Which is the correct solution?

[naveed_786110]

Ali would require a sum of Rs. 300,000 after three years from now and a sum of Rs. 500,000 after five years from now, for the purpose of education of his son. He is planning to deposit quarterly amounts in an investment scheme to get the desired amounts at the required time. If the rate of interest is 12% compounded quarterly, what amount should Ali deposit at the start of each quarter?


Sol:
This is how I solved it.

For this I made 2 accounts with investments of R1 and R2.

Since

Sn=R{(1+i)^n-1}/i x (1+i)

300,000 = R1 {(1+0.03)^12 -1 } / 0.03 x (1+0.03)
R1 = 20523

Similarly;

500,000 = R2 {(1+0.03)^20 -1 } / 0.03 x (1+0.03)

R2= 18065.87

So required sum is R1+R2 i.e; 38588.

But When I consult the solution, Its totally different approach. It says

The future value of Rs. 300,000 on completion of five years would be:
S=P(1+i)^n
= 300,000(1.03)^8=380031.02
It means if Rs. 300,000 had not been drawn after three years then at the end of investment period, Ali would had a sum of 380031.02+500,000=880,031.02

Using the same formula (as i used)

Sn=R{(1+i)^n-1}/i x (1+i)

880,031.02= R {(1+0.03)^20 -1 } / 0.03 x (1+0.03)
R = 31,797.07
Which of the solution is true? If my solution is false, what's wrong with it, what I am missing?

 

[naveed_786110]

That is correct; in Bank statement format:

QTR    TRANSACTION   INTEREST      BALANCE
 0       31,797.07        .00     31,797.07
 1       31,797.07     953.92     64,548.06
....
12       31,797.07  13,537.95    496,599.90
12     -300,000.00        .00    196,599.90
13       31,797.07   5,897.98    234,294.95
....
19       31,797.07  13,212.81    485,436.89
20                  14,563.11    500,000.00

I can't follow what/why you did, so no comments.
In case this helps:
F = 300000(1.03)^8 + 500000 = 880,031.02
The 20 quarterly payments plus interest must accumulate to F.

The problem is WHATS wrong with the first solution....

 

[naveed_786110]

The given solution uses Future Value.

You're using Present Value (which is ok to use) but NOT correctly; should be:
300000 / (1.03)^12 = 210,413.96
500000 / (1.03)^20 = 276,837.88
Total above: 487,251.84

QTR    PAYMENT    INTEREST    BALANCE
 0                          487,251.84
 1               14,617.56  501,869.40
...
12               20,234.12  694,704.62
12  -300,000.00             394,704.62
13               11,841.13  406,545.75
...
20               14,563.11  500,000.00
20  -500,000.00                    .00

Hope that helps.
Please do not request help using PM's.

Yes it really helped ..... Thanks for help

 

[Ishuda]

Ali would require a sum of Rs. 300,000 after three years from now and a sum of Rs. 500,000 after five years from now, for the purpose of education of his son. He is planning to deposit quarterly amounts in an investment scheme to get the desired amounts at the required time. If the rate of interest is 12% compounded quarterly, what amount should Ali deposit at the start of each quarter?


Sol:
This is how I solved it.

For this I made 2 accounts with investments of R1 and R2.

Since

Sn=R{(1+i)^n-1}/i x (1+i)

300,000 = R1 {(1+0.03)^12 -1 } / 0.03 x (1+0.03)
R1 = 20523

Similarly;

500,000 = R2 {(1+0.03)^20 -1 } / 0.03 x (1+0.03)

R2= 18065.87

So required sum is R1+R2 i.e; 38588.

But When I consult the solution, Its totally different approach. It says

The future value of Rs. 300,000 on completion of five years would be:
S=P(1+i)^n
= 300,000(1.03)^8=380031.02
It means if Rs. 300,000 had not been drawn after three years then at the end of investment period, Ali would had a sum of 380031.02+500,000=880,031.02

Using the same formula (as i used)

Sn=R{(1+i)^n-1}/i x (1+i)

880,031.02= R {(1+0.03)^20 -1 } / 0.03 x (1+0.03)
R = 31,797.07
Which of the solution is true? If my solution is false, what's wrong with it, what I am missing?

To me there seems to be some confusion as to what is needed for an answer so, first, lets address the formula you used. I am going to assume an immediate payment and then payments of the same amount for the given period ending, for example with the last payment three years in the future for the one account R1. Thus the quantities and formula needed are
period = quarterly for this problem
No. of payments = n [i.e. equal to 13 for R1 of one immediate and 12 future payments]
interest = i per period [= 3% per period for problem]
Principle amount needed at end of period = P
equal payment amount = p
Thus
p = P * i / [ (1+i)ⁿ - 1 ]
The difference in formulas is whether you are paying off an amount or saving to get a particular amount and whether there is an initial payment of whether the first payment starts at the beginning [immediate] or end [delayed] of the first period.

Now to the problem. Lets do your R1 and R2
p₁ = 300000 * .03 / [1.03¹³ - 1] ~ 19208.86
p₂ = 500000 * .03 / [1.03²¹ - 1] ~ 17435.89
or a total payment of
p = Rs. 36645

However, you would stop the payment for R1 at the end of 3 years and be left with the payment of Rs. 17436 for another two years. The way the answer is written, that is not what is wanted. What is wanted is an even payment for the complete five years. Thus, since you will loose the interest of that Rs. 300000 for 2 years, you need to make that up. That is the Rs. 300000*1.03⁸ addition to the total amount needed rather than just the Rs. 300000 and so, for the problem we have
period = quarterly for this problem
No. of payments = 21
interest = .03
Principle amount needed at end of period = 880031
equal payment amount = p

 

[Ishuda]

I don't see any confusion; the answer is given in the initial post:
880,031.02= R {(1+0.03)^20 -1 } / 0.03 x (1+0.03)
R = 31,797.07

There are 20 payments.

Use a spread sheet (or whatever) for a fixed payment of 31,797.07 per quarter starting with a balance of zero with a withdrawal of 300000 at the end of the third year and see what the balance is at the end of 20 payments.

Now, use the formula I gave above but with 20 payments to get a fixed payment of 32750.98 and see what the result is. I believe you will find that the correct formula is the one I gave.

The question wasn't the number of payments as I indicated in what I wrote after the formula above, it was the formula itself.

BTW: I would be interested in where the original problem says there are 20 payments. It seems to me that it mentions a number of years and a payment every quarter but nowhere mentions whether the payments begin at the beginning of the first quarter and continue through the full five years with 21 payments, begin at the end of the first quarter [beginning of the second quarter] and continue with 20 payments or ... However, it does say

...If the rate of interest is 12% compounded quarterly, what amount should Ali deposit at the start of each quarter?

So, do you make the payment at the start of quarter 1 and 21 [the end of 5 years] for 21 payments or do you make payments at the start of quarter 1 and 20 [the end of 4 years and 9 months] for 20 payments or ...?

 

[Ishuda]

Ishuda, I'm not trying to Lookagainize(!) you.
Naveed's question simply is:
the book's answer is 31,797.07: why is that the correct answer?

Since it IS the expected answer, then (to be 100% clear) the problem
should be worded something like:
Ali requires 300,000 after 3 years from now and 500,000 after 5 years from now.
In order to achieve this, he will make 20 quarterly deposits in an account at an interest rate of 12% annual compounded quarterly.
His deposits will be at the start of each quarter, the 1st one being NOW,
the final one being at the start of the 19th quarter.
3 months later, he will have 500,000 in the account.
What will be the amount of Ali's quarterly deposit?
Roger...over and out...

I agree with everything you said after your different interpretation.

Yes, working backwards you are entirely correct. I probably also should have added to the beginning payment statement an ending payment statement. That is, whether you make the final payment at the end of the total time period [there is no additional (1+i) factor] or, as in the present problem, make the final payment one period before the end of the final time period [there is the additional (1+i) factor]