# Interest calculated daily compounded monthly

1. Dec 6, 2007

Does anyone know how I can work out a formula to calculate how long it will take to pay a loan back to zero if the interest is on a loan is calculated daily but compounded at the end of the month. To add another level of complexity, assume you are making fortnightly repayments.

P (Principal) = $500,000 E (Fortnightly Repayments) =$2,500
r (annual interest rate) = 8.59%
t (daily compound interest rate) = $$\frac {0.0859}{365}$$

I started out with the following:

$$A_0 = P$$

$$A_1_4 = P-E$$

$$A_2_8 = P-2E$$

$$A_3_0 = A_2_8 + 14(A_0 * t) + 14(A_1_4 * t) + 2(A_2_8 * t)$$

$$= P - 2E + 14(Pt) + 14(Pt-Et) + 2(Pt-2Et)$$

$$= P-2E+30Pt-18Et$$

and continuing...

$$A_4_2 = A_3_0 - E = P-3E+30Pt-18Et$$

$$A_5_6 = A_4_2 - E = P-4E+30Pt-18Et$$

$$A_6_0 = A_5_6 + 12(A_3_0 * t) + 14(A_4_2 * t) + 4(A_5_6 * t)$$

$$A_1_4$$ and $$A_2_8$$ is where I make repayments. $$A_3_0$$ is where the interest is finally compounded.

Am I heading in the right direction for this?

Can anyone enlighten me on the best way to go about solving this?

2. Dec 6, 2007

### HallsofIvy

Staff Emeritus
There is no reason to "calculate the interest daily" if you don't do anything with it! If the interest is being compounded monthly, then calculate it monthly: monthly interest is 0.0859/12= 0.00716.

Can anyone enlighten me on the best way to go about solving this?[/QUOTE]
I'm not sure I understand this. The interest is compounded monthly but how often are you paying on the loan?

3. Dec 6, 2007

Let me explain. I will change from a straight 30 days being one month though...just to make things more complicated...sorry.

You start out with a loan of $500,000. Each day 8.59%/365 *$500,000 accrues in interest for 14 days until you make a payment of $2,500. So the balance at day 14 is$500,000 - $2,500 =$497,500 (with $1647.39 in interest accrued so far). For the next 14 days the loan accrues 8.59%/365 *$497,500 in interest. When you make a payment on day 28 the balance is $497,500 -$2,500 = $495,000 (with$1,639.16 interest accruing on the $497,500 for those 14 days). Lets say that day also happened to be the compound date (the day the interest is actually charged and capitilised into the loan. The balance would be:$495,000
+ $1,647.39 (interest accrued in the first 14 days) +$1,639.16 (interest accrued in the second 14 days)
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6. Dec 7, 2007