Compound Interest

**Punch** · July 2nd 2010, 04:14 AM

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

**Soroban** · July 2nd 2010, 06:13 AM

Hello, Punch!

I believe this problem is unsolvable by ordinary means.

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 an Amortization problem.

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

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

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

We want the annual interest rate, $r.$
. . The periodic interest rate is: . $i = \frac{r}{12}$

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

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

Good luck!

I'll wait in the car . . .

**Wilmer** · July 2nd 2010, 09:09 AM

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%

**skeeter** · July 2nd 2010, 09:32 AM

I love the Finance application on the TI-84 ...

**Wilmer** · July 2nd 2010, 10:52 AM

No matter how "simple and cute" the calculator appears,
the dirty work still goes on behind the scenes...

**Punch** · July 3rd 2010, 09:03 AM

The only formula I have learnt about is $A=P(1+\frac{i}{100})^n$

$A= 890 *12 * 10$
$= 106800$

therefore, $A=P(1+\frac{i}{100})^n$
$106800=72000(1+\frac{i}{100})^{10}$
$i = 4.02%$

why is my answer different from all of yours?

**Wilmer** · July 3rd 2010, 10:35 AM

Originally Posted by Punch

The only formula I have learnt about is $A=P(1+\frac{i}{100})^n$

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 ...

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