FIN3043 Investment Management. Assignment 1 solution

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1 FIN3043 Investment Management Assignment 1 solution Questions from Chapter 1 9. Lanni Products is a start-up computer software development firm. It currently owns computer equipment worth $30,000 and has cash on hand of $20,000 contributed by Lanni's owners. For each of the following transactions, identify the real and/or financial assets that trade hands. Are any financial assets created or destroyed in the transaction? a. Lanni takes out a bank loan. It receives $50,000 in cash and signs a note promising to pay back the loan over three years. b. Lanni uses the cash from the bank plus $20,000 of its own funds to finance the development of new financial planning software. c. Lanni sells the software product to Microsoft, which will market it to the public under the Microsoft name. Lanni accepts payment in the form of 2,500 shares of Microsoft stock. d. Lanni sells the shares of stock for $50 per share and uses part of the proceeds to pay off the bank loan. a. The bank loan is a financial liability for Lanni. Lanni's IOU is the bank's financial asset. The cash Lanni receives is a financial asset. The new financial asset created is Lanni's promissory note held by the bank. b. The cash paid by Lanni is the transfer of a financial asset to the software developer. In return, Lanni gets a real asset, the completed software. No financial assets are created or destroyed. Cash is simply transferred from one firm to another. c. Lanni sells the software, which is a real asset, to Microsoft. In exchange Lanni receives a financial asset, 2,500shares of Microsoft stock. If Microsoft issues new shares in order to pay Lanni, this would constitute the creation of new financial asset. d. In selling 2,500 shares of stock for $125,000, Lanni is exchanging one financial asset for another. In paying off the IOU with $50,000, Lanni is exchanging financial assets. The loan is "destroyed" in the transaction, since it is retired when paid.

2 10. Reconsider Lanni Products from the previous problem. a. Prepare its balance sheet just after it gets the bank loan. What is the ratio of real assets to total assets? b. Prepare the balance sheet after Lanni spends the $70,000 to develop its software product. What is the ratio of real assets to total assets? c. Prepare the balance sheet after Lanni accepts the payment of shares from Microsoft. What is the ratio of real assets to total assets? a. Liabilities & Assets Shareholders Equity Cash $70,000 Bank loan $50,000 Computers 30,000 Shareholders equity 50,000 Total $100,000 Total $100,000 Ratio of real to total assets = $30,000 $100,000 = 0.3 b. Liabilities & Assets Shareholders Equity Software product* $70,000 Bank loan $50,000 Computers 30,000 Shareholders equity 50,000 Total $100,000 Total $100,000 *Value at cost Ratio of real to total assets = $100,000 $100,000 = 1.0 c.

3 Liabilities & Assets Shareholders equity Microsoft shares $125,000 Bank loan $50,000 Computers 30,000 Shareholders equity 105,000 Total $155,000 Total $155,000 Ratio of real to total assets = $30,000 $155,000 = 0.2 Conclusion: When the firm starts up and raises working capital, it will be characterized by a low ratio of real to total assets. When it is in full production, it will have a high ratio of real assets. When the project "shuts down" and the firm sells it, the percentage of real assets to total assets goes down again because the product is again exchanged into financial assets. Questions from Chapter 2 1. What are the key differences between common stock, preferred stock, and corporate bonds? Common stock is an ownership share in a publicly held corporation. Common shareholders have voting rights and may receive dividends. Preferred stock represents nonvoting shares in a corporation, usually paying a fixed stream of dividends. While corporate bonds are long-term debt issued by corporations, the bonds typically pay semi-annual coupons and return the face value of the bond at maturity. 19. Consider the three stocks in the following table. Pt represents price at time t, and Qt period. represents shares outstanding at time t. Stock C splits two-for-one in the last

4 a. Calculate the rate of return on a price-weighted index of the three stocks for the first period (from t = 0 to t = 1) b. What must happen to the divisor for the price-weighted index in year 2? c. Calculate the rate of return of the price weighted index of the second period (from t = 1 to t = 2) a. At t = 0, the value of the index is: ($85 + $45 + $90)/3 = At t = 1, the value of the index is: ($90 + $40 + $100)/3 = The rate of return is: - 1 = (76.67/73.33) 1 = or 4.55% b. In the absence of a split, stock C would sell for $100, and the value of the index would be the average price of the individual stocks included in the index: ($90 + $40 + $100)/3 = $ After the split, stock C sells at $50; however, the value of the index should not be affected by the split. We need to set the divisor (d) such that: = ($90 + $40 + $50)/d d = 2.35 c. The rate of return is zero. The value of the index remains unchanged since the return on each stock separately equals zero.

5 20. Using the data in the previous problem, calculate the first-period rates of return on the following indexes of the three stocks: a. A market value weighted index b. An equally weighted index a. Total market value at t = 0 is: ($85 x 100) + ($45 x 200) + ($200 x 90) = $35,500 Total market value at t = 1 is: ($90 x 100) + ($40 x 200) + ($100 x 200) = $37,000 Rate of return = - 1 = ($37,000/$35,500) 1 = or 4.23% b. The return on each stock is as follows: R A = - 1 = ($90/$100) 1 = or 5.88% R B = - 1 = ($40/$45) 1 = or 11.11% R C = - 1 = ($100/$90) 1 = or 11.11% The equally-weighted average is: [5.88% + ( 11.11%) %]/3 = 1.96% Questions from Chapter 5 8. a. Suppose you forecast that the standard deviation of the market return will be 20% in the coming year. If the measure of risk aversion in Equation 5.17 is A = 4, what would be a reasonable guess for the expected market risk premium? b. What value of A is consistent with a risk premium of 9%? c. What will happen to the risk premium if investors become more risk tolerant?

6 a. Given that A = 4 and the projected standard deviation of the market return = 20%, we can use the below equation to solve for the expected market risk premium: A = 4 = = E(r M ) r f = Aσ M 2 = 4 (0.20) = 0.16 or 16% b. Solve E(r M ) r f = 0.09 = Aσ M 2 = A (0.20), we can get A = 0.09/0.04 = 2.25 c. Increased risk tolerance means decreased risk aversion (A), which results in a decline in risk premiums. For Problems 12 13, assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%. 12. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. a. What is the expected return and standard deviation of your client's portfolio? b. Suppose your risky portfolio includes the following investments in the given proportions: What are the investment proportions of your client's overall portfolio, including the position in T-bills? c. What is the reward-to-volatility ratio (S) of your risky portfolio and your client's overall portfolio? d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL. a. Allocating 80% of the capital in the risky portfolio P, and 20% in risk-free asset,

7 the client has an expected return on the complete portfolio calculated by adding up the expected return of the risky proportion (y) and the expected return of the proportion (1 - y) of the risk-free investment: year E(r C ) = y E(r P ) + (1 y) r f = ( ) + ( ) = or 10.40% per The standard deviation of the portfolio equals the standard deviation of the risky fund times the fraction of the complete portfolio invested in the risky fund: σ C = y σ P = = or 22.40% per year b. The investment proportions of the client s overall portfolio can be calculated by the proportion of risky portfolio in the complete portfolio times the proportion allocated in each stock. Security Investment Proportions T-Bills 20.0% Stock A % = 16.0% Stock B % = 24.0% Stock C % = 40.0% c. We calculate the reward-to-variability ratio (Sharpe ratio) using Equation For the risky portfolio: S = = = = For the client s overall portfolio: S = = =

8 E(r) 17 % P CAL ( slope=. 14 client % σ 13. Suppose the same client in the previous problem decides to invest in your risky portfolio a proportion (y) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. a. What is the proportion y? b. What are your client's investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client's portfolio? a. E(r C ) = y E(r P ) + (1 y) r f = y (1 y) 0.04 = 0.11 or 11% per year Solving for y, we get y = = Therefore, in order to achieve an expected rate of return of 11%, the client must invest 87.5% of total funds in the risky portfolio and 12.5% in T-bills.

9 b. The investment proportions of the client s overall portfolio can be calculated by the proportion of risky asset in the whole portfolio times the proportion allocated in each stock. Security Investment Proportions T-Bills 12.5% Stock A % = 17.50% Stock B % = 26.25% Stock C % = 43.75% c. The standard deviation of the complete portfolio is the standard deviation of the risky portfolio times the fraction of the portfolio invested in the risky asset: σ C = y σ P = = or 24.5% per year Questions from Chapter Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL. a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds? a. The equation for the CAL is: E(r C ) = r f + σ C = σ C Setting E(r C ) equal to 15% yields a standard deviation of %.

10 b. The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is: E(r C ) = (l y)r f + ye(r P ) = r f + y[e(r P ) r f ] = y( ) Setting E(r C ) = 15% y =.9206 (92.06% in the risky portfolio) 1 y =.0794 (7.94% in T-bills) To prevent rounding error, we use the spreadsheet with the calculation of the previous parts of the problem to compute the proportion in each asset in the complete portfolio: σ(c) W(risky portfolio) = (15% - 4.8%)/Sharpe ratio(risky portfolio) = σ(c)/σ(risky portfolio) Proportion of s tocks in comple te portfolio W(s) = W(risky portfolio)*% in stock of the risky portfolio = Proportion of bonds in comple te portfolio W(b) = W(risky portfolio)*% in bonds of the risky portfolio = If you were to use only the two risky funds and still require an expected return of 12%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimal portfolio in the previous problem. What do you conclude? Using only the stock and bond funds to achieve a mean of 12%, we solve: 12 = 15w S + 9(1 w S ) = 9 + 6w S w S =.5 Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviation of: σ P = [( ,024) + ( ) + ( )] 1/2 = 21.06% The efficient portfolio with a mean of 12% has a standard deviation of only 20.61%. Using the CAL reduces the standard deviation by 45 basis points.

11 Practice Question and Solution for Chapter 7: Questions from Chapter 7(no submission) 3. Are the following true or false? Explain. a) Stocks with a beta of zero offer an expected rate of return of zero. b) The CAPM implies that investors require a higher return to hold highly volatile securities. c) You can construct a portfolio with beta of.75 by investing.75 of the investment budget in T-bills and the remainder in the market portfolio. a. False. According to CAPM, when beta is zero, the excess return should be zero. b. False. CAPM implies that the investor will only require risk premium for systematic risk. Investors are not rewarded for bearing higher risk if the volatility results from the firm-specific risk, and thus, can be diversified. c. False. We can construct a portfolio with the beta of.75 by investing.75 of the investment budget in the market portfolio and the remainder in T-bills. 25 Suppose the yield on short-term government securities (perceived to be risk-free). is about 4%. Suppose also that the expected return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model: a) What is the expected return on the market portfolio? b) What would be the expected return on a zero-beta stock? c) Suppose you consider buying a share of stock at a price of $40. The stock is expected to pay a dividend of $3 next year and to sell then for $41. The stock risk has been evaluated at β =.5. Is the stock overpriced or underpriced? a) Since the market portfolio, by definition, has a beta of 1.0, its expected rate of return is 14%. b) β = 0 means the stock has no systematic risk. Hence, the portfolio's expected rate of return is the risk-free rate, 4%. c) Using the SML, the fair rate of return for a stock with β = 0.5 is: E(r) = 4% + ( 0.5) (14% 4%) = 1.0% The expected rate of return, using the expected price and dividend for next year:

12 E(r) = ($31 + $4)/$30 1 = = 16.67% Because the expected return exceeds the fair return, the stock must be under-priced. 27. Consider the following data for a one-factor economy. All portfolios are well diversified. Suppose another portfolio E is well diversified with a beta of 2/3 and expected return of 9%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be? a) Since the beta for Portfolio F is zero, the expected return for Portfolio F equals the risk-free rate. For Portfolio A, the ratio of risk premium to beta is: (10 4)/1 = 6 The ratio for Portfolio E is higher: (9 4)/(2/3) = 7.5 This implies that an arbitrage opportunity exists. For instance, by taking a long position in Portfolio E and a short position in Portfolio F (that is, borrowing at the risk-free rate and investing the proceeds in Portfolio E), we can create another portfolio which has the same beta (1.0) but higher expected return than Portfolio A. For the beta of the new portfolio to equal 1.0, the proportion (w) of funds invested in E must be: 3/2 = 1.5. Portfolio Weight In Asset Contribution to β Contribution to Excess Return -1 Portfolio A -1 x βa = x (10%- 4%) = -6% 1.5 Portfolio E 1.5 x βe = x (9% - 4%) = 7.5% -0.5 Portfolio F -0.5 x 0 = 0 0 Investment = 0 βarbitrage = 0 α = 1.5% As summarized above, taking a short position in portfolio A and a long position in the new portfolio, we produce an arbitrage portfolio with zero investment

13 (all proceeds from the short sale of Portfolio A are invested in the new portfolio), zero risk (because and the portfolios are well diversified), and a positive return of 1.5%.

Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory

Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory Chapter 7 Capital Asset ricing and Arbitrage ricing Theory 1. a, c and d 2. a. E(r X ) = 12.2% X = 1.8% E(r Y ) = 18.5% Y = 1.5% b. (i) For an investor who wants to add this stock to a well-diversified