1. Learning About Return and Risk from the Historical Record
Chapter 5

6. It is on the class web page, handouts, chapter 5.
Study: A new historical database for the NYSE to 1925: Performance and predictability
The previous figure came from a study: “A new historical database for the NYSE 1815 to 1925: Performance and predictability.”
It is on the class web page, handouts, chapter 5.

8. Real vs. Nominal Rates
How much was a donut in 1970? How much is a donut now?
Nominal interest rate (R):
Growth rate of your money
Real interest rate (r):
Growth rate of your purchasing power
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)
= ( ) / (1+.06)
Empirical Relationship:
Inflation and interest rates move closely together

10. Fig. 5.3 Interest and inflation rates

11. Fig 5.2 Nominal and real TW for t-bills

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

13. Rates of Return (for any asset): Single Period
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one

14. Rates of Return: Single Period Example
Ending Price = 48
Beginning Price = 40
Dividend = 2
HPR = ( )/ (40) = 25%

15. Return for Holding Period – Zero Coupon Bonds
Zero coupon Treasury Bond:
Par = $100
Price as a function of maturity T= P(T)
Total risk free return =

16. Annualized Rates of Return

17. Formulas for equivalent annual rates (EAR) and annual percentage rate (APR)

19. Characteristics of Probability Distributions
1) Mean: most likely value
2) Variance or standard deviation
3) Higher moments: Skewness and kurtosis.
Skewness – The degree of asymmetry of a distribution.
Kurtosis –The degree of peakedness of a distribution.
Stock returns are sometimes said to be “leptokurtotic” that is, peaked in the center with fat tails.
* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2.

21. 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

26. Recent total large-cap, total market, and small cap performance
Let’s add a little “spice”:
VTSMX, IJS

29. vix

30. 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 =

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
VaR normal – 5% point of a normal distribution = mean minus 1.65 s.d.
VaR actual – 5% point of a distribution
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:
The ratio of average excess returns to LPSD

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

35. Scenario or Subjective Returns: Example
State Prob. of State r in State
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15

37. Variance or Dispersion of Returns
Subjective or Scenario
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)( )2+(.2)( ) ( )2]
Var=
S.D.= [ ] 1/2 = .1095

38. Mean and Variance of Historical Returns
Arithmetic average
Variance

39. Geometric Average Returns
TV = Terminal Value of the Investment (sometimes called “terminal wealth”)
g= geometric average rate of return

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

47. Portfolio risk Previous slide return data from Ken French:
You can minimize risks by holding a combination of the assets.
It’s all about the three “C’s”…