# Thread: Random basic math question

1. Originally Posted by ptgatsby
Right, so the teacher answered and gave me... uhhh... something that made no sense (copied from the answer sheet, no doubt). This is what I got back;

1x + x(1+0.13/2)^1.5 =
1x +x(1.065)^1.5 =
1x + x(.9098621) = 1.909862x

But of course (1.065)^1.5 cannot be less than 1... unless... it is actually;

(1.065)^-1.5

Which means I may have my PV/FV wrong... although it looks right to me!
Care to share the original formal problem? There are errors in solution manuals (generally speaking).

So far, I have agreed with your solution for x, and it is clear that you can't take a number bigger than one to a positive power and get a number less than 1.

The following would make sense to me:

x + x(1+0.13/2)^-1.5 =
x + x(1.065)^-1.5 =
x + x(.9098621) = 1.909862x

So would this:

x(1+0.13/2)^1.5 =
x(1.065)^1.5 =
approx. x(1.09) = 1.09x

2. Originally Posted by ygolo
Care to share the original formal problem? There are errors in solution manuals (generally speaking).
I hesitate to post it because it is a marked paper and I'm not sure how 'random' they are... let me see what I get back and if it still isn't resolved, I'll post the problem.

3. 1.065^-1.5 = .909862128

It appears the answer sheet left out a sign.

4. Originally Posted by The_Liquid_Laser
1.065^-1.5 = .909862128

It appears the answer sheet left out a sign.
I'm hoping they accidently included the sign cause that'll get me two extra marks dammit!

(I'm pretty sure that is the case.)

5. I ended up being wrong (which in hindsight was obvious, of course). Ah well.

For all those that wanted to know the problem.... Here it is (numbers have been changed )

Three old payments on a debt were arranged at \$1800 due in 6 months paying 11.5% compounded quarterly, \$1100 due in 12 months paying 12% compounded monthly and \$500 due in 18 months paying 9.5% compounded semi-annually. These are to be replaced by two equal payments due in 9 months and 18 months respecitvely. What is the size of these payments if money is worth 13% compounded semi-annually? (Use 9 months from now as the focal date).

6. Originally Posted by ptgatsby
I ended up being wrong (which in hindsight was obvious, of course). Ah well.

For all those that wanted to know the problem.... Here it is (numbers have been changed )

Three old payments on a debt were arranged at \$1800 due in 6 months paying 11.5% compounded quarterly, \$1100 due in 12 months paying 12% compounded monthly and \$500 due in 18 months paying 9.5% compounded semi-annually. These are to be replaced by two equal payments due in 9 months and 18 months respecitvely. What is the size of these payments if money is worth 13% compounded semi-annually? (Use 9 months from now as the focal date).
The present value of the first three payments are 1700.80, 976.19, and 435.02 respectively. Total present value is \$3112.01.

You can then solve for the size of the two monthly payments with this equation:

3112.01 = x*(1 + .13/2)^-1.5 + x*(1+.13/2)^-3

If you multiply this expression by 1.065^1.5, then it is equivalent to

3420.31 = x + x(1.065)^-1.5

Solving for x yields \$1790.87 for each of the two payments. (This is assuming that I've made no mistakes which is always possible.)

7. Originally Posted by The_Liquid_Laser
The present value of the first three payments are 1700.80, 976.19, and 435.02 respectively. Total present value is \$3112.01.
I believe the PV would be 1788.72 or so (for the first).

FV=1800(1+0.115/4)^2= 1904.9878
PV=1904.9878(1+0.13/2)^-1 = \$1788.72

The contract has to be brought to completion (unless it is open, I suppose) before bringing finding today's value.

8. Originally Posted by ptgatsby
I believe the PV would be 1788.72 or so (for the first).

FV=1800(1+0.115/4)^2= 1904.9878
PV=1904.9878(1+0.13/2)^-1 = \$1788.72

The contract has to be brought to completion (unless it is open, I suppose) before bringing finding today's value.
I don't quite follow your logic. Perhaps it comes from bringing the "contract to completion". Is the debt being refinanced at time zero or 9 months from now? I was working the problem as if the debt is being refinanced at time zero.

Also I don't understand this line:
FV=1800(1+0.115/4)^2= 1904.9878
This calculation assumes that the first payment is worth \$1800 at time zero, and will be worth 1904.9878 at the time of payment.

On the other hand if you assume that the payment is worth \$1800 six months from now, then the payment is worth 1800(1 + .115/4)^-2 = 1700.80 at time zero.

9. Originally Posted by The_Liquid_Laser
On the other hand if you assume that the payment is worth \$1800 six months from now, then the payment is worth 1800(1 + .115/4)^-2 = 1700.80 at time zero.
Probably bad wording on my behalf (well, my course anyway ), but it isn't a "strip bond" value of 1800, it's 1800 + interest (at 11.5%). 1700.80 is what you would pay to get exactly 1800 at 11.5%. It should read "A debt of 1800, due in 6 months, bearing an interest rate of 11.5%" (paying in the original problem refers to interest payments being made, but is not an annuity, thus compounded.)

In either case, the value wouldn't be 1700.80 - the PV calc would have to work off the current value of money - in theory, one wouldn't pay 1700.80 if money was worth 13% (actual value of the FV 1700.80 in this situation is 1810.5). That is, someone could buy the exact same instrument at current rates to have 1810.50 rather than 1800 at 6 months. The 11.5% one would be discounted because of the change in interest rates.

(The question asks for the reference point being 9 months, but that's for the markers... should be the same answer either way, or extremely close.)

10. Originally Posted by ptgatsby
Probably bad wording on my behalf (well, my course anyway ), but it isn't a "strip bond" value of 1800, it's 1800 + interest (at 11.5%). 1700.80 is what you would pay to get exactly 1800 at 11.5%. It should read "A debt of 1800, due in 6 months, bearing an interest rate of 11.5%" (paying in the original problem refers to interest payments being made, but is not an annuity, thus compounded.)

In either case, the value wouldn't be 1700.80 - the PV calc would have to work off the current value of money - in theory, one wouldn't pay 1700.80 if money was worth 13% (actual value of the FV 1700.80 in this situation is 1810.5). That is, someone could buy the exact same instrument at current rates to have 1810.50 rather than 1800 at 6 months. The 11.5% one would be discounted because of the change in interest rates.

(The question asks for the reference point being 9 months, but that's for the markers... should be the same answer either way, or extremely close.)
Hmm...I thought that if I had made a mistake then it would probably be in the terminology. All of the math is essentially the same regardless of context, but you have to know the exact meaning of the terms in order to work the problem. For example I am still not sure what it means to have money "worth 13%". I assumed it meant that the debts were refinanced at 13%.

Anyway now that I know what you mean with regard to the debt payments. I'd work the problem this way. First get the value of each of the first three payments at 9 months.

Payment 1 at 6 months is 1800(1 + .115/4)^2 = 1904.9878
Advancing this to 9 months yields 1904.9878(1.065)^.5 = 1965.9253

Payment 2 at 12 months is 1100(1 + .12/12)^12 = 1239.5075
Discounting back to 9 months yields 1239.5075(1.065)^-.5 = 1201.0867

Payment 3 at 18 months is 500(1 + .095/2)^3 = 574.6880
Discounting back to 9 months yeilds 574.6880(1.065)^-1.5 = 522.8868

Adding these three numbers yields a value of \$3689.90 at 9 months. To find the value of the two payments to be made, solve this equation:

3689.90 = x + x(1.065)^-1.5

solving yields x = \$1932.02 per payment

(Again this assumes I understand all the terminology now, and haven't made any calculation errors.)

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