1. Risk and Return: Past and Prologue
Chapter 5
Risk and Return: Past and Prologue

2. Rates of Return: Single Period- P + D
HPR = 1 1 P
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 = ( )/ ( 20) = 25%

4. Data from Text Example p. 133
Assets(Beg.)
HPR (.20)
TA (Before
Net Flows
Net Flows (0.8)
End Assets

5. Returns Using Arithmetic and Geometric Averaging
ra = (r1 + r2 + r rn) / n
ra = ( ) / 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
= (1.3750) 1/4 -1 = = 8.29%

6. 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

7. Dollar Weighted Average Using Text Example
Net CFs
$ (mil)
Solving for IRR
1.0 = -.1/(1+r) /(1+r)2 + .8/(1+r)3 +
1.0/(1+r)4
r = 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) = 12.68%

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. Skewed Distribution: Large Negative Returns Possible
Median
Negative
Positive r

12. Skewed Distribution: Large Positive Returns Possible
Median
Negative r
Positive

13. Measuring Mean: Scenario or Subjective Returns( r ) = p s S
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states

14. Numerical Example: Subjective or Scenario Distributions
State Prob. of State rin State
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15

15. Measuring Variance or Dispersion of Returns
Subjective or Scenario
Variance = S s p ( ) [ r - E )] 2
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)( )2+(.2)( ) ( )2]
Var=
S.D.= [ ] 1/2 = .1095

16. Annual Holding Period Returns From Table 5.3 of Text
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
Lg. Stk
Sm. Stk
LT Gov
T-Bills
Inflation

17. Annual Holding Period Risk Premiums and Real Returns
Excess Real
Series Returns% Returns%
Lg. Stk
Sm. Stk
LT Gov
T-Bills
Inflation

18. 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)

19. 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)

20. 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

21. 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

22. 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%

23. Possible Combinations
E(r)
E(rp) = 15% P
rf = 7% F s
22%

24. Variance on the Possible Combined Portfolios
= 0, then s p c =
Since rf y s s

25. Combinations Without Leverage
= .75(.22) = .165 or 16.5%
If y = .75, then
= 1(.22) = .22 or 22%
If y = 1
= 0(.22) = .00 or 0%
If y = 0 s s s

26. 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

27. s E(r) CAL (Capital Allocation Line) P E(rp) = 15% E(rp) - rf = 8%s P
= 22%

28. 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

29. Quantifying Risk Aversion( ) - = s E r r .
005 A p f p
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

30. Quantifying Risk Aversion
Rearranging the equation and solving for A E ( r ) - r p f A =
.005 2 σ p
Many studies have concluded that investors’ average risk aversion is between 2 and 4