1. Risk, Return, and the Historical Record
Chapter Five
Risk, Return, and the Historical Record
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2. Chapter Overview Interest rate determinants
Rates of return for different holding periods
Risk and risk premiums
Estimations of return and risk
Normal distribution
Deviation from normality and risk estimation
Historic returns on risky portfolios

3. Interest Rate Determinants
Supply
Households
Demand
Businesses
Government’s net demand
Federal Reserve actions

4. Real and Nominal Rates of Interest
Nominal interest rate (rn):
Growth rate of your money
Real interest rate (rr):
Growth rate of your purchasing power
Where i is the rate of inflation

5. Figure 5.1 Determination of the Equilibrium Real Rate of Interest

6. Equilibrium Nominal Rate of Interest
As the inflation rate increases, investors will demand higher nominal rates of return
If E(i) denotes current expectations of inflation, then we get the Fisher Equation:

7. Taxes and the Real Rate of Interest
Tax liabilities are based on nominal income
Given a tax rate (t) and nominal interest rate (rn), the real after-tax rate is:
The after-tax real rate of return falls as the inflation rate rises

8. Rates of Return for Different Holding Periods
Zero Coupon Bond:
Par = $100
Maturity = T
Price = P
Total risk free return

9. Example 5.2 Annualized Rates of Return

10. Effective Annual Rate (EAR)
EAR: Percentage increase in funds invested over a 1-year horizon

11. Annual Percentage Rate (APR)
APR: Annualizing using simple interest

12. Table 5.1 APR vs. EAR

13. Table 5.2 T-Bill Rates, Inflation Rates, and Real Rates, 1926-2012

14. Bills and Inflation,
Moderate inflation can offset most of the nominal gains on low-risk investments
A dollar invested in T-bills from 1926– grew to $20.25 but with a real value of only $1.55
Negative correlation between real rate and inflation rate means the nominal rate doesn’t fully compensate investors for increased in inflation

15. Figure 5.3 Interest Rates and Inflation, 1926-2012

16. Risk and Risk Premiums Rates of return: Single period
HPR = Holding period return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one

17. Rates of Return: Single Period Example
Ending Price = $110 Beginning Price = $100 Dividend = $4

18. Expected Return and Standard Deviation
Expected returns
p(s) = Probability of a state
r(s) = Return if a state occurs
s = State

19. Scenario Returns: Example
State Prob. of State r in State
Excellent
Good
Poor
Crash
E(r) = (.25)(.31) + (.45)(.14) + (.25)(−.0675) + (0.05)(− 0.52)
E(r) = or 9.76%

20. Expected Return and Standard Deviation
Variance (VAR):
Standard Deviation (STD):

21. Scenario VAR and STD: Example
Example VAR calculation:
σ2 = .25(.31 − ) (.14 − .0976)2
+ .25(− − ) (−.52 − .0976)2
= .038
Example STD calculation:

22. Time Series Analysis of Past Rates of Return
True means and variances are unobservable because we don’t actually know possible scenarios like the one in the examples
So we must estimate them (the means and variances, not the scenarios)

23. Returns Using Arithmetic and Geometric Averaging
Arithmetic Average
Geometric (Time-Weighted) Average
= Terminal value of the investment

24. Estimating Variance and Standard Deviation
Estimated Variance
Expected value of squared deviations
Unbiased estimated standard deviation

25. The Reward-to-Volatility (Sharpe) Ratio
Excess Return
The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate
Risk Premium
The difference between the expected HPR on a risky asset and the risk-free rate
Sharpe Ratio

26. The Normal Distribution
Investment management is easier when returns are normal
Standard deviation is a good measure of risk when returns are symmetric
If security returns are symmetric, portfolio returns will be as well
Future scenarios can be estimated using only the mean and the standard deviation
The dependence of returns across securities can be summarized using only the pairwise correlation coefficients

27. Figure 5.4 The Normal Distribution
Mean = 10%, SD = 20%

28. Normality and Risk Measures
What if excess returns are not normally distributed?
Standard deviation is no longer a complete measure of risk
Sharpe ratio is not a complete measure of portfolio performance
Need to consider skewness and kurtosis

29. Figure 5.5A Normal and Skewed Distributions
Mean = 6%, SD = 17%

30. Figure 5.5B Normal and Fat-Tailed Distributions
Mean = .1, SD = .2

31. Normality and Risk Measures
Value at Risk (VaR)
Loss corresponding to a very low percentile of the entire return distribution, such as the fifth or first percentile return
Expected Shortfall (ES)
Also called conditional tail expectation (CTE), focuses on the expected loss in the worst-case scenario (left tail of the distribution)
More conservative measure of downside risk than VaR

32. Normality and Risk Measures
Lower Partial Standard Deviation (LPSD) and the Sortino Ratio
Similar to usual standard deviation, but uses only negative deviations from the risk-free return, thus, addressing the asymmetry in returns issue
Sortino Ratio (replaces Sharpe Ratio)
The ratio of average excess returns to LPSD

33. Historic Returns on Risky Portfolios
The second half of the 20th century, politically and economically the most stable sub-period, offered the highest average returns
Firm capitalization is highly skewed to the right: Many small but a few gigantic firms
Average realized returns have generally been higher for stocks of small rather than large capitalization firms

34. Historic Returns on Risky Portfolios
Normal distribution is generally a good approximation of portfolio returns
VaR indicates no greater tail risk than is characteristic of the equivalent normal
The ES does not exceed 0.41 of the monthly SD, presenting no evidence against the normality
However
Negative skew is present in some of the portfolios some of the time, and positive kurtosis is present in all portfolios all the time

35. Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000

36. Figure 5.8 SD of Real Equity & Bond Returns Around the World

37. Figure 5.9 Probability of Investment with a Lognormal Distribution

38. Terminal Value with Continuous Compounding
When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed