6.4: Equivalent Payments
Formula & Symbol Hub
For this chapter you used the following:
Symbols Used
- [latex]FV=[/latex] Future value or maturity value
- [latex]PV=[/latex] Present value or principal value
- [latex]i=[/latex] Periodic interest rate
- [latex]\text{C/Y}=[/latex] Compounds per year
- [latex]\text{I/Y}=[/latex] Nominal interest rate per year
- [latex]n=[/latex] Total number of compounding periods
Formulas Used
-
Formula 6.1 – Periodic Interest Rate
[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex]
-
Formula 6.2a – Number of Compound Periods
[latex]n=\text{C/Y}\times\text{Number of Years}[/latex]
-
Formula 6.2b – Future (Maturity) Value
[latex]FV=PV\times(1+i)^n[/latex]
-
Formula 6.3 – Present Value (Principal)
[latex]\begin{align*}PV=\frac{FV}{\left(1+i\right)^n}\end{align*}[/latex]
Equivalent Payments
This section explores the concept of equivalent payment streams. This involves equating two or more alternative financial streams to ensure that neither party is penalized by any choice. You then apply the concept of present value to loans and loan payments.
The Fundamental Concept of Equivalency
The fundamental concept of equivalency states that two or more payment streams are equal to each other if they have the same economic value on the same focal date. As illustrated in the figure, the two alternative financial streams are equivalent if the total of Payment Stream 1 is equal to the total of Payment Stream 2 on the same focal date. Note that the monies involved in each payment stream can be summed on the focal date because of the fundamental concept of time value of money.
HOW TO
Solve Questions Involving Equivalent Payments
Follow these steps to solve an equivalent payment question:
Step 1: Draw as many timelines as needed to illustrate each of the original and proposed agreements. Clearly indicate dates, payment amounts, and the interest rate(s). If you draw two or more timelines, align them vertically, ensuring that all corresponding dates are in the same columns. This allows you to see which payments need to be future valued and which need to be present valued to express them in terms of the chosen focal date.
Step 2: Choose a focal date to which all money will be moved. You should simplify your calculations by selecting a focal date corresponding to the date of an unknown variable.
Step 3: Calculate all needed periodic interest rates ([latex]i[/latex]) using Formula 6.1:
[latex]\begin{align*}i=\frac{\text{Nominal Rate (I/Y)}}{\text{Compound per Year (C/Y)}}\end{align*}[/latex]
Step 4: Calculate the total number of compounds, ([latex]n[/latex]) for each payment using Formula 6.2a:
[latex]n=(\text{C/Y}) \times (\text{Number of Years})[/latex]
Step 5: Perform the appropriate time value calculation for each payment using either the formula for [latex]FV[/latex] or [latex]PV[/latex].
Step 6: Equate the values of the original and proposed agreements on the focal date and solve for any unknowns.
Example 6.4.1
Assume you owe [latex]\$1,000[/latex] today and [latex]\$1,000[/latex] one year from now. You find yourself unable to make that payment today, so you indicate to your creditor that you want to make both payments six months from now instead. Prevailing interest rates are at [latex]6\%[/latex] compounded semi-annually. What single payment six months from now (the proposed payment stream) is equivalent to the two payments (the original payment stream)?
Solution
Step 1: The timeline illustrates the scenario.
[latex]\text{I/Y}=6\%[/latex]; [latex]\text{C/Y}=\text{semi-annually}=2[/latex]
Step 2: Apply the fundamental concept of time value of money, moving all of the money to the same date. Since the proposed payment is for six months from now, you choose a focal date of six months.
Step 3: Calculate the periodic interest rate, [latex]i[/latex].
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{6\%}{2}\\[1ex]i&=3\%\end{align*}[/latex]
Step 4: Calculate the total number of compounds, [latex]n[/latex].
[latex]\begin{align*}n&=\text{C/Y} \times (\text{Number of Years})\\[1ex]n&=2\times\frac{1}{2}\\[1ex]n&=1\end{align*}[/latex]
Step 5: Perform the appropriate time value calculations.
Moving today’s payment of [latex]\$1,000[/latex] six months into the future, you have
[latex]\begin{align*}FV&=\$1,\!000(1+0.03)^1\\FV&=\$1,\!030\end{align*}[/latex]
Moving the future payment of [latex]\$1,000[/latex] six months earlier, you have
[latex]\begin{align}PV&=\frac{\$1,\!000}{(1+0.03)^1}\\[1ex]PV&=\$970.87\end{align}[/latex]
| Payment | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|
| 1 | [latex]1[/latex] | [latex]6[/latex] | [latex]1,000[/latex] | [latex]0[/latex] | Answer: [latex]-\$1,030[/latex] | [latex]2[/latex] | [latex]2[/latex] |
| 2 | [latex]1[/latex] | [latex]6[/latex] | Answer:[latex]-\$970.87[/latex] | [latex]0[/latex] | [latex]1,000[/latex] | [latex]2[/latex] | [latex]2[/latex] |
Step 6: Now that money has been moved to the same focal date you can sum the two totals to determine the equivalent payment, which is [latex]\$1,030+\$970.87=\$2,000.87[/latex]. Note that this is financially fair to both parties. For making your [latex]\$1,000[/latex] payment six months late, the creditor is charging you [latex]\$30[/latex] of interest. Also, for making your second [latex]\$1,000[/latex] payment six months early, you are receiving a benefit of [latex]\$29.13[/latex]. This leaves both parties compensated equitably: Neither party is financially better or worse off because of the change in the deal.
Example 6.4.2
Johnson’s Garden Centre has recently been unprofitable and concludes that it cannot make two debt payments of [latex]\$4,500[/latex] due today and another [latex]\$6,300[/latex] due in three months. After discussions between Johnson’s Garden Centre and its creditor, the two parties agree that both payments could be made nine months from today, with interest at [latex]8.5\%[/latex] compounded monthly. What total payment does Johnson’s Garden Centre need nine months from now to clear its debt?
Solution
Step 1: The timeline illustrates the scenario.
[latex]PV_1\;\text{(today)}=\$4,500[/latex]; [latex]PV_2\;\text{(3 months from now)}=\$6,300[/latex]; [latex]\text{I/Y}=8.5\%[/latex]; [latex]\text{C/Y}=\text{monthly}=12[/latex]
Step 2: Due date for all payments [latex]=9[/latex] months from today. This is your focal date.
Step 3: Calculate the periodic interest rate, [latex]i[/latex].
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{8.5\%}{12}\\[1ex]i&=0.708\overline{3}\%\\i&=0.00708\overline{3}\end{align*}[/latex]
Step 4: Calculate the total number of compounds, [latex]n[/latex].
The first payment moves nine months into the future, or [latex]\frac{9}{12}[/latex] of a year.
[latex]\begin{align*}n_1&=\text{C/Y} \times (\text{Number of Years})\\[1ex]n_1&=12\times\frac{9}{12}\\[1ex]n_1&=9\end{align*}[/latex]
The second payment moves six months into the future, or [latex]\frac{6}{12}[/latex] of a year.
[latex]\begin{align*}n_2&=\text{C/Y} \times (\text{Number of Years})\\[1ex]n_2&=12\times\frac{6}{12}\\[1ex]n_2&=6\end{align*}[/latex]
Step 5: Perform the appropriate time value calculations.
Moving the first payment of [latex]\$4,500[/latex] nine months into the future, you have
[latex]\begin{align*}FV_1&=\$4,\!500(1+0.00708\overline{3})^9\\FV_1&=\$4,\!795.138902\end{align*}[/latex]
Moving the second payment of [latex]\$6,300[/latex] six months into the future, you have
[latex]\begin{align*}FV_2&=\$6,\!300(1+0.007083\overline{3})^6\\FV_2&=\$6,\!572.536425\end{align*}[/latex]
| Payment | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|
| 1 | [latex]9[/latex] | [latex]8.5[/latex] | [latex]4,500[/latex] | [latex]0[/latex] | Answer: [latex]-\$4,795.138902[/latex] | [latex]12[/latex] | [latex]12[/latex] |
| 2 | [latex]6[/latex] | [latex]8.5[/latex] | [latex]6,300[/latex] | [latex]0[/latex] | Answer: [latex]-\$6,572.526425[/latex] | [latex]12[/latex] | [latex]12[/latex] |
Step 6: Write as a statement.
With interest, the two payments total [latex]\$11,367.68[/latex]. This is the [latex]\$10,800[/latex] of the original principal plus [latex]\$567.68[/latex] in interest for making the late payments.
Example 6.4.3
You have three debts to the same creditor: [latex]\$3,000[/latex] due today, [latex]\$2,500[/latex] due in [latex]2\frac{1}{4}[/latex] years, and [latex]\$4,250[/latex] due in [latex]3[/latex] years [latex]11[/latex] months. Unable to fulfill this obligation, you arrange with your creditor to make two alternative payments: [latex]\$3,500[/latex] in nine months and a second payment due in two years. You agree upon an interest rate of [latex]9.84\%[/latex] compounded monthly. What is the amount of the second payment?
Solution
Determine the amount of the second payment that is due two years from today. Apply the fundamental concept of time value of money, moving all money from the original and proposed payment streams to a focal date positioned at the unknown payment. Once all money is moved to this focal date, apply the fundamental concept of equivalence, solving for the unknown payment, or [latex]x[/latex].
Step 1: With two payment streams and multiple amounts all on different dates, visualize two timelines. The payment amounts, interest rate, and due dates for both payment streams are known.
[latex]\text{I/Y}=9.84\%[/latex]; [latex]\text{C/Y}=\text{monthly}=12[/latex]
Step 2: Focal Date [latex]=2[/latex] years.
Step 3: Calculate the periodic interest rate, [latex]i[/latex].
[latex]\begin{align*}i&=\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}\\[1ex]i&=\frac{9.84\%}{12}\\[1ex]i&=0.82\%\;\text{or}\;0.0082\end{align*}[/latex]
Step 4: For each payment, calculate the number of compoundings per year by applying Formula 6.2a:
[latex]n=C/Y \times (\text{Number of Years})[/latex]
Step 5: For each payment, calculate the appropriate time value calculation. Note that all payments before the two year focal date require you to calculate future values, while all payments after the two-year focal date require you to calculate present values.
Steps 4 and 5: Using the circled number references from the timelines:
① [latex]n=12 \times 2 =24[/latex]; [latex]FV_1=\$3,\!000(1+0.0082)^{24}=\$3,\!649.571607[/latex]
② [latex]n=12 \times \frac{1}{4} =3[/latex]; [latex]\begin{align}PV_1=\frac{\$2,\!500}{1.0082^3}=\$2,\!439.494983\end{align}[/latex]
③ [latex]n=12 \times 1\frac{11}{12} =23[/latex]; [latex]\begin{align}PV_2=\frac{\$4,\!250}{1.0082^{23}}=\$3,\!522.207915\end{align}[/latex]
④ [latex]n=12 \times 1\frac{1}{4} =15[/latex]; [latex]FV_2=\$3,\!500(1+0.0082)^{15}=\$3,\!956.110749[/latex]
Step 6:
[latex]\begin{align}\$3,\!649.571607+\$2,\!439.494983+\$3,\!522.207915&=x+\$3,\!956.110749\\\$9,\!611.274505&=x+\$3,\!956.110749\\\$5,\!655.16&=x\end{align}[/latex]
| Calculation | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|
| ① | [latex]24[/latex] | [latex]9.84[/latex] | [latex]3,000[/latex] | [latex]0[/latex] | Answer: [latex]-\$3,649.571607[/latex] |
[latex]12[/latex] | [latex]12[/latex] |
| ② | [latex]3[/latex] | [latex]9.84[/latex] | Answer: [latex]-\$2,439.494983[/latex] |
[latex]0[/latex] | [latex]2,500[/latex] | [latex]12[/latex] | [latex]12[/latex] |
| ③ | [latex]23[/latex] | [latex]9.84[/latex] | Answer: [latex]-\$3,522.207915[/latex] |
[latex]0[/latex] | [latex]4,250[/latex] | [latex]12[/latex] | [latex]12[/latex] |
| ④ | [latex]15[/latex] | [latex]9.84[/latex] | [latex]3,500[/latex] | [latex]0[/latex] | Answer: [latex]-\$3,956.110749[/latex] |
[latex]12[/latex] | [latex]12[/latex] |
Section 6.4 Exercises
- A winning lottery ticket offers the following two options:
- A single payment of [latex]\$1,000,000[/latex] today or
- [latex]\$250,000[/latex] today followed by annual payments of [latex]\$300,000[/latex] for the next three years.
If money can earn [latex]9\%[/latex] compounded annually, which option should the winner select? How much better is that option in current dollars?
Solution
- The [latex]\$1,000,000[/latex] is already today.
- To fairly compare the payment plan, move all money to today as well.
[latex]\text{Focal Date}=\text{Today}[/latex]
Step 1: Find [latex]i[/latex].
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{9\%}{1}\\[1ex]i&=9\%\end{align*}[/latex]
Step 2: Find [latex]n[/latex] of the payments.
[latex]\begin{align*}n&=\text{C/Y}\times(\text{Number of Years})\\[1ex]\text{Payment #1:}\;n&=1\times 1=1\\[1ex]\text{Payment #2:}\;n&=1\times 2=2\\[1ex]\text{Payment #3:}\;n&=1\times 3=3\end{align*}[/latex]
Step 3: Find the present value of the payments.
[latex]\begin{align*}PV_1&=\frac{FV}{\left(1+i\right)^n}\\[1ex]PV_1&=\frac{\$300,000}{\left(1.09\right)^1}\\[1ex]PV_1&=\$275,229.3578\\[1ex]\\[1ex]PV_2&=\frac{FV}{\left(1+i\right)^n}\\[1ex]PV_2&=\frac{\$300,000}{\left(1.09\right)^2}\\[1ex]PV_2&=\$252,503.998\\[1ex]\\[1ex]PV_3&=\frac{FV}{\left(1+i\right)^n}\\[1ex]PV_3&=\frac{\$300,000}{\left(1.09\right)^3}\\[1ex]PV_3&=\$231,655.044\end{align*}[/latex]
Total Present Value
[latex]\begin{align*}\text{Today}&=\&250,000+\$275,229.3578+\$252,503.998+\$231,655.044\\\text{Today}&=\$1,009,388.40\end{align*}[/latex]
Conclusion: The payment plan is better by [latex]\$1,009,388.40−\$1,000,000=\$9,388.40[/latex].
Calculator instructions:
Payment #1
[latex]\text{N}=1[/latex]
[latex]\text{I/Y}=9[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{FV}=300,000[/latex]
[latex]\text{P/Y}=1=\text{C/Y}[/latex]
[latex]\text{PV}=\text{?}[/latex]
Payment #2
[latex]\text{N}=2[/latex]
[latex]\text{I/Y}=9[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{FV}=300,000[/latex]
[latex]\text{P/Y}=1=\text{C/Y}[/latex]
[latex]\text{PV}=\text{?}[/latex]
Payment #3
[latex]\text{N}=3[/latex]
[latex]\text{I/Y}=9[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{FV}=300,000[/latex]
[latex]\text{P/Y}=1=\text{C/Y}[/latex]
[latex]\text{PV}=\text{?}[/latex]
- James is a debt collector. One of his clients has asked him to collect an outstanding debt from one of its customers. The customer has failed to pay three amounts: [latex]\$1,600[/latex] eighteen months ago, [latex]\$2,300[/latex] nine months ago, and [latex]\$5,100[/latex] three months ago. In discussions with the customer, James finds she desires to clear up this situation and proposes a payment of [latex]\$1,000[/latex] today, [latex]\$4,000[/latex] nine months from now, and a final payment two years from now. The client normally charges [latex]16.5\%[/latex] compounded quarterly on all outstanding debts. What is the amount of the third payment?
Solution
[latex]\text{Focal Date}=2\;\text{years from today}[/latex]
Step 1: Find [latex]i[/latex].
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{16.5\%}{4}\\[1ex]i&=4.125\%\end{align*}[/latex]
Step 2: Find [latex]n[/latex] of the payments.
[latex]\begin{align*}n&=\text{C/Y}\times(\text{Number of Years})\\[1ex]\text{Original Payment #1:}\;n&=4\times 3.5=14\\[1ex]\text{Original Payment #2:}\;n&=4\times 2.75=11\\[1ex]\text{Original Payment #3:}\;n&=4\times 2.25=9\\[1ex]\text{Proposed Payment #1:}\;n&=4\times 1.25=5\\[1ex]\text{Proposed Payment #2:}\;n&=4\times 2=8\end{align*}[/latex]
Step 3: Find the future value of the payments.
[latex]\begin{align*}\text{Original Payment #1:}\;FV_1&=\frac{\$1,600}{\left(1+0.04125\right)^{14}}\\[1ex]FV_1&=\$2,817.670366\\[1ex]\\[1ex]\text{Original Payment #2:}\;FV_2&=\frac{\$2,300}{\left(1+0.04125\right)^{11}}\\[1ex]FV_2&=\$3,587.839398\\[1ex]\\[1ex]\text{Original Payment #3:}\;FV_3&=\frac{\$5,100}{\left(1+0.04125\right)^{9}}\\[1ex]FV_3&=\$7,337.790461\\[1ex]\\[1ex]\text{Proposed Payment #1:}\;FV_4&=\frac{\$4,000}{\left(1+0.04125\right)^{5}}\\[1ex]FV_4&=\$4,895.928462\\[1ex]\\[1ex]\text{Proposed Payment #2:}\;FV_5&=\frac{\$1,000}{\left(1+0.04125\right)^{8}}\\[1ex]FV_5&=\$1,381.783859\end{align*}[/latex]
[latex]\begin{align*}\text{Total Dated Debts}&=\text{Total Dated Payments}\\FV_1+FV_2+FV_3&=x+FV_4+FV_5\\\$13,743.30023&=x+\$6,277.712321\\x&=\$7,465.59\end{align*}[/latex]
The amount of the third payment is [latex]\$7,465.59[/latex].
Calculator instructions:
Original Payment #1
[latex]\text{N}=14[/latex]
[latex]\text{I/Y}=16.5[/latex]
[latex]\text{PV}=1,600[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Original Payment #2
[latex]\text{N}=11[/latex]
[latex]\text{I/Y}=16.5[/latex]
[latex]\text{PV}=2,300[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Original Payment #3
[latex]\text{N}=9[/latex]
[latex]\text{I/Y}=16.5[/latex]
[latex]\text{PV}=5,100[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Proposed Payment #1
[latex]\text{N}=8[/latex]
[latex]\text{I/Y}=16.5[/latex]
[latex]\text{PV}=1,000[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{CPT FV}=\$7,337.790461[/latex]
Proposed Payment #2
[latex]\text{N}=5[/latex]
[latex]\text{I/Y}=16.5[/latex]
[latex]\text{PV}=4,000[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{CPT FV}=\$4,895.928462[/latex]
- Four years ago, Aminata borrowed [latex]\$5,000[/latex] from Randal with interest at [latex]8\%[/latex] compounded quarterly to be repaid one year from today. Two years ago, Aminata borrowed another [latex]\$2,500[/latex] from Randal at [latex]6\%[/latex] compounded monthly to be repaid two years from today. Aminata would like to restructure the payments so that she can pay [latex]15[/latex] months from today and [latex]2\frac{1}{2}[/latex] years from today. The first payment is to be twice the size of the second payment. Randal accepts an interest rate of [latex]6.27\%[/latex] compounded monthly on the proposed agreement. Calculate the amounts of each payment assuming the focal date is [latex]15[/latex] months from today.
Solution
First, calculate the amounts owing under Aminata’s original loans.
Original Loan 1:
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{8\%}{4}\\[1ex]i&=2\%\\[1ex]\\[1ex]n&=\text{C/Y}\times\left(\text{Number of Years}\right)\\n&=4\times 5\\n&=20\\[1ex]\\[1ex]FV_1&=\frac{PV}{\left(1+i\right)^n}\\[1ex]FV_1&=\frac{\$5,000}{\left(1.02\right)^{20}}\\[1ex]FV_1&=\$7,429.74\;\left(\text{Due in 1 year from today}\right)\end{align*}[/latex]
Original Loan 2:
[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{6\%}{12}\\[1ex]i&=0.5\%\\[1ex]\\[1ex]n&=\text{C/Y}\times\left(\text{Number of Years}\right)\\n&=12\times 4\\n&=48\\[1ex]\\[1ex]FV_2&=\frac{PV}{\left(1+i\right)^n}\\[1ex]FV_2&=\frac{\$2,500}{\left(1.005\right)^{48}}\\[1ex]FV_2&=\$3,176.22\;\text{(Due in 2 years from today)}\end{align*}[/latex]
Calculator instructions:
Original Loan 1:
[latex]\text{N}=20[/latex]
[latex]\text{I/Y}=8[/latex]
[latex]\text{PV}=5,000[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=4=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Original Loan 2:
[latex]\text{N}=48[/latex]
[latex]\text{I/Y}=6[/latex]
[latex]\text{PV}=2,500[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=12=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Now calculate the equivalent payments under the proposed arrangement:
[latex]\begin{align*}1\;\text{year}&=12\;\text{months}\\2\;\text{years}&=24\;\text{months}\\2.5\;\text{years}&=30\;\text{months}\end{align*}[/latex]
[latex]\begin{align*}i&=\frac{6.27\%}{12}\\[1ex]i&=0.5225\%\end{align*}[/latex]
[latex]\begin{align*}\text{Total Dated Debts}&=\text{Total Dated Payments}\\FV_1+PV_1&=2x+PV_2\\[1ex]7,429.74\left(1.005225\right)^3+\frac{3,176.22}{\left(1.005225\right)^9}&=2x+\frac{x}{\left(1.005225\right)^{15}}\\[1ex]7,546.810744+3,030.686729&=2x+0.924806x\\\$10,577.49747&=2.924806x\\x&=\$3,616.48\left(\text{second payment}\right)\\\\2x&=2($3,616.48)\\2x&=\$7,232.96\left(\text{first payment}\right)\end{align*}[/latex]
The amount of each payment is [latex]\$7,232.96[/latex].
Calculator Instructions:
Original Payment 1:
[latex]\text{N}=3[/latex]
[latex]\text{I/Y}=6.27[/latex]
[latex]\text{PV}=7,429.74[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=12=\text{C/Y}[/latex]
[latex]\text{FV}=\text{?}[/latex]
Original Payment 2:
[latex]\text{N}=9[/latex]
[latex]\text{I/Y}=6.27[/latex]
[latex]\text{FV}=3,176.22[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=12=\text{C/Y}[/latex]
[latex]\text{PV}=\text{?}[/latex]
Proposed Payment 1:
[latex]\text{N}=15[/latex]
[latex]\text{I/Y}=6.27[/latex]
[latex]\text{FV}=1[/latex]
[latex]\text{PMT}=0[/latex]
[latex]\text{P/Y}=12=\text{C/Y}[/latex]
[latex]\text{PV}=\text{?}[/latex]
THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]n=\text{C/Y}\times\text{Number of Years}[/latex][latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex]
Attribution
“9.4: Equivalent Payments” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
