Question

I just got some help with a math question I just asked but I have one last question

Looking at the question, the obvious missing part is the twice-monthly payment.
The standard annuity formula only works if the interest period and the payment period are the same, here they are not.
So we have to convert the 4% compounded quarterly into a rate compounded twice a month
Let that rate be i
(1 + .04/4)^4 = (1+i)^24
take the 4th root
1.01 = (1+i)^6
now take the 6th root
1+i = 1.01^(1/6) = 1.0016597...
i = .0016597... (I stored in my calculator' memory)

304000 = paym(1 - 1.0016597..^-480)/.0016597..)
I get paym = 919.27 <----- twice monthly payment

take over to answer all that a, b, c, stuff

I was just wondering for this equation
304000 = paym(1 - 1.0016597..^-480)/.0016597..)
how the power -480 was found

apart from that Im so thankful for the help and it does answer my questions

Answer 1

[AI-generated answer — Explain Bot]

To understand how the power of -480 was found in the equation 304000 = paym(1 - 1.0016597..^-480)/.0016597..), we need to refer back to the context of the problem.

In this equation, paym represents the twice-monthly payment amount, and 304000 is the overall loan amount. The expression (1 - 1.0016597..^-480)/.0016597..) is used to calculate how many payment periods are needed to fully pay off the loan.

Now let's break it down step by step:

1. The term 1.0016597.. is the value we obtained as the interest rate per payment period (compounded twice a month). It was calculated by taking the 6th root of 1.01, which represented the interest rate per compounding period (compounded quarterly), as explained in the previous calculations.

2. The term 1.0016597..^-480 represents the cumulative effect of compound interest over 480 payment periods. The negative exponent (-480) is used because we need to find the present value of the loan amount after 480 payments have been made.

3. By subtracting 1.0016597..^-480 from 1, we find the remaining loan amount after making 480 payments.

4. Finally, dividing the remaining loan amount by the interest rate per payment period (.0016597..) gives us the payment amount (paym) required to fully pay off the loan in 480 payment periods.

So, the power of -480 in the equation represents the number of payment periods considered to calculate the remaining loan amount using compound interest.