1. Holding Period Return

2. Rates of Return: Single Period Example
Ending Price = 24
Beginning Price = 20
Dividend = 1
HPR = ( )/ ( 20) = 25%

3. Example 43
You purchased a share of stock for $20. One year later you received $2 as dividend and sold the share for $23. Your holding-period return was __________.
A) 5 percent
B) 10 percent
C) 20 percent
D) 25 percent
The holding period return on a stock was 25%. Its ending price was $18 and its beginning price was $16. Its cash dividend must have been __________.
A) $0.25
B) $1.00
C) $2.00
D) $4.00

4. Returns Using Arithmetic and Geometric Averaging
ra = (r1 + r2 + r rn) / n
ra =.10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = = 8.29%

5. Example 12 The arithmetic average of 12%, 15% and 20% is _________.
B) 15%
C) 17.2%
D) 20%
The geometric average of 10%, 20% and 25% is __________.
A) 15%
B) 18.2%
C) 18.3%
D) 23%

6. 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 =
EAR =

7. 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

8. Return Variability: The Second Lesson
1 - 8
Return Variability: The Second Lesson

9. Example What range of return would you expect to see in 95% of time?
Is it possible you can earn 65% return annually at 1% significant level?

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

11. Numerical Example: Subjective or Scenario Distributions
State Prob. of State rin State
E(r) =.15

12. 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=.01199
S.D.= [ ] 1/2 = .1095

13. Historical mean and standard deviation

14. Risk Premiums and Risk Aversion
Degree to which investors are willing to commit funds
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:
E(Rp) - Rf

15. Risk Premiums and Risk Aversion
To quantify the degree of risk aversion with parameter A:
E(Rp) – Rf = (1/2) A σp2

16. The Sharpe (Reward-to-Volatility) Measure
Sharpe ratio = Portfolio risk premium/standard deviation of the excess returns = (E(Rp) – Rf )/σp

17. Annual Holding Period Returns From Table 5.3 of Text
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
World Stk
US Lg Stk
US Sm Stk
Wor Bonds
LT Treas
T-Bills
Inflation

18. Figure 5.1 Frequency Distributions of Holding Period Returns

19. Figure 5.2 Rates of Return on Stocks, Bonds and Bills

20. 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 = (1+R) / (1 + i) - 1
2.83% = (9%-6%) / (1.06)

21. Figure 5.4 Interest, Inflation and Real Rates of Return

22. 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)
Risk premium: risk asset return-risk-free rate
Issues
Examine risk/ return tradeoff
Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

23. Example: Given: rf = 7% srf = 0% E(rp) = 15% sp = 22% y = % in p
(1-y) = % in rf

24. 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%

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

26. 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

27. 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
Reward-to-variability ratio = risk premium/standard deviation

28. Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

29. Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates

30. 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

31. Example 22
Consider the following two investment alternatives. First, a risky portfolio that pays 15% rate of return with a probability of 60% or 5% with a probability of 40%. Second, a treasury bill that pays 6%. The risk premium on the risky investment is __________.
A) 1%
B) 5%
C) 9%
D) 11%
Consider the following two investment alternatives. First, a risky portfolio that pays 20% rate of return with a probability of 60% or 5% with a probability of 40%. Second, a treasury that pays 6%. If you invest $50,000 in the risky portfolio, your expected profit would be __________.
A) $3,000
B) $7,000
C) $7,500
D) $10,000

32. Example 41
You have $500,000 available to invest. The risk-free rate as well as your borrowing rate is 8%. The return on the risky portfolio is 16%. If you wish to earn a 22% return, you should __________.
A) invest $125,000 in the risk-free asset
B) invest $375,000 in the risk-free asset
C) borrow $125,000
D) borrow $375,000
The Manhawkin Fund has an expected return of 12% and a standard deviation return of 16%. The risk free rate is 4%. What is the reward-to-volatility ratio for the Manhawkin Fund?
A) 0.5
B) 1.3
C) 3.0
D) 1.0

33. Example 422
You invest $100 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 12% and a standard deviation of 15% and a treasury bill with a rate of return of 5%. __________ of your money should be invested in the risky asset to form a portfolio with an expected rate of return of 9%
A) 87%
B) 77%
C) 67%
D) 57%
An investor invests 40% of his wealth in a risky asset with an expected rate of return of 15% and a variance of 4% and 60% in a treasury bill that pays 6%. Her portfolio's expected rate of return and standard deviation are __________ and __________ respectively.
A) 8.0%, 12%
B) 9.6%, 8%
C) 9.6%, 10%
D) 11.4%, 12%
The expected return of portfolio is 8.9% and the risk free rate is 3.5%. If the portfolio standard deviation is 12.0%, what is the reward to variability ratio of the portfolio?
A) 0.0
B) 0.45
C) 0.74
D) 1.35