# Annuity problem

• Aug 18th 2009, 07:27 PM
bruxism
Annuity problem
Hi guys. Would be much appreciated if somebody could give me a start on this problem, I'm really stuck on it.

A washing machine, cash price $850 is available on the following terms: A deposit of$ 100 followed by equal payments at the end of each month for the next 18 months.
If interest is 6% pa compounded monthly for the first 6 months and 8% pa compounded monthly thereafter, determine the size of the regular monthly payments.
• Aug 18th 2009, 09:11 PM
Wilmer
Amount borrowed = 750
Future Value of 750 (6% 6 months; 8% 12 months) = Future Value of 18 payments (similar interest)

Formula for FV of amount A: FV = A(1 + i)^n
Formula for FV of payment P: FV = P[(1 + i)^n - 1] / i

Let x = .06/12 and y = .08/12

FV of 750: [750(1 + x)^6](1 + y)^12

FV of P: {P[(1 + x)^6 - 1](1 + y)^12} / x + {P[(1 + y)^12 - 1]} / y

{P[(1 + x)^6 - 1](1 + y)^12} / x + {P[(1 + y)^12 - 1]} / y = [750(1 + x)^6](1 + y)^12

Py[(1 + x)^6 - 1](1 + y)^12 + Px[(1 + y)^12 - 1] = [750xy(1 + x)^6](1 + y)^12

P = {[750xy(1 + x)^6](1 + y)^12} / {y[(1 + x)^6 - 1](1 + y)^12 + x[(1 + y)^12 - 1]}

Substitute x = .06/12 and y = .08/12 back in to get P = 43.97989... or 43.98 rounded.

If you're confused with this: P[(1 + x)^6 - 1](1 + y)^12
it means the accumulation of 6 payments at 6% earns 8% during next 12 months.
• Aug 18th 2009, 10:58 PM
bruxism
Quote:

Originally Posted by Wilmer

If you're confused with this: P[(1 + x)^6 - 1](1 + y)^12
it means the accumulation of 6 payments at 6% earns 8% during next 12 months.

Thanks, that little bit down the end made it all very clear. I was having trouble making everything fit together (annuities, compound interest etc) but something in that last sentence made everything make sense.

Thanks for the "AHA" moment.
• Aug 19th 2009, 06:28 AM
Wilmer

I find it is a good idea with these (or anything similarly "longish") to go this way:

x = .06/12, u = (1 + x)^12
y = .08/12, v = (1 + y)^18

P = 750 x u y v / [y v (u - 1) + x (v - 1)]

Letting A = amount borrowed, n = months at 1st rate, m = months at 2nd rate,
p = 1st rate, q = 2nd rate, then you can get a general case formula:

P = A x u y v / [y v (u - 1) + x (v - 1)]
where
x = p / 12
u = (1 + x)^n
y = q / 12
v = (1 + y)^m

Of course that's restricted to monthly compounding and 2 rates,
but you get the idea, right?
• Aug 19th 2009, 10:03 AM
jonah
Just thought I'd present the equivalent alternative "bird's eye view"
Comparison date at beginning:
$
850 - 100 = {\rm{R}}\frac{{{\rm{1 - }}\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{ - \left( {{\textstyle{1 \over 2}} \times 12} \right)} }}{{{\textstyle{{0.06} \over {12}}}}} + {\rm{R}}\frac{{{\rm{1 - }}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{ - \left( {1 \times 12} \right)} }}{{{\textstyle{{0.08} \over {12}}}}}\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{ - \left( {{\textstyle{1 \over 2}} \times 12} \right)}
$

Comparison date at end:
$
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} =$
$
{\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} - 1}}{{{\textstyle{{0.06} \over {12}}}}}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} + {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} - 1}}{{{\textstyle{{0.08} \over {12}}}}}
$
• Aug 19th 2009, 10:38 AM
Wilmer
In the 2nd one, you didn't leave a "space" between the = sign and the R.
• Aug 19th 2009, 12:58 PM
jonah
Roger that.
I hope this one's more picturesque?

$
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}
$
$
= {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} - 1}}{{{\textstyle{{0.06} \over {12}}}}}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} + {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} - 1}}{{{\textstyle{{0.08} \over {12}}}}}
$
• Aug 19th 2009, 02:03 PM
Wilmer
Much better; you get a little green star (Clapping)
• Aug 19th 2009, 11:29 PM
jonah
Simpler perhaps might be the end of six months as the comprison date

$
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} = {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} - 1}}{{{\textstyle{{0.06} \over {12}}}}} + {\rm{R}}\frac{{{\rm{1}} - \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{ - \left( {1 \times 12} \right)} }}{{{\textstyle{{0.08} \over {12}}}}}
$
• Aug 20th 2009, 08:37 AM
Wilmer
I'd like to see a space on left and right of the + sign on the right;
------ + R. Thanking you in advance, I remain sincerely yours.
• Aug 20th 2009, 10:28 AM
jonah
As soon as my keyboard error gets fixed (copy and paste from old files and other sources,as opposed to good old keyboard output, seems to be generating undesirable effects with MathType output - I have no wish to learn/master latex code)