Question

Lawrence Corporation is considering these three projects whose returns are normally distributed:

Project	Investment	Exp. Ret	Std. Dev.	Corr. Coeff.
A	$20,000	12%	20%	A, B = 0.4
B	$30,000	14%	20%	A, C = 0.5
C	$50,000	16%	30%	B, C = 0.6

Find the probability that the return on the portfolio is more than 20%

Answer 1

		a	b=a/100000
		Investment	Weight
	Project A	$20,000	0.20	wa
	Project B	$30,000	0.30	wb
	Project C	$50,000	0.50	wc
		$100,000
				Ra=Return of Project A	12%
				Rb=Return of Project B	14%
				Rc=Return of Project C	16%
				Portfolio Return =wa*Ra+wb*Rb+wc*Rc
				Portfolio Return =0.2*12+0.3*14+0.5*16=	14.60%
				Sa=Standard Deviation of Project A	20%
				Sb=Standard Deviation of Project B	20%
				Sc=Standard Deviation of Project C	30%
				Corr(a,b)=Correlation of Project A and B	0.40
				Corr(a,c)=Correlation of Project A and C	0.50
				Corr(b,c)=Correlation of Project B and C	0.60
				Covariance (a,b)=Correlation(a,b)*Sa*Sb	160.00	%%	(0.4*20*20)
				Covariance (a,c)=Correlation(a,c)*Sa*Sc	300.00	%%	(0.5*20*30)
				Covariance (b,c)=Correlation(b,c)*Sb*Sc	360.00	%%	(0.6*20*30)
				Portfolio Variance =(wa^2)*(Sa^2)+(wb^2)*(Sb^2)+(wc^2)*(Sc^2)+2wa*wb*Cov(a,b)+2wa*wc*Cov(a,c)+2wb*wc*Cov(b,c)
				Portfolio Variance	464.2	%%	(0.2^2)*(20^2)+(0.3^2)*(20^2)+(0.5^2)*(30^2)+2*0.2*0.3*160+2*0.2*0.5*300+2*0.3*0.5*360
				Portfolio Standard Deviation =SQUARE ROOT (Variance)
				Portfolio Standard Deviation	21.5453	%	SQRT(464.2)
				Portfolio Mean Return	14.60%
				Probability of return of the portfolio more than 20%
								Portfolio Variance =(wa^2)*(Sa^2)+(wb^2)*(Sb^2)+(wc2)*(Sc^2)+2wa wb*Cov(a,b)+2wa wc Covía,c)+2wb*wc*Cov(b,c) 464.2 %% (0.242)* (20^2)+(0.3^2)* (20^2)+(0.5^2)* (30^2)+20.2 0.3 160+2+0.2*0.5*300+2*0.30.5*360 Portfolio Variance Portfolio Standard Deviation =SQUARE ROOT (Variance) Portfolio Standard Deviation Portfolio Mean Return Probability of return of the portfolio more than 20% SQRT(464.2) 21.54530111 % 14.60% From , Cumulative area Under Standard Normal Distribution Table D=(20%-Portfolio Mean Return)/(Portfolio Standard Distribution) D=120-14.6)/21.5453= 0.250634709 D=0.25 (rounding to two decimal places) From, Cumulative area Under Standard Normal Distribution Table For D=0.25, N(d)=0.5987 Probability of return less than 20%=0.5987 Probability of return GREATER than 20%=1-0.5987 0.4013 probability that the return on the portfolio is more than 20% 40.13%