# What is the formula for loan payoff with daily compounded interest and annual payment?

I am trying to get the loan repayment amount for below data :

Loan Amount: \$20,412.65
Payments (#): 10 Annual Interest Rate: 2.5% Loan Date: 07/02/2018 First Payment Due: 09/02/2018 (next payment will be on 09/02/2019 then 09/02/2020 and so on) Payment Frequency: Annually Compounding: Daily

My Bank is charging me \$2,285.18 for each repayment amount for 10 installments.

Using the formula derived here: Calculating interest accrued with extended Initial Payment Date

First period extension is negative: `x = -10/12` reducing the first period to 2 months.

``````pv = 20412.65
r = 2.5/100
n = 10

pv = (c (1 + r)^(-n - x) (-1 + (1 + r)^n))/r

∴ c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)

∴ c = 2284.82
``````

Which is pretty close to \$2,285.18

Reducing the granularity to days ...

62 days between 07/02/2018 and 09/02/2018

``````x = -(360 - 62)/360

∴ c = 2285.14
``````

360 day year gets closer to the repayment amount than 365.

The bank’s calculation may differ slightly, but this is the general idea.

Alternative method

Calculate interest to first payment, add to principal. Calculate payment for an annuity due.

``````s = pv (1 + r)^(62/360) = 20499.64

c = s (r/(1 - (1 + r)^-n))*1/(1 + r) = 2285.14
``````