Question

Your boss has asked for your assistance regarding the company’s investments. One of their current investments has $10 million invested in long-term corporate bonds. This bond portfolio's expected annual rate of return is 9%, and the annual standard deviation is 10%.

An external firm was contacted to see if the company could improve their return. Their recommendation has the company invested in an index fund that closely tracks the Standard & Poor's 500 Index. The index has an expected return of 14%, and its standard deviation is 16%.

After consideration, you suggest the company puts all these funds into a combination evenly split between the index fund and corporate fund. Your boss likes the suggestion but wants to see proof that your return is better than the outside organization. She all wants to make certain that you considered risk when making your suggestion.

How do you determine the correlation coefficient?

What is the rate of return for a 50/50 mix of the two portfolios?

What is the standard deviation for a 50/50 mix of the two portfolios?

Answer 1

In the question, followings values are given,

Expected return of bond portfolio (let us assume it to ER_X) = 9%,

Standard Deviation of bond portfolio (SD_X) = 10%,

Expected return of index portfolio (let us assume it to ERy) = 14%,

Standard Deviation of index portfolio (SDY) = 16%

Let denote Weightage of bond portfolio as W_X and index portfolio as W_Y. Therefore, W_X = W_Y = 0.5

Now, the correlation coefficient can be found by using the following formula,

Correlation Coefficient_(X,Y) = COV_(X,Y)/ (SD_X*SD_Y) ; where COV_(X,Y) = Covariance of X and Y; SD_X and SD_Y refers to Standard Deviation of X and Y respectively

And formula for COV_(X,Y) is,

COV_(X,Y) = ER_X * ERY / (Number of Assets) = (9%*14%)/2 = 0.0063

Correlation Coefficient = COV_(X,Y)/ (SD_X*SD_Y). = 0.0063/ (10%*16%) = 0.39

Rate of return or a 50/50 mix of the two portfolios = W_X * ER_X + W_Y * ER_Y = 0.5*9% + 0.5 *14%

Or, Rate of return or a 50/50 mix of the two portfolios = 11.5%

Standard deviation for a 50/50 mix of the two portfolios = (W_X² * SD_X² + 2* W_X * W_Y * SD_X * SD_Y * Coefficient Correlation +W_Y² * SD_Y²)^(1/2) = (0.5^2*10%^2 + 2*0.5*0.5*10%*16%*0.39 + 0.5^2 * 16%^2)^(1/2) = (0.0025 + 0.00312 + 0.0064)^(1/2)

Or, Standard deviation for a 50/50 mix of the two portfolios = 10.96%