Just how to establish yearly settlements on a partly settled financing?

A 10-year financing of $500 is settled with settlements at the end of yearly. The lending institution costs passion at an yearly reliable price of 10%. Each of the first 10 settlements is 150% of the quantity of passion due. Each of the last 10 settlements is X. Calculate X.

I found this technique inquiry while researching for my actuary test. I attempted it on my very own and also obtained stuck:

$\$ $500 will certainly gain $\$ $50 passion yearly, so each of the first 10 settlements have to be $\$ $75.

After that after 10 years, a total amount of 750 has actually been settled.
In 10 years, I can locate the gathered financial debt by claiming PV= 500, I/Y= 10, N= 10, offering me FV= $\$ $1296.87. So the equilibrium would certainly be $\$ $1296.87- $\$ $750= $\$ $546.97.

Currently I am stuck! Just how do I figure out what the last settlements should be? I recognize I can not simply separate $\$ $546.97/ 10= $\$ $54.697, due to the fact that the lending institution is still billing passion while the consumer repays this continuing to be financial debt, so there would certainly still be the passion left over.

This scenario isn't stated throughout my calculator guidebook! Can among you offer me some description concerning what is taking place to make sure that I can do it by hand?

I attempted working with it some extra, and also thought of an actually wonderful suggestion! Given that the settlements are 1.5 x the 10% passion, it's similar to repaying 5% of the major yearly! This conserves me a great deal of time, due to the fact that I can simply set I/Y=-5 and also get FV=598.74 on my calculator. I did it the lengthy means by computing the future value of each passion settlement (ends up they were not all $75, due to the fact that the superior principal obtained smaller sized), and also they coincided. Is this constantly mosting likely to function, or did I simply get fortunate below?

An additional upgrade!
I assume I addressed it. All I required to do was to set FV=0, I/Y=10, N=10, PV=598.74, and afterwards I obtained PMT=97.44. I never ever made use of the PMT switch in the past, however, so exists a few other means I can examine the solution is right?

2019-05-07 07:37:48
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Answers: 1

This trouble remains in 2 phases. For the initial stage, notification that you are paying 150% passion, yet winding up owing extra. This is due to the fact that you deducted $\$ $750 from the future value, when actually each $\$ $75 quantity was paid at once in the past and also requires be transformed to a future value also. The settlement in the nth year has a future value of $75\times(1.1)^{10-n}$. The complete future value of the settlement is : $$75(1.1^9)+75(1.1^8)+\cdots+75(1.1^0).$$

Note that I have actually thought that the passion is billed prior to the settlements are made. This series is a geometric progression. We consider it as a geometric series backwards to make the mathematics less complicated. It has first term ($a$) 75, each term 1.1 times the previous ($r$) and also 10 terms ($n$). The amount is offered by the formula : $$\frac{a{r^{n-1}}}{r-1}=\frac{75{1.1^{10-1}}}{0.1}\approx\$1195.31$$

After we have actually addressed this first component, after that it is simply a typical interest with repayments problem..

2019-05-08 23:54:02