John loaned $72000 from the bank to be paid over 10 years. Given that the bank charges compound interest, calculate the annual interest rate if his monthly instalments over the 10 years was $890

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- Jul 2nd 2010, 04:14 AMPunchCompound Interest
John loaned $72000 from the bank to be paid over 10 years. Given that the bank charges compound interest, calculate the annual interest rate if his monthly instalments over the 10 years was $890

- Jul 2nd 2010, 06:13 AMSoroban
Hello, Punch!

I believe this problem is unsolvable by ordinary means.

Quote:

John loaned $72,000 from the bank to be paid over 10 years.

The bank charges compound interest,

Calculate the annual interest rate if his monthly installments was $890.

This is anproblem.*Amortization*

$\displaystyle \text{Formula: }\;A \;=\;P\,\dfrac{i(1+i)^n}{(1+i)^n-1}$

. . $\displaystyle \text{where: }\;\begin{Bmatrix}A &=& \text{periodic payment} \\ P &=& \text{principal} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}$

We are given: .$\displaystyle A = \$890,\;P = \$72,\!000,\;n = 120\text{ months}$

We want the annual interest rate, $\displaystyle r.$

. . The*periodic*interest rate is: .$\displaystyle i = \frac{r}{12}$

Substitute: . $\displaystyle 890 \;=\;72,\!000\,\dfrac{i(1+1)^{120}}{(1+i)^{120}-1} $

Now*all you have to do*is solve for $\displaystyle i$ . . . then multiply by 12.

Good luck!

I'll wait in the car . . . - Jul 2nd 2010, 09:09 AMWilmer
Easier to use P = Ai / (1 - f) where f = 1 / (1 + i)^n

Rearrange to get P/A = i / (1 - f)

"Hit and miss" (iteration) required to get i.

If i / (1 - f) > P/A then i is too high...

Keep adjusting i until both sides are reasonably equal...

Get my drift?

For this one, i = .702492.... is VERY close, hence .702492 * 1200 = ~8.43% - Jul 2nd 2010, 09:32 AMskeeter
I love the Finance application on the TI-84 ...

- Jul 2nd 2010, 10:52 AMWilmer
No matter how "simple and cute" the calculator appears,

the dirty work still goes on behind the scenes... - Jul 3rd 2010, 09:03 AMPunch
The only formula I have learnt about is $\displaystyle A=P(1+\frac{i}{100})^n$

$\displaystyle A= 890 *12 * 10$

$\displaystyle = 106800$

therefore, $\displaystyle A=P(1+\frac{i}{100})^n$

$\displaystyle 106800=72000(1+\frac{i}{100})^{10}$

$\displaystyle i = 4.02%$

why is my answer different from all of yours? - Jul 3rd 2010, 10:35 AMWilmer
That's the formula to calculate the Future value of an amount invested now:

NOT of a series of deposits or payments.

Like, if $3000 is deposited NOW in an account paying 9% annually,

after 8 years the $3000 will have accumulated to:

3000(1 + 9/100)^8 = 5977.68792...$5977.69 rounded

That formula does NOT apply in your problem's case.

You were given the formula previously by Soroban and myself.

If you are unable to realise that the rate i CANNOT be solved for directly,

then you need classroom help which cannot be provided here ...