Risk Measurement
Risk = Actual return deviated from Expected Return
=
ROI
i
E(R)
Outcome
Probability
R
P
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)
PR
0
R=E(R)=
PR=0
R-R
(R-R)2
P(R-R)2
P(R-R)2
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  P ( R  R ) 2
Standard Deviation, Sigma, Expected Return
()(1)R
- 100
0
(1)()
+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)
Std. Dev. ()
Coef. of Var. (CV)
Risk (b)
8.0
0.0
0
0.0
17.4% 1.7%
20.0
13.4
1.1
7.9
1.29 -0.86
13.8%
18.8
1.4
0.68
15.0% 9.6%
15.3
3.3
1.0
0.3
1.00
Returns on Alternative Investments
II. Continuous Probability Distribution
Probability of Occurrence
0.4
Market Portfolio
-45
-30
-15
0
15
k
30
45
60
75
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
CVT-bills
CVHighTech
CVCollections
CVU.S. Rubber
CVMarket

CV =
k
= 0.0% / 8.0% = 0.0
= 20.0% / 17.4% = 1.1
= 13.4% / 1.7% = 7.9
= 18.8% / 13.8% = 1.4
= 15.3% / 15.0% = 1.0
Ranking of Investment Alternatives
Expected
Return
Security
k
High Tech
17.4%
Market
15.0
U.S. Rubber 13.8
T-bills
8.0
Collections
1.7
1 = Least risky
5 = Most risky
Risk


Ranking
20.0%
5
15.3
3
18.8
4
0.0
1
13.4
2
CV
1.1
1.0
1.4
0.0
7.9
CV
Ranking
3
2
4
1
5
Portfolio Return & Standard Deviation
2-Stock Portfolio Return: 50% High Tech and 50% Collections
n
kp =

wiki
i=1
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%
Standard Deviation:
State of the
Economy
Recession
Below average
Average
Above average
Boom
Expected Return
Prob. High Tech Collections 2-Stk Portfolio
0.10
-22.0%
28.0%
3.0%
0.20
-2.0
14.7
6.4
0.40
20.0
0.0
10.0
0.20
35.0
-10.0
12.5
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 & Collections
optional question integrated case
Rate of Return (%)
20
16
12
kP
8
4
0
0
20
40
Standard Deviation P (%)
20
16
12
8
4
0
0
60
80
100
% in High Tech
60
80
100
% in High Tech
P
20
40
Portfolio Size & Risk
Density
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 Beta
Return on Stock i,ki (%)
High Tech (slope = beta = 1.29)
Market (slope = beta = 1.0)
40
U.S. Rubber (slope = beta = 0.68)
20
-20
20
40
Return on the Market, kM (%)
-20
Year HighTech
1990
-12.3%
1991
14.1
1992
17.4
1993
20.6
1994
47.0
mean
17.0%
beta
1.29
T-Bills Collections U.S.Rubber The Market
8.0% 21.6%
-1.9%
-8.0%
8.0
3.9
12.1
12.5
8.0
1.8
13.8
15.0
8.0
-0.6
15.5
17.5
8.0
-18.1
29.4
38.0
8.0%
1.7%
13.8%
15.0%
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 (%)
22
20
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-2
SML: ki = krf + (kM - kRF) bi
= 8% + 7%(b i)
High Tech
kM
U.S. Rubber
kRF
Collections
-1
0
1
2
Changes in the Security Market Line
Required & Expected
Rates of Return (%)
Ki
30
Increased Risk Aversion
25
20
Increased Inflation
15
10
5
0
0.00
Original Situation
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%