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Zeno’s Paradox and the Folly of the Monthly Minimum Payment

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Suppose you want to walk from the Empire State Building in New York City to the corner of Hollywood and Vine in Los Angeles—a distance of about 2775 miles; and, on the first day, your journey, you walk 2% of the total distance to your destination, 55.5 miles. Continuing your journey at a constant rate of 55.5 miles/day would result in a traveling time of 50 days.

Alternately, let’s say that, on each day of your journey, you travel 2% of the remaining distance to your destination. Theoretically, you would never actually reach the Empire State Building; but, after about 800 days of walking, you’d be just over 1 foot from your destination—certainly close enough to take one last heroic step before collapsing from exhaustion in the lobby.

Zeno’s Dichotomy Paradox

Your imaginary journey is a variation of a mathematical conundrum first proposed by the Greek philosopher Zeno of Elea sometime in the 5th Century B.C. According to Zeno’s Dichotomy Paradox, a person traveling from one point to the next must first travel half the distance, which takes a certain finite amount of time. He then must travel half of the remaining distance; which takes a proportionately smaller amount of time. Now three-quarters of the way to his destination, the traveler continues his trip in increments equivalent to half of the remaining distance. Each step takes him closer to his destination; but since the remaining distance can, theoretically, always be divided in half, his journey will never actually end.

Zeno’s hypothetical journey assumes a traveler walking from one fixed point to the next; and, in our version of the Dichotomy Paradox, the total distance between Los Angeles and New York City remains constant. What if, on the other hand, the Empire State Building slowly moved further away from you at the rate of about 15% of the remaining distance each year? How long would your trip be then?

Or what if you occasionally extended your trip a little further—arbitrarily adding a few hundred miles to your journey every now and then—until the very idea of a final destination recedes beyond all meaning, and life becomes an endless, exhausting march. It’s a lousy way to travel.

It’s an even worse way to manage debt.

All consumer credit cards indicate a minimum monthly payment—a percentage of the card-holder’s outstanding balance that must be paid each month to avoid a late fee. Although methods differ from one consumer lending institution to the next, federal law requires that credit card issuers calculate a user’s minimum monthly payment so that it includes a portion of the principle amount—thus avoiding Zeno’s paradox. Cardholders who pay the minimum will eventually pay off their debt, provided they don’t make additional purchases on credit; but applying the minimum monthly payment toward a consumer debt significantly extends the period of debt, often well beyond the life of the item purchased.

Credit card issuers determine a card-holder’s monthly minimum payment using one of the following formulas.

Percent + fee
Using this formula, credit card issuers calculate a minimum monthly payment by taking a percentage of a card-holder’s outstanding balance (between 2% and 5%, depending on the card) and adding it to any fees or penalties for late payment. For example, a card-holder with an outstanding balance of $5000 whose credit card issuer requires a minimum payment of 2%, must pay at least $100 to avoid a late fee.

$5000 x 2% = $100

Assuming the same card carries an annual percentage rate (APR) of 15%, the user’s next statement will show an outstanding balance of approximately $4961.25. The lower outstanding balance will result in a marginally lower minimum payment on the card-holder’s next statement; but as the monthly minimum decreases, so too does the payment’s overall affect on the amount of debt the card-holder carries. Submitting the minimum payment of $99.23 on an outstanding balance of $4961.25 at 15% APR will result in a new balance of $4922.80; and so on.

At this rate, and assuming the card-holder accrues no further debt, it will take over twenty-seven-and-a-half years to pay off the initial debt of $5000; and the total amount of interest paid over that period will be over $7500.

NOTE: For the sake of simplicity, I’ve calculated the monthly finance charge by dividing the APR by twelve—the number of months in a year. Thus, an APR of 15% would yield a monthly finance charge of 1.25% of the outstanding balance. Since months vary in length, a more accurate method of calculating monthly finance charges would be to divide APR by 365, then multiply the result by the number of days in a specific billing cycle. Moreover, card issuers typically use a card-holder’s average daily balance—the sum of each day’s outstanding balance divided by the number of days in given month—when calculating monthly interest. As a result, the figures in these calculations are rough estimates; which, nonetheless, provide a reasonably accurate assessment of the long-term consequences of poor debt management.

Percent + fee + finance charge

Another common method of determining minimum monthly payment is to add a card-holder’s monthly finance charge to 1% of the outstanding balance, plus any fees. Using this formula, a card-holder with an outstanding balance of $5000 and no late fees on a credit card with an 15% APR is required to pay at least $112.50 to avoid incurring any penalties.

($5000 x (15%/12)) + ($5000 + 1%) = $112.50

As with the percent plus fee method, this formula results in a minimum monthly payment that gradually decreases over the life of the debt; which means that the rate at which the card-holder pays off his or her debt gets slower each month. At this rate; and, again, assuming the card-holder takes on no new debt, it will take over twenty-two years to pay off the principle plus the nearly $6000 in interest accrued over that period.

In both cases, making a fixed monthly payment of $150 will pay off the principle (plus $1508 interest) in less than four years; a fixed monthly payment of $200 will pay off the principle (plus $1033 in interest) in just over two-and-a-half years; and a fixed monthly payment of $250 will pay off the principle (plus $790) in two years. While none of these three options is a bargain, a moderate fixed monthly payment seems positively frugal next to the folly of the monthly minimum.

The hidden consequences

The costs of making the minimum allowable payment on an outstanding balance reach far beyond the monthly account statement. In addition to spending thousands of dollars on finance charges and carrying debt for several decades, consistently submitting the minimum monthly payment on credit cards lowers your credit score; which can jeopardize your ability to borrow money in the future. In some cases, a weak credit rating can have adverse affect on everything from your annual insurance premiums, to your monthly rent; since these amounts are frequently raised for individuals deemed poor credit risks.

And then there’s the sheer absurdity of continuing to pay for an item long after it has become useless or obsolete. A top-of-the-line Nikon digital SLR camera, retailing for around $7000, will end up taking twenty-five years (15% APR, with a 2% minimum monthly payment) to pay off, and wind up costing $15,230. At the rate consumer technology goes obsolete, you may wind up paying off the debt decades after the camera has become a paperweight.

About Tim Dewey

Tim Dewey is a literature professor and freelance writer living in Minneapolis, MN.

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