<p>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)</p>

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

<p><span class="math-container">$$100,000=\frac{5,000}{0.05}\left[1-\frac1{(1.05)^5}\right]+\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$</span> and solve for <span class="math-container">$n$</span>. </p>

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

<p><span class="math-container">$$78,352.61665=\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$</span></p>

<p>However, hopefully you can see that you failed to account for the fact that the <span class="math-container">$10,000$</span> 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:</p>

<p><span class="math-container">$$n=-\frac{\ln\left(1-\frac{(78,352.62)(0.05)(1.05)^6}{10,000}\right)}{\ln(1.05)}=14.2067$$</span></p>

<p>That means that they will finish paying the debt in period "<span class="math-container">$19.2067$</span>", i.e. in period 20. That is after 5 years of paying <span class="math-container">$5,000$</span>, 14 periods of paying <span class="math-container">$10,000$</span> and paying the remainder (less than 10,000) on period 20. </p>

CC BY-SA 4.0
2020-01-08T18:42:12.613Z
2020-01-08T18:42:12.613Z
Regio
<p>I'm solving the following problem:</p>

<blockquote>
  <p>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%.</p>
  
  <p>After how many years will the debt be fully paid?</p>
</blockquote>

<p>After 5 years I got:</p>

<p><span class="math-container">$78352,61655 = 5000 * \frac{1-(1+0,05)^{-5}}{0,05}$</span></p>

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

<p><span class="math-container">$n = \frac{ln(\frac{78352,61655*0,05}{10000}+1)}{ln(1+0,05)}$</span></p>

<p><span class="math-container">$n=6,78$</span> years</p>

<p>Are my steps correct?</p>

<p>Thanks!</p>

<interest-rate>
After how many years will loan be paid with constant payment
Daniel
# After how many years will loan be paid with constant payment

(Asked by: username_0 on 2020-01-08)

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!

**Tags:** interest-rate

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## 1 answers

### Answer by username_1 on 2020-01-08. Score: 3 | Accepted answer

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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