1Chapter 5Risk and Return – Part I3/1/2006 FIN3710 - Investment - Professor Rui Yao 2Single Period Return Holding Period Return: ¾ percentage gain during a period¾ HPR: holding period return¾ P0: beginning price¾ P1: ending price¾ D1: cash dividend (or bond coupon)0011PPDPHPR−+=23/1/2006 FIN3710 - Investment - Professor Rui Yao 3Probability Distribution of Returns Scenario analysis¾ A list of possible economic scenarios, the likelihood and HPR associated with each scenario Probability distribution of HPR describes the chances of returns falling to different levels An example of return distributionEconomy State (s) Prob: p(s) HPR: r(s)Boom 1 0.25 44%Normal 2 0.50 14%Bust 3 0.25 -16%3/1/2006 FIN3710 - Investment - Professor Rui Yao 4Risk Measure Expected Return (µ)¾ Probability-weighted mean value of HPR Return Variance (σ2)¾ Probability-weighted squared deviation from mean HPR Standard Deviation (σ)¾ Square root of the variance%14%)]16(25.0%145.0%4425.0[)()(][ =−×+×+×===∑ssrsprEµ045.0)14.16.(25.0)14.14(.5.0)14.44(.25.0])[)()((][22222=−−×+−×+−×=−==∑rEsrsprVarsσ%21.212121.0045.0][][ ====≡ rVarrSDσ33/1/2006 FIN3710 - Investment - Professor Rui Yao 5Multi-period Return Consider a mutual fund¾ Net inflow when the fund does well¾ Net outflow when the fund does poorly Question - how do we characterize the fund’s historical performance over the year?1Q 2Q 3Q 4QAssets at the start 1.0 1.2 2.0 0.8HPR (%) 0.10 0.25 -0.20 0.25Assets before net inflow 1.1 1.5 1.6 1.0Net Inflow 0.1 0.5 -0.8 0.0Assets in the end 1.2 2.0 0.8 1.03/1/2006 FIN3710 - Investment - Professor Rui Yao 6Multi-period Return Arithmetic Average¾ Sum of returns in each period divided by the number of periods ra: arithmetic return ri: HPR in the i-th period N: number of periods¾ Best forecast of future single-period return An estimate of expected return∑==+++=NiiNarNNrrrr1211...43/1/2006 FIN3710 - Investment - Professor Rui Yao 7Example: Arithmetic Average Return  Calculate the arithmetic average return of the fund3/1/2006 FIN3710 - Investment - Professor Rui Yao 8Multi-period Return Geometric Average¾ A single period return that gives the same cumulative performance as a sequence of actual returns for a buy-and-hold strategy¾ Also called time-weighted average return rg: geometric average return ri: HPR in the i-th period N: number of periods¾ Better measure for buy-and-hold investing[]1)1(1)1(...)1()1(11121−⎥⎦⎤⎢⎣⎡+=−+××+×+=∏=NNiiNNgrrrrr53/1/2006 FIN3710 - Investment - Professor Rui Yao 9Example: Geometric Average Return Calculate the geometric average return of the mutual fund If you did not add or withdraw any funds during the year, what is the ending asset value for an initial investment of $1000?[]%29.81%)251(%)201(%)251(%)101(41=−+×−×+×+=gr3/1/2006 FIN3710 - Investment - Professor Rui Yao 10Multi-period Return Dollar-weighted return (IRR) ¾ The discount rate that sets the present value of the future cash flows equal to the amount of initial investment ¾ Same as Internal Rate of Return (IRR)∑=+=+++++++=NiiiNNIRRCFIRRCFIRRCFIRRCFCF02210)1()1(...)1(1063/1/2006 FIN3710 - Investment - Professor Rui Yao 11Multi-period Return Dollar-weighted return (continued)¾ Take into account changes in the amount of asset under investment¾ Conventions (from investor’s viewpoint) Initial investment as outflow (negative) Ending value from liquidation as inflow (positive) Additional investment as outflow (negative) Withdrawal as inflow (positive)¾ Good measure with intermediate contribution/withdrawal3/1/2006 FIN3710 - Investment - Professor Rui Yao 12Example: Dollar-weighted Return Calculate IRR based on our mutual fund data¾ By definition¾ Using Excel function1Q 2Q 3Q 4QAssets at the start 1.0 1.2 2.0 0.8HPR 10.0% 25.0% -20.0% 25.0%Assets before net inflow 1.1 1.5 1.6 1.0Net Inflow 0.1 0.5 -0.8 0.0Assets in the end 1.2 2.0 0.8 1.0t =1 t =2 t =3 t =4CF0 = -1t =0CF1 = -.1 CF2 = -.5 CF3 = .8 CF4 = 1.0432)1(01)1(8)1(511010IRR.IRR.IRR.IRR.+++++−++−+−=Time 0 1 2 3 4 IRRCF -1.0 -0.1 -0.5 0.8 1.0 4.17%73/1/2006 FIN3710 - Investment - Professor Rui Yao 13Real vs. Nominal Returns Historical returns are usually measured in nominal dollar terms¾ 10% nominal return does not mean you can buy 10% more goods and services¾ In the course of the investment, the prices of “consumption basket” might also change Inflation (i ) is usually measured by CPI (CPI-U)¾ A bundle of goods considered representative of consumption pattern of a typical family Investors ultimately are interested in purchasing power, i.e. real returns3/1/2006 FIN3710 - Investment - Professor Rui Yao 14Real vs. Nominal Returns Nominal return measures the growth rate of your money (R) Real return measures the growth rate of your purchasing power (r) An exact formular = (R – i ) / (1 + i )  An approximate formulaR ≅ r + i or r ≅ R - i Tax is levied on nominal returns (R), not real returns (r)¾ Good or bad for an investor?83/1/2006 FIN3710 - Investment - Professor Rui Yao 15An Example on Return Taxation Over the past year, you observed the following:¾ Inflation rate is 10%¾ Your investment generates 14% return¾ You are in the 35% marginal tax bracket Q: What is your before-tax real return? Q: What is your after-tax real return?3/1/2006 FIN3710 - Investment - Professor Rui Yao 16Wrap-up Expected return and variance calculation under scenario analysis Different summary measures of realized returns over multiple holding period Real vs. nominal returns¾ Tax