# Interest paid on annuity?

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.

• I don't understand the "financial" terms: what does it mean to have a "$5$ year payment freedom" and did you take that into account? – MoebiusCorzer Dec 7 '16 at 16:37
• Can you draw the cashflows( in terms of A ) for each payment date? – Canardini Dec 7 '16 at 16:38
• @MoebiusCorzer Sorry, 5 year payment freedom means that the repayments could be postponed by 5 years. But in these 5 years, interest on the loan would accumulate. – Jane Doe Dec 7 '16 at 16:39

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

• Yes, thank you @Satish, this is the exact same way i would have solved it yet the answer is wrong (according to my professor). The reason being is that since we are calculating with 5 years payment freedom, the annuity will start right at the 60th month. Therefore the we can't solve it by discounting the first payment. Honestly if you are confident in your thinking, (i agree with you) then i will take this problem up with my teacher, i think there is a mistake on his part. – Jane Doe Dec 10 '16 at 19:27
• @Jane Doe, five year payment freedom means that you only pay interest and at the end of the 60th month, you will have a balance that is equal to the mortgage amount plus all the interest due up until the 60th payment. Now the repayament starts on the 61st month up until end of 240th month. ( it is an "ordinary Annuity" where the payment of principal happens at the end of the 61st, 62nd,.... months. – Satish Ramanathan Dec 11 '16 at 1:11
• @Jane Doe, contd.. If it were "annuity due" then the payment of interest and principal would start at the beginning of the 61st, 62nd ,.. months which might affect the annuity amount. But I think clarify with your professor on the payment period if it allows the customer to not pay till the end of the 60th month. Otherthan that this is the way to go. Goodluck – Satish Ramanathan Dec 11 '16 at 1:11

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 $$I=n A-L=\\, 1,030,653.14$$