Blog Entry #101
The Balance of an Annuity
If you have a financial calculator, chances are that you have a Amortization function, which assists you in finding the interest and principal paid and the balance after a set of payments.
Another quick way way to find the balance is to calculate the future value (FV).
Example:
A 20-year loan for $100,000 is issued at 5% annual interest. Find the monthly payment.
Known:
PV (Present Value) = 100,000
I/YR (Annual Interest) = 5%
Periodic Interest = 5/12 %
N (number of payments) = 20 * 12 = 240
FV (Future Value) = 0
Computed (HP 10bII+):
PMT (Payment) = -659.955739217 (-659.96 rounded to 2 places)
Two Ways to Find the Balance
Amortization Function
Enter the range of payments you want to amortize and execute the amortization function/spreadsheet.
For the HP 10bII+, to amortize the first k payments, press:
1 [INPUT] k [Down shift key] [FV] (AMORT)
The HP 10bII+ will display the range of payments to be amortized. Confirm this by pressing the equals key. [ = ].
The total principal paid is given. Press [ = ] to get the total interest paid. Press [ = ] one more time to get the balance.
In short, for the balance of the first n payments:
1 [INPUT] k [Down shift key] [FV] (AMORT) [ = ] [ = ] [ = ]
Future Value Method
To find the balance after the first k payments, enter k as N and compute future value (FV).
For example, if I want to find the balance of a loan after 12 payments, I enter 12 as N, and compute FV.
This is an alternate method, and a possible work-around for anyone having a finance calculator sans an amortization feature.
Remember: The TVM functions uses the cash-flow convention: positive numbers for receipts and negative numbers for payments.
A Comparison of Methods
A recap of the data in our example:
Known:
PV (Present Value) = 100,000
I/YR (Annual Interest) = 5%
Periodic Interest = 5/12 %
N (number of payments) = 20 * 12 = 240
FV (Future Value) = 0
Computed (HP 10bII+):
PMT (Payment) = -659.955739217 (-659.96 rounded to 2 places)
*Note: Signs will be ignored.
Balance after 60 payments:
Amortization Method: 83,454.57
FV Method: 83,454.86
Balance after 120 payments:
Amortization Method: 62,220.85
FV Method: 62,221.52
Balance after 180 payments:
Amortization Method: 34,970.35
FV Method: 34,971.35
I find similar results by using the Hewlett Packard HP 10bII+, Texas Instruments BA II Plus, and the Casio FC-200V.
Why is there a difference?
The TVM module uses a master equation involving N, I/YR, P/Y, PMT, PV, and FV and the financial calculator calls on this master equation to solve for the desired variable.
When using the amortization method, the financial calculator uses a different set of formulas. The formulas have this format:
Interest for the Period = Previous Balance * I/YR% * P/Y^-1
Principal for the Period = Payment - Interest for the Period
New Balance = Previous Balance - Principal for the Period
Prior to calculation, payment and interest are rounded.
For the exact formulas, consult the calculator financial manuals. Casio, Texas Instruments, and Hewlett Packard all provide formulas for TVM and amortization.
Link to the HP 10BII+ Manual: Check the Appendix
Thank you and take care!
Eddie
2012