B.A. 130 Fall 1996
Midterm 1, Version
A
Part A (20 minutes): Identifications. Write one sentence on
each.
1. What is a corporate bond?
- A security issued by private corporations: it pays interest (usually
in the form of a semi-annual coupon payment), and upon maturity the issuing
corporation pays the bond's principal value to the bond's owner. A bond
is a negotiable security--that is, you don't need the issuing company's
permission to sell or buy it.
2. What is a required rate of return?
- The required rate of return for a security of a given systematic risk
is that expected rate of return at which demand and supply for the security
are balanced. It is equal to the risk-free rate plus the security's beta
times the risk premium.
3. What is meant by the phrase: "the present value of the firm's growth
opportunities"?
- The present value of a firm's anticipated opportunities to make positive
present-value investments: investments that yield more than the market rate
of return.
4. What is meant by the term "beta"?
- The proportional change in the value of a security that accompanies
a proportional change in the value of the market as a whole. A security
that on average went up 2% for every 1% marginal swing in the market would
have a beta of 2.
5. What is a dividend?
- The periodic payment (usully quarterly) made at the Board of Directors'
discretion to owners of common stock, a security issued by private corporations
that carries the rate to vote for the Board of Directors at the firm's annual
meeting.
6. What is a certainty-equivalent value?
- The certain cash flow to be received at some future time t
that you would value as exactly worth the uncertain distribution of cash
flow outcomes to be received at that future time t.
7. What is a Treasury bill?
- A discount bond issued by the government for a term of 3 to 6 months.
Part B (20 minutes): Net Present Values. Show your work.
1. The net present value today at a discount rate of ten percent per year
of a security that will pay you $1,100,000 two years from now, and $1,100,000
a year thereafter.
- The perpetuity piece has a value of $11,000,000 when evaluated two
years from now, so the whole thing is equivalent to a value of $12,100,000
payable two years from now, or $11,000,000 payable one year from now, or
a present value of $10,000,000 today.
2. The net present value today, at a discount rate of eight percent per
year, of a nine-year annuity of $100,000 a year.
- 9 years x 8% per year = 0.72; so the "rule of 72" can be
applied: a perpetuity of $100,000 at 8% per year is worth $1,250,000; the
annuity is worth half that--$625,000
5. The approximate present value, at a discount rate of 6% per year, of
$100,000,000 payable seventy-two years from now.
- From the "rule of 72", a sum discounted at six percent per
year halves in value every twelve years. So this sum halves six times--from
100 to 50 to 25 to 12.5 to 6.25 to 3.125 to 1.5625 million.
6. The approximate present value, at a discount rate of 6% per year,
of $200,000,000 payable seventy-two years from now.
- From the "rule of 72", a sum discounted at six percent per
year halves in value every twelve years. So this sum halves six times--from
200 to 100 to 50 to 25 to 12.5 to 6.25 to 3.125 million.
2. The net present value today, at a discount rate of ten percent per year,
of a security that will pay you $1,650,000 two years from now, and $1,650,000
a year thereafter.
- A value of $16,500,000 in year 1, or a present value of $15,000,000
today.
3. The approximate net present value today, at a discount rate of
eight percent per year, of a nine-year annuity of $200,000 a year.
- 9 years x 8% per year = 0.72; so the "rule of 72" can be
applied: a perpetuity of $200,000 at 8% per year is worth $2,500,000; the
annuity is worth half that--$1,250,000
4. The net present value today, at a discount rate of 25% per year, of a
security that will pay you $1,250,000 two years from now, and $1,250,000
a year thereafter.
- The perpetuity is worth $5,000,000 in year one, or a present value
of $4,000,000 (full credit for an answer of $1,440,000)
5. The net present value, at a discount rate of 10% per year, of a cash
flow that is $1,210,000 in year two, $605,000 in year three, and that grows
at a rate of 5% per year thereafter.
- the perpetuity is worth $605000/(.10-.05)= $12,100,000 in year two;
so the entire package is worth $13,310,000 in year two. Discount at 10%
back to year two, and the package is worth $12,100,000 in year one, or $11,000,000
in present value today.
Part C (20 minutes): Risk, Return, and the Capital Asset Pricing
Model. Show your work.
1. Suppose the covariance of the Betatron Company's annual return with the
market is .04, and the standard deviation of the market return is 20% per
year. Suppose further that half of the market value of the Betatron Company
is held in the form of bonds, and half is held in the form of common stocks.
What is the beta of the Betatron Company's common stock?
- Betatron as a company has a beta of 1. Since stocks have half its
value, if the stocks carry all the risk than the common stock has a beta
of 2. (If the bonds are risky as well--so that the stocks do not carry all
the risk--the beta will be between 1 and 2; full credit for either answer).
2. Suppose we have two stocks with uncorrelated returns. Suppose further
that the standard deviation of each of their annual returns is 20%. What
is the standard deviation of the return on a portfolio that puts half of
its value into each of these two stocks?
3. Suppose we have a risk-free rate of 5% per year, and a market rate of
10% per year. What is the value of a stock with an expected dividend next
year of $5, an expected dividend growth rate thereafter of 2% per year,
and a beta of one?
- The required rate of return is 10%, so D/(r-g) = 5/(.10-.02)=5/(.08)=$62.50
4. Suppose that the risk-free rate is 5% per year, and the market rate is
10% per year. Suppose that we have two stocks with returns that are perfectly
negatively correlated. Suppose further that the standard deviation of each
of their annual returns is 20%. What is the expected return on a portfolio
that puts half of its value into each of these two stocks?
- Because the correlation is -1, if you put half your wealth in each
security your portfolio has no risk. Thus it must yield a return of 5% pe
year.
5. Suppose that the risk-free rate is 5% per year, and that we have two
stocks with returns that are perfectly positively correlated. Suppose further
that the standard deviation of each of their annual returns is 20%. What
is the standard deviation of a portfolio that puts half of its value into
each of these two stocks?
- 20%. No benefits from diversification when security returns are perfectly
positively correlated.
Part D (20 minutes): Further Topics.
1. Suppose that we have five portfolios:
- A offers an expected return of 5% per year, with a standard deviation
of 15% per year.
- B offers an expected return of 10% per year, with a standard deviation
of 20% per year.
- C offers an expected return of 15% per year, with a standard deviation
of 30% per year.
- D offers an expected return of 20% per year, with a standard deviation
of 50% per year.
- E offers an expected return of -5% per year, with a standard deviation
of 30% per year.
A client asks if you can construct a portfolio that has an expected return
of 7.5% per year and a standard deviation of no more than 17.5% per year.
Can you? Explain your reasoning.
A client asks if you can construct--out of A through D alone--a portfolio
with a standard deviation of less than 50% per year and an expected return
of 20% per year. Can you? Explain your reasoning. Suppose that you also
had access to a risk-free asset that paid an expected return of 5% per year.
Would it change your answer?
- Answer: Suppose that the returns to securities A and B were
perfectly correlated. Then 50% of your wealth in A and 50% in B would generate
a portfolio with an expected return of 7.5% and a standard deviation of
17.5%. Suppose that the returns to A and B were not perfectly correlated.
Then the standard deviation of a 50/50 portfolio would be less than 17.5%,
because risk would be reduced through diversification. Therefore you can
do it.
- Answer: As far as the second part is concerned, you really
cannot tell; maybe (if the covariances are right) but maybe not.
- Answer: Adding a risk-free security allows you to combine a
position long 1.5 times your wealth in C with a short position of -.5 times
your wealth in the risk-free asset, for an expected return of 20% per year
and a standard deviation of 45%.