Annuity Problem - Assistance required

[naveed_786110]

An individual plans to borrow Rs. 400,000 to buy a new car. The loan will be for 3 years at 12% annual rate compounded monthly. He can pay Rs. 12500 per month during the first year. What amount would he be required to pay during he next two years in order to repay the loan?

I tried it and got result "Rs. 12,207" but its wrong..... the correct answer is around Rs. 13750..... Kindly help
Here is how I solved it.

 

[Ishuda]

An individual plans to borrow Rs. 400,000 to buy a new car. The loan will be for 3 years at 12% annual rate compounded monthly. He can pay Rs. 12500 per month during the first year. What amount would he be required to pay during he next two years in order to repay the loan?

I tried it and got result "Rs. 12,207" but its wrong..... the correct answer is around Rs. 13750..... Kindly help
Here is how I solved it.
View attachment 5300

For annuity problems, payment problems, etc where the 'payments' are on a regular basis of some period, with interest i charged per period, and for n payments starting at the end of the first period the basic formula is
p = P i xⁿ / (xⁿ - 1)
where p is the periodic payment/annuity, P is the principal amount, and x=1+i.

In this problem we first note that the i is 12%/12 = 1% and x=1.01 since it is a 12% annual and thus 1% monthly interest. Next we would use the formula with p=Rs. 12500 and n=12 [payments at the end of months 1, 2, 3, ..., 12] to find out how much we had reduced the principal in the first year. You did that.

You have used the reduced principal for P and an n=24 to compute the payments for the next two years. However, how about the rest of the interest on the balance. That is you have paid off the Rs. 140688.50 and interest on that amount (as you computed) but what about the interest on the remaining Rs. 259311.50 which has accumulated during the year. If you add that back in to your reduced principal, what is the principal balance for the remaining two years and what would the payment then be?

 

[Ishuda]

u = owing after 1 year =
400000(1.01^12) - 12500(1.01^12 - 1)/.01 = 292198.72

You now treat u as a new loan over 24 months;
p = payment next 2 years =
u(.01) / (1 - 1/1.01^24) = 13754.81

Yes and that is what you get if you add back in the interest on the balance you didn't pay off.

 

[naveed_786110]

Understood.....
I add 1 year interest to remaining balance i.e; Rs. 259311 and add it to next present value.....that makes 292198.12 that would be treated as new present value.
Thanks very much for your help

 

[Ishuda]

What does that mean, Ishuda?

He subtracted from 400,000; should be from 400000(1.01)^12

Your Payment formula is interesting; however, the standard one is:
P = Ai / (1 - v) where v = 1 / (1+i)^n

Doing the problem the way the op did it initially and using the formula I put up
p = P i xⁿ / (xⁿ - 1) = P i / (1 - 1/xⁿ)
[which is the A i /(1 - v) of Denis] we have the amount paid off in the first year
P₁ = p (xⁿ - 1) / (i xⁿ ) = 12500 * (1.01¹² - 1)/0.1/1.01¹² = 140688.50
That leaves a balance owing of 259311.5 which has been accumulating interest for a year or the amount now left to finance is
P₂ = (P - P₁) xⁿ = 259311.50 * 1.01¹² = 292198.72

Note that
(P - P₁) xⁿ = [P - p (xⁿ - 1) / (i xⁿ )] xⁿ = P xⁿ - p (xⁿ - 1) / i
which is the formula Denis used

 

[jonah2.0]

WARNING: Beer soaked rambling/opinion/observation/reckoning ahead. Read at your own risk. Would be readers can take it seriously or take it with a grain of salt. In no event shall the wandering math knight-errant Sir jonah in his inebriated state (usually in his dead tired but mentally revived inebriated state) be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his "enhanced" beer (and tequila/absinthe) powered views.

u = owing after 1 year =
400000(1.01^12) - 12500(1.01^12 - 1)/.01 = 292198.72

You now treat u as a new loan over 24 months;
p = payment next 2 years =
u(.01) / (1 - 1/1.01^24) = 13754.81

Sorry I'm late; doing a lot of driving, drilling, and painting lately.
Love your positivity Sir D.
As always, negativity is my preferred outlook.
Thus, with the present as the focal date, the corresponding equation of value would be

\(\displaystyle 400,000 = 12,500{{1 - (1.01)^{ - 12} } \over {.01}} + X{{1 - (1.01)^{ - 24} } \over {.01}}(1.01)^{ - 12} \)

Same result.
Sir D's Legendary Bank statement format should erase any lingering doubt.

 

[Ishuda]

Ishuda, you're giving me a headache

This is a SIMPLE problem.

We need 3 simple steps:
1: u = 400000 * 1.01^12 = 450730.012...

2: v = 12500 * (1.01^12 - 1) / .01 = 158531.287...

3: p = .01(u - v) / (1 - 1/1.01^24) = 13,754.808...

Formulas are "known as":
1: future value of $1

2: future value of $1 deposited periodically

3: periodic payment required to pay off $1

Yes, but that's not the way the OP did the problem. The question was something to the effect of "why doesn't this work". Using the same methods that the OP used, I was trying to show what the OP didn't account for.

In addition, yes it is great if you can remember the specific formulas for periodic payment to get either a present value of periodic payments, the P= part of the equation, or a periodic payment for a particular present value, the p= part of the equation. However, you really only need to remember the one side (as the OP and I used) and convert back and forth.

I also admit the use of a v=1/xⁿ makes for a simpler equation, but the use of the xⁿ makes for a crazy way to get future value where you let the n in (xⁿ -1)/i equal 1 to get FV=p=P xⁿ [I told you it was crazy]

 

[naveed_786110]

Yes, but that's not the way the OP did the problem. The question was something to the effect of "why doesn't this work". Using the same methods that the OP used, I was trying to show what the OP didn't account for.

In addition, yes it is great if you can remember the specific formulas for periodic payment to get either a present value of periodic payments, the P= part of the equation, or a periodic payment for a particular present value, the p= part of the equation. However, you really only need to remember the one side (as the OP and I used) and convert back and forth.

I also admit the use of a v=1/xⁿ makes for a simpler equation, but the use of the xⁿ makes for a crazy way to get future value where you let the n in (xⁿ -1)/i equal 1 to get FV=p=P xⁿ [I told you it was crazy]

Right Ishuda,
I was wondered why its not working and what I am missing.... Now its clear.....Thanks so much both of you.....

 

[Ishuda]

Good stuff!

I see you use unwieldy formulas, like (your present value):
p[(1.01^12 - 1) / (.01 * 1.01^12)]

Suggest you use the "standard" form:
p[(1 - 1.01^(-12)) / .01]

Saves time...important on timed tests...

That's actually the one I use if I am doing it myself, especially if I'm doing a 'guestimate' in my head.

 

[Ishuda]

Is that why your head is so beat up :twisted:

Just a little soft.