Bored? Financially-inclined? Please Check My Lump Sum vs. Loan Payoff Homework

[Lackland]

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?

[marty998]

Paying the lump sum means you save $3,728 in interest. (I get $3,810, but the difference is trivial).

However you cannot invest the 10K for 10 years, because you need to draw down on it to pay off your loan month by month.

So you would need to draw down $1,373 per year out of your 10k investment to pay off the loan. I think you'll find that equates to a neat 6.8% return required. And in a weird circular way I've proved nothing but there you go

If your investment beats 6.8% you win. If not you lose.

[JamesAt15]

I think OP's saying he has $10K saved he can either dump towards his student loans, or invest to try to make more. I assume he either doesn't have to make payments against his student loans, or he will make the minimum payments with other income.

Numerically I believe you come out ahead by putting your lump sum into investments. Most folks assume that the market will return a bit less than 10% yearly on average, minus a bit less than 3% for inflation, giving you around a 7% inflation adjusted return.

In the real world, though, numbers don't always  have the final say. Don't these student loans incur considerable fees and penalties if you miss required payments or have other similar issues? Most of the horror stories I hear about those with student loans involve the amount owed expanding to outrageous amounts due to fees and penalties well above the base interest rate.

You could be taking on a big risk of incurring these fees and penalties with your loans by putting your money to investments instead of paying them down. I will have to take your word for it that conventional wisdom says to pay them off, but it sounds like a good "sleep better at night" tradeoff to get them paid off quickly instead of going for another couple of percentage points of return.

[JamesAt15]

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?

I plugged these figures into my investment calculator and got the same result.

I also tried out paying off 10K of your 30K loan and then let it grow at 6.8% for 10 years with no additional payments, and got a final amount of about $39,400.

If instead you left the 30K loan untouched, and put the 10K into investments that you figure will give a 10% return (leaving out inflation), after 10 years you now owe $59,100 on the loans but have $27K in your investments, so you could pay off the loan and owe about $32K. Somewhat better off than the scenario above.

But for sleep-at-night reasons above, I think you'd be better off just paying off the loan.

[marty998]

I think OP's saying he has $10K saved he can either dump towards his student loans, or invest to try to make more. I assume he either doesn't have to make payments against his student loans, or he will make the minimum payments with other income.

I hear what you're saying but I don't think you can take into account other income for this exercise. This is a binomial choice we are talking about. If you invest, then you are hit with higher than otherwise interest on your student loan repayments.

I also forgot about tax. You need to pay tax on your investment earnings, but loan repayments are tax free. Hence you need a much bigger investment return all things being equal.

[JamesAt15]

Well, I think the OP was trying to get an idea of equivalency between putting 10K against his student loans, versus investing it at X% return, to get an idea of what X is and if it's as much of a no-brainer decision to put it towards the loans as people seem to say it is.

I had also forgotten about tax, although maybe for this thought experiment you could assume he could dump the funds into a Roth or other tax-advantaged account. If not, I assume he'd need to pay 15% long term cap gains on the $17K profit in my second scenario, which drops the profit down to $14,400. Narrows the gap even further - I'd still go for the loan payoff.

[pom]

It is the drawdown that makes the difference, you could not invest the 10k for 10 years because of that drawdown.

Imagine that you earn that 3.2% (or 0.267% per month). The first month you start with 10k, pay the monthly amount of 114 and earn 26 in interest. For the second month you only start wih 9912, pay 114 again and earn 26 again, ending up with 9824.

According to my calculation, you would run out of money after 99 payment and you will be on the hook for the last 13.

Basically, barring any tax differential, the interest rate that you must earn is exactly the interest rate that you pay.