Consider the following information about three stocks Rate of Return If State Oc

Question

Consider the following information about three stocks Rate of Return If State Occurs State of Economy Boom Normal Bust Probability of State of Economy Stock A Stock C 30 .40 30 .27 23 01 Stock B 32 18 15 -48 -.32 a-1 If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) Portfolio expected return 11.54 % a-2 What is the variance? (Do not round intermediate calculations and round your answer to 5 decimal places, e.g., 32.16161.) Variance a-3 What is the standard deviation? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) Standard deviation b. If the expected T-bill rate is 4.90 percent, what is the expected risk premium on the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places e.g, 32.16.) Expected risk premium

Explanation / Answer

If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return?

This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:

Boom: E(Rp) = .40(.27) + .40(.32) + .20(.55) = .346 or 34.60%

Good:    E(Rp) = . 40(.23) + .40(.18) + .20(.15) = 0.194 or 19.40%

Poor:    E(Rp) = .40(.01) + .40(–0.32) + .20(–.48) = –0.22 or -22.00%

And the expected return of the portfolio is:

E(Rp) = .30(.346) + .40(.194) + .30(–.22) = 0.1154 or 11.54%

What is the variance?

To find the variance, we find the squared

deviations from the expected return. We then multiply each possible squared deviation by its probability, and then sum.

The result is the variance. So, the variance of the portfolio is:

sp2 = .30(.346 – .1154)2 + .40(.194 –0.1154)2 + .30(–.22 – .1154)2 = 0.049513

What is the standard deviation?

sp = (0.049513).5 = .222516 or 22.25%

If the expected T-bill rate is 4.40 percent, what is the expected risk premium on the portfolio?

Risk premium =E(RP ) - Rf = 11.54-4.90 = 6.64%

If the expected inflation rate is 4.40 percent, what are the approximate and exact expected real returns on the portfolio?

Approximate Expected real return = 11.54-4.40 = 7.14

To find the exact real return, we will use the Fisher equation. Doing so, we get:

1 + E(Ri) = (1 + h)[1 + e(ri)]

1.1154 = (1.044)[1 + e(ri)]

e(ri) = (1.1154/1.044) – 1 = .068391 or 6.84%

What are the approximate and exact expected real risk premiums on the portfolio?

The approximate real risk premium is the expected return minus the risk-free rate, so:

Approximate expected real risk premium = .1154 – .049 = .6.64%

The exact expected real risk premium is the approximate expected real risk premium, divided by one plus the inflation rate, so:

Exact expected real risk premium = .0664/1.049 = 6.33%

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