Interest Calculation and Compounding
- keithwoodlock (Unlicensed)
Introduction
This page runs through interest calculation and compounding examples for Passbook style savings account.
There are two methods to calculate the interest on passbook savings:
- The daily balance method; and
- The average daily balance method.
The examples below use end-of-day-balances and refer to the interest posting & compounding periods used.
Terms
Compounding Period: is the span of time which at the end of the interest earned over this period on the savings account balance in accumulated with the balance of the account is applicable for interest calculations in future periods. As a result the interest is compounded. The following compounding period frequencies are supported:
- Daily: compounding occurs on the balance each day and is accumulated with the balance so that the next days interest calculation takes it into account.
- Monthly: compounding occurs on the interest earned over the entire month. Next months interest calculation is on the savings account blance + any interest earned in previous compounding period.
Posting Period: is the span of time which at the end of the interest earned over this period on savings account is credited or posted to the clients account. A posting period may include many compounding periods. The following posting period frequencies are supported:
- Monthly: The period mirrors the calendar month so periods are 01 Jan - 31 Jan, 01 Feb - 28 Feb, 01 Mar - 31 Mar etc
- Quarterly: The period mirrors the calendar quarter so periods are 01 Jan - 31 Mar, 01 Apr - 30 Jun etc
- Annually: Ther period mirrors the calendar year so perdios are 01 Jan - 31 Dec for each year.
Nominal annual interest rate %: Also known as nominal APR, its nominal as this rate doesnt reflect influences such as inflation or compounding. The percentage is represented as a number e.g. 20 for 20% NOT as a fraction 0.2 = 20%
Interest Calculation Method: We support Daily Balance and Average Daily Balance
Days in Year: Some people want their interest calculations to be done over a 360 day year instead of 365/366 day year.
Account Activity
The client makes the following transactions throughout the month on an account opened on 01 March 2013:
| Date | Transaction | Amount |
|---|---|---|
| 01 Mar 2013 | Deposit | 1200 |
| 02 Mar 2013 | Withdrawal | 100 |
| 10 Mar 2013 | Withdrawal | 400 |
| 15 Mar 2013 | Deposit | 200 |
| 16 Mar 2013 | Withdrawal | 900 |
| 18 Mar 2012 | Deposit | 200 |
| 21 Mar 2012 | Deposit | 700 |
| 31st Mar 2013 | Withdrawal | 100 |
Daily Balance Method
Compound Interest Forumula using in Daily Compounding:
The future value of a savings account balance on a given end of day is calculated using the following:
FV = PV x (1 + r)^n where,
PV = The end of day balance e.g. 1200
r = periodic interest rate so in this case daily interest rate which is worked out by dividing 1 by 365 (number of days in year) and mulitplying by 0.05 (5%)
n = number of compounding periods the balance/PV will compound - in the same period as the periodic interest rate - so in this case Days. So if a balance exists for 5 days we use n=5 to calculate what the balance produces in interest when compounding.
Summary
The frequency of the posting period makes no difference (or neglible difference) to the interest earned on the savings account when using compound interest. See the break down of the account activity with monthly, quarterly and yearly posting frequencies.
Daily Compounding - Monthly Posting Using Daily Balance Method
The account activity breaks down into the following End of Day Balances:
Monthly Posting so we want to post interest to the account for periods (01 Mar - 31 Mar), (01 Apr - 30 Apr), (01 May - 31 May), (01 Jun - 30 Jun)
| Date | Up to & (inclusive) | Days | Type | Amount | Opening Bal | Closing Bal | Interest on Bal | Interest To Date | Bal for Compounding | (I) Posted | Gain/Loss Due To Rounding |
|---|---|---|---|---|---|---|---|---|---|---|---|
01/03/2013 | 01/03/2013 | 1 | Deposit | 1200 | 0 | 1200 | 0.164383562 | 0.164383562 | 1200.164383562 | ||
| 02/03/2013 | 09/03/2013 | 8 | Withdrawal | 100 | 1200 | 1100 | 1.206237813 | 1.370621375 | 1,101.370621375 | ||
| 10/03/2013 | 14/03/2013 | 5 | Withdrawal | 400 | 1100 | 700 | 0.480522469 | 1.851143844 | 701.851143844 | ||
| 15/03/2013 | 15/03/2013 | 1 | Deposit | 200 | 700 | 900 | 0.123541253 | 1.974685097 | 901.974685097 | ||
| 16/03/2013 | 17/03/2013 | 2 | Withdrawal | 900 | 900 | 0 | 0.000541047 | 1.975226144 | 1.975226144 | ||
| 18/03/2013 | 20/03/2013 | 3 | Deposit | 200 | 0 | 200 | 0.083014888 | 2.058241032 | 202.058241032 | ||
| 21/03/2013 | 30/03/2013 | 10 | Deposit | 700 | 200 | 900 | 1.236458229 | 3.294699261 | 903.294699261 | ||
| 31/03/2013 | 31/03/2013 | 1 | Withdrawal | 100 | 900 | 800 | 0.110040370 | 3.404739631 | 803.404739631 | 3.40 | -0.004739631 |
| 01/04/13 | 30/04/2013 | 30 | Interest Posting | 3.40 | 800 | 803.40 | 3.308210288 | 3.308210288 | 806.708210288 | 3.31 | +0.001789712 |
| 01/05/13 | 31/05/2013 | 31 | Interest Posting | 3.31 | 803.40 | 806.71 | 3.432803347 | 3.432803347 | 810.142803347 | 3.43 | -0.002803347 |
| 01/06/13 | 30/06/2013 | 30 | Interest Posting | 3.43 | 806.71 | 810.14 | 3.335964006 | 3.335964006 | 813.475964006 | 3.34 | +0.004035994 |
| 122 | 813.48 | 13.48 | Loss of 0.003506984 |
Daily Compounding - Quarterly Posting Using Daily Balance Method
The account activity breaks down into the following End of Day Balances:
Quarterly Posting so we want to post interest to the account for periods (01 Jan - 31 Mar), (01 Apr - 30 Jun)
| Date | Up to & (inclusive) | Days | Type | Amount | Opening Bal | Closing Bal | Interest on Bal | Interest To Date | Bal for Compounding | (I) Posted | Gain/Loss Due To Rounding |
|---|---|---|---|---|---|---|---|---|---|---|---|
01/03/2013 | 01/03/2013 | 1 | Deposit | 1200 | 0 | 1200 | 0.164383562 | 0.164383562 | 1200.164383562 | ||
| 02/03/2013 | 09/03/2013 | 8 | Withdrawal | 100 | 1200 | 1100 | 1.206237813 | 1.370621375 | 1,101.370621375 | ||
| 10/03/2013 | 14/03/2013 | 5 | Withdrawal | 400 | 1100 | 700 | 0.480522469 | 1.851143844 | 701.851143844 | ||
| 15/03/2013 | 15/03/2013 | 1 | Deposit | 200 | 700 | 900 | 0.123541253 | 1.974685097 | 901.974685097 | ||
| 16/03/2013 | 17/03/2013 | 2 | Withdrawal | 900 | 900 | 0 | 0.000541047 | 1.975226144 | 1.975226144 | ||
| 18/03/2013 | 20/03/2013 | 3 | Deposit | 200 | 0 | 200 | 0.083014888 | 2.058241032 | 202.058241032 | ||
| 21/03/2013 | 30/03/2013 | 10 | Deposit | 700 | 200 | 900 | 1.236458229 | 3.294699261 | 903.294699261 | ||
| 31/03/2013 | 31/03/2013 | 1 | Withdrawal | 100 | 900 | 800 | 0.110040370 | 3.404739631 | 803.404739631 | 3.40 | -0.004739631 |
| 01/04/13 | 30/06/2013 | 91 | Interest Posting | 3.40 | 800 | 803.40 | 10.08 | ||||
| 122 | 813.48 | 13.48 |
Daily Compounding - Annual Posting Using Daily Balance Method
The account activity breaks down into the following End of Day Balances:
Annual Posting so we want to post interest to the account for periods (01 Jan - 31 Dec)
| Date | Up to & (inclusive) | Days | Type | Amount | Opening Bal | Closing Bal | Interest on Bal | Interest To Date | Bal for Compounding | (I) Posted |
|---|---|---|---|---|---|---|---|---|---|---|
01/03/2013 | 01/03/2013 | 1 | Deposit | 1200 | 0 | 1200 | 0.164383562 | 0.164383562 | 1200.164383562 | |
| 02/03/2013 | 09/03/2013 | 8 | Withdrawal | 100 | 1200 | 1100 | 1.206237813 | 1.370621375 | 1,101.370621375 | |
| 10/03/2013 | 14/03/2013 | 5 | Withdrawal | 400 | 1100 | 700 | 0.480522469 | 1.851143844 | 701.851143844 | |
| 15/03/2013 | 15/03/2013 | 1 | Deposit | 200 | 700 | 900 | 0.123541253 | 1.974685097 | 901.974685097 | |
| 16/03/2013 | 17/03/2013 | 2 | Withdrawal | 900 | 900 | 0 | 0.000541047 | 1.975226144 | 1.975226144 | |
| 18/03/2013 | 20/03/2013 | 3 | Deposit | 200 | 0 | 200 | 0.083014888 | 2.058241032 | 202.058241032 | |
| 21/03/2013 | 30/03/2013 | 10 | Deposit | 700 | 200 | 900 | 1.236458229 | 3.294699261 | 903.294699261 | |
| 31/03/2013 | 31/03/2013 | 1 | Withdrawal | 100 | 900 | 800 | 0.110040370 | 3.404739631 | 803.404739631 | |
| 01/04/13 | 30/06/2013 | 91 | No activity | 3.40 | 800 | 800 | 10.077033616 | 13.481773247 | 813.481773247 | Interest earned but not yet posted |
Compound Interest Forumula used in Monthly Compounding:
The interest earned on a savings account balance on a given end of day is calculated using the following:
A= Prn where,
P = The end of day balance e.g. 1200
r = periodic interest rate so in this case daily interest rate which is worked out by dividing 1 by 365 (number of days in year) and mulitplying by 0.05 (5%)
n = number of compounding periods the balance/PV will compound - in the same period as the periodic interest rate - so in this case Days. So if a balance exists for 5 days we use n=5 to calculate what the balance produces in interest when compounding.
Compounding Periods Matter
The frequency of the posting period makes no difference (or neglible difference) to the interest earned on the savings account when using compound interest. But the compounding periods do. Not the slight different in interest earned for monthly compounding versus daily.
Monthly Compounding - Monthly Posting using Daily Balance Method
The account activity breaks down into the following End of Day Balances:
Monthly Posting so we want to post interest to the account for periods (01 Mar - 31 Mar), (01 Apr - 30 Apr), (01 May - 31 May), (01 Jun - 30 Jun)
| Date | Up to & (inclusive) | Days | Type | Amount | Opening Bal | Closing Bal | Interest on Bal | Interest To Date | Balance For Compounding |
|---|---|---|---|---|---|---|---|---|---|
01/03/2013 | 01/03/2013 | 1 | Deposit | 1200 | 0 | 1200 | 0.164383562 | 0.164383562 | |
| 02/03/2013 | 09/03/2013 | 8 | Withdrawal | 100 | 1200 | 1100 | 1.206237813 | 1.370621375 | |
| 10/03/2013 | 14/03/2013 | 5 | Withdrawal | 400 | 1100 | 700 | 0.480522469 | 1.851143844 | |
| 15/03/2013 | 15/03/2013 | 1 | Deposit | 200 | 700 | 900 | 0.123541253 | 1.974685097 | |
| 16/03/2013 | 17/03/2013 | 2 | Withdrawal | 900 | 900 | 0 | 0.000541047 | 1.975226144 | |
| 18/03/2013 | 20/03/2013 | 3 | Deposit | 200 | 0 | 200 | 0.083014888 | 2.058241032 | |
| 21/03/2013 | 30/03/2013 | 10 | Deposit | 700 | 200 | 900 | 1.236458229 | 3.294699261 | |
| 31/03/2013 | 31/03/2013 | 1 | Withdrawal | 100 | 900 | 800 | 0.110040370 | 3.404739631 | 803.404739631 |
| End of posting period | 31 | 3.40 to be posted | |||||||
| 01/04/13 | 30/04/2013 | 30 | Interest Posting | 3.40 | 800 | 803.40 | 3.308210288 | 3.308210288 | 806.797260274 |
| 01/05/13 | 31/05/2013 | 31 | none | 0 | 803.40 | 803.40 | 3.432803347 | 6.727356165 | 810.001643836 |
| 01/06/13 | 30/06/2013 | 30 | none | 0 | 803.40 | 803.40 | 3.335964006 | 10.056657535 | 813.555712329 |
| End of posting period | 91 | 10.06 to be posted | |||||||
| Total | 122 | 13.46 posted | |||||||
Average Daily Balance
Average Balance Steps
The accumulated end-of-day balance of $24,800 is divided by the total number of days in the period (31) to find the average daily balance of $800.
The average daily balance should be rounded to five or more decimals, in this case $800.00000.
The periodic interest rate would be 0.00424657534 (0.5 x (1/365) x 31)
Step 1 5.0 divided by 100 Result: 0.05
Step 2 0.50 times 1/365 Result: 0.000136986
Step 3 0.000136986 times 31 Result: 0.004246575
Step 4 0.004246575 times $800 Result: 3.397260274
Step 5 Rounded Result: $3.40