Question

(15) The investment department of Big Bucks Businesses (BBB) is examining several different strategies. The forecasting department has indicated that each economic state is equally likely to occur. The estimated return for each security under each state is below. Tarragon Inc. manufactures decorative bottles, and Vintner Corp. is a mining firm. The following table summarizes the data: Economic State Tarragon Vintner Recession -11% 4% Average 20% 8% Boom 40% 6% a. Sara believes that BBB should invest 80% in Tarragon and 20% in Vintner. What is the return and variance on this portfolio? Use the population variance for the securities. b. Can Sara find a combination of Tarragon and Vintner that has no risk? If so, how much should BBB invest in each one? If not, explain why not. Hint: Check the correlation.

Answer 1

Portfolio weightes =		Wt =	80%
		Wv =	20%
Lets first calculate the Return and standard deviation of both the stocks -
Expected return =		Sum (Prob. X Return)
Standard Deviation =		Sq. root of =	Sum (Prob x (x-mean)^2)
(i) Tarragon
State	Probability(P)	Return(x)	P x R	X - Mean	(X - Mean)^2	P x (X - Mean)^2
Recession	0.33	-11	-3.666667	-27.3333	747.1111	249.037
Average	0.33	20	6.666667	3.666667	13.44444	4.481481
Boom	0.33	40	13.33333	23.66667	560.1111	186.7037
			16.33333			440.2222
	Expected return =		16.33333
	(Mean)
	Standard Deviation =		Sq. Root of(440.22)
			20.98147
(ii) Vintner
State	Probability(P)	Return(x)	P x R	X - Mean	(X - Mean)^2	P x (X - Mean)^2
Recession	0.333333	4	1.333333	-2	4	1.333333
Average	0.333333	8	2.666667	2	4	1.333333
Boom	0.333333	6	2	0	0	0
			6			2.666667
	Expected return =		6
	(Mean)
	Standard Deviation =		Sq. Root of(2.6667)
			1.632993
(A)	Return of a portfolio =		Weighted average of returns
			0.8 x 16.33 + 0.2 x 6
			14.267
	Variance of portfolio =		[Wa^2 x Sda^2 + Wb^2 x SDb^2 + 2 x wa x wb x COV(a,b) ]
	Lets calculate COV(t,v)
State	Probability(P)	Return (T)	Return (V)	(T-Mean)	(V-Mean)	P x (T-Mean) x (V-Mean)
Recession	0.333333	-11	4	-27.3333	-2	18.22222
Average	0.333333	20	8	3.666667	2	2.444444
Boom	0.333333	40	6	23.66667	0	0
						20.66667
	COV(T,V) =		P x (T-Mean) x (V-Mean)			20.66667
	Variance =	0.8^2 x 20.98^2 + 0.2^2 x 1.63^2 + 2 x 0.8 x 0.2 x 20.667
		281.74 + 0.10 + 6.61
		288.46
(B)	No risk means NO variance or SD.
	Portfolio may diversify the risk but the risk can never be zero,
	Because the formula for portfolio Variance =
	[Wa^2 x Sda^2 + Wb^2 x SDb^2 + 2 x wa x wb x COV(a,b) ]
	Here the all 3 terms are positive therefore the sum can never be 0