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I'm solving the following problem:

A young couple took a loan 100 000 USD. For the first 5 years they pay 5000 USD yearly. After 5 years they pay 10 000 USD yearly. The interest rate is 5%.

After how many years will the debt be fully paid?

After 5 years I got:

$78352,61655 = 5000 * \frac{1-(1+0,05)^{-5}}{0,05}$

Then I calculated how many years do they need to pay with constant payment of 10 000 USD:

$n = \frac{ln(\frac{78352,61655*0,05}{10000}+1)}{ln(1+0,05)}$

$n=6,78$ years

Are my steps correct?

Thanks!

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Some of your equations are wrong and you need to account for the time value of money correctly in some places. It might be useful to write down a timeline and make sure to bring all payments to the same point in time (let's take time 0 as our reference)

Therefore, what you receive should have the same present value to what you pay (I'll use American conventions for punctuations):

$$100,000=\frac{5,000}{0.05}\left[1-\frac1{(1.05)^5}\right]+\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$ and solve for $n$.

So the first calculation was correct (though your formula is not), since doing the first operation you get:

$$78,352.61665=\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$

However, hopefully you can see that you failed to account for the fact that the $10,000$ payments start coming at period 6 (so the annuity formula will bring the payoffs to period 5 and you have to further discount them 5 more periods), further, you also did something funny with the signs when you solved for n. Instead I get that:

$$n=-\frac{\ln\left(1-\frac{(78,352.62)(0.05)(1.05)^6}{10,000}\right)}{\ln(1.05)}=14.2067$$

That means that they will finish paying the debt in period "$19.2067$", i.e. in period 20. That is after 5 years of paying $5,000$, 14 periods of paying $10,000$ and paying the remainder (less than 10,000) on period 20.

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  • $\begingroup$ Thank you so much $\endgroup$ – Daniel Jan 8 '20 at 19:17

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