1.
Consider the following two mutually exclusive projects:
Year
0
1
2
3
Cash Flow (X)
$
(24,000)
$
10,620
$
10,900
$
10,500
Cash Flow (Y)
$ (24,000)
$
12,100
$
9,360
$
10,400
The required rate of return is 10%.
a.
Explain what it means when we say that these two projects are mutually exclusive. (1 points)
Mutually exclusive investment decisions are situations in which taking one investment prevents
the taking of another. For example only one building can be constructed on a given plot of land.
b.
Explain why these cash flows are conventional. What is the problem with nonconventional cash
flows? (2 points)
Conventional cash flows are when we start out with an outflow and future projects have positive
cash flows. There is only one time when the cash flows switch signs from negative to positive.
This means that the project has only one unique IRR.
Nonconventional cash flows are when we have multiple changes in the signs of successive cash
flows. The number of signs changes gives the number of possible IRR’s that can be associated
with a project that has non-conventional cash flows.
c.
If you apply the payback criterion, which project will you choose? Please show your calculations.
(3 points)
Year
Cash Flow (X)
0
($24,000)
1 $
10,620
2 $
10,900
3 $
10,500
Cash Flow (Y)
($24,000)
$
12,100
$
9,369
$
10,400
Cumulative Cash Flows
Cash Flow (X) Cash Flow
($24,000) ($24,000)
($13,380) ($11,900)
($2,480) ($2,531)
$8,020
$7,869
2.236
2.243
The first step is to calculate the cumulative sums that are given in columns 4 and 5. After the end
of the third year, neither project has recovered the initial investment. This means that the
payback occurs in the third year for both projects. Both projects payback at approximately 2.24
years. The 0.24 is calculated by dividing the remaining amount to be paid back by the size of the
cash flow in the third year.
d.
Explain a major concern you might have if you used the payback criterion. (1 points)
The major concern about the payback rule is that it doesn’t take into account the time value of
money because it ignores the concept of present value.
e.
Calculate the NPV for each project. If you apply the NPV criterion, which investment will you
choose? Why? (3 points)
Required Rate of Return
Year
Cash Flow (X)
0
($24,000)
1 $
10,620
2 $
10,900
3 $
10,500
NPV
$2,551.62
10%
Cash Flow (Y)
($24,000)
$
12,100
$
9,369
$
10,400
Present Value
Cash Flow (X) Cash Flow (Y)
($24,000)
($24,000)
$9,655
$11,000
$9,008
$7,743
$7,889
$7,814
$2,556.65 $
2,551.62 $ 2,556.65
The present values are calculated by using the Present Value Interest Factor (PVIF) or the PV
Excel function which implements the formula
PV =
FV
(1 + r )
t
Project Y has a slightly larger present value so that is the one that we should choose. This is
easier in this case because both projects require the same initial cash outflow of $24,000. The
largest present value increases wealth by the most.
f.
Explain how you could calculate and use NPV profiles to determine the IRR’s of these projects. (3
points)
The NPV profile for a project puts the discount rate on the x-axis and the NPV that corresponds
to that discount rate on the y-axis. For conventional cash flows, this means that the NPV for the
projects starts positive and then becomes negative as the discount rate increases. The IRR is the
point on the graph where the NPV goes to zero or crosses the x-axis.
g.
Explain how you would use the NPV profiles to determine the crossover rate. (2 Points)
The crossover rate is where the NPV’s of the two projects are equal to each other. One project
dominates the other by the NPV rule until the crossover discount rate. After the cross over rate,
the preference between the two projects switches. At the crossover rate, the NPV’s of the two
projects are identical and the decision maker is indifferent between the two.
h.
Calculate the profitability index and then explain which investment you should choose? In this
case, why is your answer almost identical to just comparing NPV’s? (3 points)
Required Rate of Return
Year
Cash Flow (X)
0
($24,000)
1 $
10,620
2 $
10,900
3 $
10,500
NPV
$2,551.62
Present
Value of
Future
Profitability
Index
$26,551.62
1.106
10%
Cash Flow (Y)
($24,000)
$
12,100
$
9,369
$
10,400
Present Value
Cash Flow (X) Cash Flow (Y)
($24,000)
($24,000)
$9,655
$11,000
$9,008
$7,743
$7,889
$7,814
$2,556.65 $
2,551.62 $ 2,556.65
$26,556.65
1.107
The profitability index is the ratio of the present value of the future cash flows divided by the
initial cash outlay. The answer is identical to the first present value calculation because both
projects share the same initial cash outflow.
2.
You are buying a home for $200,000 and have $40,000 as a down payment so that you are able to avoid
private mortgage insurance (PMI). Your mortgage banker makes you a fully amortized 15 year loan with
4% APR.
a.
Draw the cash flows and explain how you would calculate your payments. Your answer should specify
all of the arguments that you would substitute into the appropriate Excel functions. (4 points)
b.
If your mortgage banker requires a balloon payment at the end of 3 years, diagram the cash flows
and explain how you would calculate the size of the balloon. Your answer should specify all of the
arguments that you would substitute into the appropriate Excel functions. (4 points)
c.
If your mortgage broker quotes you an interest rate of 3.5% with 1 point, explain how you would
calculate the EAR for this loan. Your answer should include cash flow diagrams and an
explanation of the formula that the Excel function uses to calculate the EAR. (4 points)
First we need to calculate the payment using the quoted rate or 3.5% annually or 3.5% / 12.
Next we calculate how much we will actually receive from the bank. In this problem 1 point costs
of 1% of $160,000 so we actually only receive $158,400 from the bank. We now calculate the
APR implied by the cash flows shown below:
The APR must be converted into an EAR by taking into consideration the compounding using the
following formula that is implemented in the EFFECT function:
12
APR 

EAR =
1 +
 −1
12 

12
 3.65 
= 1 +
 − 1 = 3.71%
12 

3.
The 2/15/2026 US Treasury Bond has a 6 percent coupon rate. The asked price is $138.61 and has a YTM
of 1.76%. A similar TIPS which matures on 1/15/2026 has a yield to maturity of 0.51% and price of
$114.12.
a. Draw a cash flow diagram and use equations to explain the relationship between the YTM and
the price of the US Treasury Bond. (4 points)
The price of the bond is the present value of the future cash flows discounted at the YTM divided by
two. The YTM is that discount rate that causes the price to be equal to the present value of the
future cash flows. The price and YTM divided by two are determined simultaneously so that
Price
=
20
∑
i =1
$138.61
=
(
1 + ytm
20
∑
i =1
b.
Coupon
2
+
) (
i
FaceValue
1 + ytm
30
+
2
)
20
100
(1 + 0.88% ) (1 + 0.88% )
i
20
Explain why you would be willing to pay a premium for the Treasury Bond? (2 points)
With a YTM of 1.76%, the coupon payments per year would be $17.60 for a bond trading at par.
If a bond were issued today with a coupon rate of 6%, then it would pay $60 per year. Therefore,
the old bond with the coupon rate of 6% is very attractive to everyone and the price would be
bid up until it also has a YTM of 1.76%.
c.
Explain how TIPS protect bond holders from inflation. (2 Points)
Every month the face value of TIPS are adjusted for inflation. This means that the buying power
of the bonds stays constant. Also the coupon payment is recalculated as the new face value
multiplied by the original coupon rate. Therefore, the coupon payments are also protected
against inflation. For this reason, the YTM on TIPS are real interest rates.
d.
What is the expected rate of inflation implied by these two US Treasury securities? Explain your
calculations. (2 Points)
The Fischer Equation states that the nominal interest rate is the sum of the real interest rate and
expected inflation. This means that
Expected Inflation = Nominal Rate - Real Rate
= 1.76 − 0.51 = 1.25
4.
The current value of the core CPI (January 2016) is 245. Two years ago (January 2014) the core CPI was
236.
a. Explain the general process used by the Bureau of Labor Statistics or other groups to construct
price indexes. (2 points)
The BLS specifies a basket of goods and then purchases the same basket of good over and over
during each time period (monthly). The price index is a transformed or scaled value of the cost
of this basket of goods as it changes over time with reference to a base year.
b.
Explain how the core CPI differs from the CPI for all Urban Consumers: All Items. Why is it
important to use the core CPI rather than the CPI: All Items? (2 points)
Because food and energy prices are so volatile, they are excluded from the basket used to
calculate the price index. The core CPI is a much better measure of inflation because it doesn’t
increase and decrease radically when the price of oil or price of food change.
c.
Calculate and explain how much the aggregate price level increased (inflation rate) during the
past two years. (2 points)
The inflation rate over this two year period is:
245 − 236
= 3.8%
236
This gives an approximate annual interest rate of 1.9% because we divide the above by 2. We
could figure out the compounded rate if we wanted to be exact.
d.
If price of gasoline was $2.00 in January 2014, what is the equivalent price in 2016 dollars?
Explain and show your calculations. (2 points)
We convert the January 2014 price into 2016 dollars by multiplying by the ratio of the CPI in 2016
to the CPI in 2014.
$2.00 ×
245
≈ $2.08
236