Return and Risk

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

2. 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. Single Period Return • Holding Period Return: • Percentage gain during a period • HPR: holding period return • P0: beginning price • P1: ending price • D1: 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? P0 P1+D1 t = 0 t = 1

4. 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. 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. 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. Measuring Risk and Return • You decide to invest in IBM, what will be your return over next year? • Scenario Analysis vs. Historical Record • Scenario Analysis: • Historical Record: • What time period historical data should you use? • What data is relevant now? 1930s? 1980s? 2008?

8. Risk and Return Measures • Scenario Analysis and Probability Distribution • Expected Return • Return Variance • Standard Deviation (“Risk”)

9. Risk and Return Measures • More Numerical Analysis • Using Excel

10. 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. Accounting for Risk - Sharpe Ratio • Reward-to-Variability (Sharpe) Ratio • E[r] – rf - Risk Premium • r – rf - Excess Return • rf - Risk-free rate, i.e. 1 month T-Bill rate • Sharpe ratio for a portfolio: or

12. Risk and Horizon • S&P 500 Returns 1970 – 2005 • How do they compare* ? • Mean 0.0341*260 = 8.866% • Std. Dev. 1.0001*260 = 260.026% SURPRISED??? * There is approximately 260 working days in a year

13. Consecutive Returns It is accepted that stock returns are independent across time • Consider 260 days of returns r1,…, r260 • Means: E(ryear) = E(r1) + … + E(r260) • Variances vs. Standard Deviations: s(ryear) ¹s(r1) + … + s(r260) Var(ryear) = Var(r1) + … + Var(r260)

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

15. 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).

16. 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”

17. 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?