# Portfolio Optimization Questions

1. Evaluate whether the following statements are true or false.

a) Even if a risky security has a return lower than the risk-free rate, this security could be held for diversification purposes.

b) The Glass-Steagall Act of 1933 separated commercial banking from investment banking.

c) If returns on two stocks are perfectly positively correlated, you can build a riskless portfolio with those stocks when short-selling is not allowed.

d) A risk-free security has a variance of zero.

e) A security with a beta of zero is risk free.

f) The CAPM implies that investors demand high returns for holding securities with low betas.

g) If a stock lies above the securities market line, it is overvalued.

h) There is an overwhelming majority of evidence against market efficiency: if you find a profitable trading strategy, you immediately tell the world how to trade.

i) The small firm in January effect is strongest at the beginning of the month.

j) When a risk-free asset is available, the minimum variance portfolio is the optimal portfolio for all risk-averse investors.

See attached for the rest of the questions.

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#### Solution Preview

Please find the detailed solutions in the attached word document and excel file. Attaching the documents, since plain text would not support the equations and the calculations.

1. Evaluate whether the following statements are true or false.

a) Even if a risky security has a return lower than the risk-free rate, this security could be held for diversification purposes. - True (For example, Gold is considered a negative beta investment during times of stock market crashes)

b) The Glass-Steagall Act of 1933 separated commercial banking from investment banking. - True

c) If returns on two stocks are perfectly positively correlated, you can build a riskless portfolio with those stocks when short-selling is not allowed. - False

d) A risk-free security has a variance of zero. - True

e) A security with a beta of zero is risk free. - True

f) The CAPM implies that investors demand high returns for holding securities with low betas. - False

g) If a stock lies above the securities market line, it is overvalued. - False

h) There is an overwhelming majority of evidence against market efficiency: if you find a profitable trading strategy, you immediately tell the world how to trade. - True

i) The small firm in January effect is strongest at the beginning of the month. - False

j) When a risk-free asset is available, the minimum variance portfolio is the optimal portfolio for all risk-averse investors. - False

2. Consider the following scenario analysis involving asset A and asset B:

Economic Condition Probability Return on X Return on Y

Boom 0.1 30% 12%

Accelerated Growth 0.2 20% 10%

Normal Growth 0.4 15% 9%

Slowdown 0.2 10% 8%

Recession 0.1 -50% -4%

a) Compute the expected returns and the standard deviation of returns for both assets. Then compute the correlation coefficient between the returns on the two assets.

E(RX) = (0.1*30%)+(0.2*20%)+(0.4*15%)+(0.2*10%)+(0.1*-50%)

= 10%

E(RY) = (0.1*12%)+(0.2*10%)+(0.4*9%)+(0.2*8%)+(0.1*-4%)

= 8%

σ(RX) =sqrt( 0.1*(30%-10%)2+0.2*(20%-10%)2+0.4*(15%-10%)2+0.2*(10%-10%)2+0.1*(-50%-10%)2)

...

#### Solution Summary

5 questions are answered, one of which is a multi-part true/false style one while the others require the reader to look over tables of information. Attached in Word and Excel.

Portfolio Optimization Problem

Create an Excel optimization file with good modeling techniques. Using Solver and Solver Table.

Not sure how to get started on this problem any help would be appreciated.

Thank you.

Here is the problemt:

Portfolio Optimization Problem

The annual returns for three companies over the last 12 years are given below, where the return for year n is defined as:

(closing price,n) - (closing price,n-1) + (dividends,n) / (closing price,n-1)

Year IBC NMC NBS

1 11.2% 8.0% 10.9%

2 10.8% 9.2% 22.0%

3 11.6% 6.6% 37.9%

4 -1.6% 18.5% -11.8%

5 -4.1% 7.4% 12.9%

6 8.6% 13.0% -7.5%

7 6.8% 22.0% 9.3%

8 11.9% 14.0% 48.7%

9 12.0% 20.5% -1.9%

10 8.3% 14.0% 19.1%

11 6.0% 19.0% -3.4%

12 10.2% 9.0% 43.0%

Part a: Using Excel, determine for each company the estimated mean return and standard deviation. Also, calculate the estimated correlations and covariances between the companies' returns.

Hint: Use the Excel functions AVERAGE, STDEV, and CORREL to determine the means, standard deviations and correlations. The covariances can then be calculated from the correlations (as in the class example). Click on fx in the Formula Bar for help with using these Excel functions.

Part b: Using the three stocks, a financial planner would like to create the least risky portfolio with minimum of a 12% expected return. Implement a spreadsheet model to determine the desired portfolio. Discuss the solution and highlight exactly how risk is minimized in the optimal portfolio, while achieving the desired return.

Part c: Either by repeatedly re-running the model, or by using SolverTable, create the chart for the efficient frontier for this portfolio optimization problem. Explain the usefulness of such a chart.