Payment Plans

Stev381 · #1 $Old$ October 12th, 2009, 10:22 AM

Hi to you,

I don't know how to solve this problem:

Smith Wishes to buy a TV set and is offered a time payment plan whereby he makes 24 monthly payments of 30 each starting now. Smith wants the payments to start in 2 months rather than now. If interest is at a one-month interest rate of 1%, what is the present value now of the savigs to Smith if the seller agrees to Smith's terms?

answer :12.68$

What I made :

First find what he would have paid :
$30* \frac{(1+0.01)^{24}-1}{0.01} = 771,49$

Secondly find what he would pay if the seller agrees :

$30*\frac{(1+0.01)^{22}-1}{0.01} = 683,44$

Then one minus the other and bring back to time 0 but this is:

771,49 - 683,44 = 88,05,

$88,05*(1+0.01)^{-24} = 69.35$

Which is not the good answer.

Any help would be greatly appreciated!
Stev381

Wilmer · #2 $Old$ October 12th, 2009, 11:25 AM

If payments start NOW (as is specified in problem), then the PV
is calculated using immediate payment formula:
this results in 643.67

If he waits 2 months, then the PV in 2 months will be 637.30;
this needs to be discounted for a further 2 months:
637.30 / 1.01^2 = 624.74

So difference is 643.67 - 624.74 = 18.93

The given answer of 12.68 is not correct: it does not assume
an immediate payment.
It would be correct if 1st payment was not immediate, but in 1 month.

jonah · #3 $Old$ October 12th, 2009, 03:19 PM

Quote:

Originally Posted by Stev381 $View Post$

Hi to you,
Smith Wishes to buy a TV set and is offered a time payment plan whereby he makes 24 monthly payments of 30 each starting now. Smith wants the payments to start in 2 months rather than now. If interest is at a one-month interest rate of 1%, what is the present value now of the savigs to Smith if the seller agrees to Smith's terms?

answer :12.68$

If the seller agrees to Smith’s terms, then the selling price or present value of the TV set as specified by the time payment plan whereby he makes 24 monthly payments each starting now is represented by

$\ddot A = 30\frac{{1 - \left( {1.01} \right)^{ - 24} }}{{0.01}}\left( {1.01} \right) \approx {\rm{643}}{\rm{.674633906125}}...$

After two months, Smith’s debt to the seller balloons to

$30\frac{{1 - \left( {1.01} \right)^{ - 24} }}{{0.01}}\left( {1.01} \right)\left( {1.01} \right)^2 \approx {\rm{656}}{\rm{.612494047638}}...$

From what I could tell of the given answer of $12.68, it was calculated thus:

$\left[ {30\frac{{1 - \left( {1.01} \right)^{ - 24} }}{{0.01}}\left( {1.01} \right)\left( {1.01} \right)^2 - 30\frac{{1 - \left( {1.01} \right)^{ - 24} }}{{0.01}}\left( {1.01} \right)} \right]\left( {1.01} \right)^{ - 2}$
$\approx {\rm{12}}{\rm{.6829331845045}}...$

I don’t really see that as “the present value now of the savings to Smith if the seller agrees to Smith's terms”.
I see that more as an additional burden/payment on Smith’s part if Smith insists on such terms.
If Smith does pay $30 at the end of two months and $30 monthly thereafter, he will definitely make more than 24 payments for such a TV set.

Stev381 · #4 $Old$ October 13th, 2009, 09:22 AM

Ok thanks to both of you, I'll ask some clarification about this problem and I'll come back to you with the answer given by me professor.

Wilmer · #5 $Old$ October 13th, 2009, 09:58 AM

Just make sure your teacher agrees that IF the arrangement (as example)
is 24 monthly payments of $30 with interest of 1% (.01) per month, the
1st payment being 1 month later, then the amount borrowed (or TV cost)
is $637.30

Stev381 · #6 $Old$ October 20th, 2009, 07:51 AM

So this is what he said (and it makes sense to me) but it is true that the question is confusing:

This is a deffered annuity question. The present value of both plan are equal. (PV = 643.67$). The arrangement that Smith want to make is that he isn't charge interest during the two months he is waiting to make the first payment. So the PV of the second plan is equal to the first one but discounted two period so = 630.99$. This is the amound he will have to invest right now at the same interest rate to have 643.67$ in two months so he is saving (643.67$ - 630.99$ = 12.68$).

Wilmer · #7 $Old$ October 20th, 2009, 10:12 AM

Quote:

Originally Posted by Stev381 $View Post$

So this is what he said (and it makes sense to me) but it is true that the question is confusing:

This is a deffered annuity question. The present value of both plan are equal. (PV = 643.67$). The arrangement that Smith want to make is that he isn't charge interest during the two months he is waiting to make the first payment. So the PV of the second plan is equal to the first one but discounted two period so = 630.99$. This is the amound he will have to invest right now at the same interest rate to have 643.67$ in two months so he is saving (643.67$ - 630.99$ = 12.68$).

Disagree.

Taking the 1st plan, he makes a payment NOW, so owes 613.67.
23 more payments will pay it off: total interest = 76.33
(23 * 30 - 613.67 = 76.33)

Taking 2nd plan (with interest charged for 1st 2 months), interest of
6.44 (.01*643.67) will be added a month later (new balance = 650.11),
then the 1st payment will be received a month later: balance 626.61.

23 more payments plus a 24th payment of 16.42 will pay it off;
interest will total 92.75 (including the 1st 2 months).

92.75 - 76.33 = 16.42 is the amount he saves if no interest for 1st 2 months.

Notice that this equals the 24th payment of 16.42: that is the only
difference between the 2 plans.

Hope Sir Jonah has a look at this; he's GOT to agree!

Stev381 · #8 $Old$ October 20th, 2009, 01:35 PM

Oh I do understand your point of view,

The thing is that it is "implied" (but very, very uncleary) that he won't be charged interest during the two months that he waits to make the payment.

He wants both the PV of the payments plans to be equal...but one deffered by two months...

jonah · #9 $Old$ October 21st, 2009, 12:37 PM

Quote:

Originally Posted by Stev381 $View Post$

The thing is that it is "implied" (but very, very uncleary) that he won't be charged interest during the two months that he waits to make the payment.

Can you highlight for us where in the following original problem statement was this "implied"?

Quote:

Originally Posted by Stev381 $View Post$

Smith Wishes to buy a TV set and is offered a time payment plan whereby he makes 24 monthly payments of 30 each starting now. Smith wants the payments to start in 2 months rather than now. If interest is at a one-month interest rate of 1%, what is the present value now of the savigs to Smith if the seller agrees to Smith's terms?

Your original problem statement, as far as I can tell, contained no explicit provision, which says that Smith “won't be charged interest during the two months that he waits to make the payment.”

At any rate, I don’t think any sensible businessman would agree to Smith’s absurd offer of no interest for two months as you proposed. No interest (or rent) for two months despite the fact that the TV gets used for two months? To paraphrase Homey the clown: I don’t think so.

Any businessman would probably tell Smith to come back in two months instead when he has the first $30 down payment.