Annuity Problem - Help required

[naveed_786110]

A firm will need Rs. 300,000/- at the end of 3 years to repay a loan. The firm decided that it would deposit Rs. 20,000 at the start of each quarter during these 3 years into an account. The account would yield 12% per annum compounded quarterly during the first year. What rate of interest should it earn in the remaining 2 years to accumulate enough amount in this account to pay the loan at the end of 3 years?

Here is how I just tried to solve it...but unable to solve the equation.... Kindly guide....Thanks

 

[Ishuda]

A firm will need Rs. 300,000/- at the end of 3 years to repay a loan. The firm decided that it would deposit Rs. 20,000 at the start of each quarter during these 3 years into an account. The account would yield 12% per annum compounded quarterly during the first year. What rate of interest should it earn in the remaining 2 years to accumulate enough amount in this account to pay the loan at the end of 3 years?

Here is how I just tried to solve it...but unable to solve the equation.... Kindly guide....Thanks
View attachment 5314

Why didn't you use the same formula (with a different number of payments) for FV2 as you did for FV1?

I would suggest you use x = 1 + i as your variable [and write i as x - 1]. That way, at the end, you will get a 10th degree equation in x which you will need to solve by a 'guess and correct' method such as Newton's method. For a partial list see
http://en.wikipedia.org/wiki/Root-finding_algorithm
for example.

 

[naveed_786110]

Why didn't you use the same formula (with a different number of payments) for FV2 as you did for FV1?

I would suggest you use x = 1 + i as your variable [and write i as x - 1]. That way, at the end, you will get a 10th degree equation in x which you will need to solve by a 'guess and correct' method such as Newton's method. For a partial list see
http://en.wikipedia.org/wiki/Root-finding_algorithm
for example.

I don't understand Ishuda. I have just used the same formula for the next two years i.e; for FV2.
FV2=20,000(1+r2/4)^8 * (1+r2) .... Is there something wrong with this formula....I have kept in mind that payment of RS. 20,000 is made at the START of each quarter.
'Guess and correct' can be used but Its a multiple choice question, i just need equation to PUT and check ANSWER.

Thankful to you for instant reply.

 

[Ishuda]

I don't understand Ishuda. I have just used the same formula for the next two years i.e; for FV2.
FV2=20,000(1+r2/4)^8 * (1+r2) .... Is there something wrong with this formula....I have kept in mind that payment of RS. 20,000 is made at the START of each quarter.
'Guess and correct' can be used but Its a multiple choice question, i just need equation to PUT and check ANSWER.

Thankful to you for instant reply.

You did
FV = p\(\displaystyle \frac{x\, (x^n\, -\, 1)}{x\, -\, 1}\)
which, for FV1 is p=20000, i=.03 (x=1.03), and n=4
FV1 = 20000 \(\displaystyle \frac{1.03\, (1.03^n\, -\, 1)}{0.03}\)
which is what you have.

For FV2 we have p=20000, i=?, (x=?) and n=8 or
FV2 = 20000 \(\displaystyle \frac{x\, (x^8\, -\, 1)}{i}\)
So that the total is
FV = FV1 x⁸ + FV2 = 20000 [ \(\displaystyle \frac{1.03\, (1.03^n\, -\, 1)}{0.03}\) x⁸ + \(\displaystyle \frac{x\, (x^8\, -\, 1)}{i}\)]

Oh, and that was a 9th degree equation, not a 10th degree.

 

[naveed_786110]

You did
FV = p\(\displaystyle \frac{x\, (x^n\, -\, 1)}{x\, -\, 1}\)
which, for FV1 is p=20000, i=.03 (x=1.03), and n=4
FV1 = 20000 \(\displaystyle \frac{1.03\, (1.03^n\, -\, 1)}{0.03}\)
which is what you have.

For FV2 we have p=20000, i=?, (x=?) and n=8 or
FV2 = 20000 \(\displaystyle \frac{x\, (x^8\, -\, 1)}{i}\)
So that the total is
FV = FV1 x⁸ + FV2 = 20000 [ \(\displaystyle \frac{1.03\, (1.03^n\, -\, 1)}{0.03}\) x⁸ + \(\displaystyle \frac{x\, (x^8\, -\, 1)}{i}\)]

Oh, and that was a 9th degree equation, not a 10th degree.

Please check the last step which i have just done....Isn't it the same which you are saying? OR Am I missing something?....But the problem is at r=13.8% equation is not satisfied....

 

[naveed_786110]

Naveed, after you use the "correct!" formula,
you can ONLY solve by iteration...as Ishuda told you.

Are you allowed to use a financial calculator?

No Denis.... Financial calculator is not allowed..... I just need to know about the last step of my solution....If it's a true equation, then why r=13.8% is not satisfying the equation....And if the last step(the equation i made) is incorrect, whats wrong with it....

 

[naveed_786110]

Naveed, after you use the "correct!" formula,
you can ONLY solve by iteration...as Ishuda told you.

Are you allowed to use a financial calculator?

No....financial calculator is not allowed....I just wanted to know if my last step(equation) is true or there is some issue with that.....I have four options for 'r' and correct one is r=13.8% but it is not satisfying the equation.

 

[naveed_786110]

Ishuda told you your last equation is WRONG...did you not "listen"?
OK, let's do it once and for all:
a = 86182.71
p = 20000
i = .138 / 4 = .0345
n = 8

u = a(1 + i)^n = 113048.3183... ; you got that part right

It is the next equation that's wrong; should be:

v = [p((1 + i)^n - 1) / i](1 + i) = 186946.7537...

u + v = 299995.072

Are you ok now?

OK OK I got it
Thanks a lot