-diversifiable) and unsystematic (unique, diversifiable)

• Stating that required return on any investment is the risk free return plus a risk premium measured by its systematic risk

E(ri)=rf+[E(rm)-rf]β where β = covariance risk of security i E(rm)-rf = market risk premium

13

Expected portfolioReturn Kp

Risk, σp

E

D

C

B

A

Feasible, or Attainable, Set

Feasible Set of Risky Portfolios

14

Expected portfolioReturn Kp

Risk, σp

E

D

C

B

A

Optimal Portfolio Selection

Optimal PortfolioInvestor B

Optimal PortfolioInvestor A

15

Expected portfolioReturn Kp

Risk, σp

E

D

C

B

A

Efficient Frontier with Risk-Free Asset

rm

kRF

M

KM

Z

new efficient portfolio

Y=mx+bKi=Krf+σi/σm(Km-Krf)b = interceptm = slope = Km-Krf/σm