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Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Chapter 6
Risk and Return: Past
and Prologue
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
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Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Rates of Return: Single Period
P D
P
HPR 
P
1
0
1
0
HPR = Holding Period Return
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
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Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Rates of Return: Single Period
Example
Ending Price =
Beginning Price =
Dividend =
24
20
1
HPR = ( 24 - 20 + 1 )/ ( 20) = 25%
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© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
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Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Data from Text Example p. 154
1
1.0
.10
Assets(Beg.)
HPR
TA (Before
Net Flows
1.1
Net Flows
0.1
End Assets
1.2
Irwin / McGraw-Hill
2
1.2
.25
3
2.0
(.20)
4
.8
.25
1.5
0.5
2.0
1.6 1.0
(0.8) 0.0
.8 1.0
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
5
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Returns Using Arithmetic and
Geometric Averaging
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25 - .20 + .25) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1
= (1.5150) 1/4 -1 = .0829 = 8.29%
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
6
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Dollar Weighted Returns
Internal Rate of Return (IRR) - the
discount rate that results present value
of the future cash flows being equal to
the investment amount
•
•
•
•
•
Considers changes in investment
Initial Investment is an outflow
Ending value is considered as an inflow
Additional investment is a negative flow
Reduced investment is a positive flow
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
7
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Dollar Weighted Average Using Text
Example
Net CFs
$ (mil)
1 2
- .1 - .5
3
.8
4
1.0
Solving for IRR
1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 +
1.0/(1+r)4
r = .0417 or 4.17%
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
8
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Quoting Conventions
APR = annual percentage rate
(periods in year) X (rate for period)
EAR = effective annual rate
( 1+ rate for period)Periods per yr - 1
Example: monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01)12 - 1 = 12.68%
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
9
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Characteristics of Probability
Distributions
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately normal,
the distribution is described by
characteristics 1 and 2
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
10
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Normal Distribution
s.d.
s.d.
r
Symmetric distribution
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© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
11
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Skewed Distribution: Large Negative
Returns Possible
Median
Negative
Irwin / McGraw-Hill
r
Positive
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
12
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Skewed Distribution: Large Positive
Returns Possible
Median
Negative
Irwin / McGraw-Hill
r
Positive
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
13
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Measuring Mean: Scenario or
Subjective Returns
Subjective returns
E(r) = S p(s) r(s)
s
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
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© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
14
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Numerical Example: Subjective or
Scenario Distributions
State
Prob. of State
rin State
1
.1
-.05
2
.2
.05
3
.4
.15
4
.2
.25
5
.1
.35
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
15
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Measuring Variance or Dispersion of
Returns
Subjective or Scenario
Variance = S p(s) [rs - E(r)]
2
s
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
16
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Real vs. Nominal Rates
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R - i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9% - 6%
Fisher effect: Exact
r = (R - i) / (1 + i)
2.83% = (9%-6%) / (1.06)
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
17
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Annual Holding Period Returns
From Figure 6.1 of Text
Geom
Series
Mean%
Lg Stk
11.01
Sm Stk 12.46
LT Gov
5.26
T-Bills
3.75
Inflation
3.08
Irwin / McGraw-Hill
Arith Stan.
Mean% Dev.%
13.00
20.33
18.77
39.95
5.54
7.99
3.80
3.31
3.18
4.49
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
18
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Annual Holding Period Risk
Premiums and Real Returns
Risk
Series
Premiums%
Lg Stk
9.2
Sm Stk
14.97
LT Gov
1.74
T-Bills
--Inflation
--Irwin / McGraw-Hill
Real
Returns%
9.82
15.59
2.36
0.62
---
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
19
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Allocating Capital Between Risky &
Risk-Free Assets
• Possible to split investment funds
between safe and risky assets
• Risk free asset: proxy; T-bills
• Risky asset: stock (or a portfolio)
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© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
20
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Allocating Capital Between Risky &
Risk-Free Assets (cont.)
• Issues
– Examine risk/ return tradeoff
– Demonstrate how different degrees of risk
aversion will affect allocations between
risky and risk free assets
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© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
21
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Example Using the Numbers in
Chapter 6 (pp 171-173)
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
22
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
23
Bodie • Kane • Marcus
Essentials of Investments
E(r)
Possible Combinations
E(rp) = 15%
rf = 7%
0
Irwin / McGraw-Hill
Fourth
Edition
P
F
22%
s
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
24
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Variance on the Possible Combined
Portfolios
Since
s r = 0, then
f
sc = y s p
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
25
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Combinations Without Leverage
If y = .75, then
s c = .75(.22) = .165 or 16.5%
If y = 1
s c = 1(.22) = .22 or 22%
If y = 0
sc = 0(.22) = .00 or 0%
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
26
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Using Leverage with Capital
Allocation Line
Borrow at the Risk-Free Rate and invest
in stock
Using 50% Leverage
rc = (-.5) (.07) + (1.5) (.15) = .19
sc = (1.5) (.22) = .33
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
27
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
CAL
(Capital
Allocation
Line)
E(r)
P
E(rp) = 15%
E(rp) - rf = 8%
rf = 7%
0
Irwin / McGraw-Hill
) S = 8/22
F
P = 22%
s
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
28
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
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
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
29
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Quantifying Risk Aversion
Erp   rf  .005  A sp
E(rp) = Expected return on portfolio p
rf = the risk free rate
.005 = Scale factor
A x sp = Proportional risk premium
The larger A is, the larger will be the
added return required for risk
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
30
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Quantifying Risk Aversion
Rearranging the equation and solving for A
E ( rp )  rf
A
.005  σ 2
p
Many studies have concluded that
investors’ average risk aversion is
between 2 and 4
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.