INVESTMENTS Risk and Risk aversion
INVESTMENTS Fourth Edition Risk and Risk Aversion Risk Aversion Chapter 2 Chapter 2
Risky investments -ree lryestrrier p=6 W,=150 Profit=50 Risky iny 1-p=4 Wo=80 Profit=-20 Risk Free t-bills Profit =5 Risk Premium =17
INVESTMENTS Fourth Edition W1 = 150 Profit = 50 W2 = 80 Profit = -20 p = .6 1-p = .4 100 Risky Inv. Risk Free T-bills Profit = 5 Risk Premium = 17 Risky Investments Risky Investments with Risk with Risk-Free Investment Free Investment
IIITOTMATNTO Asset reture a Asset returns over a given period are often uncertain D t p P R P D + p Po is the price at the beginning of period PI is the price at the end of the period--Uncertain DI is the dividend at the end of the period--Uncertain So return on a asset is a random variable, characterized by All possible outcomes Probability of each outcome
INVESTMENTS Fourth Edition Asset Returns Asset Returns Asset returns over a given period are often uncertain. 1 0 1 1 0 1 1 0 − + = + − = P D P P D P P R P 0 is the price at the beginning of period P 1 is the price at the end of the period--Uncertain D 1 is the dividend at the end of the period—Uncertain So return on a asset is a random variable, characterized by: • All possible outcomes • Probability of each outcome
INVESTMENTS Expected rate of return on a! inyestment E。R E0(D1+P1)-P E。D,+E。P Expected rate of return compensates for time- k value and r1s」 Eo(R=R+ Risk premium For example E R E(D1+P1) 122 1=22
INVESTMENTS Fourth Edition Expected rate of return on a investment Expected rate of return on a investment 1 ( ) 0 0 1 0 1 0 0 1 1 0 0 − + = + − = P E D E P P E D P P E R Expected rate of return compensates for timevalue and risk E0 (R) = R f + Risk premium For example: 1 22 % 100 122 ( ) 0 0 1 1 0 0 = − = + − = P E D P P E R
INVESTMENTS Variance on a inyestment Fro example (R)=0.6×(50%-22%)2+0.4×(-20%-22 =0.1176 (R)=06×(50%-299+04×(20%-292 0.3429
INVESTMENTS Fourth Edition Variance on a investment Variance on a investment Fro example 0.1176 ( ) 0.6 (50 % 22%) 0.4 ( 20 % 22%) 2 2 2 = σ R = × − + × − − 0.3429 ( ) 0.6 (50 % 22%) 0.4 ( 20 % 22%) 2 2 = σ R = × − + × − −
INVESTMENTS Measuring expected return and risk z Moments of return distributions state ean Asset probability 1/3 Asset RO() 5 Asset R1 10 20 556 555 Asset 2 R2 Asset R3 55 4.55 Between asset 0 and 1, which one would you choose Between asset 1 and 2 Between asset 2 and 3
INVESTMENTS Fourth Edition Measuring expected return and risk Measuring expected return and risk-- example example Moments of return distributions 5 5 5 5 5 20 15 14.5 5 5 5 6 5 -10 -5 -5.5 R0(%) R1 R2 R3 Asset 0 Asset 1 Asset 2 Asset 3 Asset probability 1/3 1/3 1/3 state 1 2 3 Mean Between asset 0 and 1,which one would you choose? Between asset 1 and 2…… Between asset 2 and 3……
INVESTMENTS These returns have the following ean St d Skewness R0(%) 0.00 RI R2 5555 12.25 8.16 R3 8.16 0.57 Skewness =E[x-Ex u/St Dof x
INVESTMENTS Fourth Edition These returns have the following These returns have the following moments moments R3 5 8.16 -0.57 R2 5 8.16 0 R1 5 12.25 0 R0(%) 5 0.00 0 Mean St D Skewness Skewness {E[( x Ex ] } / St .D of x 1/ 3 3 = −
INVESTMENTS Pis业 A version&[ili Investor' s view of risk Risk averse Risk neutral Risk Seeking a Utility ■ Utility Function U=E(R)-005A0 A measures the degree of risk aversion
INVESTMENTS Fourth Edition Investor’s view of risk - Risk Averse - Risk Neutral - Risk Seeking Utility Utility Function U = E ( R) - .005 A σ 2 A measures the degree of risk aversion Risk Aversion & Utility Risk Aversion & Utility
INVESTMENTS To measure risk, we need to know how investors react to uIncerrillfry-上 ey i135uInnprIon 0I [103 node Higher mean in return is preferred ■ER a Higher standard deviation in return is disliked E(R-ER a Investor care only about mean and st. D Investor don't care about higher moments, such as skewness a Under the assumption 1-4. Std give a measure of risk In general, other moments may matter
INVESTMENTS Fourth Edition To measure risk, we need to know how investors react t To measure risk, we need to know how investors react t o uncertainty uncertainty—Key assumption of most model Key assumption of most model Higher mean in return is preferred: E R Higher standard deviation in return is disliked Investor care only about mean and st. D Investor don’t care about higher moments, such as skewness Under the assumption 1-4. Std give a measure of risk In general, other moments may matter. 2 σ = E ( R − ER )
INVESTMENTS Risk aversion and value Using the sample iriyestrment U=E(R)-.005A02 22-005A(34%)2 Risk aversion a value High 5-6.90 34.66 T-bill= 5% Low 11622
INVESTMENTS Fourth Edition Risk Aversion and Value: Risk Aversion and Value: Using the Sample Investment Using the Sample Investment U = E ( R ) - .005 A σ 2 = .22 - .005 A (34%) 2 Risk Aversion A Value High 5 -6.90 3 4.66 Low 1 16.22 T-bill = 5%
