Dr. P.V. Viswanath
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Superquiz

Notes:

1. If your answers are not legible or are otherwise difficult to follow, I reserve the right not to give you any points.
2. If you cheat in any way, I reserve the right to give you no points for the exam, and to give you a failing grade for the course.
3. You may bring in sheets with formulas, but no worked-out examples, or definitions, or anything else.
4. You must explain all your answers. For the quantitative questions, you must show your formulas and youc computation, else you may get no credit at all.

1. Tamako wants to buy a car, but the sort of car that she wants costs \$45,000. Unfortunately, she only has \$23,000. Fortunately, she has access to a banker, who is willing to lend her the additional money at a stated interest rate of 10% per year. Tamako has to start making payments on the loan, two years after she has taken out the loan. If she wants to be done with repaying the loan in five years (that is, she will make 36 equal payments), what is the amount of her monthly payment? (10 points)

Another banker (let's call him Lazard) wants to compete with the first banker (let's call him JP). Lazard is willing to allow Tamako to make bi-monthly payments (once in 2 months), which Tamako finds attractive. However, the stated interest rate is 11%. What will Tamako's bi-monthly payments be, and will she go with JP or Lazard? Assume that the payments are not delayed as in the JP case (that is, there will be 30 payments, starting in two months). (10 points)

2. (40 points) Gerber Scientific Inc. (GRB) and SPSS Inc. (SPSS) are two stocks that Punita has been considering for her portfolio. Although she knows that diversification reduces risk and she would be better off investing in many assets, she believes that GRB and SPSS are good deals, right now, and she decides to put all of her money into these two stocks. Since the two stocks are in different industries (SPSS in Software and GRB in Scientific machinery), the correlation coefficient between the two stocks is quite low, on the order of 0.25. Punita has, in addition, come up with the following additional estimates for the two stocks:
 SPSS GRB Expected Return 22% 18% Standard deviation of returns 10% 23%

1. Punita wants an expected return of 19% on her portfolio. How much should she invest in SPSS and how much in GRB?
2. What would the standard deviation of returns on this portfolio be?
3. Punita also has the opportunity to lend at a guaranteed rate of 5%. If she likes higher expected returns and lower standard deviation of returns, how should she combine the two risky stocks? What is the expected return on this portfolio of risky stocks?
4. If Punita wanted to invest in the riskfree asset, as well as in the risky portfolio, how much should she invest in both, in order to get an expected reutrn of 10%?

3. (10 points each) Define and write two sentences about any four of these terms:

1. moral hazard
3. volatility
4. actuaries
5. hedgers
6. diversifying

4. (Bonus; 10 points): You expect to close on a house purchase in 2 months. You have already agreed on a price of \$300,000. You only have \$50,000 of the money needed for the purchase, but a bank has agreed to lend you the additional \$250,000. The interest rate that you will have to pay on the loan depends on market conditions in 2 months, when you will actually need the money. You are worried that interest rates might go up, and you sell futures contracts on 10 year US Treasury Notes, with a nominal value of \$250,000. This futures contract is a commitment to sell U.S. Treasury notes having a face value at maturity of \$100,000 with a maturity of 10 years.

Will this increase or decrease your risk of interest rates changing between now and the time you buy your house? Why?

Superquiz Solutions

1. Tamako needs to borrow 45000 - 23000 = \$22000. The stated interest rate is 10%, with monthly payments. Hence the effective interest rate is (1+0.1/12)12 -1 or ((1.008333)12 -1 or 10.47%. The face loan will, therefore, rise to (22000(1.1047)2 or \$26848.60 after two years. Now, we can compute her monthly payments by looking at the formula for the present value of an annuity, i.e. her monthly payment will be C, where 26848.60 = (C/0.00833)[1-(1.00833)-36]. Solving, we find C = \$866.33

Banker Lazard has a stated interest rate of 11% with bi-monthly payments. Hence the bi-monthly non-annualized rate is 11/6 or 1.8333% per two months, for an EAR of (1.01833)6 - 1 or 11.517%. We can compute her bi-monthly payments by solving for C in the equation, 22000 = (C/0.01833)[1-(1.01833)-30], or C = 959.93 every two months.

Banker JP is preferable because her effective rate of interest is lower with him, 10.47% versus 11.517% for Lazard.

2.

1. If Punita wants a return of 19%, and she invests a% in SPSS and (1-a)% in GRB, we get 22a + 18(1-a) = 19, or a = 25%.
2. The variance of returns on this portfolio are [(0.25)(0.1)]2 + [(0.75)(.23)]2 + 2(0.25)(0.75)(.10)(.23)(0.25) = 0.032538. The standard deviation is the square root of this, or 18.038%.
3. The tangent portfolio is given by the formula , where the two risky assets are labelled D and E. Using this formula, we get the weight for SPSS for the tangent portfolio to be 68.61%; the weight on Gerber Scientific, then is the remainder, i.e. 31.39%. The expected return on this portfolio is (68.61)(22) + (31.39)(18) = 20.74%
4. To get an expected return of 10%, she should invest a% in the riskkfree asset, where 5a + 20.74(1- a) = 10. Solving, we get a = 68.23%

4. This futures contract is a commitment to sell U.S. Treasury notes at a pre-determined price. If interest rates go up, the actual price of the notes will go down, and you will make money, since you have already agreed on the price, upfront. If interest rates go down, you will lose money. On the other hand, with the house, if interest rates go up, it will cost you more to get the loan, while it's good for you if interest rates drop. This is the opposite of what will happen with the futures contract. Hence, entering into the futures contract will decrease your risk.

Final Exam

Notes:

1. If your answers are not legible or are otherwise difficult to follow, I reserve the right not to give you any points.
2. If you cheat in any way, I reserve the right to give you no points for the exam, and to give you a failing grade for the course.
3. You may bring in sheets with formulas, but no worked-out examples.
5. You have 2 hours to complete the exam; please make sure to attempt all the questions, so I can give you partial credit, if necessary.

1. (8 points) Bodie and Merton's textbook, on p. 347 in section 13.2 has the following statement:
In the CAPM, the equilibrium risk premium on the market portfolio is equal to the variance of the market portfolio times a weighted average of the degree of risk aversion of the holders of wealth (A): E(rM) - rf = A sM2
Keeping in mind that risk tolerance is simply the opposite of risk aversion, would you say that the VIX is a measure of risk tolerance? If not, what does it measure?
2. (8 points) Why is the VIX used to measure the market risk premium?
3. (7 points) Why is the amount of leverage used by investors a measure of investor confidence?
4. (7 points) Why is a measure based on the prices of call and put options called a "Volatility" Index?

Investors' View Of Risk Returns To Normal
Wall Street Journal, September 10, 2007; Page C1

For investors hoping the markets will settle down, here is a simple, sobering message: Don't hold your breath.

After a long period of unusual stability in stock and bond markets, the wrenching losses of the past few months may have merely brought investors' perceptions of risk back to where they should have been in the first place.

"People were pricing things as if there was never going to be another recession, or even a financially difficult period or corporate default," says Byron Wien, chief investment strategist at hedge fund Pequot Capital Management. "We're moving toward normal."

And normal might not be as stable as investors had come to believe.

The Chicago Board Options Volatility Index, or VIX, which tracks the prices of call and put options on the S&P 500 index, is a popular measure of risk tolerance. Options can be used as insurance against losses from market swings, because an option allows an investor to buy or sell a stock at a preset price. The more worried investors get about losing money on investments, the more options cost, and the higher the VIX goes.

On the heels of Friday's disappointing jobs report and the nearly 250-point loss in the Dow Jones Industrial Average, the VIX rose 2.24 points to 26.24. That puts it well above the multiyear low of 9.89 it hit in January. But it is far from unusually high by historical standards.

"If the VIX was at 40, I think you could comfortably say this risk aversion is extreme. But the VIX at 26? This is sort of the midpoint in 1999 and 2000," says Douglas Cliggott, chief investment officer at money manager Dover Management.

Similarly, junk-bond prices have fallen sharply in recent months, driving yields higher. At the beginning of June, the Merrill Lynch High Yield Bond Index yielded 2.41 percentage points more than comparable Treasurys. Friday, that "spread" over low-risk Treasurys had widened to 4.73 percentage points. That's a massive shift. But interest-rate spreads on junk bonds are still a bit smaller than the historical norm.

The risk tolerance of investors had been rising for many months, in part because there was a growing perception that the economy was becoming more stable. With a recession now possibly in the cards, that view will be reassessed.

Hedge funds and other large investors appear to have been reducing the amount of leverage -- or borrowed money -- they use to invest. Leverage is safe when the economy is stable, but dangerous in a downturn. If investors use less of it, that will cut back on the flow of cash into the markets.

"People are waiting for credit spreads and volatility to go to the levels of the past three years, but that shouldn't be the base comparison," says Brett Gallagher of Julius Baer Investment Management.

Short-term credit markets, on the other hand, are showing extreme levels of risk aversion. The London interbank offered rate, or Libor, has risen far above the Fed's overnight target rate, a rare occurrence that suggests banks around the world are very wary of what lurks on each other's balance sheets.

Maybe they know something everyone else should be more worried about.

2. (10 points) Ford Corporation has a lot of convertible bonds on its balance sheet. Suppose one of them is a 10 year bond with a coupon rate of 3.5% selling at par. You know, however, that comparable bonds that are not convertible actually sell at a yield-to-maturity of 8% per annum. What is the value of the convertibility feature? Assume bi-annual coupon payments, as well as a face value of \$1000 per bond.

3. (10 points) In 2005, Ford Motor Company paid dividends of 40 cents per share to common stockholders (10 cents per quarter). However, in the third quarter of 2006, this was reduced to 5 cents, and in the last quarter, the dividends were eliminated altogether. A footnote in the firm's 10-K has the following information:

On December 15, 2006, we entered into a new secured credit facility which contains a covenant prohibiting us from paying any dividends (other than dividends payable solely in stock) on our Common and Class B Stock, subject to certain limited exceptions. As a result, it is unlikely that we will pay any dividends in the foreseeable future.

If Ford Motor Company does not expect to pay any dividends in the foreseeable future, would it be right to conclude that the stock is worthless? Why or why not?

4. a. (10 points) According to Yahoo (http://finance.yahoo.com/q/ks?s=goog), Google's beta is 1.21. Assume that the market risk premium is 6% per annum. The 10 year T-bond currently yields 4.15% (http://finance.yahoo.com/). What is the rate of return that investors should require to invest in Google, according to the CAPM?
b. ( 15 points) The Free Cash Flow to Equity for Google is \$1.09b., according to Yahoo (actually, this is Levered Free Cash, as defined at http://help.yahoo.com/l/us/yahoo/finance/tools/research-12.html). This is the cashflow that Google had available to it, last year, after it took care of its short-term and long-term investment needs, and after adjusting for payments to and from bondholders. According to some researchers, this could be considered a measure of how much the company could afford to pay out in dividends. Google currently has 312.84m. shares outstanding, and it sold for \$718.42 as of the end of trading on Monday, December 10, 2007. If you believe that Google's current price is correct, what is the implied rate of growth of dividends, assuming that Google is going to grow at the same rate of growth forever? (Hint: If you use the usual formula to compute the growth rate, you should get a surprising answer.)

5. a. (7 points) Options are traded on the CBOE on Ford Motor Company stock with an exercise price of 8 with an expiration date six months from now. The current stock price is around \$7. Assume that the stock price can rise to \$9 in six months or drop to \$6.50. What should the price of the call be if the risk-free rate is 5% per six months?
b. (7 points) What should the price of a put option on Ford stock be, with an exercise price of \$8?
c. (6 points) Show that your answer satisfies the put-call parity relationship.

Final Solutions

1.a. VIX tracks, that is, is (approximately) an index of the prices of all and put premiums on the market portfolio. Now we know that call and put prices are increasing in the volatility of the returns on the underlying assets. Hence, VIX is directly a measure of market uncertainty, that is, of sM2, rather than of A. Of course, since the puts and calls are assets, their prices are inversely related to their required rate of return and hence on the market risk premium and therefore on the market risk aversion as well. But in this, they are no different from any other asset. Hence, VIX primarily measures market uncertainty, rather than the market risk aversion.

b. If risk tolerance is relatively constant over time (and since it is a characteristic of individual preferences, there is no reason to believe that it changes a lot), then changes in the market risk premium would be tracked by changes in the market uncertainty parameter. Hence, changes in the market risk premium can be measured by movements in the VIX.

c. If investors desire a certain variance of portfolio returns, this can be reached with a lower level of leverage, the higher the volatility of returns on the underlying assets. (Consider, for example, an investor who is investing in the market portfolio as well as borrowing and lending -- the greater the variance of returns of the market portfolio, the greater the amount of leverage to attain a given variance of portfolio returns.) Hence the amount of leverage used by investors can be considered a negative measure of their forecast variance of the return on the market, and hence, of their confidence in the market.

d. The prices of calls and puts are increasing in the volatility of the underlying stocks. Hence, an index of calls and puts also tracks volatility.

2. If the Ford Motor Company bond were not convertible, with a 3.5% coupon, it would sell at a price of (17.5/0.04)[1-(1.04)-20] + 1000/(1.04)-20 = 694.22. Since the bond is actually selling at par, the value of the convertibility option is 1000-694.22 or \$305.78.

3. If the firm were not expected to pay any dividends at all in the future, the stock would, of course, be worthless. However, what is meant is that the firm is not expecting to pay dividends for the foreseeable future -- that is, the future that can be forecast with some degree of accuracy. This might be, say, three or four or five years. However, the firm by not paying dividends right now and keeping cash on hand instead would have more flexibility and more funds available to reinvest in the firm. Consequently, it would be more likely to reach profitability eventually, at which time, it would, indeed, pay dividends. Hence the firm's shares would not be worthless in spite of the footnote in the 10-K.

4.a. The required rate of return in Google would be, using the CAPM formula, 4.15% + 1.21(6%) = 11.41%

b. Free Cash Flow to Equity per share for 2006 (assuming the calendar year is the fiscal year), using the numbers provided would be 1090/312.84 or \$3.484. If we use this as a measure of Google's dividends last year, and assume that Google's dividends are growing at the same rate forever, we would have 718.42 = 3.484(1+g)/(0.1141-g). Solving, we find that 718.42(0.1141) - 718.42g = 3.484+3.484g. In other words, g = 10.87%.

c. It is unlikely that Google will continue to grow forever at a rate of 10.87%. Since the economy as a whole does not grow that rapidly, if Google does grow so rapidly, it will eventually constitute the entire economy! We must, therefore assume that Google will grow very rapidly for a finite number of years, but then grow at a more sedate pace thereafter. It will therefore end up being a large part of the economy but not the bulk of the economy, as would transpire if we assumed a perpetual growth rate of 10.87%.

5.a. Assuming that the replicating portfolio consists of s units of stock and lending with a face value of L, we solve the equations 9s+L=1 and 6.5s+L=0. We find that s = 1/2.5 = 0.4 and L = -2.6. Hence the value of the call today is 0.4(7) - 2.6/1.05 = \$0.3238.

b.We know, according to the put-call parity formula, that S+P = C+PV(E). Hence P = 0.3238 + 8/1.05 - 7 = \$0.9429

c. The answer in (b) clearly satisfies the put-call parity formula, since it was deduced using the formula. However, the put price could also have been computed directly.

Assuming a replicating portfolio consisting of S units of stock and lending with a face value of L, we solve the equations 9s+L=0 and 6.5s+L=1.5. We find that s = -0.6 and L = 5.4. Hence the value of the call today is -0.6(7) + 5.4/1.05 = \$0.9429