1) The document calculates unanticipated risk, expected returns, and factor sensitivities for three stocks (A, B, C) based on growth factors and CAPM. 2) It finds a risk-free rate of 8% and factor premium of 1 when using stocks A and B. However, using stocks A and C, it calculates a risk-free rate of 9% and factor premium of 1.5. 3) This inconsistency between the results for the risk-free rate and factor premium using different stock combinations presents an arbitrage opportunity for investors.

1. a) In thiswe are requiredtocalculate the valuesof Fi.e.unanticipatedrisk:
Consideringthe growthfactor:( 1/3*0%) + (1/3*5%) +(1/3*10%)
= 0+.017+ .033 = 5%
Nowthe unanticipatedriskforeachsituation canbe calculatedasfollows:
F (ugly) = 0% - 5% = -5%
F (bad) = 5% - 5% = 0%
F (good) = 10%-5% = 5%
b) In thiswe needtocalculate the returnon assetand the factor sensitivity:
Stock A, badstate can be usedto calculate the expectedreturn:
The formulafor assetreturnis:RA = E (ri) + βi (F) + e
0.06 = E (ri) + βi * 0% => E (ri) = 0.06
Factor sensitivity:foruglystate:0.16 = 0.06 + βi*-5% => βi = -2%
For goodstate:- 0.04 = 0.06 + βi * 5% => βi = -2%
Stock B, bad state can be usedtocalculate the expectedreturn:
The formulafor assetreturnis:RA = E (ri) + βi (F) + e
0.09 = E (ri) + βi * 0% => E (ri) = 0.09
Factor sensitivity:foruglystate:0.04 = 0.09 + βi*-5% => βi = 1%
For goodstate:0.14 = 0.09 + βi * 5% => βi = 1%
Stock C, bad state can be usedto calculate the expectedreturn:
The formulafor assetreturnis:RA = E (ri) + βi (F) + e
0.12 = E (ri) + βi * 0% => E (ri) = 0.12
Factor sensitivity:foruglystate:0.02 = 0.12 + βi*-5% => βi = 2%
For goodstate:0.22 = 0.12 + βi * 5% => βi = 2%

2. c) In thiswe are requiredtocalculate the risk-free rate andfactorpremium.Thiscanbe done using
CAPMmodel:
E (ri) = Rf+ β*riskpremium
Takingonly 2 securitiesi.e.A andB
Stock A, E (ri) = 0.06, β = -0.02 alsothe riskpremiumis1.
Therefore,riskfree rate:E(ri) = Rf + β * riskpremium=> 0.06 = Rf + -0.02*1
Rf = 0.08
Stock B, E (ri) = 0.09, β = 0.01 alsothe riskpremiumis1.
Therefore,riskfree rate:E(ri) = Rf + β * riskpremium=> 0.09 = Rf + 0.01*1
Rf = 0.08
Therefore,the risk free rate is 8% and factor premiumis 1.
d) In thiswe are requiredtocalculate the risk-free rate andfactorpremium.Thiscanbe done using
CAPMmodel:
E (ri) = Rf+ β*riskpremium
Takingonly2 securitiesi.e.A andC
Stock A, E (ri) = 0.06, β = -0.02 alsothe riskpremiumis1.5
Therefore,riskfree rate:E(ri) = Rf + β * riskpremium=> 0.06 = Rf + -0.02*(1.5)
Rf = 0.09
Stock C, E (ri) = 0.12, β = 0.02 alsothe risk premiumis1.5
Therefore,riskfree rate:E(ri) = Rf + β * riskpremium=> 0.12 = Rf + 0.02*1.5
Rf = 0.09
Therefore,the risk free rate is 9% and factor premiumis 1.5
e) Comparingthe resultsof c and d inthe problemgiven,we cansaythat there isan arbitrage
opportunityforthe investor.