Explanation of calculations for Problem 6 on Spring 96 exam.

I assume you can calculate all of the results for Part A.

For B: You need to first determine whether you want to use Asset B to construct your portfolio instead of using Asset A (the Market Portfolio). Test Asset B against the SML to see if it lies above or below the line.

The SML predicted return on Asset B is   6.5% + .1772(10.65%-6.5%) = 7.2353%

According to your personal expectation, you expect a return of 10.65%, so Asset B lies above the line. You therefore want to buy it and not buy Asset A. (If it plotted on or below the line you would not want Asset B and would instead use Asset A to construct your portfolio.)

Now you need to construct a portfolio that will have an expected return of 22% using Asset B and T-Bills (borrowing or investing). Since the expected return on your position is larger than the expected return of Asset B, you know that you must need to be borrowing to move out to the right on the line connecting the riskless asset and Asset B.

Since the expected return of a 2-asset portfolio is the weighted average of the returns of the assets in the portfolio, you know that

            22% = Wt(B) x 10.65% + Wt(T) x 6.5%

and you need to solve for the weights. This will be a lot easier if you remember that the weights must sum to 1, so we can use [1-Wt(B)] = Wt(T), and therefore solve for Wt(B) in

            22% = Wt(B) x 10.65% + [1-Wt(B)] x 6.5%

Doing this gives Wt(B) = 3.735   and  [1-Wt(B)] = Wt(T) = - 2.735

So, for every dollar of your own money you are investing you are borrowing $2.735 so you can invest $3.735 in Asset B. Plug these weights back in and make sure that 22% return on the position is the result.

NOTE: The answers that were listed on the web page were wrong. I have changed them now. I apologize for this error.

For C:   First figure the expected return and risk of the combination of Asset A and Asset B without worrying about the third asset (T).

With $20,000 in Asset B and $40,000 in Asset A, you will have a portfolio worth $60,000 with Wt(A) = 2/3 and Wt(B) = 1/3. Using the usual formulas, this portfolio has E(R) = 10.65%, SD = 5.768%, and Beta = .7257

Note that Beta for the portfolio is just the weighted average of the assets in the portfolio, and the Beta of Asset A is 1.0

Let's give this portfolio a name, Asset C, to avoid confusion in the following discussion.

Now you need to consider the position composed of a $60,000 investment in Asset C that is financed with $35,000 of your own money and $25,000 that was borrowed at 6.5%. So, we need to determine Wt(C) and Wt(T) in this position.

Wt(C) will be the total investment in C divided by the amount or your own money in the position, or $60,000/$35,000 = 1.714

Since Wt(C) and Wt(T) need to sum to 1, Wt(T) must be -.714

Once you have these weights, you can compute the parameters for the portfolio as E(R) = 13.61%,  SD = 9.89%, and Beta = 1.244.