Risk and Return: Past and Prologue

1
Chapter 5

• Risk and Return Past and Prologue

2
Rates of Return Single Period

P
P
D
-

HPR

1
0
1
P
0
HPR Holding Period Return P1 Ending price P0
Beginning price D1 Dividend during period
one
3
Rates of Return Single Period Example

• Ending Price 24
• Beginning Price 20
• Dividend 1
• HPR ( 24 - 20 1 )/ ( 20) 25

4
Returns over Multiple Periods

• 1 2 3 4
• Assets(Beg.) 1.0 1.2 2.0 .8
• HPR .10 .25 (.20) .25
• TA (Before
• Net Flows 1.1 1.5 1.6 1.0
• Net Flows 0.1 0.5 (0.8) 0.0
• End Assets 1.2 2.0 .8 1.0

5
Returns Using Arithmetic and Geometric Averaging

• Arithmetic
• ra (r1 r2 r3 ... rn) / n
• ra (.10 .25 - .20 .25) / 4
• .10 or 10
• Geometric
• rg (1r1) (1r2) .... (1rn) 1/n - 1
• rg (1.1) (1.25) (.8) (1.25) 1/4 - 1
• (1.5150) 1/4 -1 .0829 8.29

6
Dollar Weighted Returns

• Internal Rate of Return (IRR) - the discount rate
  that makes the present value of future cash flows
  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

7
Dollar Weighted Average Using Text Example

• Net CFs 1 2 3 4
• (mil) - .1 - .5 .8 1.0
• Solving for IRR
• 1.0 -.1/(1r)1 -.5/(1r)2 .8/(1r)3
• 1.0/(1r)4
• r .0417 or 4.17

8
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

9
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

10
Normal Distribution
s.d.
s.d.
r
Symmetric distribution
11
Empirical Rule

• For data having a bell-shaped distribution
• Approximately 68 of the data values will lie
  within one std. dev. of the mean
• Approximately 95 of the data values will lie
  within two std. dev. of the mean.
• Almost all of the data will be within three std.
  dev. of the mean.

12
Measuring Mean Scenario or Subjective Returns
Subjective returns
p(s) probability of a state r(s) return if a
state occurs 1 to s states
13
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
14
Measuring Variance or Dispersion of Returns

• Subjective or Scenario

Standard deviation variance1/2
Using Our Example
Var (.1)(-.05-.15)2(.2)(.05- .15)2...
.1(.35-.15)2 Var .01199 S.D. .01199 1/2
.1095
15
Annual Holding Period ReturnsFrom Table 5.3 of
Text

• Geom. Arith. Stan.
• Series Mean Mean Dev.
• Lg. Stk 10.51 12.49 20.30
• Sm. Stk 12.19 18.29 39.28
• LT Gov 5.23 5.53 8.18
• T-Bills 3.80 3.85 3.25
• Inflation 3.06 3.15 4.40

16
Annual Holding Period Risk Premiums and Real
Returns

• Excess Real
• Series Returns Returns
• Lg. Stk 8.64 9.34
• Sm. Stk 14.44 15.14
• LT Gov 1.68 2.38
• T-Bills --- 0.60
• Inflation --- ---

17
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)

18
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)

19
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

20
Example Using the Numbers in Chapter 6 (pp
171-173)
21
Expected Returns for Combinations
22
Variance on the Possible Combined Portfolios
23
Combinations Without Leverage
s
s
s
24
Capital Allocation Line
E(r)
E(rp) 15
P
rf 7
F
0
s
22
25
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

26
CAL (Capital Allocation Line)
E(r)
P
E(rp) 15
E(rp) -
rf 8
) S 8/22
rf 7
F
0
s
22
P
27
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

28
Quantifying Risk Aversion
(
)
s



-
A
r
r
E
5
.
p
f
p
E(rp) Expected return on portfolio p rf the
risk free rate .5 Scale factor A x sp
Proportional risk premium The larger A is, the
larger will be the added return required for risk
29
Quantifying Risk Aversion
Rearranging the equation and solving for A
-
r
r
E
)
(
f
p

A

2
.5
s
p
Many studies have concluded that investors
average risk aversion is between 2 and 4