# Thread: Annuity Problem - deferred

1. ## Annuity Problem - deferred

A mother borrowed 100000 with 2% interest monthly and promised to pay the amount by 30 equal monthly installments starting 1 year from now. Determine the monthly payment.

a. 5551.66 c. 7551.66
b. 6551.66 d. 8551.66

thanks....

2. ## Re: Annuity Problem - deferred

First you need to get what the cash balance would be 11 months from now using $\displaystyle F = P(1 + i)^{(m*n)}$. Then use the formula $\displaystyle A = P*\dfrac{(\dfrac{r}{m})(1+\dfrac{r}{m})^{(m*n)}}{( 1+\dfrac{r}{m})^{(m*n)} - 1}$ where r is the nominal rate of interest which from your problem is 2%, n is the time in years, m is the number of times compounded in one year so in this case 12. However the answer I have computed (3483.34) is not among your choices.. so either I am wrong or your choices or given are wrong.

3. ## Re: Annuity Problem - deferred

Originally Posted by catenary
However the answer I have computed (3483.34) is not among your choices..
It shouldn't be.
Originally Posted by catenary
so either I am wrong or your choices or given are wrong.
The given is just fine.

Accordingly, we have the following equation of value :

100000=R*[1-(1.02)^(-30)]/.02*(1.02)^(-11)

where R is the monthly payment.

4. ## Re: Annuity Problem - deferred

Originally Posted by catenary
First you need to get what the cash balance would be 11 months from now using $\displaystyle F = P(1 + i)^{(m*n)}$. Then use the formula $\displaystyle A = P*\dfrac{(\dfrac{r}{m})(1+\dfrac{r}{m})^{(m*n)}}{( 1+\dfrac{r}{m})^{(m*n)} - 1}$ where r is the nominal rate of interest which from your problem is 2%, n is the time in years, m is the number of times compounded in one year so in this case 12.
Just out curiosity, how did you apply your formulas?

5. ## Re: Annuity Problem - deferred

Originally Posted by jonah
Just out curiosity, how did you apply your formulas?
Well neither does your equation seem to give a value among the choices..

Here is how it is applied:
$\displaystyle F = 100000(1+\frac{.02}{12})^{12*\frac{11}{12}} = 101,848.6878$ It is 11 months because we start computing the payment of the first annuity based off this which will be a month after the initial principal for which it is computed.
Then,
$\displaystyle 101,848.6878*\dfrac{(\frac{.02}{12})(1+\frac{.02}{ 12})^{(12*2.5)}}{(1+\frac{.02}{12})^{(12*2.5)} - 1} = 3483.365176$

6. ## Re: Annuity Problem - deferred

Originally Posted by catenary
Well neither does your equation seem to give a value among the choices..
Actually, it does. It really does. It's practically a dead giveaway to the OP minus the analysis of course. It's up to him or her to calculate the monthly payment as depicted in my equation of value. Since you were not able to come up with one of the choices using my equation

Originally Posted by catenary
Here is how it is applied:
$\displaystyle F = 100000(1+\frac{.02}{12})^{12*\frac{11}{12}} = 101,848.6878$ It is 11 months because we start computing the payment of the first annuity based off this which will be a month after the initial principal for which it is computed.
Then,
$\displaystyle 101,848.6878*\dfrac{(\frac{.02}{12})(1+\frac{.02}{ 12})^{(12*2.5)}}{(1+\frac{.02}{12})^{(12*2.5)} - 1} = 3483.365176$
Alas Sir catenary, I must object on how you misapplied the fundamental compound interest formula. You clearly (as far as my happy inebriated brain could tell) plugged in incorrect values. You may need a review of nominal and effective rates. And what about that other formula? Whoever gave you that formula must be out of his mind.

Cheers.

7. ## Re: Annuity Problem - deferred

The answer in book is "a", but I cannot come up with a correct solution. thanks for helping...

8. ## Re: Annuity Problem - deferred

Originally Posted by lynyrdz
The answer in book is "a", but I cannot come up with a correct solution. thanks for helping...
Is this post an indication that you already worked out the simple algebra of my equation of value (see my 1st post) to match your book.

Come to think of it, I guess my 1st post was practically a spoon fed solution.

9. ## Re: Annuity Problem - deferred

I realize that I mistook the 2% stated in the problem to be the nominal rate of interest(r). Turns out it is actually the rate of interest per period(i). you can replace all of the $\displaystyle \frac{r}{m}$ in the formula I gave with just 'i' now, because $\displaystyle i = \frac{r}{m}$ and it now gives "a" from the choices.

Im not certain, but I think that when interpreting interest problems, if you have a statement like "with 2% interest monthly" then it means that 2% is the rate of interest per period while the number of compounding periods in one year is 12, but if you have a statement like "with interest at 2% compounded monthly" then it means that 2% is the nominal rate of interest while the number of compounding periods is still 12. credits to sir jonah for nudging me towards this direction.

10. ## Re: Annuity Problem - deferred

compound amount after 1 year
CA =100,000* (1+.02)^12 = 100,000* (1.268) = 126800
capital recovery in 30 months =126800* (.02 ) (1.02)^30/(1.02)^30-1= 126800*(.0447)=5668 per month

11. ## Re: Annuity Problem - deferred

[deleted what I wrote, sorry for unintentionally bumping the thread; accident]

12. ## Re: Annuity Problem - deferred

WARNING: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall 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.

Originally Posted by bjhopper
compound amount after 1 year
CA =100,000* (1+.02)^12 = 100,000* (1.268) = 126800
Must object to this intermediate rounding. We might as well use all the digits of a calculator's display (or stored value) to achieve the highest possible accuracy. Thus,
100,000*(1+.02)^12=126,824.17945625... or 126,824.18
Rounding to the nearest cent might be acceptable in some circles but shaving \$24.18 is way too much rounding.
Originally Posted by bjhopper
capital recovery in 30 months =126800* (.02 ) (1.02)^30/(1.02)^30-1= 126800*(.0447)=5668 per month
Not entirely sure what kind of grouping went on here but by WolframAlpha (go here),
(.02 ) (1.02)^30/(1.02)^30-1 is definitely not equal to .0447

Clearly, the end of one year was used as the comparison date.
It should have been (using compound interest formula and annuity due version of the amortization formula)
100000(1.02)^12=R*[1-(1.02)^(-30)]/.02*(1.02)
Whatever comparison date you choose, be it the present, the end of 11 months, the end of 1 year, etc., the monthly payment will be always be the same.
With the end of the 41st month (11 + 30) as comparison date, we would have
100000(1.02)^41=R[(1.02)^30-1]/.02