Chapter 5 Risk and Return – Part II - City University of New...
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1 Chapter 5 Risk and Return – Part II 3/16/2006 FIN3710 - Investment - Professor Rui Yao 2 Estimation Based on a Historical Sample Estimating (arithmetic) mean return Estimating Return Variance (σ 2 ) Estimating Standard Deviation (σ) Estimating Geometric Mean Return ∑ = = + + + = N i i N r N N r r r r 1 2 1 1 ... 1 ) ( ] [ 2 2 − − = = ∑ N r r r Var i i σ ] [ ] [ r Var r SD = = σ [ ] 1 ) 1 ( 1 ) 1 ( ... ) 1 ( ) 1 ( 1 1 1 2 1 − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = − + × × + × + = ∏ = N N i i N N g r r r r r
Transcript of Chapter 5 Risk and Return – Part II - City University of New...
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Chapter 5Risk and Return – Part II
3/16/2006 FIN3710 - Investment - Professor Rui Yao 2
Estimation Based on a Historical SampleEstimating (arithmetic) mean return
Estimating Return Variance (σ2)
Estimating Standard Deviation (σ)
Estimating Geometric Mean Return
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Historical 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 -2001
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Risk Premium and Risk AversionRisk-free rate
The rate of return that can be earned with certaintyRisk premium
An expected return in excess of risk-free rateAlso called excess return
Example: what is the historical risk premium for small and large stocks?
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Real Returns vs. Excess ReturnsExcess Real
Series 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 )
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Risk Premium and Risk AversionRisk-aversion (A)
A measure of the reluctance of investor to take risksMore risk-averse investor wants more reward (higher risk premium) for bearing the same risk
A mathematical relationship linking risk aversion and risk to required risk premium
E[rp] – rf = ½ A σ2p
A – Risk aversion coefficientE[rp] – Expected return of the portfolioσp
2 – Variance of the portfolio
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Risk Aversion and Indifference Curve
E[rp]
σp0
Indifference curve for two investors
X
Y
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Example: Risk AversionS&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?
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Example: Risk AversionCan 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:
(a). E(rp) = 10%; σp =20%(b). E(rp) = 15%; σp =40%
(c). What about someone with A = 1?
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