Question

Consider the following scenario analysis:

		Rate of Return
Scenario	Probability	Stocks		Bonds
Recession	0.3	-4	%	12	%
Normal economy	0.4	13		7
Boom	0.3	22		3

Assume a portfolio with weights of 0.60 in stocks and 0.40 in bonds.

a. What is the rate of return on the portfolio in each scenario? (Enter your answer as a percent rounded to 1 decimal place.)

	Rate of Return Recession % Normal economy % Boom %

b. What are the expected rate of return and standard deviation of the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

	Expected return % Standard deviation %

c. Would you prefer to invest in the portfolio, in stocks only, or in bonds only? Explain the benefit of diversification.

Answer 1

Step 1: Calculation of Expected Return & Variance of Stock

Scenario	Probability	Return (Rs)	Rs * Probability	Deviation (Ds)	Ds^2	Ds^2 * Probability
Recession	0.3	-4%	-0.012	-0.1460	0.0213	0.0063948
Normal	0.4	13%	0.052	0.0240	0.0006	0.0002304
Boom	0.3	22%	0.066	0.1140	0.0130	0.0038988
		Total	0.106		Total	0.0105240

a. Calculation of Expected return of Stock

where Rs = Return of Stock

ER(s) = 0.106 or 10.6%

b. Calculation of Variance of Stock

where Ds = Rs - ER(S)

0.010524

Step 2: Calculation of Expected return & Variance of Bond

Scenario	Probability	Return (Rb)	Rb * Probability	Deviation (Db)	Db^2	Db^2 * Probability
Recession	0.3	12%	0.036	0.0470	0.0022	0.0006627
Normal	0.4	7%	0.028	-0.0030	0.0000	0.0000036
Boom	0.3	3%	0.009	-0.0430	0.0018	0.0005547
		Total	0.073		Total	0.0012210

a. Calculation of Expected return of Bond

where Rb = Return of Bond

ER(b) = 0.073 or 7.3%

b. Calculation of Variance of Bond

where Db = Rb - ER(B)

0.001221

Step 3 : Calculation of Covariance

Scenario	Probability	Deviation (Ds)	Deviation (Db)	Ds * Db	Ds * Db * Probability
Recession	0.3	-0.146	0.047	-0.006862	-0.002059
Normal	0.4	0.024	-0.003	-0.000072	-0.000029
Boom	0.3	0.114	-0.043	-0.004902	-0.001471
				Total	-0.003558

where Ds = Rs - ER(s)
Db = Rb - ER(b)

COV (s,b) = -0.003558

A) Calculation of rate of return on the portfolio in each scenario

Rp = Ws* R(s)+ Wb * R(b)
where Ws & Wb = weight of Stock & Bond respectively
  R(s) & R(b) = Return of Stock & Bond respectively

	Expected Return	Return of Portfolio for each year
	Stock	Bond	Working
Weights	Ws = 0.6	Wb = 0.4	Rp = Ws* R(s)+ Wb * R(b)	Rp
Recession	-4%	12%	0.6*-4% + 0.4*12% =	2.40%
Normal	13%	7%	0.6*13% + 0.4*7% =	10.60%
Boom	22%	3%	0.6*22% + 0.4*3% =	14.40%

B) Calculation of expected rate of return and standard deviation of the portfolio

(i) Calculation of expected rate of return of the portfolio
Ws = 0.6
Wb = 0.4
ER(b) = 7.3%
ER(s) = 10.6%

ERp = Ws* ER(s)+ Wb * ER(b)
ERp = 0.6 *10.6% + 0.4*7.3%
ERp = 9.28%

(ii) Calculation of standard deviation of the portfolio

0.010524
0.001221
COV (s,b) = -0.003558
Ws = 0.6
Wb = 0.4

0.047709 or 4.77%

c) Would you prefer to invest in the portfolio, in stocks only, or in bonds only? Explain the benefit of diversification:

	Stock	Bond	Portfolio
Return	10.60%	7.30%	9.28%
Risk	10.26%	3.49%	4.77%

(Note: Risk = )

Keeping in mind the above data ,

• In case we invest only in stock the rate of return will be higher but the corresponding risk will also be high.
• In case we invest only in bond we will be able to reduce the risk but the corresponding return will also reduce.

Therefore, we should not invest only in stock or bond but in a portfolio with proportionate investment made in stock and bond in such a manner that it satisfies our risk and return appetite.
Investing in a portfolio provides the benefits of diversification.

Benefit of Diversification: Diversification helps in achieving desired rate of return while reducing the risk levels to the minimum. A diversified portfolio helps in distribution of the risk. In diversification by combing assets in appropriate manner subject to degree of correlation between the assets, the portfolio risk can be minimized.

In our case, the risk of stock = 10.26% and risk of bond 3.49% while the portfolio risk is 4.77% which is less than the high risk of stock.
Also, the Return of stock = 10.60% and return of bond 7.3% while the portfolio return is 9.28% which is more than the low return of bond.
By combining two negatively correlated stocks we have reduced our portfolio risk while return of the portfolio is higher