Question

QUESTION 2

MONTHLY Expected Returns, Standard Deviations, and Correlations July 1990 through December 2014, 294 months, returns are total stock market returns for the region)
			Correlation with:
Asset Class	Expected Return	Std. Deviation	US	Japan	Asia (non-Japan)	Europe
US	0.89%	4.34%	1.00	0.41	0.71	0.79
Japan	0.21%	5.97%	0.41	1.00	0.47	0.50
Asia (non-Japan)	0.96%	6.01%	0.71	0.47	1.00	0.75
Europe	0.74%	5.04%	0.79	0.50	0.75	1.00

4. Suppose you are currently invested 100% in U.S. stocks and you CANNOT short:

A. Find the portfolio that maximizes expected return if you want the same risk of U.S. stocks.

B. What is the expected return of this portfolio and what are the portfolio weights?

Answer 1

Please refer to the excel sheet I attached.

I have solved the above question using excel since manual calculation would take too long.In the excel sheet , first you can write all the information given in the question as I did by making the asset table.

Then to to calculate the weights for the portfolio which will give you the maximum return with the risk the same as that of the US asset class,

I created a weights table, manually writing equal weights for all the asset classes and then calculating its square in the other column for an easier calculation.

I also created a cell called SUM for the sum of all the weights which should be 1. and another cell writing the variance of the portfolio. Since in the question we need the risk of the portfolio to be the same as US asset

we need the standard deviation or variance of the portfolio to be equal to that of the US asset class

therefore, Standard deviation of the portfolio = 4.34% or 0.0434 OR

variance of portfolio = square of the standard deviation = 0.0434*0.0434=0.00188356

This is the value you need to enter in the variance cell.

you will also calculate the value of expected return of portfolio, E(r) = W1*R1+W2*R2+W3*R3+W4*R4

since there are 4 assets, W1= weight of asset 1 in the portfolio

R1 = return of asset 1

asset 1 = US, asset 2= JAPAN, Asset 3= ASIA, Asset 4 = Europe

Then after inputting all these values we will use Excel Solver to determine the weights for the portfolio we want .

you will find solver in the data tab of excel. after opening solver, you need to set the target cell which is the value you want to maximise/minimize , in our case we want to maximise our return , so we will input the cell in which the expected return has been calculated.

and we will click on maximise .

In the "By changing cells column" we will select the all the 4 weight cells that we entered manually. because these are the values we want to change to alter to determine the portfolio weights we want keeping the conditions we are given.

In the "Subject to constraints column "

we will input all the constraints we want to be maintained in order to get the answer

First we will click on the cell where the sum of weights is calculated and then we would put in the constaint as

this should always = 1

then another constraint given in question is that the standard deviation/variance should be the same as US asset.

so we will again enter another constarint by clicking on add, and then select the cell containing the value of variance we calculated and then selecting the constraint that this should be equal to 0.00188356 ( same as that of US asset)

and then we finally click on solve.

after solving you will see the weights changed as above.

(i) The weights of the portfolio will be

w1 = -0.0526

w2 =0.3338

w3 = 0.4422

w4 = 0.2765

these weights then will give you an expected return = 0.0581 = 5.81% ( calculated by the same formula mentioned earlier.