Denis said:
(1250+0*x)(1+i)^359 + (1250+1*x)(1+i)^358 + (1250+2*x)^357 + .... + (1250+358*x)(1+i)^1 + (1250+359*x)(1+i)^0 =
200000(1+i)^360
I wish I thought of this first. There's a slight typo though. The 3rd term from the left was missing the (1+i). It should have been
(1250+0*x)(1+i)^359 + (1250+1*x)(1+i)^358 + (1250+2*x)
(1+i)^357 + .... + (1250+358*x)(1+i)^1 + (1250+359*x)(1+i)^0 =
200000(1+i)^360
tkhunny said:
Well, sure, but can our questioner add that up?
Sure he can.
It just needs a little tweaking.
Accordingly,
(1250+0*x)(1+i)^359 + (1250+1*x)(1+i)^358 + (1250+2*x)(1+i)^357 + .... + (1250+358*x)(1+i)^1 + (1250+359*x)(1+i)^0 =
200000(1+i)^360
?
[(1250)(1+i)^359 + (0*x)(1+i)^359] + [(1250)(1+i)^358 + (1*x)(1+i)^358] + [(1250)(1+i)^357 + (2*x)(1+i) ^357] + … + [(1250)(1+i)^1 + (358*x)(1+i)^1] + [(1250)(1+i)^0 + (359*x)(1+i)^0] =
200000(1+i)^360
?
[(1250)(1+i)^359 + (1250)(1+i)^358 + (1250)(1+i)^357 + … + (1250)(1+i)^1 + (1250)(1+i)^0] + [(0*x)(1+i)^359 + (1*x)(1+i)^358 + (2*x)(1+i) ^357 + … + (358*x)(1+i)^1 + (359*x)(1+i)^0] =
200000(1+i)^360
?
[(1250)(1+i)^359 + (1250)(1+i)^358 + (1250)(1+i)^357 + … + (1250)(1+i)^1 + (1250)] +
[(1*x)(1+i)^358 + (2*x)(1+i) ^357 + … + (358*x)(1+i)^1 + (359*x)] =
200000(1+i)^360
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 1,250 \cdot\)\(\displaystyle s_{\left. {\overline {{360}}}\! \right| } _i\)\(\displaystyle + x \cdot\)\(\displaystyle \frac{{(1 + i) \cdot s_{\left. {\overline {{359}}}\! \right| } _i - 359}}{i}\)\(\displaystyle = 200,000(1 + i)^{360}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle 1,250 \cdot \frac{{(1 + i)^{360} - 1}}{i}\)\(\displaystyle + x \cdot \frac{{(1 + i) \cdot \frac{{(1 + i)^{359} - 1}}{i} - 359}}{i}\)\(\displaystyle = 200,000(1 + i)^{360}\)
\(\displaystyle \Leftrightarrow\)
\(\displaystyle x \approx 1.265892746\)
A monthly payment increase of 1.27 (last payment = 1705.93 = 1,250 + 1.27*359) in theory should do it. Rounding all computations to the nearest cent, however, will result in a final outstanding liability (principal + interest) of approximately 1,057.02 on an amortization schedule.
tkhunny said:
I just created an amortization schedule for a large medical bill. I get 10 months to pay it off. I created a simple schedule with exponentially increasing payments. I'm sure my creditor thinks I'm nuts.
Is it something like the growing annuity mentioned in Wikipedia entry for the Time Value of money?