Question

Weight of Security A	Weight of Security B	Portfolio Return	Portfolio Standard Deviation
0%	100%	12	225.0
20%	80%	10.4	121.0
40%	60%	8.8	49.0
60%	40%	7.2	9.0
80%	20%	5.6	1.0
100%	0%	4.0	25.0
How can you choose the right combination (i.e.,, the weights on A and B) to create a risk-free portfolio? Based on the result in part , could the equilibrium rf be greater than 6%? Explain.

Answer 1

return of security B=portfolio return when weight of security B is 100%=12%

return of security A=portfolio return when weight of security A is 100%=4%

standard deviation of security B=sqrt(portfolio standard deviation when weight of security B is 100% ^2)=sqrt(225)=15%

standard deviation of security A=sqrt(portfolio standard deviation when weight of security A is 100% ^2)=sqrt(25)=5%

when weight of security A is 20%
portfolio standard deviation=sqrt((20%*5%)^2+(80%*15%)^2+2*20%*5%*80%*15%*correlation)

sqrt(121)=sqrt((20%*5%)^2+(80%*15%)^2+2*20%*5%*80%*15%*correlation)
=>correlation is -1

we see correlation is -1

portfolio standard deviation is wa*5%-wb*15%

risk free means portfolio standard deviation is zero

=>wa/wb=3/1

So, wa=0.75
wb=0.25

Security A will be 75% in portfolio and Security B will be 25% in portfolio

Portfolio returns=75%*4%+25%*12%=6.000%

Risk free rate=portfolio returns when standard deviation is zero=6%

So, equilibrium risk free rate must be 6% it cannot be greater tor less than 6% otherwise arbitrage would occur