Interest paid on annuity?

Question

I'm vexed after a question about annuities which i seem to misunderstand, any help would be greatly appreciated.

A mortgage of $\$3,000,000$ is given as an annuity loan with nominal interest rate $2.40\%$ per annum. With a payment periode of one month, and a $20$ year repayment period. It is agreed that there should be a $5$ year payment freedom, thereafter a fixed amount $A$, will be paid every month for the remaining $240$ months.

What are the total interest payments?

The way i solve this problem is first to take out the most important parts of the question listing them up as so.

• $20$ years = $240$ months
• $r = \frac{0.0240}{12}+1 = 1.002$
• $\$3,000,000$ needs to be moved forward in time by 60 months, because interest will be accumulated.

Now i use the finite geometric sequence to find $A$ the annuity setting it as $a_1$. My equation looks like this.

$$\frac{A}{1.002}\left(\frac{1-\left(\frac{1}{1.002}\right)^{240}}{1-\frac{1}{1.002}}\right)=3000000 \dot{} (1.002)^{60}$$

Now here i get a value for $A$ as something around $17751$ i then multiply it by $240$ the amount of months, and then subtract the loan amount from that value, leaving me with the interest paid. Yet, somehow my answer is wrong. Could anyone explain to me what i'm doing wrong here. Also if there is any mathematical concepts which would help me further solve these kind of questions easier i greatly appreciate hearing about them. As of now i know of calculus, algebra, and some trigonometry, ala basic college math.

Thank you greatly, i have nowhere else to turn. Therefore i very much appreciate the help i've received on this forum.

Answer 1

Answer:

For the first five years, you pay the interest only on the Mortgage amount of 3,000,000 (M). Let us say that it is $S_1$

$S_1 = M*(1+.024/12)^{60}$

You are right in calculating the Annuity Amount. You could choose to create an amort schedule like the below and calculate the interest amount.

Good luck

Answer 2

Let be $L=\$\,3,000,000$, $i^{(12)}=2.40\%$, $m=5\times 12=60$, $n=15\times 12=180$. The monthly interest rate is $i=\frac{2.40\%}{12}=0.2\%$.

We assume all payments are made at the end of each period.

The loan is repayed as a annuity (immediate) of value $A$ payed each month for $n$ months but deferred by $m$ months, so we have $$ L=A\,_{m|}a_{\overline{n}|i} $$ where $$v=\frac{1}{1+i},\qquad _{m|}a_{\overline{n}|i}=v^m a_{\overline{n}|i}, \qquad a_{\overline{n}|i}=\frac{1-v^n}{i}$$

So we have $$ A=\frac{_{m|}a_{\overline{n}|i}}{L}=\frac{v^m\,a_{\overline{n}|i}}{L}= \$ \,22,392.52 $$

The interest paid is $$ I=n A-L=\$\, 1,030,653.14 $$