To Pay Weekly or Monthly?

I mentioned I was going to switch to a weekly loan pay-off because it was mathematically a better move.  I received this comment yesterday:

I’m wondering what kind of formula you used to determine that you would pay off your loans 10 weeks earlier by making weekly loan payments. I’ve wanted to do this for a while, but only if it’s going to make a huge impact on my pay off time. I’ve asked all of my finance friends for help with this but so far nothing!

For background I currently pay $1700/month on a ~42K loan with a 3.15% interest rate.

Any help you could give me would be greatly appreciated!!

Here’s the long-story short: The more you can pay-off immediately, the less you pay in interest.  Let’s do some math.

Let’s say you have $1000 to put towards student loans in 1 month.  In order of most effective, here’s the best way to pay them:

  1. $1000 at the beginning of the month
  2. $250 each week
  3. $1000 at the end of the month

The idea is simple: pay now and your balance is smaller and, thus, accumulates less interest.  From everything I’ve researched, federal loans compound interest daily.  That means you’re looking at this formula to figure out how much interest is accumulating and added to the balance (P) after d days (r = interest rate):

compound-interest-formula

Let’s do the math on those 3 options:

$1,700 Paid on Balance of 42k at 3.15%
Simple.  $42,000 – 1,700 = $40,300.  Now, let’s add the interest that will accumulate in the next 28 days (let’s say this is February to keep things simple.
New Balance after Interest Accumulation = 40,300[1+(0.0315/365.25)]^7 = $40,397.43

$425 Paid Weekly for Four Weeks on Balance of 42k at 3.15%
A little trickier.  Let’s assume we pay at the beginning of the week.  By week:

  1. $42,000 – $425 = $41,575 (New Balance)
  2. New Balance after Interest Accumulation = 41,575[1+ (0.0315/365.25)]^7 = $41,600.11
    Balance after payment = $41,600.11 – $425 = $41,175.11
  3. New Balance after Interest Accumulation = 41,175.11[1+ (0.0315/365.25)]^7 = $41,199.97
    Balance after payment = $41,199.97 – $425 = $40,774.97
  4. New Balance after Interest Accumulation = 40,774.97[1+ (0.0315/365.25)]^7 = $40,799.59
    Balance after payment = $40,799.59 – $425 = $40,374.59
  5. Add in the remaining 7 days of interest to close out the 28 days:
    40,374.59[1+(0.0315/365.25)]^7 = $40,398.97

$1,700 Paid on Balance of $42k at 3.15% after a month of interest accumulation

  1. Interest Accumulation = 42,000[1+(0.0315/365.25)]^30 = $42,108.80
    Balance after payment = $42,101.54 – 1,700 = $40,401.54

Notice the difference in allowing that interest to sit and grow for just a month – a difference of about $4.  You’ve paid the same amount of money, just you haven’t waited and allowed interest to compound.

Of course, this is all relative depending on what you consider the beginning of the month.  The best course of action is to pay into your loan as soon as you have the money to do it.  If you want to get ahead of the game, what you could do is “borrow” money from yourself (from a savings account, perhaps) and pay off your loan in smaller weekly intervals and then replenish your savings when your paycheck comes in.  Again, this is relative though.  This was my thought process at first – I would “borrow” the money from myself and then pay back into my savings on pay day.  But what I realized is that option 1 is still the best option.  If I have the money, it’s best to put it immediately at the loans.  Therefore, last month I added a “week” or paid an additional 1/4 of what I normally pay a month and went back to making a payment as soon as my paycheck clears.  As often as I can, I’ll add a week to get the effect I describe below:

Where the big 10 week difference comes into play is when you hit those months that have “5” weeks.

Paying monthly = 12 payments per year.
Paying weekly = 52 payments per year.

If you pay $2,000 a month, for example, that means you pay $24,000 in a year. If you pay $500 a week (which for most months is the same amount), you actually pay $26,000 in a year.

That’s the big difference – those “extra” weeks in a calendar year.  You actually are paying at a more rapid rate to get that 10 week difference I originally mentioned.

The really, really short version is simply: Pay off as much as you can as soon as you can so there’s less balance to compound interest on.

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