Question

In the following year, there is a 10% chance of a bear market, a 20% chance of a bull market, and a 70% chance of a neutral market.
Debt will return -2%, 5%, and 5% in the bear, bull and neutral market, respectively.
Equity will return -15%, 15%, and 8% in the bear, bull and neutral market , respectively .

What is the expected return of debt in %?  (round to 1 decimal place)

What is the volatility of equity in %? (round to 1 decimal place)

What is the covariance of debt and equity? (leave as a decimal; round to 5 decimal places)

What is the correlation of debt and equity?   (leave as a decimal; round to 3 decimal places)

What is the expected return of a risky portfolio made up of 70% bonds and 30% stock?  (units of %; round to 2 decimal places)  

What is the volatility of a risky portfolio made up of 70% bonds and 30% stock?    (units of %; round to 1 decimal places)   


What is the Sharpe ratio of the risky portfolio made up of 70% bonds and 30% stock? (round to 2 decimal places)

Answer 1

Bond
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Bear	0.1	-2	-0.2	-6.3	0.0003969
Bull	0.2	5	1	0.7	9.8E-06
Normal	0.7	5	3.5	0.7	0.0000343
	Expected return %=	sum of weighted return =	4.3	Sum=Variance Bond=	0.00044
			Standard deviation of Bond%	=(Variance)^(1/2)	2.1
Stock
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Bear	0.1	-15	-1.5	-22.1	0.0048841
Bull	0.2	15	3	7.9	0.0012482
Normal	0.7	8	5.6	0.9	0.0000567
	Expected return %=	sum of weighted return =	7.1	Sum=Variance Stock=	0.00619
			Standard deviation of Stock%	=(Variance)^(1/2)	7.9
Covariance Bond Stock:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
Bear	0.1	-6.3	-22.1	0.0013923
Bull	0.2	0.7	7.9	0.0001106
Normal	0.7	0.7	0.9	0.0000441
			Covariance=sum=	0.00155
		Correlation A&B=	Covariance/(std devA*std devB)=	0.936
	Expected return%=	Wt Bond*Return Bond+Wt Stock*Return Stock
	Expected return%=	0.7*4.3+0.3*7.1
	Expected return%=	5.14
	Variance	=( w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB))
	Variance	=0.7^2*0.021^2+0.3^2*0.07867^2+2*0.7*0.3*0.021*0.07867*0.9364
	Variance	0.00142
	Standard deviation=	(variance)^0.5
	Standard deviation=	3.8%

Sharpe ratio(reward to variability)

=(Return-risk free rate)/std dev

=(5.14-4.3)/3.77

=0.2228 = 22.3%