Question

The economy has a 10 percent chance of booming, 60 percent chance of being normal, and 30 percent chance of going into a recession. A stock is expected to return 16 percent in a boom, 11 percent in a normal, and lose 8 percent in a recession. The return on real estate is 10% in a normal and a boom, and -10% in a recession. What is the correlation between the return of the stock and the return of real estate?

		0.758
		0.941
		0.980
		0.987

Answer 1

0.987

Step-1:Calculation of expected return, variance and standard deviation of Stock's return
		Prob.	Return	Expected return-Stock	Variance
		a	b	c=a*b	d=((b-5.800%)^2)*a
	Booming	10%	16%	1.600%	0.104%
	Normal	60%	11%	6.600%	0.162%
	Recession	30%	-8%	-2.400%	0.571%
	Total			5.800%	0.838%
	Standard deviation of return			=	Variance ^ (1/2)
				=	0.838%	^ (1/2)
				=	9.152%
Step-2:Calculation of expected return, variance and standard deviation of real estate's return
		Prob.	Return	Expected return-Real Estate	Variance
		a	b	c=a*b	d=((b-4.000%)^2)*a
	Booming	10%	10%	1.00%	0.00036
	Normal	60%	10%	6.00%	0.00216
	Recession	30%	-10%	-3.00%	0.00588
				4.000%	0.840%
	Standard deviation of return			=	Variance ^ (1/2)
				=	0.840%	^ (1/2)
				=	9.165%
Step-3:Calculation of covariance of return between stock and real estate
		Prob.	Stock's return	Real estate's return	Covariance of return between stock and real estate
		a	b	d	f=((b-5.800%)*(d-4.000%))*a
	Booming	10%	16%	10%	0.061%
	Normal	60%	11%	10%	0.187%
	Recession	30%	-8%	-10%	0.580%
	Total				0.828%
Step-4:Calculation of correlation between the return of the stock and the return of real estate
	Correlation		=	Covariance between return/(Standard deviation of stock*Standard deviation of real estate)
			=	0.8280%	/	(9.152%*9.165%)
			=	0.8280%	/	0.8388%
			=	0.987