# Simple Interest Problem: Payments of $4000 each due in 4, 9, #### Jason ##### New member Hi Everyone, I am having a small problem here im not realy looking for the answer to this question rather a little push in the right direction. Here is my question: 1. Payments of$4000 each due in four, nine, and eleven months from now are to be settled by three equal payments due today, six months from now, and twelve months from now. What is the amount of equal payments if interest is 7.35% and the agreed focal date is today.

So do i bring all the amounts back to the focal date(today) by using 4000/1+7.35%(4/12) then bring all of the amounts back up to the new dates? then where do i go from there. I realize this is probaly a very simple question but its all new to me. I appreciate any help that can be given to me.

Jason

#### tkhunny

##### Moderator
Staff member
Re: Simple Interest Problem

Jason said:
1. Payments of $4000 each due in four, nine, and eleven months from now are to be settled by three equal payments due today, six months from now, and twelve months from now. What is the amount of equal payments if interest is 7.35% and the agreed focal date is today. Let's just build the payment streams. Worry about the interest rate later. 4000*(v^4 + v^9 + v^11) = Pmt*(v^0 + v^6 + v^12) You must see where this comes from. Now for the interest rate. As used above, 'v' is a monthly interest discount. Convert that 7.35% annual to a monthly rate and you should be close to done. #### Denis ##### Senior Member Re: Simple Interest Problem Yes, you start by calculating present value of each; not sure what you mean by "simple interest"; I'll assume no compounding at any time; calculate the present values this way (yours looks kinda messy!): 4000 / (1 + .0735/12*4) 4000 / (1 + .0735/12*9) 4000 / (1 + .0735/12*11) These will add up to 11442.88 Since interest does not compound, but a payment is made in 6 months, then the 7.35 must be reduced to it's semi-annual equivalent: (1 + r)^2 = 1.0735 r = .0366098... : ~7.21969% Using 11442.88 as amount borrowed, .0366098 as semi-annual rate, and 3 as number of payments (1st at beginning) will result in 3 payments of 3950.33 I assume you're aware of the loan payment formula. #### Jason ##### New member Thank you very much, this was exactly what i was looking for. I really appreciate your help. Jason #### jonah2.0 ##### Junior Member DISCLAIMER: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall the wandering math knight-errant Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views. Whilst googling for equations of value as it applies to simple interest, Google led me to this 9 year old post. I thought perhaps I give it another opinion, late and useless as it is to Jason, just to let the readers be aware of another possible interpretation. 1. Payments of$4000 each due in four, nine, and eleven months from now are to be settled by three equal payments due today, six months from now, and twelve months from now. What is the amount of equal payments if interest is 7.35% and the agreed focal date is today.
Let's just build the payment streams. Worry about the interest rate later.

4000*(v^4 + v^9 + v^11) = Pmt*(v^0 + v^6 + v^12)

You must see where this comes from.

Now for the interest rate. As used above, 'v' is a monthly interest discount. Convert that 7.35% annual to a monthly rate and you should be close to done.
By Sir tkhunny's suggestion back then, it would seem that with
$$\displaystyle v = \frac{1}{{1 + i}} = \left( {1 + i} \right)^{ - 1}$$
and IF $$\displaystyle i = \left( {1.0735} \right)^{\tfrac{1}{{12}}} - 1$$
then
$$\displaystyle 4000*\left( {v^4 + v^9 + v^{11} } \right) = Pmt*\left( {v^0 + v^6 + v^{12} } \right)$$
$$\displaystyle \Leftrightarrow Pmt = {\text{3951}}{\text{.9387222968}}$$

Yes, you start by calculating present value of each; not sure what you mean
by "simple interest"; I'll assume no compounding at any time; calculate the
present values this way (yours looks kinda messy!):
4000 / (1 + .0735/12*4)
4000 / (1 + .0735/12*9)
4000 / (1 + .0735/12*11)

These will add up to 11442.88

Since interest does not compound, but a payment is made in 6 months,
then the 7.35 must be reduced to it's semi-annual equivalent:
(1 + r)^2 = 1.0735
r = .0366098... : ~7.21969%

Using 11442.88 as amount borrowed, .0366098 as semi-annual rate,
and 3 as number of payments (1st at beginning) will result in 3
payments of 3950.33

I assume you're aware of the loan payment formula.
In contrast, by Sir Denis' suggestion back then, it would seem that
IF $$\displaystyle i = \left( {1.0735} \right)^{\tfrac{1}{2}} - 1$$ then
$$\displaystyle {\text{11442}}{\text{.88 = }}Pmt*\left( {v^0 + v^1 + v^2 } \right)$$
$$\displaystyle \Leftrightarrow Pmt = {\text{3950}}{\text{.3273087763}}$$

By Sir jonah's suggestion, this would be
$$\displaystyle {\text{4000*}}\left( {\frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{\text{4}}}{{{\text{12}}}}}} + \frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{\text{9}}}{{{\text{12}}}}}} + \frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{{\text{11}}}}{{{\text{12}}}}}}} \right)$$
$$\displaystyle {\text{ = }}Pmt*\left( {\frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{\text{0}}}{{{\text{12}}}}}} + \frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{\text{6}}}{{{\text{12}}}}}} + \frac{{\text{1}}}{{{\text{1 + }}{\text{.0735*}}\tfrac{{{\text{12}}}}{{{\text{12}}}}}}} \right)$$

$$\displaystyle \Leftrightarrow Pmt = {\text{3951}}{\text{.152948515}}$$

Hey Bob (Barker), I mean Drew (Carey), which one wins the showcase?

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#### suppressive

##### New member
I got my assignment back, it had the same question on it with different values. I used the steps Jonah suggested and it was marked as correct.

1. Payments of \$5100 each due in five, ten, and fourteen months from now are to be settled by three equal payments due today, seven months from now, and eleven months from now. What is the size of the equal payments if interest is 8.75% and the agreed focal date is today?

P = 5100 / (1+0.0875/12*5) =4920.6
P = 5100 / (1+0.0875/12*10)= 4753.4
P = 5100 / (1+0.0875/12*14)= 4627.599

Adding these up, we get:14301.6

Calculating the unknownequivalent payment for the next set of months:

14301.6 = x +x/(1+0.0875/12*7) + x/(1+0.0875/12*11)

14301.6 = 0.95143675x +0.925747634x + x

14301.6 = 2.877184384x

4970.692904 = x

Thanks again guys 