Loan Interest Payment Puzzle / Question

[Tiberius]

This was a puzzle / question I encountered earlier today and I found it quite interesting.

Suppose that you have a $2,000 loan which accrues 5% interest each month, and a separate $20,000 loan that accrues 3% interest each month. At the end of each month (after interest has been accrued) you have $2,000 in savings that you can use on one of the loans. Assume that interest accumulates before you make your payment. Also assume that if you have less than $2000 left on one of the loans, you can use the amount left after interest and paying it off on the other loan.

The question is: which loans should you pay off first in order to reduce the total amount of interest you accrue.

A follow-up question (which I don't have the answer to) is: what is the optimal strategy for paying off the debt to reduce the total amount of interest accrued, assuming that you can split your $2,000 between the loans each month (i.e. put $100 on one, and $1900 on the other).

[mlmooney89]

I personally would pay off the smaller loan first cause it would be easy to get rid of and you can then focus on the bigger one. Before you pay off one you will be getting $100 and $600 interest so by paying off the smaller one first you get rid of interest it generates.

[Jackalope]

My financial planner has always advised me to always pay down higher interest rate debt first. That may not be strictly mathematically correct, however.

Also - as I can pretty reliably get significantly better than 5% ROI, I wouldn't exactly be in a big hurry to pay either off, compared to investing.

[Pat Mustard]

Ok I did a quick excel working of the problem.

First paying the small loan first:
Small

Code:

Principle Interest Lump   payment balance
2000      1.05     2100   2000         100
100        1.05      105   2000       -1895

Big

Code:

Principle Interest Lump     payment balance
20,000.00 1.03    20,600.00 0.00       20,600.00
20,600.00 1.03    21,218.00 1,895.00   19,323.00
19,323.00 1.03    19,902.69 2000       17,902.69
17,902.69 1.03    18,439.77 2000       16,439.77
16,439.77 1.03    16,932.96 2000       14,932.96
14,932.96 1.03    15,380.95 2000       13,380.95
13,380.95 1.03    13,782.38 2000       11,782.38
11,782.38 1.03    12,135.85 2000       10,135.85
10,135.85 1.03    10,439.93 2000        8,439.93
8,439.93 1.03     8,693.13 2000        6,693.13
6,693.13 1.03     6,893.92 2000        4,893.92
4,893.92 1.03     5,040.74 2000        3,040.74
3,040.74 1.03     3,131.96 2000        1,131.96
1,131.96 1.03    1,165.92 2000         -834.08

Ok so Small before big gives you 16 monthly payments of $2,000 plus a final one of $1,165.92 totalling $33,165.92 total payment.

Let's do it the other way shall we.
Big

Code:

Principle Interest Lump        payment balance
20,000.00 1.03     20,600.00 2,000.00 18,600.00
18,600.00 1.03     19,158.00 2,000.00 17,158.00
17,158.00 1.03     17,672.74 2,000.00 15,672.74
15,672.74 1.03     16,142.92 2,000.00 14,142.92
14,142.92 1.03     14,567.21 2,000.00 12,567.21
12,567.21 1.03     12,944.23 2,000.00 10,944.23
10,944.23 1.03     11,272.55 2,000.00 9,272.55
9,272.55 1.03      9,550.73 2,000.00 7,550.73
7,550.73 1.03      7,777.25 2,000.00 5,777.25
5,777.25 1.03      5,950.57 2,000.00 3,950.57
3,950.57 1.03      4,069.09 2,000.00 2,069.09
2,069.09 1.03      2,131.16 2,000.00 131.16
131.16 1.03         135.09 2,000.00 -1,864.91

Small

Code:

Principle Interest Lump payment balance
2,000.00 1.05 2,100.00 0.00 2,100.00
2,100.00 1.05 2,205.00 0.00 2,205.00
2,205.00 1.05 2,315.25 0.00 2,315.25
2,315.25 1.05 2,431.01 0.00 2,431.01
2,431.01 1.05 2,552.56 0.00 2,552.56
2,552.56 1.05 2,680.19 0.00 2,680.19
2,680.19 1.05 2,814.20 0.00 2,814.20
2,814.20 1.05 2,954.91 0.00 2,954.91
2,954.91 1.05 3,102.66 0.00 3,102.66
3,102.66 1.05 3,257.79 0.00 3,257.79
3,257.79 1.05 3,420.68 0.00 3,420.68
3,420.68 1.05 3,591.71 0.00 3,591.71
3,591.71 1.05 3,771.30 0.00 3,771.30
3,771.30 1.05 3,959.86 2,000.00 1,959.86
1,959.86 1.05 2,057.86 2,000.00 57.86
57.86 1.05 60.75 2,000.00 -1,939.25

Second one comes out at 16 payments of $2,000 and a final payment of $60.75 giving $32,060.75.

My workings say that you will be better off by $1,000 or there abouts if you pay off the bigger loan first of all before paying the smaller one. Despite the fact the smaller one accures more interest the differential between the interests isn't enough to ovetake the principle differentials in the timeframe of full payment.

I'll run a half and half scenario in a minute, just want to post this first.

[brewer]

Being a lazy shit, I'd pay them off with the least numbers of checks to write.

[Pat Mustard]

Ok the half and half

Small

Code:

Principle Interest Lump payment balance
2,000.00 1.05 2,100.00 1,000.00 1,100.00
1,100.00 1.05 1,155.00 1,000.00 155.00
155.00 1.05 162.75 1,000.00 -837.25

Big

Code:

Principle Interest Lump payment balance
20,000.00 1.03 20,600.00 1,000.00 19,600.00
19,600.00 1.03 20,188.00 1,000.00 19,188.00
19,188.00 1.03 19,763.64 1,837.25 17,926.39
17,926.39 1.03 18,464.18 2,000.00 16,464.18
16,464.18 1.03 16,958.11 2,000.00 14,958.11
14,958.11 1.03 15,406.85 2,000.00 13,406.85
13,406.85 1.03 13,809.06 2,000.00 11,809.06
11,809.06 1.03 12,163.33 2,000.00 10,163.33
10,163.33 1.03 10,468.23 2,000.00 8,468.23
8,468.23 1.03 8,722.27 2,000.00 6,722.27
6,722.27 1.03 6,923.94 2,000.00 4,923.94
4,923.94 1.03 5,071.66 2,000.00 3,071.66
3,071.66 1.03 3,163.81 2,000.00 1,163.81
1,163.81 2.03 2,362.54 2,001.00 361.54
361.54 3.03 1,095.45 2,002.00 -906.55

Ok that's 15 monthly payments of $2,000 each and one final payment of $1,093.45 giving a total payment of $31,093.45 (if my ready reckoning is best). So half and half is better than my previous two scenarios.

Here's a link to the excel sheet with the formulae marked out for you. The small loan is the first sheet and the big loan the second. You can tweak the numbers how you want, and I've assumed the compounding is monthly rather than daily (most banks compound their interest daily, but it's a bitch to work down to the multiplying factor, and the numbers won't be more than a few pennies off at most).

Hope this helps.

[Fireball]

I'd invest $6.50 in a bag of quick-setting cement for the loan shark holding the $2k loan, and drop him in the ocean.

This is a perfect example of a problem that could use differential calculus for a solution, but I'm too lazy and rusty.

[BrianSoddingBoru4]

I would finally ring up the Nigerian Finance Minister and get him to lend me $22 000.

Boru

[Pat Mustard]

I would finally ring up the Nigerian Finance Minister and get him to lend me $22 000.

Boru

You mean this guy?



http://www.irregularwebcomic.net/2026.html

He's surely good for it.

[SteelCurtain]

You should pay off the small loan first.

You'll pay about $2000 less in interest if you do.