After how many years will loan be paid with constant payment

Question

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:

$$78352.61655 = 5000 \cdot \frac{1-(1+0.05)^{-5}}{0.05}$$

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

$$n = \frac{\ln\left(\frac{78352-61655\cdot 0.05}{10000}+1\right)}{\ln(1+0.05)}$$

$n=6.78$ years

Are my steps correct?

Thanks!

Answer 1

In the first $5$ years, the loan will not decrease. If we look at the first year, then we can see why. First we find out how much interest was added in the year by calculating \begin{align}&\color{white}=\text{initial loan value} \times \text{interest rate}\\ &=100000*1.05\\ &=105000\end{align}

We then subtract the payment for the year $$105000-5000=100000$$

We can clearly see that the payments in the first 5 years merely keep the interest at bay.

We can then use the formula $$P=\frac{r(PV)}{1-(1+r)^{-n}}$$ to calculate how long it will take to pay off the remaining balance. In this equation, \begin{align}P&=\text{yearly payment}=10000\\ PV&=\text{starting value of the loan}=100000\\ r&=\text{interest rate}=0.05\\ n&=\text{number of years}\end{align}

\begin{align}10000&=\frac{0.05\times 100000}{1-(1+0.05)^{-n}}\\ 1-(1.05)^{-n}&=\frac{5000}{10000}\\ 1-(1.05)^{-n}&=\frac 12\\ \frac12&=(1.05)^{-n}\\ n&=-\log_{1.05}\left(\frac12\right)\\ &=14.2067\end{align}

(we did the last step using Wolfram|Alpha but you could use a calculator if yours supports different logarithm bases)

Therefore, the total repayment time for the loan is the original $5$ years plus these $14.2067$ years, making a total of $19.2067$ years, or $19$ years and about $2.5$ months.

Answer 2

After five years the couple still owes $\$100,000$ since their payment size of $\$5,000$ is exactly equal to the amount of interest per year.

Let $n$ be the additional years needed to pay off the loan. We have $$100000=10000\cdot \frac{1-1.05^{-n}}{.05}$$

This gives $$.5=1-1.05^{-n}$$

So $n=-\frac{\ln .5}{\ln 1.05}\approx14.2$

So it will take an additional $14.2$ years (probably $15$ years with an adjusted final payment).

So the total length of the loan is $19.2$ years (or $20$ years depending on how the payments are structured).