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CHAPTER 5
Risk and Rates of Return
5-1
5.1 Rates of Return

Holding Period Return: Rates of Return over a given period
Ending price  Beginning price  Cash dividend
HPR 
Beginning price


Suppose the price of a share is currently $100, and your time
horizon is one year. You expect the cash dividend during the
year to be $4. Your best guess about the price of the share is
to be $110 after one year from now. So,
$110  $100  $4
HPR 
 14%
$100
5-2
Measuring Investment Return over Multiple Period
Year
Annual Holding Period Return
2000
0.2
2001
0.05
2002
-0.05
2003
0.15
2004
0.3
Arithmetic Mean
(20+5-5+15+30)/5
Geometric Mean
(1.2*1.05*.95*1.15*1.3)1/5-1
=0.13
=0.1
5-3
5.2 Risk and Return

Risk is the concept of fluctuations. This fluctuations can
be
(i) a deviation of the actual return from the expected
return, or
(ii) a deviation of average return from the year to year
return. Higher the fluctuations, higher is the risk.
Measures of risk are:
 i. Standard Deviation
 ii. Coefficient of Variance
 iii. Beta
5-4
Calculation of Risk-Return
R

Expected Return  E ( R)  R 
  (R * P )
n
i
i
Risk   
Risk   
2
(
R

R
)
 i
n 1
i
( For time series data)
2
(
P
)
(
R

R
)
( for probabilit y distribution)
 i i
5-5
Calculation of Risk-Return
(Historical Data)
Year
2000
2001
2002
2003
2004
Mean Return=
Stand.Deviation2
Stand.Deviation=
Return (%)
20
5
Dev. (Ri-E(R))
7
-8
Dev. Square
49
64
-5
15
30
13%
-18
2
17
324
4
289
730
Sum of Dev sq=
730/(5-1)=
Square root (182.5)=
182.5
13.5%
5-6
Calculation of Risk-Return: Scenario Analysis
(Probability Distribution)
State of
Economy
Boom
Normal
Recession
Probability Return (Ri)
(Pi)
(%)
0.25
Exp. Value
(Pi*Ri)
25
0.5
14
0.25
0
i 1
Return =13.25%
Risk=8.9%
Deviation
Square
Dev sq* Pi
6.25
(25-13.25)
=11.75
(11.25)2
=138.0625
(138*.25)
=34.52
7
(14-13.25)
=0.75
(0.75)2
=0.5625
(.56*.5)
=0.28
0
(0-13.25)
=-13.25
(-13.25)2
=175.5625
(175*.25)
=43.89
m
m
E ( R)   Pi Ri
Deviation
(Ri-E(R))
   pi {[( Ri  E ( Ri )]2 }
2
13.25%
i 1
Stand Dev
78.69
8.87
5-7
Figure:5.3:Normal Distribution
A normal distribution looks like a bell-shaped curve.
Probability
Mean=13.25%
Standard Deviation=8.87%
– 3
– 13.36%
– 2
– 4.5%
– 1
4.4%
0
13.25%
+ 1
22.12%
+ 2
31%
+ 3
39.86%
68.26%
95.44%
99.74%
5-8
5.3 THE HISTORICAL RECORD
Bills, Bonds, and Stocks:1926-2006
5-9

Figure 5.1 Frequency Distributions of Holding Period Returns
5-10
Probability distributions

With the same average return more standard deviation
means more risk. Shown graphically. Note that as risk
increases height goes down and width increases.
Firm X
 x  10%
 y  40%
Firm Y
-70
0
15
Expected Rate of Return
100
Rate of
Return (%)
5-11
Effectiveness of Diversification of Portfolio
Climate
Probability
Rainy
0.25
Moderate
Dry
E(R) =
Risk  
Return
Ice-cream
Portfolio
Return
25
-5
10
0.5
14
10
12
0.25
0
15
7.5
13.3%
7.5%
10.4%
8.9%
7.5%
1.9%
 R * P 
i
i
2
(
R

R
)
 i Pi 
Return of
Umbrella
5-12
Portfolio Effects on Risk and Return
14
12
Return
10
8
Series1
6
4
2
0
0
2
4
6
8
10
Risk
5-13
Optimum portfolio and CML:
Given the feasible set highest possible utility
function gives us O.P. and the tangency is CML
.
U1
O.P
CML
Return
Borrowing
Feasible set
Lending
Risk (σ)
5-14
Investor attitude towards risk


Risk aversion – assumes investors dislike risk
and require higher rates of return to
encourage them to hold riskier securities.
Risk premium – the difference between the
return on a risky asset and less risky asset,
which serves as compensation for investors to
hold riskier securities.
5-15
 P2
Risk Premiums and Risk Aversion

If T-Bill denotes the risk-free rate, rf, and variance,
denotes volatility of returns then:
The risk premium of a portfolio is:
 2,
To quantify the degree of risk aversion with parameter A:
 P2


If the risk premium of a portfolio is 8%, and the standard deviation is
20%, then risk aversion of the investor is: A=.08/(.5x.22)=4.
Compare this with risk premium of 10%, and A=.1/(.5x.22)=5
5-16
The Sharpe (Reward-to-Volatility)
Measure
5-17
Coefficient of Variation (CV)
A standardized measure of dispersion about
the expected value, that shows the risk per
unit of return. When, both return and risk
increase then coefficient of variance (CV)
should be used.
Std dev 
CV 
 ^
Mean
k
5-18
Use of coefficient of variance

Example: We have 2 alternatives to invest. Security A
has a mean return of 10% and a standard deviation of
6%, and security B has a mean return of 13% with a
standard deviation of 8%. Which investment is better.
6%
CVA 
*100  60%
10%
8%
CVB 
*100  61.5%
13%

So, security A is better as the Coefficient of variance
of A is less than the that of B.
5-19
5.4 INFLATION AND REAL RATES OF RETURN
5-20
Real vs. Nominal Rates
Fisher effect: Approximation
Let,
nominal rate=R
Real rate=r
Inflation rate (CPI)=i
nominal rate = real rate + inflation rate: R ≈ r + i
or r = R - i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9% - 6%
Fisher Effect:

2.83% = (9%-6%) / (1.06)
5-21
5.5 ASSET ALLOCATION ACROSS RISKY
AND RISK-FREE PORTFOLIOS
5-22
Risk tolerance:
Slope of Indifference curves

Return
Conservative
Normal
15%
Aggressive
10%
7%
6%
Risk (σ)
5-23
Optimum portfolio with
different risk tolerance

Return
CML
Efficient Frontier
Rm
σm
Risk (σ)
5-24
Risk Aversion and Allocation



Greater levels of risk aversion lead to larger proportions
of the risk free rate
Lower levels of risk aversion lead to larger proportions of
the portfolio of risky assets
Willingness to accept high levels of risk for high levels of
returns would result in leveraged combinations
5-25
Allocating Capital




Possible to split investment funds between safe and risky
assets
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
Issues
 Examine risk/ return tradeoff
 Demonstrate how different degrees of risk aversion will
affect allocations between risky and risk free assets
5-26
The Risky Asset:
Text Example (Page 149)
Total portfolio value
= $300,000
Risk-free value
= 90,000
Risky (Vanguard and Fidelity) = 210,000
Vanguard (V) = 54% =$113,400
Fidelity (F)
= 46% = $96,600
Total
=$210,000
y=210,000/300,000
=0.7 (portfolio in risky assets)
1-y=90,000/300,000
=0.3 (proportion of Risk-free investment)
Vanguard
Fidelity
Portfolio P
Risk-Free Assets F
Portfolio C
113,400/300,000 = 0.378
96,600/300,000 = 0.322
210,000/300,000 = 0.700
90,000/300,000 = 0.300
300,000/300,000 = 1.000
5-27
Change in risk exposure (p.150)








Suppose, the investor decides to decrease the risk exposure
from 0.7 to 0.56.
Now, the risky portfolio would be=0.56*300,000=$168,000
This requires a sale of (210,000-168,000)=$42,000 of risky
holdings, and use the sale proceeds to purchase risk-free asset
How would the investment of risky asset change?
Sale of Vanguard=42,000*.54=$22,680;
 New Vanguard holding=113,400-22,680=$90,720
Sale of Fidelity=42,000*.46=$19,320
 New Fidelity holding=96,600-19,320=$77,280
Total Vanguard & Fidelity=90,720+77,280=$168,000
y=168,000/300,000=.56
5-28
Capital Allocation Line (CAL)
(Test of linearity)
rf  7%
 rf  0%
E (rp )  15%  p  22%
y  % in risky portfolio; (1  y )  % in rf
E (r c )  Expected return of complete portfolio  y.E (rp )  (1  y )rf ;  c  y. p ; slope 
E (rc )  rf
c
Example,
If ,
y  1; E (rc )  15%,  c  1* .22  .22; slope 
Lending
E (rp )  rf
p

.15  .07
 0.36
.22
case : 50 : 50
y  .5; E (rc )  (0.5 * .15)  (.5 * .07)  .11,  c  y *  p  (0.5 * .22)  .11; slope 
.11  .07
 0.36
.11
Borrowing case; If , leverage 50% :
y  1.5; E (rc )  (1.5 * .15)  (.5 * .07)  .19;  c  1.5 * .22  .33; slope 
.19  .07
 .036
.33
5-29
Figure 5.5 Investment Opportunity Set
with a Risk-Free Investment
5-30
return
CAL and CML

Efficient Frontier
Rm
rf
m
P

CAL1 is dominant over CAL2, CAL2 is dominant over CAL3.

CML is dominant over CAL1.
5-31
5.6 PASSIVE STRATEGIES AND THE
CAPITAL MARKET LINE
5-32
Table 5.5 Average Rates of Return,
Standard Deviation and Reward to Variability
5-33
Use of historical data to predict CML




Use of old data is popular
Old data may not be representative for future
Weight of old data keeps changing
Correction of old data for future is largely subjective,
but customary.
5-34
Costs and Benefits of Passive Investing



Active strategy entails costs
Free-rider benefit
Involves investment in two passive portfolios
 Short-term T-bills
 Fund of common stocks that follows a broad
market index. Diversification can be based on
asset allocation like industry classification,
large and small firms, local and foreign firms
5-35