Simple Interest Question involving focal dates and equivalent payments.

[suppressive]

Hey all, first time poster here.

I'm working on a problem to which I have the answer to, but I can't seem to figure out how I get there.

Payments of 4000 each due in four, eight and twelve months respectively are to be
settled by five equal payments due today, three months from now, six months from now, nine
months from now and twelve months from now. What is the size of the equal payments if
interest is 12.75% and the agreed focal date is today?

Answer: 2351.17

So what I did so far was calculate the present value payments for each of the first three time periods listed in the question.

Present value for four months:
=4000/(1+0.1275/12*4) = 3836.93

Present value for eight months:
=4000/(1+0.1275/12*8) = 3686.636

Present value for twelve months:
=4000(1+0.1275/12*12) = 3547.672

Adding these together gives me:

11071.24


I'm not sure where to go from here however. Dividing that sum by 5 gives me the wrong answer so I'm trying to see what else I can do from here. Any advice/tips would be appreciated

 

[Ishuda]

Hey all, first time poster here.

I'm working on a problem to which I have the answer to, but I can't seem to figure out how I get there.



Answer: 2351.17

So what I did so far was calculate the present value payments for each of the first three time periods listed in the question.

Present value for four months:
=4000/(1+0.1275/12*4) = 3836.93

Present value for eight months:
=4000/(1+0.1275/12*8) = 3686.636

Present value for twelve months:
=4000(1+0.1275/12*12) = 3547.672

Adding these together gives me:

11071.24


I'm not sure where to go from here however. Dividing that sum by 5 gives me the wrong answer so I'm trying to see what else I can do from here. Any advice/tips would be appreciated

You did a present value assuming simple interest. Were that the correct way to do it, you now need to use the amount to compute a payment going the other way taking into account the interest you will own. That is, you will make a payment immediately, will owe interest on the remainder for 3 months, then make a payment and owe interest on the remainder for 3 months and make a payment, etc.

However, I would have done it a different way. Consider that you are paying off a loan. Use the proper PV formula [the one with payments at the end of the period] to compute the value of what you have 'borrowed'. Now use the proper PV formula [the one with payments at the beginning of the period] to compute your payments.

 

[jonah2.0]

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.

Hey all, first time poster here.

I'm working on a problem to which I have the answer to, but I can't seem to figure out how I get there.

Payments of 4000 each due in four, eight and twelve months respectively are to be
settled by five equal payments due today, three months from now, six months from now, nine
months from now and twelve months from now. What is the size of the equal payments if
interest is 12.75% and the agreed focal date is today?

Answer: 2351.17

So what I did so far was calculate the present value payments for each of the first three time periods listed in the question.

Present value for four months:
=4000/(1+0.1275/12*4) = 3836.93

Present value for eight months:
=4000/(1+0.1275/12*8) = 3686.636

Present value for twelve months:
=4000(1+0.1275/12*12) = 3547.672

Adding these together gives me:

11071.24


I'm not sure where to go from here however. Dividing that sum by 5 gives me the wrong answer so I'm trying to see what else I can do from here. Any advice/tips would be appreciated

Standard equations of value for simple interest.
The next step would be to equate 11071.24 with the present value of the 5 equal unknown payments.
Proceed as you did with when you "calculated the present value payments for each of the first three time periods". Thus, X + X/(1+.1275*3/12) + ... etc.
Equivalently, you can't could also write X + X(1+.1275*3/12)^(-1) + ... etc.
You should have no trouble getting your desired answer provided that you don't do any intermediate rounding for any coefficient of X.

 

[suppressive]

Standard equations of value for simple interest.
The next step would be to equate 11071.24 with the present value of the 5 equal unknown payments.
Proceed as you did with when you "calculated the present value payments for each of the first three time periods". Thus, X + X/(1+.1275*3/12) + ... etc.
Equivalently, you can't could also write X + X(1+.1275*3/12)^(-1) + ... etc.
You should have no trouble getting your desired answer provided that you don't do any intermediate rounding for any coefficient of X.

Hey guys thanks for the response!

By equate do you mean do something like this:

11071.24 = E1+E2+E3+E4+E5

E1 = 4000/(1+0.1275/ ? )
E2 =4000/(1+0.1275/12*3)
E3 = 4000/(1+0.1275/12*6)
E4 = 4000/(1+0.1275/12*9)
E5 = 4000/(1+0.1275/12*12)

Just a few questions, since the first payment out of the 5 is due today, what would I use for the time variable in that formula? Would it just be zero? Or is that what I am solving for?

I tried solving for E1 using the equation I listed above however I got a different answer. (3764.04)

 

[jonah2.0]

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.

By equate do you mean do something like this:

11071.24 = E1+E2+E3+E4+E5

E1 = 4000/(1+0.1275/ ? )
E2 =4000/(1+0.1275/12*3)
E3 = 4000/(1+0.1275/12*6)
E4 = 4000/(1+0.1275/12*9)
E5 = 4000/(1+0.1275/12*12)

Just a few questions, since the first payment out of the 5 is due today, what would I use for the time variable in that formula? Would it just be zero? Or is that what I am solving for?

I tried solving for E1 using the equation I listed above however I got a different answer. (3764.04)

You didn't read my absinthe powered post carefully.
No worries. I still love you.
Your post:
4000/(1+0.1275/ ? ) + 4000/(1+0.1275/12*3) + ...
My post:
X + X/(1+.1275*3/12) + ...

 

[suppressive]

You didn't read my post carefully.
No worries. I still love you.
Your post:
4000/(1+0.1275/ ? ) + 4000/(1+0.1275/12*3) + ...
My post:
X + X/(1+.1275*3/12) + ...

I think I love you more cause I was able to get to the answer!!! I understand why now too! 3 equal payments of 4000 is not going to add up to 5 equal payments of 4000, so we put x there and solve for the unknown equivalent payment for 5 payments

Thanks so much buddy <3 this question had me stumped lol

 

[Ishuda]

The way I would have done it:
Let
PV(i,n) = \(\displaystyle \dfrac{i}{1-\dfrac{1}{(1+i)^n}}\)

A payment p at the end of the period on principle amount borrowed P is given by
p = P * PV(i,n)
where i is the interest rate per period and n is the number of payments. This type of payment is sometimes called payment due or payment annuity due.

We are given three payments of $4000 at a period of 1/3 year with an annual rate of 12.75%. To determine P we have i =.1275/3=.04125. Thus
P =4000 / PV(0.04125,3) = 11074.08

A payment p at the beginning of the period on principle amount borrowed P is given by
p = P * PV(i,n) / (1+i)
where i is the interest rate per period and n is the number of payments. This type of payment is sometimes called payment immediate or payment annuity immediate.

We now want to know what would be for five payments on 11074.08 if we started payments immediately at a period of 1/4 year at an annual interest rate of 12.75%. To determine p we have i = .1275/4 = 0.030938. Thus
p = 11074.08 * PV(0.030938,5) / 1.030938 = 2351.79

4 months	Payment	Interest	Balance	3 months	Payment	Interest	Balance
		0.04125				0.030938	11074.08
0			11074.08	0	2351.79	0.00	8722.29
4	4000.00	456.81	7530.89	3	2351.79	269.85	6640.35
8	4000.00	310.65	3841.54	6	2351.79	205.44	4494.00
12	4000.00	158.46	0.00	9	2351.79	139.03	2281.24
				12	2351.79	70.58	0.03

 

[suppressive]

@Denis

I was doing some practice questions from this pdf on a random university's website and it had the solutions posted. Here's the link: http://www.mif.vu.lt/matinf/asm/bg/pas/1FMa.pdf
Press ctrl+f on the pdf and type in "payments of 4000" and it should take you to the question #22

We haven't learned that formula in class yet, so I'm not familiar of it as of now. I'm not really sure how often the interest is compounded; the question doesn't really say so :/ I guess the 'default' option to assume would be annually? But then again since there's 5 months it may suggest that it's compounded quarterly. Hmm.

P.s thanks for the bank account format, that really makes it easier to visualize what's going on.


@Ishuda

That way seems a lot less tedious. But I still have yet to be introduced to that formula so I think my professor wants us to use whatever tools we currently have. The answers slightly off from what the solution suggests (.79) but it could be due to rounding.



@Denis

I haven't been taught the formula, but when I was playing around with the question earlier, that was one of the things I thought of doing. I ended up looking for a formula to find the quarterly rate which was:

(1+0.1275)^0.25 - 1 = 0.030455253
0.1275 being the annual rate, which is converted to the quarterly rate hence the 0.25.

I'm not sure how to answer the question you gave me though to be honest. I would think you'd have to convert the trimesterly rate to the annual rate, and then using the formula I have used above, you would convert it to find the quarterly rate.

I just haven't learned how to convert from trimesterly -> annual. :/


Btw, thanks for the replies, you guys are super helpful.

 

[jonah2.0]

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.
\(\displaystyle 4000\left( {1 + .1275*\tfrac{4}{{12}}} \right)^{ - 1} + 4000\left( {1 + .1275*\tfrac{8}{{12}}} \right)^{ - 1} + 4000\left( {1 + .1275*\tfrac{{12}}{{12}}} \right)^{ - 1}\)
\(\displaystyle = X + X\left( {1 + .1275*\tfrac{3}{{12}}} \right)^{ - 1} + X\left( {1 + .1275*\tfrac{6}{{12}}} \right)^{ - 1} + X\left( {1 + .1275*\tfrac{9}{{12}}} \right)^{ - 1} + X\left( {1 + .1275*\tfrac{{12}}{{12}}} \right)^{ - 1}\)

\(\displaystyle X \approx {\text{2351}}{\text{.17083466542}}...\)

As I said from the beginning, standard equations of value for simple interest.
Thread title says "Simple Interest Question involving focal dates and equivalent payments."
Thus, simple interest was used. The problem and the given answer has all the indications of having been taken from a finance math textbook.
For additional information on this not so often encountered equation of value involving simple interest, see
Equations of Value for Simple Interest, Not so simple.

Notwithstanding all this business, management gives profuse thanks to Sir Denis The Sober and Sir Ishuda for their comments, opinions and reckonings. Rest assured that while their computations came very close to the given answer, management has decided to go in Sir jonah's direction.

 

[jonah2.0]

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.

I beg to disagree Sir Jonah.
I had tried simple interest (amongst a few other ways) and was
not able to hit 2351.17 with any of them.

I love you Sir Denis!!!
I guess we'll just have to agree to disagree then.

2351.17 is indeed what your equations yield, but I'm ready to
bet the formula you use is faulty for the purpose of simple interest;

You would lose that bet should I care to accept it. What would you care to bet anyway?
I'm thinking you should write "I love you" back to me if you should get convinced over time that "my" formula is not faulty. Knowing you however, it's a fairly good bet that it may take some time (and an open mind on your part) before you're convinced by "my" formula.

not sure why (yet) but I think because of the use of ^(-1).

Last time I checked
\(\displaystyle F = P(1 + rt) \Leftrightarrow P = \frac{F}{{(1 + rt)}} \Leftrightarrow P = F\frac{1}{{(1 + rt)}} \Leftrightarrow P = F(1 + rt)^{ - 1} \)

Your formula results in a PV of 11071.24.
11071.24 * 1.1275 = 12482.83 ; that should be 12510.00

4000 @ 12.75% (8 months) = 4340.00
4000 @ 12.75% (4 months) = 4170.00
4000 @ 12.75% (0 months) = 4000.00
add 'em up: 12510.00

Your contention is duly noted.

So present value should be 11095.34:
11095.34 * 1.1275 = 12510.00

Agreed.

Such calls for the 5 payments to be 2352.0564... each.

How on earth did you manage to conjure that?

r = .1275
2352.0564[5 + r + (3/4)r + (1/2)r + (1/4)r] = 12510.00

They do tally but I'm afraid you've lost me.

If that formula comes from another site, then that site is wrong!

I'm afraid that "formula" is apparently quite endemic in not just one site but in several others and (shocking) several other books as well. I respectfully suggest that you upgrade your background knowledge in equation of value as it applies to compound interest to include applications in simple interest as well.

No matter how we "cook it", BOTH cases must have FV of 12510.00

Indeed. But only if the focal or comparison date is at the end of 12 months.
Accordingly,under a simple interest scenario for equations of value, the unknown payment is not as constant as they are in a compound interest setting. They vary a little. At the end of 12 months we'd have
\(\displaystyle 12510 = X\left( {1 + .1275*\frac{{12}}{{12}}} \right) + X\left( {1 + .1275*\frac{9}{{12}}} \right) + X\left( {1 + .1275*\frac{6}{{12}}} \right) + X\left( {1 + .1275*\frac{3}{{12}}} \right) + X\)
\(\displaystyle \Leftrightarrow X \approx {\text{2352}}{\text{.05640423032}}\)

Management thanks you profusely for your comments, Sir D.T. Sober. Rest assured
that such will be duly noted in your Personnel (yes, 2 n's) File, and seriously taken in
consideration at your forthcoming Annual Performance Review (not to be confused with
the APR associated with financial jargon).

 

[jonah2.0]

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.

I "know" mine's the correct(!) result, but still
can't see exactly why yours is "so close yet so far"

What?!!!
Even with suppressive's quoted pdf source complete with discussion and answer keys?
And still you hold out hope that you're correct?
Well you know what they say about hope from The Shawshank Redemption.
Andy Dufresne: [in letter to Red] Remember Red, hope is a good thing, maybe the best of things, and no good thing ever dies.
You must be sipping on some good coffee.
Civet coffee perhaps?
Heard elephant coffee is the rage these days.

 

[Ishuda]

...
The "answer" of 2351.17 you provided is not quite right;
should be 2348.86
...
You said you got the 2351.17 as your solution:
I don't believe you...show me

SIMPLE INTEREST: PV= P/[1+M * i]
Monthly Interest	0.010625		0.010625
Payment	4000		2351.17
Months	PV	Months	PV
4	3836.93	0	2351.17
8	3686.64	3	2278.54
12	3547.67	6	2210.27
		9	2145.96
		12	2085.29
TOTAL	11071.24		11071.23

 

[jonah2.0]

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.

Seriously Ishuda, I always thought that simple interest worked this way:

00                              .00
04    4000.00    (   .00)   4000.00
08    4000.00    (170.00)   8000.00
12    4000.00    (340.00)  12000.00
12                510.00   12510.00

In other words, absolutely no compounding...
Guess I was wrong...but secretly right

No worries.
I still love you.

Doubt thou the stars are fire;
Doubt that the sun doth move;
Doubt truth to be a liar;
But never doubt I love.

William Shakespeare, Hamlet

 

[Ishuda]

Seriously Ishuda, I always thought that simple interest worked this way:

00                              .00
04    4000.00    (   .00)   4000.00
08    4000.00    (170.00)   8000.00
12    4000.00    (340.00)  12000.00
12                510.00   12510.00

In other words, absolutely no compounding...
Guess I was wrong...but secretly right

And Denise, you are oh so right if one is looking forward to the future but, in this case, I am a backward child and must work backwards from the future.