So, I've been paying down student loans left and right, mostly for the wonderful feeling of freedom that comes with it. 3M and this forum have helped me get aggressive about it. But there's a financial math question mark floating around in the back of my mind that's been causing me difficulty; and, as I barely know anything about financial math (after all, I took out over $30K at 6.8% to invest in a marginally useful graduate education), I was wondering if anyone can talk about something that's been bothering me for weeks.

A question that comes up often on these forums is whether it's better to pay off a loan with a lump-sum or invest that lump sum in an asset. Most people here are adamant about paying off high interest loans as fast as possible; especially the 6.8% educational loans that so many hopeful, bright young Americans screwed themselves over with. The phrase is usually that you're getting "a guaranteed 6.8% return" when you pay one of these loans down. But I think this is kind of a false comparison, because the principal on a loan that is being paid down is disappearing, month by month.

So, to compare I used Excel to calc out the total cost of a ten year loan at 6.8% on a principal of $10,000, compounded monthly. You get $13,728 total after ten years. I wanted to know if what's the annual rate on a lump-sum investment of $10K that would be at least as as good as this over the same period? I tried solving for an r in the simple interest formula:

P * (1 + r)^n = Total where P is principal, r is interest rate, n is number of years

$10,000 * (1 + r)^10 = $13,728

(1 + r) ^ 10 = 1.3728

1 + r = (1.3728)^(1/10)

r = 0.032191 or 3.2%

Conceptually, then, can we say that paying off $10K at 6.8% all at once is like investing $10,000 at 3.2% for ten years?