# Final Exam FIN 529 Investments Mini

Philip H. Dybvig
Washington University in Saint Louis
February 27, 1997

This is a closed-book examination. Answer all questions as directed. Mark your answers directly on the examination. There are no trick questions on the exam. There are some formulas from the course (including some you will not need) at the end of the exam. Good luck!

A. General Concepts Short Anwer: 20 points (Answer each question in no more than one sentence of ordinary length.)

1. Name three types of investment. (For example, Treasury Bond would be one type.)

There are lots of correct answers, e.g., any three of domestic equities, foreign equities, Treasury Bills, Municipal Bonds, Warrants, Real Estate, Corporate Bonds.

2. What would be choice variables in a typical simple portfolio problem?

There are several good answers. Three are number of shares of each security, investment proportions in each security, and dollar investment in each security.

3. What is the main trade-off in the CAPM?

The main trade-off is between risk and return, where risk is measured by the variance and return by the mean.

4. What is ERISA?

ERISA, the Employee Retirement Income Security Act of 1974, is the main body of law regulating pension plans.

5. Name a common performance benchmark.

There are many good answers; the S&P 500 will probably be the most common answer.

B. Return Computations 40 points

1. single security A stock had a price a month ago of \$50. Today the price is \$45 and the stock has paid a dividend of \$2 during the month. What was the rate of return over the month?

Return = (45 + 2 - 50)/50 = -6%

2. portfolio return A month ago, a portfolio was invested 40% in equities and 60% in bonds. Over the month, the bonds returned 1% and the equities returned 10%. What was the portfolio return over the month?

Return = 40% x 10% + 60% x 1% = 4.6%

3. unitization At the beginning of a quarter, a portfolio was worth \$300,000. Two months into the quarter, the market was up and the fund had grown to \$315,000. At that time, there was a cash infusion of \$185,000, increasing the fund to \$500,000. In the final month of the quarter, the market was down and the portfolio value fell to \$495,000. What was the unitized return over the quarter?

(315,000/300,000) x (495,000/500,000) - 1 = 3.95%

4. after-tax return What is the break-even tax rate for a comparison between an untaxed municipal bond yielding 6% and a taxable Treasury bond yielding 8%?

6% = 8%(1-t) or t=25%

C. Portfolio Insurance Computation 30 points

A Simple Example in One Period

Riskless bond (interest rate is 50%):

```100 --- 150
```

Stock (underlying portfolio):

```    |- 125
50 -|
|-  50
```

Insured portfolio and put (floor = 95, insurance paid for separately):

```    |-  125              |- 0
? -|                 ? -|
|-  95               |- 45
```
1. What is the cost of the insurance (the put)?

r = 150/100 = 1.5, u = 125/50 = 2.5, d = 50/50 = 1
value = V_u(r-d)/(r(u-d)) + V_d(u-r)/(r(u-d)) = 0 x 2/9 + 45 x 4/9 = 20

2. What portfolio of the stock and the bond replicates the put? (I suggest that you verify your answer by making sure the replication works whether the stock goes up or down.)

stock = (0 - 45)/(2.5-1) = -30 or short \$30 worth (= short .6 shares)
bond = 20 - (-30) = 50 or long \$50 worth (= long .5 bonds)

3. What are the holdings in stocks and bonds of the insured portfolio including both the underlying portfolio and the put?

stock = 50 (underlying) - 30 (put replication) = \$20 long (equals .4 shares long)
bond = 0 (underlying) + 50 (put replication) = \$50 long (equals .5 bonds long)

D. True and False 10 points (Note: my answers have explanations but explanations are not needed on the exam.)

1. Using historical mean returns for individual stocks is the most reliable method for obtaining mean returns to be used in mean-variance analysis.

false. Because of the luck of the draw, historical returns differ significantly from any reasonable estimate of expected return, and tend to lead to a prescription of going long and short extreme amounts.

2. The main benefit of diversification is an increase in average expected return above that of the typical stock in the portfolio.

false. Diversification means the variance is small. The expected return of a portfolio is always a weighted average of the individual stock returns, which means that the expected return of the portfolio is the expected return of a typical stock in the portfolio, practically by definition.

3. According to the CAPM, investors are rewarded for taking on market level risk but not for taking on idiosyncratic (security-specific) risk.

true. This is exactly one of the main results in the CAPM.

4. To first approximation, the optimal response to transaction costs is to trade less to avoid the costs.

true. Costs are first order and the benefits of holding the optimal level of risk is second order.

5. In the paper ``Why Should Older People Invest Less in Stocks than Younger People?'' (by Jagannathan and Kocherlakota), it is argued that the sensible answer to the question in the title is that over a long horizon, substantial amounts of risk can be eliminated.

false. The main point of the paper is that this and most other traditional answers to the question given by practitioners are fallacious.

## Some useful formulas

Black-Scholes call option pricing formula:

```SN(x ) - BN(x ),
1        2
```

where
```                    2           2
x  = log(S/B)/sqrt(s T) + sqrt(s T)/2
1
```

and
```                    2           2
x  = log(S/B)/sqrt(s T) - sqrt(s T)/2.
2
```

Binomial option pricing:

```Value = p V  + p V
u u    d d
```

where
```      r-d
p  = ------
u   r(u-d)
```

and
```      u-r
p  = ------.
d   r(u-d)
```

Binomial replicating portfolio:

```        V  - V
u    d
stock = -------
u-d
```

and
```        uV  - dV
d     u
bond =  ---------
r(u-d)
```