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 1926

Source: Ibbotson Associates

0.1

10

1000

1925 1933 1941 1949 1957 1965 1973 1981 1989 1997

S&PSmall CapCorp BondsLong BondT Bill

Inde

x

Year End

1

5520

1828

55.38

39.07

14.25

0.1

10

1000

1925 1933 1941 1949 1957 1965 1973 1981 1989 1997

S&PSmall CapCorp BondsLong BondT Bill

The Value of an Investment of $1 in 1926

Source: Ibbotson Associates

Inde

x

Year End

1

613

203

6.15

4.34

1.58

Real returns

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

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

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?