How to calculate total interest from annuity-immediate?

[Eclair_de_XII]

Homework Statement

"A loan of 10,000 is repaid by twenty level installments at the end of every six months. The nominal interest rate is 8%, convertible half-yearly. Find the total amount of interest payment made over the ten-year period."

Homework Equations

The Attempt at a Solution

##10000=k[\frac{1-(1.04)^{-20}}{0.04}]##
##k=10000[\frac{0.04}{1-(1.04)^{-20}}]=217.45##

I think I calculated the amount of each of the twenty level installments, but I don't know how to proceed to calculate total interest accrued from here.

 

[StoneTemplePython]

Homework Statement

"A loan of 10,000 is repaid by twenty level installments at the end of every six months. The nominal interest rate is 8%, convertible half-yearly. Find the total amount of interest payment made over the ten-year period."

Homework Equations

The Attempt at a Solution

##10000=k[\frac{1-(1.04)^{-20}}{0.04}]##
##k=10000[\frac{0.04}{1-(1.04)^{-20}}]=217.45##

I think I calculated the amount of each of the twenty level installments, but I don't know how to proceed to calculate total interest accrued from here.

can you show a bit more work? I didn't understand this.

##20\big(k\big) = 20\big(217.45\big) \lt 10,000 \lt \text{total cash pmt obligation }##

so that doesn't seem to be the amount for the 20 level installments.

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The idea, like in a lot of math, is to set something up then run it forward and run it backward. The forward part is you have a present value of payments that is ##10,000##. So you have

##10,000 = \sum_{k=1}^{20} \frac{x}{(1+r)^k} = x \sum_{k=1}^{20} \delta^k##

where ##\delta := \frac{1}{1+r}##, ##x = \text{level payment amount}## and I think ##r = 0.04##, though using the term that interest is 'convertible' seems nonstandard at least compared to what I'm used to, so there may be fine tuning need on compounding periods.

My guess is that you accidentally used ##\delta = (1+r)## instead.
- - - -
solve for ##x## and that's first part. The second part is a simple decomposition of the total payments.

 

[Eclair_de_XII]

Okay, I ran through the numbers in Excel and came up with ##x=735.82##. Then I multiplied that by twenty, and subtracted by ##10,000## for the interest accrued, which would be ##I=4716.35##. In any case, I initially used a formula in my textbook that expressed the present value of ##n## level payments with interest rate ##i## as ##a(n,i)=\frac{1-(1+i)^{-n}}{i}##, which is how I got that first answer.

 

[lurflurf]

^you flipped it over you should use
##k=\frac{i}{1-(1+i)^{-n}}a##

edit: I see that is what you used. Arithmetic error?

 

[Eclair_de_XII]

Yes, I suppose so.