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
