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# Calculator blog

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## HP12c accuracy in AMORT calculations

While preparing some explanations on how to use the TVM feature of the HP12c, I found a disconcerting difference:

Loan calculation: monthly payment, 200.000€ initial amount, 20 years, complete payment, 6% annual interest, END mode. In brackets, actual keystrokes.

n => 20*12 = 240 [ 20 g n]
i =>  6/12 = 0,5% [ 6 g i ]
PV => 200000 [ 200000 PV]
FV => 0 [ 0 FV]
Calculate monthly payment: [ PMT ] => -1432,86

Loan calculation 2: with the above data, how much will it be left to pay at the end of year 10 ?

n => 10*12 = 120 [ 10 g n]
Calculate future value with same interest and payment: [ FV ] => -129062,84 of principal.

Loan calculation 3: how much have we paid of interest and principal in the first 10 years:

Follow the loan calculation 1 steps to make sure everything is in its place with the original numbers;
[ 120 f AMORT] => -101006,34 (interest paid)
[ x<>y ] => 70936,86 (capital paid)
Pressing now [ RCL PV ] shows what is left to pay: 129063,14. This number is different by 0,301 to the number above. It however adds up to 200000 with the capital paid above.

This took me a long time to understand, and I also checked with other calculators: the original 12c, HP17bII+, HP19bII...all of them reproduced this difference.

I ended up asking Gene Wright, probably the most knowledgeable in the crossroad of HP calculators and Finance math. He came back with the most likely explanation: the calculation depends on how many decimals you set. I had set 2 decimals for the HP12c, and the firmware takes that as the exact amount paid in each monthly, payment, not using the hidden accuracy of the full mantissa. This was proven by setting the number of decimals to 9: both calculations coincide.

There is an additional comment: the AMORT calculation for 120 periods was instantaneous in the latest HP12c, but took more than half a minute in the original HP12c !! I was not used to wait ! I will now continue with other HP12c features, like IRR and NPV