Basic Concepts of Finance: Discounting Payments made in Arrears vs. in Advance
In my previous post, we examined a payment schedule in which $100 was paid on January 1st for ten years. We introduced the concept of discounting and determined that setting aside $916.22 and investingat a 2% annual interest rate would be sufficient to fund ten payments.
This time, we will shift the payment to the end of the year, to December 31st.
Our original schedule will be updated as below.
| INSTALLMENT | DATE OF PAYMENT | ANNUAL INSTALLMENT | CUMULATIVE AMOUNT PAID |
|---|---|---|---|
| Column (1) | (2) | (3) | (4) |
| 1 | 12/31/2025 | $ 100 | $ 100 |
| 2 | 12/31/2026 | $ 100 | $ 200 |
| 3 | 12/31/2027 | $ 100 | $ 300 |
| 4 | 12/31/2028 | $ 100 | $ 400 |
| 5 | 12/31/2029 | $ 100 | $ 500 |
| 6 | 12/31/2030 | $ 100 | $ 600 |
| 7 | 12/31/2031 | $ 100 | $ 700 |
| 8 | 12/31/2032 | $ 100 | $ 800 |
| 9 | 12/31/2033 | $ 100 | $ 900 |
| 10 | 12/31/2034 | $ 100 | $ 1 000 |
The twelve-months provides additional time to earn extra interest. Investing $100 at an annual rate of 2% would result in the following returns.
| Year | $100 invested on 1/1/2025 at an annual rate of 2% |
|---|---|
| (1) | (2) |
| 1 | $ 102.00 |
| 2 | $ 104.04 |
| 3 | $ 106.12 |
| 4 | $ 108.24 |
| 5 | $ 110.41 |
| 6 | $ 112.62 |
| 7 | $ 114.87 |
| 8 | $ 117.17 |
| 9 | $ 119.51 |
| 10 | $ 121.90 |
In our example however, we are due to make a payment of only $100 each year. We don’t need the excess cash.
We need to restructure the schedule above and determine how much we must invest at a 2% annual rate, on January 1st, 2025, so that every year on December 31st, $100 can be withdrawn and paid to the client.
- Year one: We set aside an $x amount on January 1st, 2025, which will grow to $100 by December 31st. Since $x earns interest, the equation is:
$x*(1+2%)=$100 and solving for $x gives $98.
- Year two: With two years to earn interest, the equation is:
$x*(1+2%)*(1+2%)=$100. Solving for $x gives around $96.
- Year three: With three years to earn interest, the equation is:
$x*(1+2%)(1+2%)(1+2%)=$100. Solving for $x gives roughly $94.
The amounts for subsequent years can be calculated similarly. Our updated payment schedule is shown below.
| INSTALLMENT | DATE OF PAYMENT | ANNUAL INSTALLMENT | CUMULATIVE AMOUNT PAID | AMOUNT INVESTED ON 1/1/2025 |
|---|---|---|---|---|
| (1) | (2) | (3) | (4) | (5) |
| 1 | 12/31/2025 | $ 100 | $ 100 | $ 98.04 |
| 2 | 12/31/2026 | $ 100 | $ 200 | $ 96.12 |
| 3 | 12/31/2027 | $ 100 | $ 300 | $ 94.23 |
| 4 | 12/31/2028 | $ 100 | $ 400 | $ 92.23 |
| 5 | 12/31/2029 | $ 100 | $ 500 | $ 90.75 |
| 6 | 12/31/2030 | $ 100 | $ 600 | $ 88.80 |
| 7 | 12/31/2031 | $ 100 | $ 700 | $ 87.06 |
| 8 | 12/31/2032 | $ 100 | $ 800 | $ 85.35 |
| 9 | 12/31/2033 | $ 100 | $ 900 | $ 83.68 |
| 10 | 12/31/2034 | $ 100 | $ 1 000 | $ 82.03 |
| Grand Total | $898.26 |
In total, the present value is $898.26. When the payments were made at the start of each year, the total was $916.22 - a difference of $17.96.
| INSTALLMENT | Amount invested on 1/1/2025 with payments made on 12/31 each year | Amount invested on 1/1/2025 with payments made on 1/1 each year |
|---|---|---|
| (1) | (2) | (3) |
| 1 | $ 98.04 | $ 100.00 |
| 2 | $ 96.12 | $ 98.04 |
| 3 | $ 94.23 | $ 96.12 |
| 4 | $ 92.23 | $ 94.23 |
| 5 | $ 90.75 | $ 92.23 |
| 6 | $ 88.80 | $ 90.75 |
| 7 | $ 87.06 | $ 88.80 |
| 8 | $ 85.35 | $ 87.06 |
| 9 | $ 83.68 | $ 85.35 |
| 10 | $ 82.03 | $ 83.68 |
| Grand Total | $898.26 | $ 916.22 |
The final payment schedule, showing annual payments and including all earned interest.
| Installments | Date of payment | Amount of money in the pot before drawdown | Amount of money in the pot after drawdown | Payment |
|---|---|---|---|---|
| (1) | (2) | (3) | (4) | (5) |
| 1 | 12/31/2025 | $ 898.26 | $ 816.22 | $ 100 |
| 2 | 12/31/2026 | $ 816.22 | $ 732.55 | $ 100 |
| 3 | 12/31/2027 | $ 732.55 | $ 647.20 | $ 100 |
| 4 | 12/31/2028 | $ 647.20 | $ 560.14 | $ 100 |
| 5 | 12/31/2029 | $ 560.14 | $ 471.35 | $ 100 |
| 6 | 12/31/2030 | $ 471.35 | $ 380.77 | $ 100 |
| 7 | 12/31/2031 | $ 380.77 | $ 288.39 | $ 100 |
| 8 | 12/31/2032 | $ 288.39 | $ 194.16 | $ 100 |
| 9 | 12/31/2033 | $ 194.16 | $ 98.04 | $ 100 |
| 10 | 12/31/2034 | $ 98.04 | $ 0.00 | $ 100 |
What happens in the schedule above is the following:
- we start with a present value of $898.26 on January 1st, 2025.
- this amount earns an annual interest of 2% and on December 31st, 2025 increases to $816.22.
- we withdraw $100 to make the first payment, which leaves us with $816.22.
- $816.22 earns interest and increases to $732.55 on December 31st, 2026.
- the schedule continues in similar fashion until the final payment is made on December 31st, 2034.
A monetary difference between payments made on December 31st and January 1st not very large when dealing with small amounts. However, when amounts increase to thousands - or even hunderds of thousands of dollars - twelve months can make a significan impact.
Until next time !
M | K
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