# After how many years will loan be paid with constant payment

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!

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.

• Thank you so much Jan 8 '20 at 19:17