Question

Expected Return, Varianoe & Standard Deviation Suppose your expectations of the stock market are shown as follows: Prob r(s) Boom 0.2 30% Normal 0.5 20% Recession 0.3 -20% (Expected Return(E(r)), variance(o2) and standard deviation (o) Compute the mean of the rate of return on stocks. Risk Premium & Capital Allocation Consider when you decide to invest in a risky portfolio P (with the expected return of 20% and standard deviation of 30 %) and a Treasury bill (The rate of the Treasury bill is 5%). Assume that in this investment, you use all the money you have on your own without borrowing from the outside. 1. For the following 3 cases, compute the expected return(E(r)), standard deviation -r) of each case separately (o) and risk premium (E(r) Case 1: You put all your money into the Treasury bill Case 2: You put all your money into the risky portfolio. Case 3: You put half of your money into the Treasury bill and half of the money into the risky portfolio 2. Compute the Sharpe Ratio of Case 2 and Case 3, compare and explain the result

Answer 1

		P	R	P*R	D=R-10	E=D^2	F=E*P
		Probability	Return(%)	Probability*Return	Deviation fromMean	Deviation Squared	Deviation squared*Probability
	Boom	0.2	30	6	20	400	80
	Normal	0.5	20	10	10	100	50
	Recession	0.3	-20	-6	-30	900	270
			Total	10		Total	400
	Mean (Expected) Return	10	%
	Variance	400
	Standard Deviation =Square Root of Variance
	Standard Deviation	20%	(SQRT(400)
	R1=Return of Risky Folio	20%
	S1=Standard Deviation of risky folio	30%
	R2=Return of Treasury Bill	5%
	S2=Standard Deviation of treasury bill	0%
	W1=Weight of Risky Folio
	W2=Weight of Treasury Bill
	Rp=Portfolio Return=W1*R1+W2*R2=W1*20+W2*5
	Vp=PortfolioVariance =(W1^2)*(S1^2)+(W2^2)*(S2^2)+2W1*W2*Covariance1,2
	Since , S2=0, Covariance1,2=0
	Vp=(W1^2)*(30^2)=900*(W1^2)
	Sp=PortfolioStandard Deviation=Square Root of Vp
	Sharpratio =(Rp-Riskfree rate)/Sp=(Rp-5)/Sp
		w1	w2	Rp=w1*20+w2*5	Vp=(w1^2)*900	Sp=Square root of Vp	SR=(Rp-5)/Sp
		Weight of	Weight of	NewPortfolio	New Portfolio	New Portfolio	SHARP
		Portfolio	T-Bill	Return(%)	Variance	Std. Deviation(%)	RATIO
	CASE1:AllMoney in treasury Bill	0	1	5	0	0
	case 2:Half in Portfolio,Half in treasury bill	0.5	0.5	12.5	225	15	0.5
	Case 2:All Money in Risky portfolio	1	0	20	900	30	0.5
		Standard deviation=Sp	Risk Premim (Rp-5)	Sharp Ratio
	CASE1	0	0
	CASE2	15%	7.50%	0.5
	CASE3	30%	15%	0.5