Home work 1
1. You run a construction firm. You have just won a contract to build a government office building. Building it will
require one year and require an investment of $10.23 million today and $5.00 million in one year. The
government will pay you $22.50 million upon thebuilding’s completion. Suppose the cash flows and their times
of payment are certain, and the risk-free interest rate is 11%.
a. What is the NPV of this opportunity?
Subtract the present value of the costs from the benefits to get the NPV.
The present value of the benefits one year from now is:
PV=FV/(1+rf)
PV=$22.50 million/(1+0.11)= $20.27 million
NPER
RATE
PV
Given
1
11%
Solve for PV
20.27
We can also compute this result using a spreadsheet:
PMT
FV
0
−22.5
Excel Formula
=PV(0.11,1,0,−22.5)
The present value of the costs is:
PV =PV (This year's cost)+PV (Next year's cost)
PV=$10.23 million+$5.00 million(1+0.11)=$14.73 million
Finally, subtract the present value of the costs from the present value of the benefits to get the NPV:
NPV=PV (Benefits)−PV (Costs)
NPV=$20.27 million−$14.73 million=$5.54 million
b. How can your firm turn this NPV into cash today?
The firm can borrow $20.27 million today, and pay it back with 11% interest using the $22.50 million it will receive from
the government ($20.27 million×1.11=$22.50 million). The firm can use
$10.23 million of the $20.27 million to cover its costs today and save $4.50 million in the bank to earn 11% interest to
cover
its
cost
of
$4.50 million×1.11=$5.00million
next
year.
This
leaves
$20.27 million−$10.23 million−$4.50 million=$5.54 million in cash for the firm today.
2. Suppose Bank One offers a risk-free interest rate of 6.5% on both savings and loans and Bank Enn offers a
risk-free interest rate of 7.0% on both savings and loans.
a. What arbitrage opportunity is available?
There is an arbitrage opportunity available. You should take a loan from Bank One at the lower rate of
6.5% and then deposit it into savings at Bank Enn at the higher 7.0% rate.
b. Which bank would experience a surge in demand for loans? Which bank would receive a surge
in deposits?
Many investors would recognize this arbitrage opportunity, so Bank One would experience a surge in demand for
loans and Bank Enn would receive a surge in deposits
c. What would you expect to happen to the interest rates the two banks are offering?
This surge in demand for loans would force Bank One to increase its loan rate, while the surge in deposits would
cause Bank Enn to decrease its savings rate.
3. Consider two securities that pay risk-free cash flows over the next two years and that have the current market
prices shown here:(Click on the following icon in order to copy its contents into a spreadsheet.)
Security
B1
B2
Price Today Cash Flow in One Year
$190
$200
$172
0
Cash Flow in Two Years
0
200
Solution :
a. What is the no-arbitrage price of a security that pays cash flows of $200 in one year and $200 in two years?
This security has the same cash flows as a portfolio of one share of B1 and one share of B2.Therefore, its noarbitrage price is the price of B1 plus B2. That is, the price is:
P=$190+$172=$362
b. What is the no-arbitrage price of a security that pays cash flows of $200 in one year and $1,200 in two
years?
This security has the same cash flows as a portfolio of one share of B1 and 6 shares of B2.Therefore, its no-arbitrage
price is the price of one share of B1 and 6 shares of B2:
P=$190+6×$172=$1,222
c. Suppose a security with cash flows of $100 in one year and $200 in two years is trading for a price of $260.
What arbitrage opportunity is available?
There is an arbitrage opportunity because the no-arbitrage price should be $267
($190/2+$172) which is greater than the security price of $260.
One should buy two shares of the security at $260 per share and sell one share of B1 and two shares of B2. Total
profit would be $14 ($190+$172×2−$260×2).
4. The table here shows the no-arbitrage prices of securities A and B that we calculated
Cash Flow in One Year
Security
Market Price Today
Weak Economy
Strong Economy
Security A
$238
$5
$610
Security B
$352
$610
$5
Solution:
a. What are the payoffs of a portfolio of one share of security A and one share of security B?
The payoff of a portfolio of one share of security A and one share of security B will be the sum of the cash flows in
either scenario.
Portfolio
A+B pays $615 in both cases (i.e., it is risk free).
b. What is the market price of this portfolio?
The market price of this portfolio will be:
Market price=Market price A+Market price B
Market price=$238+$352=$590
What expected return will you earn from holding this portfolio?
The expected return is:
Expected return=Portfolio payoff−Market priceMarket price
Expected return=$615−$590$590=4.24%
5. Suppose a risky security pays an expected cash flow of $84 in one year. The risk-free rate is 4.3%, and the
expected return on the market index is 9.8%
6.
7.
8.
9.
10.
11.
12.
13.
14.
b.
15.