ordinary annuity...

It would seem that we are to put some money aside in order that we are able to meet a liability of $3000 that we will have in three years time. We do this by putting away $P now and $P in two years time. At the end of the three year period, we will have $2P plus interest and that will have to equal $3000.

A monthly rate of r%  equates to a yearly rate of 6% if (1 + r/100)^12 = 1.06.

That means that r = 0.486755.

So, if we invest $P now, in two years we have $P(1 + 0.486755/100)^24 = $1.1236P.

Investing a further $P takes us to $2.1236P,

and then after another twelve months we will have $2.1236P(1 + 0.486755/100)^12 = $2.251P.

Equating that to $3000 gets us P = $1332.73.

The difference of about $4 between this and the book answer due to rounding errors ?

Tiggsy.

Hi vest4r

If the debt is due in 3 years why would you have to make the final payment in 2 years ???

sorry for the delayed response, 

yeah I wrote it correctly,

this is what the textbook has the solution as,

I was just hoping someone might explain the solution better.

Thanks for your help.

This is what your textbook meant:

1)- You have a loan due in 3 years @ 6% comp. monthly.

2)- Its PV as of today is: $2,506.93

3)-They want you to equate this PV to 2 payments, 1 due today and 1 due 2 years from now.

4)-To do that, you find the PV of 1 dollar two years from now, which comes to: 0.887185668891.

5)-You add the amount in 4 above to 1 dollar due today and you ge: 1.887185668891.

6)-Then you take the PV in 2 above and divide it by the value in 5 above: $2,506.93/1.887185668891.

7)-The result in 6 above is: $1,328.40, slightly different from your calculation due to calculating the PV of 1 dollar due in 2 years to an accuracy of 12 decimal places.

Finally, in your calculations, you have used 5% compounded monthly instead of 6% compounded monthly, and yet got accurate results, which means you must have copied it from your book.

This is relatively common in the business world, but is often stated or expressed more accurately in calling the $3,000 loan, a "promissory note". So, basically what the question is, is this:

You have a promissory note due in 3 years and is to be replaced with 2-equal-amount promissory notes, one due now and the other due in 2 years from now. But the PV of the $3000 will equate the PV of the other two notes.

Tiggsy:

You got everything correct, except the interest rate of 6% compounded monthly., which means you simply divide 6% by 1200 =0.005 exactly. If you replace your calculated monthly interest rate with 0.005, your final calculation will be exactly the same as in "Guest #4" calculation of $1,328.40.

To me, (and I admit to not being into these money/finance problems), it looks like the 0.005 is an approximation to my 0.00486755

It possibly comes from the binomial expansion of (1 + r/100)^12.

\(\displaystyle \left(1+\frac{r}{100}\right)^{12}=1+12\frac{r}{100}+\frac{12.11}{2!}\left(\frac{r}{100}\right)^{2}+\frac{12.11.10}{3!}\left(\frac{r}{100}\right)^{3}+\dots\) .

Provided r/100 is small, we can say that

\(\displaystyle 1+12\frac{r}{100} \approx 1.06\), so \(\displaystyle \frac{r}{100} \approx \frac{0.06}{12}=0.005\) .

Tiggsy.

Since it states that 6% is compounded monthly, it can easily be converted to compounded annually:

[1 + 6/[12. 100]]^12 =1.005^12 =1.0616778 -1 x 100 =6.16778%- This is called the EFFECTIVE annual rate. 6% compounded monthly is called the "NOMINAL" annual rate. You took 6% as the "effective" annual rate instead of "nominal" annual rate,  and found its "nominal" monthly rate by: 1.06^(1/12) =0.486755.
 

The question stated 6% per annum compounded monthly