Risk Measurement

Risk = Actual return deviated from Expected Return

= ROI i E(R)

Outcome Probability

R P PR R-R (R-R)2 P(R-R)2

Heads +100 50% .5(100) 100-0=100 10,000 .5(10,000)

Tails -100 50% .5(-100) -100-0=-100 10,000 .5(10,000)

0 P(R-R)2

R=E(R)=PR=0

2 = 10,000 = +/- 100

E(R) = PiRi = R = Expected Return

Ex-ante = Future Events

Ex-post = Historical Data

= Standard Deviation = risk

= Variance = P(R-R)2

2 2 P R R( )__

Standard Deviation, Sigma, Expected Return

R - 100 0 +100

Returns on Alternative Investments

I. Discrete Probability Distribution

Estimated Rate of Return

State of the T- High U.S. Market 2-Stock

Economy Probability Bills Tech Collections Rubber Portfolio Portfolio

Recession 0.1 8.0% -22% 28.0% 10.0% -13.0% 3.0%

Below average 0.2 8.0 -2.0 14.7 -10.0 1.0 6.4

Average 0.4 8.0 20.0 0.0 7.0 15.0 10.0

Above average 0.2 8.0 35.0 -10.0 45.0 29.0 12.5

Boom 0.1 8.0 50.0 -20.0 30.0 43.0 15.0

Expected Ret (k) 8.0 17.4% 1.7% 13.8% 15.0% 9.6%

Std. Dev. ( 0.0 20.0 13.4 18.8 15.3 3.3

Coef. of Var. (CV) 0 1.1 7.9 1.4 1.0 0.3

Risk (b) 0.0 1.29 -0.86 0.68 1.00

Returns on Alternative Investments

II. Continuous Probability Distribution

Probability of Occurrence

0.4

Market Portfolio

-45 -30 -15 0 15 30 45 60 75

k Rate of Return

(%)

Calculation of k

n

k = Piki

i=1

kHigh Tech = 0.10(-22.0%) + 0.20(-2.0%)

+ 0.40(20.0%) + 0.20(35.0%)

+ 0.10(50.0%) = 17.4%

kT-bills = 8.0%

kCollections = 1.7%

kU.S.Rubber = 13.8%

kM = 15.0%

Calculation of

n

= VARIANCE = 2 = (ki - k)2Pi

i=1

High Tech = [(-22.0 - 17.4)2 0.10 + (-2.0 - 17.4)2 0.20

+ (20.0 - 17.4)2 0.40 + (35.0 - 17.4)2 0.20

+ (50.0 - 17.4)2 0.10]1/2 = (401.1)1/2 = 20.0%

T-bills = 0.0%

Collections = 13.4%

U.S.Rubber = 18.8%

M = 15.3%

Continuous Probability Distributions: High Tech, U.S. Rubber, & T-Bills

Probability of Occurrence

T-Bills

High Tech

U.S. Rubber

-45 -30 -15 0 8 15 30 45 60

Rate of Return (%)

Calculation of CV

CV =

k

CVT-bills = 0.0% / 8.0% = 0.0

CVHighTech = 20.0% / 17.4% = 1.1

CVCollections = 13.4% / 1.7% = 7.9

CVU.S. Rubber = 18.8% / 13.8% = 1.4

CVMarket = 15.3% / 15.0% = 1.0

Ranking of Investment Alternatives

Expected

Return Risk CV

Security k Ranking CV Ranking

High Tech 17.4% 20.0% 5 1.1 3

Market 15.0 15.3 3 1.0 2

U.S. Rubber 13.8 18.8 4 1.4 4

T-bills 8.0 0.0 1 0.0 1

Collections 1.7 13.4 2 7.9 5

1 = Least risky

5 = Most risky

Portfolio Return & Standard Deviation

2-Stock Portfolio Return: 50% High Tech and 50% Collections

kp = n wiki

i=1

kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%

Standard Deviation:State of the Expected Return

Economy Prob. High Tech Collections 2-Stk Portfolio

Recession 0.10 -22.0% 28.0% 3.0%

Below average 0.20 -2.0 14.7 6.4

Average 0.40 20.0 0.0 10.0

Above average 0.20 35.0 -10.0 12.5

Boom 0.10 50.0 -20.0 15.0

Portfolio Return & Standard Deviation

By considering the portfolio return in each state of the economy, we have another way of calculating kp:

kp = 0.10(3.0%) + 0.20(6.4%) + 0.40(10.0%)

+ 0.20(12.5%) + 0.10(15.0%) = 9.6%

Given the distribution of returns for the portfolio, we can calculate the portfolio’s p and CV:

p = [(3.0 - 9.6)2 0.10 + (6.4 - 9.6)2 0.20

+ (10.0 - 9.6)2 0.40 + (12.5 - 9.6)2 0.20

+ (15.0 - 9.6)2 0.10]1/2 = 3.3% and

CVp = 3.3% / 9.6% = 0.34

Portfolio Returns & Risk: High Tech & Collectionsoptional question integrated case

Rate of Return (%)

20

16

12 kP

8

4

0

0 20 40 60 80 100 % in High Tech

Standard Deviation P (%)

20

16

12 P

8

4

0

0 20 40 60 80 100 % in High Tech

Portfolio Size & RiskDensity

Portfolio of

Stocks with K p=16%

One Stock

0 16 Percent

1. gets smaller as more stocks are combined.

2. kp remains constant.

3. So, if you don’t like risk, hold a portfolio (or a mutual fund).

Portfolio Risk, p (%)

33

30 Minimum attainable risk

in a portfolio of average stocks

25 Diversifiable, Risk

sM = 20.6

15 Stand-alone

Risk Market Risk

10

0 10 20 30 40 1,500+ # of stocks in portfolio

Chapter 6 The Concept of BetaReturn on Stock i,ki (%)

High Tech (slope = beta = 1.29)

40 Market (slope = beta = 1.0)

U.S. Rubber (slope = beta = 0.68)

20

-20 20 40

Return on the Market, kM (%)

-20

Year HighTech T-Bills Collections U.S.Rubber The Market

1990 -12.3% 8.0% 21.6% -1.9% -8.0%

1991 14.1 8.0 3.9 12.1 12.5

1992 17.4 8.0 1.8 13.8 15.0

1993 20.6 8.0 -0.6 15.5 17.5

1994 47.0 8.0 -18.1 29.4 38.0

mean 17.0% 8.0% 1.7% 13.8% 15.0%

beta 1.29 0.00 -0.86 0.68 1.00

Security Market Line Equation

kRF = T-Bill reate = 8%

kM = km = 15%

ki = kRF +(kM - kRF)bi

kHigh Tech = 8.0% + (15.0% - 8.0%)1.29

= 8.0% + (7.0%) 1.29

= 8.0% +9.0% = 17.0%

kM = 8.0% + (7.0%) 1.00 = 15.0%

kU.S.Rubber = 8.0% + (7.0%) 0.68 = 12.8%

kT-bills = 8.0% + (7.0%) 0.00 = 8.0%

kCollections = 8.0% + (7.0%) (-0.86) = 2.0%

Security Market Line Graph

Required & Expected

Rates of Return (%) SML: ki = krf + (kM - kRF) bi

22 = 8% + 7%(bi)

20

18 High Tech

16 kM

14 U.S. Rubber

12

10

8 kRF

6

4

2 Collections

0

-2

-4

-6

-2 -1 0 1 2

Beta

Changes in the Security Market Line

Required & Expected

Rates of Return (%)

Ki

30 Increased Risk Aversion

25

20 Increased Inflation

15

10

5 Original Situation

0

0.00 0.50 1.00 1.50 2.00 Beta

Portfolio size and risk

Large company stock : 12.6% + 20% = 32.5%

12.6% - 20% = -7.5%

Small company stock : 17.7% + 34.4% = 52.1%

17.7% - 34.4% = -16.7%

Long term bonds : 6% + 8.7%

6% - 8.7%

U.S bill : 3% + 3.3%

3% - 3.3%