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

2. 5.1 RATES OF RETURN

3. Single Period Return
Measured by the
Holding Period Return

4. Rates of Return: Multi-Period Example Text (Page 118)
Data from Table 5.1
Beginning Assets
HPR (0.20)
Assets (Before
Net Flows)
Net Flows (0.8)
Ending Assets

5. Cumulative HPR over the Multiperiod
Cumulative HPR over 4-year Period
= (1+HPR1) x (1+HPR2) x (1+HPR3) x (1+HPR4) - 1
= (1+0.10) x (1+0.25) x (1+ (-02)) x (1+0.25) – 1
= 37.5%

6. What are the average returns over the 4-Year Period?
1. Arithmetic mean return
ra = (r1 + r2 + r rn) / n
ra = ( ) / 4 = .10 or 10%
2. Geometric mean return
(=Time-weighted average return)
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 = = 8.29%
3. Internal Rate of Return
(=Dollar-weighted average return)

7. Dollar Weighted Average Return (=IRR) Using Text Example (Page 119)
Net CFs
$ (mil)
1.0
IRR = 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. 5.2 RETURN AND RISK

10. Normal Distribution
s.d.
s.d. r
Symmetric distribution

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

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

13. Measuring Mean: Given Subjective Probability Distribution
Subjective returns
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 (t) Prob. of State (t) r in state t
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
= .15 or 15%

15. Measuring Variance or Dispersion of Returns
Subjective or Scenario

16. Measuring Variance or Dispersion of Returns
Using Our Example:
Var =[(.1)( )2+(.2)( ) ( )2] =
S.D. = [ ]1/2 = or 10.95%

17. The Sharpe (Reward-to-Volatility) Measure

18. When you have the past historical data,
When you have the past historical data, we can use them to estimate the proxy for future return. In this case, the return is defined by the arithmetic mean return.
FIMA Research Center, University of Hawaii

19. When the past historical data are available:
FIMA Research Center, University of Hawaii

20. 5.3 THE HISTORICAL RECORD

21. Annual Holding Period Returns (1926 – 2006) From Table 5.3 of Text
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
World Stock
US Large Stock
US Small Stock
World Bonds
LT Treasury
T-Bills
Inflation

22. Annual Holding Period Excess Returns From Table 5.3 of Text
Risk Stan. Sharpe
Series Prem. Dev.% Measure
World Stock
US Large Stock
US Small Stock
World Bonds
LT Treasury

23. Figure 5.3 US Large Common Stocks: Normal Distribution with Mean of 12% and St Dev of 20%

24. The probability that an annual return
will fall within:
-8% and 32% will be 68%
-28% and 52% will be 95%
-48% and 72% will be 99.7%

25. Figure 5.2 Rates of Return on Stocks, Bonds and T-Bills

26. Table 5.4 Size-Decile Portfolios

27. 5.4 INFLATION AND REAL RATES OF RETURN

28. Real vs. Nominal Rates Fisher effect: 2.83% = (9%-6%) / (1.06)
R = nominal rate of return; i = inflation rate; and r = real rate of return
2.83% = (9%-6%) / (1.06)

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

30. 5.5 ASSET ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS

31. Allocating Capital
Possible to split investment funds between safe and risky assets
Proxy for Risk free asset: T-bills
Risky asset: stock (or a portfolio)

32. The Risky Asset: Text Example (Page 134)
Total portfolio value = $300,000
Risk-free value = ,000
Risky (Vanguard and Fidelity) = 210,000
Vanguard (V) = 54%
Fidelity (F) = 46%

33. The Risky Asset: Text Example (Page 143)
Vanguard 113,400/300,000 =
Fidelity ,600/300,000 =
Portfolio P 210,000/300,000 =
Risk-Free Assets F 90,000/300,000 =
Portfolio C 300,000/300,000 =

34. Calculating the Expected Return Text Example (Page 145)
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf

35. 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%

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

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

38. 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

39. 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

40. 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

41. 5.6 PASSIVE STRATEGIES AND THE CAPITAL MARKET LINE

42. Table 5.5 Average Rates of Return, Standard Deviation and Reward to Variability

43. 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 mimics a broad market index

44. HW Assignment for Chapter 5
Do Problems 14, 19, and 20
Due Date: February 6