# What's the APR for this in combination with the Balloon Payment of 6525 USD?

I am trying to solve the following:

``````Finance Amount: 24.951,82 USD
Interest Rate: 2.29%
Residual Value: 6.525 USD
Number of Months: 60
Fee: 374.28
Years: 5
``````

What's the APR for this in combination with the Balloon Payment of 6525 USD?

Similar to this: How to calculate APR on asset appreciation

As shown here, the sum of the discounted cash flows equals the loan amount.

Assuming the periodic payments end in month `n` and the balloon is paid in month `n + 1`.

With

``````s = present value of loan
m = periodic repayment
r = periodic rate
b = balloon payment
n = number of periodic payments
``````

``````∴ b = ((1 + r) (m + (1 + r)^n (r s - m)))/r

and m = (r ((1 + r)^(1 + n) s - b))/((1 + r) ((1 + r)^n - 1))
``````

To have the balloon paid at the end of the 5th year, assuming the interest rate is nominal 2.29% compounded monthly

``````s = 24951.82 + 374.28 = 25326.10
r = 2.29/100/12
n = 5*12 - 1
b = 6525

m = (r ((1 + r)^(1 + n) s - b))/((1 + r) ((1 + r)^n - 1)) = 349.893
``````

So 59 monthly payments of \$349.89 followed by \$6525 at the end of month 60.

Confirmed by the site in the earlier link, (with slight rounding difference)

`n m + b = 27168.71`

and the total interest paid is `n m + b - s = 1842.61`

To calculate the equivalent monthly rate `x`

``````s (1 + x)^60 = n m + b

∴ x = ((n m + b)/s)^(1/60) - 1 = 0.00117119
``````

So the equivalent APR is `12 x = 1.4%` compounded monthly

Note, if the balloon payment is to coincide with the final periodic payment the formula for `s` can be slightly modified (with a final `n` instead of `n + 1`)

``````s = (m - m (1 + r)^-n)/r + b/(1 + r)^n
∴ m = (r ((1 + r)^n s - b))/((1 + r)^n - 1)

n = 60
m = 344.384
x = 0.00118305
APR = 1.42% compounded monthly
``````