Introduction of Risk and Return

Text: Chapter 9

Introduction to Risk and Return

Common stocks 13.0% 9.2% 20.3%

Small-company stocks 17.7 13.9 33.9

Long-termcorporate bonds 6.1 2.3 8.7

Long-termgovernment bonds 5.6 1.8 9.2

Intermediate-termgovernment bonds 5.4 1.6 5.7

U.S. Treasury bills 3.8 3.2

Inflation 3.2 4.5

Risk premium Arithmetic (relative to U.S. Standard

Series mean Treasury bills) deviation

-90% 90%0%

The Value of an Investment of $1 in 1900

$1

$10

$100

$1,000

$10,000

$100,000

Start of Year

Doll

ars

Common Stock

US Govt Bonds

T-Bills

15,578

14761

20

04

The Value of an Investment of $1 in 1900

Real returns

$1

$10

$100

$1,000

Start of Year

Do

lla

rs

Equities

Bonds

Bills

719

6.81

2.80

20

04

Rates of Return 1926-1997

Source: Ibbotson Associates

-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Common Stocks

Long T-Bonds

T-Bills

Year

Per

cent

age

Ret

urn

Measuring Risk

1 1 24

12 1113

1013

3 20123456789

10111213

-50

to -

40

-40

to -

30

-30

to -

20

-20

to -

10

-10

to 0

0 to

10

10 t

o 20

20 t

o 30

30 t

o 40

40 t

o 50

50 t

o 60

Return %

# of Years

Histogram of Annual Stock Market ReturnsHistogram of Annual Stock Market Returns

Average Market Risk Premia (by country)

4.3 4.7 5.1 5.3 5.8 5.9 5.9 6.3 6.4 6.67.6 8.1 8.2 8.6 9.3

10 10.7

0123456789

1011

Den

ma

rk

Bel

giu

m

Sw

itze

rla

nd

Sp

ain

Ca

na

da

Irel

an

d

Ger

ma

ny

UK

Av

era

ge

Net

her

lan

ds

US

A

Sw

eden

So

uth

Afr

ica

Au

stra

lia

Fra

nce

Ja

pa

n

Ita

ly

Risk premium, %

Country

Variance and Standard Deviation

VAR (ri~) = E [ ri

~ - E(ri~) ]2 = ri

2 ,

COV( r1~, r2

~ ) = E [(r1~ - E(r1

~)) (r2~ - E(r2

~))]

where ri~ is actual return governed by probability distribution

EX: The return of asset i next period is ether .2 with prob. 60% or -.1 with prob. 40%

E(ri~) = .6*.2 + .4*(-.1) = .08

Var(ri~) = .6*(.2-.08)2 + .4*(-.1-.08)2 = .0216

Return and Variance of Two Assets

Calculating Portfolio risks of two stocks

E(r~) Weight SD(r~)Stock A .15 .6 18.6Stock B .21 .4 28

E(rp~) = x1*E(r1

~) + x2*E(r2~), where x1 + x2 = 1

E(rp~) = .6*.15 + .4*.21 = .174

What about variance? x1*12 + x22

2?

No!!!

Covariance of a Portfolio of Two Assets

p2 = E [ rp

~ - E(rp~) ]2

= E [ x1r1~ + x2r2

~ - x1*E(r1~) - x2*E(r2

~) ]2

= E[ x1* (r1~ - E(r1

~)) + x2* (r2~-E(r2

~)) ]2

= E[ x12(r1

~-E(r1~))2 + x2

2(r2~-E(r2

~))2 +

x1x2(r1~-E(r1

~))(r2~-E(r2

~)) + x1x2(r1~-E(r1

~))(r2~-E(r2

~))]

= x121

2 + x222

2 + 2x1x2COV(r1~, r2

~)

Define COV(r1~, r2

~) = E[(r1~-E(r1

~)) (r2~-E(r2

~))] = 12

Correlation Coefficient

To get rid of the unit, we define Correlation coefficient

12 = COV(r1~, r2

~) / 12 , where -1<= <= 1

Thus, p2 = x1

212 + x2

222 + 2x1x21212

If 12 = 1, then p = X11 + X22

If 12 < 1, then p < X11 + X22

Stock 1 Stock 2

Stock 1 x121

2 x1x2COV(r1~, r2

~)

Stock 2 x1x2COV(r1~, r2

~) x222

2

Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

%12)1540(.)1060(. ReturnExpected

Portfolio Risk

2222

22

211221

2112212221

21

)3.27()40(.σx3.272.181

60.40.σσρxxCola-Coca

3.272.181

60.40.σσρxx)2.18()60(.σxMobil-Exxon

Cola-CocaMobil-Exxon

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

Portfolio Risk

% 18.3 9.333 DeviationStandard

9.33318.2x27.3)2(.40x.60x

]x(27.3)[(.40)

]x(18.2)[(.60) Variance Portfolio22

22

The Composition of Portfolio Variance

Two risky assets Three assets Four assets N risky assets

Variance of a Diversified Portfolio

What is the variance of portfolio if the number of stock increases?

General Formula: a portfolio with equally weighted N stocks

Portfolio variance:

= N (1/N)2 * average var. + (N2-N)(1/N)2 * average cov.

= 1/N * average var. + (1-1/N) * average cov.

As N increases, the variance of each individual stock becomes less

important, and the average covariance becomes dominant.

How Does Diversification Reduce Risks?

The central message: total risk can be decomposed into two parts: systematic and unsystematic risks.

Therefore diversification can only eliminate unique risks (or unsystematic risks, diversifyable risks), can not eliminate market risk (systematic risks, undiversificable risk)

What is unsystematic risks? RD program, new product introduction, labor relations,

personal changes, lawsuits.

The risk of a well-diversified portfolio depends on the market risk of the securities included in the portfolio.

Measuring Risk

0

5 10 15

Number of Securities

Po

rtfo

lio s

tan

da

rd d

ev

iati

on

Market risk

Uniquerisk

How Individual Securities Affect Portfolio Risk?

A B

Row 1 A . .62*18.62 .6*.4*.2*18.6*28 Row 2 B .6*.4*.2*18.6*28 .42*282

Row 1 = .6 * [.6*18.62 + .4*.2*18.6*28] = .6 * 249

Row 2 = .4 * [.6*.2*18.6*28 + .4*282] = .4 * 376

Total = 300

The contribution of stock A to portfolio risk is WEIGHT * COVARIANCE WITH ALL THE SECURITIES IN THE PORTFOLIO (INCLUDING ITSELF)

The risk of a stock not only depend on its own risks, but also its contribution to the risk of whole portfolio.

Stock’s Beta

If the portfolio is the market portfolio, then we have the formal definition of Beta

Beta - Sensitivity of a stock’s return to the return on the market portfolio.

= Cov (ri~, rm

~) / Var(rm~)

= i,mi m/ m2

= i,m [i/ m]

Conclusions

Markets risk accounts for most of the risk of a well-diversified portfolio.

The beta of an individual security measure its sensitivity to market movement.

A nice property of Beta:

p = Xii, where Xi is the weight of market value of asset I

Does corporate diversification add value for shareholders?