How to calculate total interest from annuity-immediate?

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It's always the small things. Thank you!In summary, the conversation discussed a loan of $10,000 being repaid in twenty level installments at the end of every six months with a nominal interest rate of 8% convertible half-yearly. The total amount of interest paid over the ten-year period was calculated using the formula a(n,i)=\frac{1-(1+i)^{-n}}{i}, resulting in an answer of $4,716.35. However, after some adjustments and calculations, the correct amount of interest accrued was found to be $4,716.35.
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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.
 
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  • #2
Eclair_de_XII said:

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.

- - - -
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.
 
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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.
 
  • #4
^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?
 
  • #5
Yes, I suppose so.
 

1. How do I calculate total interest from an annuity-immediate?

The formula for calculating total interest from an annuity-immediate is: Total Interest = (Annual Interest Rate/Number of Payments per Year) x Principal Amount x Number of Years. This formula takes into account the annual interest rate, the number of payments made per year, the principal amount, and the number of years the annuity will be paid out for.

2. What is an annuity-immediate?

An annuity-immediate is a type of financial product that involves a series of equal payments made at regular intervals, typically monthly or yearly, for a specified period of time. Annuities are commonly used for retirement planning, as they provide a steady stream of income during retirement.

3. How does an annuity-immediate differ from an annuity-due?

An annuity-immediate makes payments at the beginning of each period, while an annuity-due makes payments at the end of each period. This means that with an annuity-immediate, the first payment is made immediately, while with an annuity-due, the first payment is made at the end of the first period.

4. Is the total interest from an annuity-immediate taxable?

The total interest earned from an annuity-immediate is generally taxable as income. However, if the annuity is held in a tax-advantaged account such as a traditional IRA or 401(k), the interest will not be taxed until it is withdrawn from the account.

5. Can I calculate the total interest from an annuity-immediate without knowing the annual interest rate?

No, the annual interest rate is a crucial component in calculating the total interest from an annuity-immediate. Without this information, it is not possible to accurately determine the total interest earned. If you do not know the interest rate, you can contact the issuer of the annuity or refer to the contract for this information.

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