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Risk and Return

Risk and Return. Holding Period Return Multi-period Return Return Distribution Historical Record Risk and Return. 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

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Risk and Return

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  1. Risk and Return Holding Period Return Multi-period Return Return Distribution Historical Record Risk and Return

  2. 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

  3. Multi-period Return • What’s the return over a few periods? • Consider a mutual fund story • Net inflow when the fund does well • Net outflow when the fund does poorly • Question: • How would we characterize the fund’s performance over the year?

  4. Multi-period Return • Arithmetic Average • Sum of each period return scaled by the number of periods • ra: arithmetic return • ri: HPR in the ith period • N: number of periods • Example: • Calculate the arithmetic return of the fund

  5. Multi-period Return • Geometric Average • Single period return giving the same cumulative performance as the sequence of actual returns • rg: geometric return • ri: HPR in the ith period • N: number of periods • Example: • Calculate the geometric return of the fund

  6. Multi-period Return: Dollar-weighted • Internal Rate of Return (IRR) • The discount rate that sets the present value of the future cash flows equal to the amount of initial investment • Considers change in the initial investment • Conventions (from investor’s viewpoint) • Initial investment as outflow (negative) • Ending value as inflow (positive) • Additional investment as outflow (negative) • Reduced investment as inflow (positive)

  7. Multi-period Return: Dollar-weighted • Example: IRR = ? (assets in million dollars) • By definition • Using Excel t =0 t =1 t =2 t =3 t =4 CF0 = -1 CF1 = -.1 CF2 = -.5 CF3 = .8 CF4 = 1.0

  8. 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

  9. 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%

  10. 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

  11. 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:

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

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

  14. 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?

  15. Historical Record • Annual HPR of different securities • Risk premium = asset return – risk free return • Real return = nominal return – inflation • From historical record 1926-2006 Risk Premium and Real Return are based on APR, i.e. arithmetic average

  16. 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:

  17. 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

  18. 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)

  19. 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!

  20. Accounting for Risk - Sharpe Ratio • Reward-to-Variability (Sharpe) Ratio • E[r] – rf - Risk Premium • r – rf - Excess Return • Sharpe ratio for a portfolio: or

  21. 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?

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