Question

Consider the following information for Hope Co.:

State            Probability X	Y
Boom          .35              15%	10%
Normal        .50              10%	8%
Recession .15                5%	10%

1. Compute using 2 different ways the expected return for a portfolio with an investment of $8,000 in asset X and $2,000 in asset Y.

2. Based on your previous answer, what is the standard deviation for the same portfolio.

3. If the expected inflation rate is 4.5% and the nominal expected return of the previous portfolio is the one computed in question Q1), what are the exact and approximate expected real returns?

Answer 1

Please note the solution :

1) Calculation

Particulars	Probablity	Returns X	Returns Y	(X- Mean X)	(Y - Mean Y)	(X- Mean X)2	(Y - Mean Y)2	P * (X- Mean X)2	P * (Y - Mean Y)2	P(X- Mean X)(Y - Mean Y)
Boom	0.35	15	10	4	1	16	1	5.6	0.35	1.4
Norma	0.50	10	8	-1	-1	1	1	0.50	0.50	0.5
Recession	0.15	5	10	-6	1	36	1	5.4	0.15	-0.9
								11.50	1	1

Mean X = P * X

             = 0.35 * 15 + 0.50 * 10 + 0.15 * 5

             = 11 %

Mean Y = P * Y

             = 0.35 * 10 + 0.50 * 8 + 0.15 * 10

             = 9 %

Standard Deviation X = [P * (X- Mean X)2 ]1/2

                                   = [11.50] 1/2

                                   = 3.39%

Standard Deviation Y = [P * (Y- Mean Y)2 ]1/2

                                   = [1] 1/2

                                   = 1%

Covariance (X. Y) = P(X- Mean X)(Y - Mean Y)

                              = 11.50 %2

Correlation = Covariance (X. Y) / (Standard Deviation X ) * (Standard Deviation Y)

                   = 1 / (3.39 * 1 )

                   = 0.295

2) Answers

> Expected return of portfolio = 11% * 8000/10000 + 9% * 2000/10000

                                            = 10.60 % Answer

> Standard deviation of portfolio =

Thus , SD = [ (0.8)2 * (3.39)2 + (0.20)2 * (1)2 + (2 * 0.80 * 0.20 * 3.39 * 1 * 0.295)]1/2

                 = 2.78 % Answer

> Exact Real rate of return = (1+Nominal rate ) / (1+inflation rate )

                               = (1.1060) / (1.045) - 1

                               = 5.84 % Answer

> Approximate real rate = Nominal rate - inflation rate

                                    = 10.60 - 4.50

                                    = 6.10 % Answer