1. Return and Risk Returns – Nominal vs. Real Holding Period Return Multi-period Return Return Distribution Historical Record Risk and Return

2. Investments 72 Real vs. Nominal Rate  Real vs. Nominal Rate – Exact Calculation:  R: nominal interest rate (in monetary terms)  r: real interest rate (in purchasing powers)  i: inflation rate  Approximation (low inflation):  Example  8% nominal rate, 5% inflation, real rate?  Exact:  Approximation:

3. Investments 73 Single Period Return  Holding Period Return:  Percentage gain during a period  HPR: holding period return  P 0 : beginning price  P 1 : ending price  D 1 : cash dividend  Example  You bought a stock at $20. A year later, the stock price appreciates to $24. You also receive a cash dividend of $1 during the year. What’s the HPR? P0P0 P 1 +D 1 t = 0t = 1

4. Investments 74 Multi-period Return: APR vs. EAR  APR – arithmetic average  EAR – geometric average  T: length of a holding period (in years)  HPR: holding period return  APR and EAR relationship

5. Investments 75 Multi-period Return - Examples  Example 1  25-year zero-coupon Treasury Bond  Example 2  What’s the APR and EAR if monthly return is 1%

6. Investments 76 Return (Probability) Distribution  Moments of probability distribution  Mean: measure of central tendency  Variance or Standard Deviation (SD): measure of dispersion – measures RISK  Median: measure of half population point  Return Distribution  Describe frequency of returns falling to different levels

7. Investments 77 Risk and Return Measures  You decide to invest in IBM, what will be your return over next year?  Scenario Analysis vs. Historical Record  Scenario Analysis:

8. Investments 78 Risk and Return Measures  Scenario Analysis and Probability Distribution  Expected Return  Return Variance  Standard Deviation (“Risk”)

9. Investments 79 Risk and Return Measures  More Numerical Analysis  Using Excel

10. Investments 710 Risk and Return Measures  Example  Current stock price $23.50.  Forecast by analysts:  optimistic analysts (7): $35 target and $4.4 dividend  neutral analysts (6): $27 target and $4 dividend  pessimistic analysts (7): $15 target and $4 dividend  Expected HPR? Standard Deviation?

11. Investments 711 Historical Record  Annual HPR of different securities  Risk premium = asset return – risk free return  Real return = nominal return – inflation  From historical record 1926-2005 Risk Premium and Real Return are based on APR, i.e. arithmetic average

12. Investments 712 Risk and Horizon  S&P 500 Returns 1970 – 2005  How do they compare* ?  Mean0.0341*260 = 8.866%  Std. Dev.1.0001*260 = 260.026% SURPRISED??? DailyYearly Mean0.0341%Mean8.9526% Std. Dev.1.0001%Std. Dev.15.4574% * There is approximately 260 working days in a year

13. Investments 713 Consecutive Returns It is accepted that stock returns are independent across time  Consider 260 days of returns r 1,…, r 260  Means: E(r year ) = E(r 1 ) + … + E(r 260 )  Variances vs. Standard Deviations:  (r year )   (r 1 ) + … +  (r 260 ) Var(r year ) = Var(r 1 ) + … + Var(r 260 )

14. Investments 714 Consecutive Returns Volatility Daily volatility seems to be disproportionately huge!  S&P 500 Calculations  Daily:Var(r day ) = 1.0001^2 = 1.0002001  Yearly:Var(r year ) = 1.0002001*260 = 260.052  Yearly:  Bottom line: Short-term risks are big, but they “cancel out” in the long run!

15. Investments 715 Accounting for Risk - Sharpe Ratio  Reward-to-Variability (Sharpe) Ratio  E[r] – r f - Risk Premium  r – r f - Excess Return  Sharpe ratio for a portfolio: or

16. Investments 716 Normality Assumption  The normality assumption for simple returns is reasonable if the horizon is not too short (less than a month) or too long (decades).

17. Investments 717 Other Measures of Risk - Value at Risk  Term coined at J.P. Morgan in late 1980s  Alternative risk measurement to variance, focusing on the potential for large losses VaR statements are typically made in $ and pertain to a particular investment horizon, e.g. –“Under normal market conditions, the most the portfolio can lose over a month is $2.5 million at the 95% confidence level”

18. Investments 718 Wrap-up  What is the holding period return?  What are the major ways of calculating multi-period returns?  What are the important moments of a probability distribution?  How do we measure risk and return?