if you were paying quarterly, that's what your quarterly payments would be.
since you are not paying quarterly, but paying at the end of the year, then each quarterly payment does not reduce the remaining balance, but remains in the account until the end of the year.
the quarterly payment, remaining in the account, would accrue interest at the rate of the quarterly interest rate, which is 13.8% / 4 = 3.45% per quarter.
the payments made at the end of the year would then have to be:
quarter 1 payment * (1 + .0345) ^ 3
quarter 2 payment * (1 + .0345) ^ 2
quarter 3 payment * (1 + .0345) ^ 1
quarter 4 payment * (1 + .0345) ^ 0
the future value of these payments would become one payment of 3835.101166 at the end of the year rather then 4 payments at the end of each quarter of 910.560636 that add up to an annual total of 3642.242544.
i arrived at these figures in the following manner.
first i calculated the payment required at the end of each quarter.
my inputs to my calculator were:
present value = 13000
future value = 0
interest rate per quarter = 13.8% per year / 4 = 3.45%
number of quarters = 5 years * 4 quarters per year = 20 quarters.
payments made at the end of each quarter were calculated to be 910.560636.
i then assumed end of year payments rather than end of quarter payments.
in doing so, i still had to deal with quarterly compounding, therefore i had to use the effective interest rate per year and not the nominal interest rate per year.
the effective interest rate per year was calculated as follows:
divide the annual interest rate by 4 to get .138 / 4 = .0345
***** note that i'm using the rate for this calculation, and not the rate percent.
the rate is equal to the percent / 100.
add 1 to it to get 1.0345
raise that to the 4th power to get 1.0345 ^ 4 = 1.1453807171.
subtract 1 from that to get .1453807171.
that's the effective annual interest rate, as opposed to the nominal annual interest rate of .138.
multiply it by 100 to get 14.53807171%.
i then made the following inputs into my calculator.
present value = 13000
future value = 0
interest rate per year = 14.53071712%
number of years = 5
payments made at the end of each year were 3835.101166.
this was not equal to the sum of 4 quarterly payments of 910.560636 which were equal to 3642.242544.
the reason was that, if i did not pay at the end of each quarter, then the amount i would have paid was not subtracted from the remaining balance, and was therefore being charged interest at the quarterly interest rate of 3.45%.
bottom line is the end of year payment was not equal to the sum of the quarterly payments, but was somewhat higher because of accrued interest on those payments that were not made.
the following excel printout should show this to be true.
your solution is that the annual payment required is 3835.101166 which is equal to 3835.10 when rounded to two decimal places.