1Chapter 5Risk and Return – Part II3/16/2006 FIN3710 - Investment - Professor Rui Yao 2Estimation Based on a Historical Sample Estimating (arithmetic) mean return Estimating Return Variance (σ2) Estimating Standard Deviation (σ) Estimating Geometric Mean Return∑==+++=NiiNrNNrrrr1211...1)(][22−−==∑NrrrVariiσ][][ rVarrSD ==σ[]1)1(1)1(...)1()1(11121−⎥⎦⎤⎢⎣⎡+=−+××+×+=∏=NNiiNNgrrrrr23/16/2006 FIN3710 - Investment - Professor Rui Yao 3Historical ReturnsGeom. Arith. Stan.Series Mean% Mean% Dev.%Sm. Stk 12.19 18.29 39.28Lg. Stk 10.51 12.49 20.30LT Gov 5.23 5.53 8.18T-Bills 3.80 3.85 3.25Inflation 3.06 3.15 4.40 Based on data from 1926 -20013/16/2006 FIN3710 - Investment - Professor Rui Yao 4Risk Premium and Risk Aversion Risk-free rate¾ The rate of return that can be earned with certainty Risk premium¾ An expected return in excess of risk-free rate¾ Also called excess return Example: what is the historical risk premium for small and large stocks?33/16/2006 FIN3710 - Investment - Professor Rui Yao 5Real Returns vs. Excess ReturnsExcess RealSeries Returns% Returns%Sm. Stk 14.44 15.14Lg. Stk 8.64 9.34LT Gov 1.68 2.38T-Bills 0.00 0.60 Excess return: extra return over a riskfreeinvestment opportunity (re= R – rf) Real return calculated using approximate formula (r = R – i )3/16/2006 FIN3710 - Investment - Professor Rui Yao 6Risk Premium and Risk Aversion Risk-aversion (A)¾ A measure of the reluctance of investor to take risks More risk-averse investor wants more reward (higher risk premium) for bearing the same risk¾ A mathematical relationship linking risk aversion and risk to required risk premiumE[rp] – rf= ½ A σ2pA – Risk aversion coefficientE[rp] – Expected return of the portfolioσp2– Variance of the portfolio43/16/2006 FIN3710 - Investment - Professor Rui Yao 7Risk Aversion and Indifference CurveE[rp]σp0Indifference curve for two investorsXY3/16/2006 FIN3710 - Investment - Professor Rui Yao 8Example: Risk Aversion S&P 500 index over the coming year is predicted to have an expected return of 10% and standard deviation of 18%. If one-year T-bill rate is 5%, what is the suggested risk aversion for an average S&P 500 investor?53/16/2006 FIN3710 - Investment - Professor Rui Yao 9Example: Risk Aversion Can use certainty equivalent return to rank different risky portfolios and riskfree security EU = rce= E(rp) – ½ A * σp2 If your risk aversion is 4, which one of the following portfolio will you prefer:¾ (a). E(rp) = 10%; σp =20%¾ (b). E(rp) = 15%; σp =40%¾ (c). What about someone with A =
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