Learning About Return and Risk from the Historical Record Chapter 5

It is on the class web page, handouts, chapter 5. Study: A new historical database for the NYSE 1815 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.

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) 0.0283 = (0.09 - 0.06) / (1+.06) Empirical Relationship: Inflation and interest rates move closely together

Fig. 5.3 Interest and inflation rates

Fig 5.2 Nominal and real TW for t-bills

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

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

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

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 =

Annualized Rates of Return

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

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.

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

Recent total large-cap, total market, and small cap performance https://finance.yahoo.com/chart/%5EGSPC Let’s add a little “spice”: VTSMX, IJS

vix http://finance.yahoo.com/q/bc?s=%5EVIX&t=5y&l=on&z=m&q=l&c

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 =

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

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

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

Scenario or Subjective Returns: Example State Prob. of State r in State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) = .15

Variance or Dispersion of Returns Subjective or Scenario Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095

Mean and Variance of Historical Returns Arithmetic average Variance

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

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

Portfolio risk Previous slide return data from Ken French: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html You can minimize risks by holding a combination of the assets. It’s all about the three “C’s”…