Drawing a timeline will help solve this problem. Investing in a troubles business. You are buying a business that has been losing money. Next year (at t = 1) you will need to invest $100 into the business (to cover losses). You will be making contributions at t = 2, t = 3, t = 4, t = 5. The contributions are declining at 20% from year to year (for example, your contribution at t = 2 is $100*(1-0.2) = $80). At t = 6 the business will have zero profits (your cash flow at t = 6 is zero). At t = 7 you will receive first profits of $90. Profits will be paid forever and will grow at an annual rate of 7%. The discount rate is 12% per year.

Stage one. What is the Present Value (at t = 0) of all your investments during the first stage? (You are calculating the present value of payments made at t = 1, 2, 3, 4, 5).

Stage two. What is the Present Value (at t = 0) of all your profits. (You are calculating the present value of payments received, which are profits, from t =6 to infinity).

Total. What is the value of this business at t = 0?

Question 2

You are valuing a company that will experience two stages. During the first stage, the revenues and expenses will grow quickly. During the second stage, the revenues and the expenses will grow at slower rates. The revenues will be received annually and the expenses will be paid annually. The discount rate is 12% per year. For this problem, you may find it useful to consult the two-stage valuation diagram (red and blue) that was handed out in class.

Stage 1. Revenues at t = 1 will equal $1 million. The revenues will grow at 10% per year until t = 7. That is, the last revenue collected during the first stage is at t = 7. The expenses at t = 1 will equal $0.8 million. The expenses will grow at 12% per year until t = 7. That is, the last expenses paid during the first stage are at t = 7. Hint: Revenues in Stage 1 are a growing annuity. Expenses in Stage 1 are another growing annuity.

What is the present value, at t = 0, of all revenues collected during Stage 1?

What is the present value, at t =0, of all expenses paid during Stage 1?

What is the net value of Stage 1 (at t = 0)?

Stage 2. Remember that the last payment in Stage 1 is payment at t = 7. Immediately after this payment the second stage begins. The second stage lasts forever. The revenues will grow at 3% per year in the second stage. The expenses in the second stage will grow at 2% per year.

What is the present value, at t = 0, of all revenues collected during stage 2?

What is the present value, at t = 0, of all expenses paid during stage 2?

What is the net value of Stage 2 (at t = 0)?

Value of the business. You have now computed the present value of revenues in stage 1, expenses in stage 1, revenues in stage 2, and expenses in stage 2. What is the total value of the business today (at t = 0)?

PART 2. RISK, RETURN and COST OF CAPITAL.

Question 3. CAPM: Stock. The Capital Asset Pricing Model (CAPM) is a model used to compute returns (expected returns) for different assets. In the model, rf is the risk-free rate (the rate paid on a government bond or on a bank deposit); rM is the market rate of return (the rate of return on a large, diversified, collection of assets, where the un-systematic risk has been diversified away); beta coefficient A for the asset, for which we would like to compute the rate of return.

Return on the market is 10% per year. Risk-free rate is 2% per year. Consider a stock that has expected return of rA=-6% per year. What is this stocks beta?

Question 4. CAPM: Stock. A stock has expected returns of 12% per year. The standard deviation of returns on this stock is 50% (annual). The market return is 8.5% per year, and the risk-free rate is 0.5% per year. The standard deviation of returns on the market is 30% (annual). What is the correlation of returns on this stock with market returns?

Question 5. Portfolio Risk and Return.

[Portfolio] You have the following investment portfolio: $5,000 is invested in the stock of Johnson & Johnson (JNJ), and $10,000 is invested in the stock of Intel Corp. (INTC).

1. What are the portfolio weights (wJNJ and wINTC)?

2. The return on the JNJ stock is 9% per year; the return on INTC stock is 11% per year. What is the return on your portfolio?

3. The variance of returns on JNJ is JNJ2=0.40, the variance of returns on INTC is INTC2=0.60; The correlation coefficient between these two stocks is JNJ,INTC=0.7. What is the variance of your portfolio?

Question 6. You are interested in building a portfolio of two stocks. The first stock has a beta of 0.5, 1=0.5. The second stock has a beta of 2.0, 2=2.0. You want to build a portfolio that has exactly as much systematic risk as the market. What percentage of your wealth must be invested in stock 1 and what percentage in stock 2?