1. 5 Risk and Return: Past and Prologue Bodie, Kane and Marcus
Essentials of Investments 9th Global Edition

2. Holding-Period Return (HPR) HPR= [PS − PB + CF] / PB
5.1 Rates of Return
Holding-Period Return (HPR)
Rate of return over given investment period
HPR= [PS − PB + CF] / PB
PS = Sale price
PB = Buy price
CF = Cash flow during holding period

3. Table 5.1 Quarterly Cash Flows/Rates of Return of a Mutual Fund
1st Quarter
2nd Quarter
3rd Quarter
4th Quarter
Assets under management at start of quarter ($ million) 1
1.2 2
0.8
Holding-period return (%) 10 25
−20 20
Total assets before net inflows
1.1
1.5
1.6
0.96
Net inflow ($ million)
0.1
0.5
−0.8
0.6
Assets under management at end of quarter ($ million)
1.56

4. Measuring Investment Returns over Multiple Periods
5.1 Rates of Return
Measuring Investment Returns over Multiple Periods
Arithmetic average
Sum of returns in each period divided by number of periods
Geometric average
Single per-period return; gives same cumulative performance as sequence of actual returns
Compound period-by-period returns; find per-period rate that compounds to same final value
Dollar-weighted average return
Internal rate of return on investment

5. 5.1 Rates of Return Conventions for Annualizing Rates of Return
APR = Per-period rate × Periods per year
1 + EAR = (1 + Rate per period)
1 + EAR = (1 + Rate per period)n = ( )n
APR = [(1 + EAR)1/n – 1]n
Continuous compounding: 1 + EAR = eAPR
APR n

6. Scenario Analysis and Probability Distributions
5.2 Risk and Risk Premiums
Scenario Analysis and Probability Distributions
Scenario analysis: Possible economic scenarios; specify likelihood and HPR
Probability distribution: Possible outcomes with probabilities
Expected return: Mean value of distribution of HPR
Variance: Expected value of squared deviation from mean
Standard deviation: Square root of variance

7. Spreadsheet 5.1 Scenario Analysis for the Stock Market

8. 5.2 Risk and Risk Premiums

9. Figure 5.1 Normal Distribution with Mean Return 10% and Standard Deviation 20%

10. Normality over Time 5.2 Risk and Risk Premiums
When returns over very short time periods are normally distributed, HPRs up to 1 month can be treated as normal
Use continuously compounded rates where normality plays crucial role

11. Deviation from Normality and Value at Risk
5.2 Risk and Risk Premiums
Deviation from Normality and Value at Risk
Kurtosis: Measure of fatness of tails of probability distribution; indicates likelihood of extreme outcomes
Skew: Measure of asymmetry of probability distribution
Using Time Series of Return
Scenario analysis derived from sample history of returns
Variance and standard deviation estimates from tim series of returns:

12. 5.2 Risk and Risk Premiums
Suppose you’ve estimated that the fifth- percentile value at risk of a portfolio is – 30%. Now you wish to estimate the portfolio’s first-percentile VaR (the value below which lie 1% of the returns). Will the 1% VaR be greater or less than –30%?

13. Risk Premiums and Risk Aversion
5.2 Risk and Risk Premiums
Risk Premiums and Risk Aversion
Risk-free rate: Rate of return that can be earned with certainty
Risk premium: Expected return in excess of that on risk-free securities
Excess return: Rate of return in excess of risk- free rate
Risk aversion: Reluctance to accept risk
Price of risk (A): Ratio of risk premium to variance

14. The Sharpe (Reward-to-Volatility) Ratio
5.2 Risk and Risk Premiums
The Sharpe (Reward-to-Volatility) Ratio
Ratio of portfolio risk premium to standard deviation
Mean-Variance Analysis
Ranking portfolios by Sharpe ratios

15. 5.2 Risk and Risk Premiums
11. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $ 70,000 or $ 195,000, with equal probabilities of .5. The alternative riskless investment in T-bills pays 4%.
a.   If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
b.   Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be?
c.   Now suppose you require a risk premium of 11%. What is the price you will be willing to pay now?
d.   Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

16. World and U.S. Risky Stock and Bond Portfolios
5.3 The Historical Record
World and U.S. Risky Stock and Bond Portfolios
World Large stocks: 24 developed countries, about stocks
U.S. large stocks: Standard & Poor's 500 largest cap
U.S. small stocks: Smallest 20% on NYSE, NASDAQ, and Amex
World bonds: Same countries as World Large stocks
U.S. Treasury bonds: Barclay's Long-Term Treasury Bond Index

17. Figure 5.3 Frequency Distribution of annual, Continuously Compounded rates of Return, 1926-2010

18. Figure 5.4 Rates of Return on Stocks, Bonds, and Bills

19. Table 5.2 Annual rate-of-return statistics for diversified portfolios for and three subperiods (%)

20. 5.4 Inflation and Real Rates of Return
Equilibrium Nominal Rate of Interest
Fisher Equation
R = r + E(i)
E(i): Current expected inflation
R: Nominal interest rate
r: Real interest rate

21. 5.4 Inflation and Real Rates of Return
U.S. History of Interest Rates, Inflation, and Real Interest Rates
Since the 1950s, nominal rates have increased roughly in tandem with inflation
1930s/1940s: Volatile inflation affects real rates of return

22. Figure 5.5 Interest Rates, Inflation, and Real Interest Rates 1926-2010

23. 5.5 Asset Allocation across Portfolios
Portfolio choice among broad investment classes
Complete Portfolio
Entire portfolio, including risky and risk-free assets
Capital Allocation
Choice between risky and risk-free assets
John Bogle of Vanguard Says:
The most fundamental decision of investing is the allocation of your assets: How much should you own in stock? How much should you own in bonds? How much should you own in cash reserves? That decision [has been shown to account] for an astonishing 94% of the differences in total returns achieved by institutionally managed pension funds There is no reason to believe that the same relationship does not also hold true for individual investors.

24. 5.5 Asset Allocation across Portfolios
The Risk-Free Asset
Treasury bonds (still affected by inflation)
Price-indexed government bonds
Money market instruments effectively risk- free
Risk of CDs and commercial paper is miniscule compared to most assets

25. 5.5 Asset Allocation Across Portfolios
Portfolio Expected Return and Risk
P: portfolio composition
y: proportion of investment budget
rf: rate of return on risk-free asset
rp: actual rate of return
E(rp): expected rate of return
σp: standard deviation
E(rC): return on complete portfolio
E(rC) = yE(rp) + (1 − y)rf
σC = yσrp + (1 − y) σrf

26. Figure 5.6 Investment Opportunity Set

27. 5.5 Asset Allocation across Portfolios
Capital Allocation Line (CAL)
Plot of risk-return combinations available by varying allocation between risky and risk-free
Risk Aversion and Capital Allocation
y: Preferred capital allocation

28. 5.5 Asset Allocation across Portfolios

29. 5.5 Asset Allocation across Portfolios
13) Assume that you manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28% (Weights: Stock A 20%; Stock B 30%; Stock C 50%(. The T-bill rate is 4%. Suppose your client decides to invest in your risky portfolio a proportion (y) of his total investment budget so that his overall portfolio will have an expected rate of return of 11%.
What is the proportion, y?
What are your client’s investment proportions in your 3 stocks and the T-bill fund?
What is the standard deviation of the rate of return on your client’s portfolio?

30. 5.5 Asset Allocation across Portfolios
14) Assume that you manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 4%. Suppose your client prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%.
What is the investment proportion, y?
What is the expected returns of the overall portfolio?

31. 5.5 Asset Allocation across Portfolios
17) Assume that you manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. There is also a passive fund that mimics the S&P Index with an expected return of 13% and a standard deviation of 25%. The T-bill rate is 7%.
Explain to your client the disadvantage of the switch.
Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one.

32. 5.5 Asset Allocation across Portfolios
You’ve just decided upon your capital allocation for the next year, when you realize that you’ve underestimated both the expected return and the standard deviation of your risky portfolio by 4%. Will you increase, decrease, or leave unchanged your allocation to risk-free T-bills?

33. 5.6 Passive Strategies and the Capital Market Line
Passive Strategy
Investment policy that avoids security analysis
Capital Market Line (CML)
Capital allocation line using market-index portfolio as risky asset

34. Table 5.4 Excess Return Statistics for S&P 500
Average
Std Dev.
Sharpe Ratio
5% VaR
8.00
20.70
.39
−36.86
11.67
25.40
.46
−53.43
5.01
17.58
.28
−30.51
7.19
17.83
.40
−42.28

35. Cost and Benefits of Passive Investing
5.6 Passive Strategies and the Capital Market Line
Cost and Benefits of Passive Investing
Passive investing is inexpensive and simple
Expense ratio of active mutual fund averages 1%
Expense ratio of hedge fund averages 1%-2%, plus 10% of returns above risk-free rate
Active management offers potential for higher returns

36. My Problems 1 VaR 4 Capital Allocation 6 E(r) and SD for a stock & CP
11 Risk Premium & Price of Portfolio
13 CAL (y based on r, w, SD)
14 CAL (Max r st SD, what is y? r?)
17 CAL comparison passive vs active (fees effect)

37. TA Problems 5 Mean r and SD for Scenario 7 Arithmetic Avg, GA, DWRR
8 Risk Premium & Risk Aversion
9 Calculate TR for S&P500
12 CAL (r, w, Sharpe Ratio)
CFA 1 Portfolio Ending Value Using Which Avg