End-of-Chapter Comprehensive/Spreadsheet Problem
a. Find the FV of $1,000 invested to earn 10% after 5 years. Answer this question by using a math formula and also by using the Excel function wizard.
Inputs: PV = $1,000
I = 10%
N = 5
Formula: FV = PV(1+I)^N =
Wizard (FV):
Note: When you use the wizard and fill in the menu items, the result is the formula you see on the formula line if you put the pointer on cell E13. Put the pointer on E13 and then click the function wizard (fx) to see the completed menu. Finally, it is generally easiest to fill in the wizard menus by clicking on one of the menu slots to activate the cursor in that slot and then clicking on the input cell where the item is given. Then, hit the tab key to move down to the next menu slot.
Experiment by changing the input values to see how quickly the output values change.
b. Now create a table that shows the FV at 0%, 5%, and 20% for 0, 1, 2, 3, 4, and 5 years. Then create a graph with years on the horizontal axis and FV on the vertical axis to display your results.
Begin by typing in the row and column labels as shown below. We could fill in the table by inserting formulas in all the cells, but a better way is to use an Excel data table as described in 05model. We used the data table procedure. Note that the Row Input Cell is D10 and the Column Input Cell is D11, and we set Cell B33 equal to Cell E12. Then, we selected (highlighted) the range B33:E39, then select Data tab > What-If Analysis > Data Table, and filled in the menu items to complete the table.
Years (D11) Interest Rate (D10)
$- 0% 5% 20%
0
1
2
3
4
5
To create the graph, first select the range C34:E39. Then click Insert tab > Scatter. Then follow the menu. It is easy to make a chart, but a lot of detailed steps are involved to format it so that it's "pretty." Pretty charts are generally not necessary to get the picture, though. We put the chart right on the spreadsheet so we could see how changes in the data lead to changes in the graph.
Note that the inputs to the data table, hence to the graph, are now in the row and column heads. Change the 20% in Cell E33 to 0.3 (or 30%), then to 0.4, then to 0.5, etc., to see how the table and the chart change.
c. Find the PV of $1,000 due in 5 years if the discount rate is 10%. Again, work the problem with a formula and also by using the function wizard.
Inputs: FV = $1,000
I = 10%
N = 5
Formula: PV = FV/(1+I)^N =
Wizard (PV):
Note: In the wizard's menu, use zero for Pmt because there are no periodic payments. Also, set the FV with a negative sign so that the PV will appear as a positive number.
d. A security has a cost of $1,000 and will return $2,000 after 5 years. What rate of return does the security provide?
Inputs: PV = -$1,000
FV = $2,000
I = ?
N = 5
Wizard (Rate):
Note: Use zero for Pmt since there are no periodic payments. Note that the PV is given a negative sign because it is an outflow (cost to buy the security). Also, note that you must scroll down the menu to complete the inputs.
e. Suppose California's population is 36.5 million people, and its population is expected to grow by 2% annually. How long will it take for the population to double?
Inputs: PV = -36.5
FV = 73
I = growth rate 2%
N = ?
Wizard (NPER): = Years to double.
f. Find the PV of an ordinary annuity that pays $1,000 each of the next 5 years if the interest rate is 15%. Then find the FV of that same annuity.
Inputs: PMT $(1,000)
N 5
I 15%
PV: Use function wizard (PV) PV =
FV: Use function wizard (FV) FV =
g. How will the PV and FV of the annuity change if it is an annuity due rather than an ordinary annuity?
For the PV, each payment would be received one period sooner, hence would be discounted back one less year. This would make the PV larger. We can find the PV of the annuity due by finding the PV of an ordinary annuity and then multiplying it by (1 + I).
PV annuity due = × 1.15 =
Exactly the same adjustment is made to find the FV of the annuity due.
FV annuity due = × 1.15 =
h. What will the FV and the PV for problems a and c be if the interest rate is 10% with semiannual compounding rather than 10% with annual compounding?
Part a. FV with semiannual compounding: Orig. Inputs: New Inputs:
Inputs: PV = $1,000 $1,000
I = 10% 5%
N = 5 10
Formula: FV = PV(1+I)^N =
Wizard (FV):
Part c. PV with semiannual compounding: Orig. Inputs: New Inputs:
Inputs: FV = $1,000 $1,000
I = 10% 5%
N = 5 10
Formula: PV = FV/(1+I)^N =
Wizard (PV):
i. Find the annual payments for an ordinary annuity and an annuity due for 10 years with a PV of $1,000 and an interest rate of 8%.
Inputs: N 10
I 8%
PV -$1,000
PMT: Use function wizard (PMT) PMT =
PMT (Due): Use function wizard (PMT) PMT =
j. Find the PV and the FV of an investment that makes the following end-of-year payments. The interest rate is 8%.
Year Payment
1 100
2 200
3 400
Rate = 8%
To find the PV, use the NPV function: PV =
Excel does not have a function for the sum of the future values for a set of uneven payments. Therefore, we must find this FV by some other method. Probably the easiest procedure is to simply compound each payment, then sum them, as is done below. Note that since the payments are received at the end of each year, the first payment is compounded for 2 years, the second for 1 year, and the third for 0 years.
Year Payment x (1 + I )^(N - t) = FV
1 100
2 200
3 400
Sum of FV's =
"An alternative procedure for finding the FV would be to find the PV of the series using the NPV function, then compound that amount for 3 years at 8%, as is done below:"
PV =
FV of PV =
k. 5 banks offer nominal rates of 6%, but differ in their compounding frequency.
A = annually; B = semiannually; C = quarterly; D = monthly; and E = daily.
I NOM 6%
Deposit $5,000
(1) A B C D E
(i) EAR
(ii) Deposit $5,000. What is FV1?
(iii) Deposit $5,000. What is FV2?
(2) Would they be equally able to attract funds? No. People would prefer more compounding to less.
(i) What nominal rate would cause all banks to provide same EAR as Bank A?
A B C D E
I NOM
Each of these nominal rates based on the frequency of compounding will provide an EAR of 6%.
(3) You need $5,000 at the end of the year. How much do you need to deposit annually for A, semiannually, for B, etc. beginning today, to have $5,000 at the end of the year?
A B C D E
PMT
(4) Even if the banks provided the same EAR, would a rational investor be indifferent between the banks? Probably not. An investor would probably prefer the bank that compounded more frequently.
l. Suppose you borrow $15,000. The interest rate is 8%, and it requires 4 equal end-of-year payments. Set up an amortization schedule that shows the annual payments, interest payments, principal repayments, and beginning and ending loan balances.
Original amount of mortgage: $15,000
Term to maturity: 4
Interest rate: 8%
Annual payment (use PMT function):
Beginning Ending
Year Balance Payment Interest Principal Balance
1
2
3
4
Extension: (i) Create a graph that shows how the payments are divided between interest and principal repayment over time.