Sunday Function

Death and taxes. And dead is dead, but taxes come in a huge panoply of forms. There's property taxes, excise taxes, sin taxes and income taxes. There's gas taxes and sales taxes and VAT taxes (yeah, I know) and death taxes.

With my politics I'm not a huge fan of any of them, but they are what they are. And every once in a while they produce some interesting math. One of the more interesting taxes, mathematically speaking, is the capital gains tax. It works like this: you invest your money in stocks or bonds or some other investment vehicle. In an ideal world your investment grows in value and eventually you'll take your profits and go on vacation.

But hold on! Uncle Sam wants a cut. If you invested $100 and you ended up with $120, Uncle Sam will take a cut of that $20 profit in the forms of a capital gains tax. At the moment the capital gains tax rate is 15%, so you'll send $3 to the treasury and end up with $117 in your pocket.

Now hold on again! A dollar when you opened your investment is not the same thing as a dollar when you took your profits. Inflation has eroded the value of the dollar and thus your purchasing power vis a vis your original investment is not well described by the difference in the raw dollar amount. You have to adjust for the rescaling of dollars to stuff caused by inflation.

If your investment grows at a constant percentage rate, its growth is given by:


Before inflation and capital gains tax, that's the growth of your money where r is your interest rate and t is the amount of time that has passed. For convenience we've scaled the initial investment to 1. But we're taxed on our profit, so f(t) becomes:


The second term is the profit, and we multiply it by the tax rate T to find the tax we owe. We subtract that from our previously accumulated total. And that's the money we have left over. But if we want to know how much this will buy in terms of what dollars bought when you opened your investment, we have to multiply by the factor that adjusts dollars now into dollars then. We'll pretend inflation is constant as well and occurs at a rate ri. This gives:


So let's give that a shot with the interest and inflation rates both equal to 4%.i-ec8d2bc9698de2bf6063e486cb0cc968-g1.png

Over time the purchasing power declines to 85% of its original magnitude, because a larger and larger portion of our investment is profit and Washington takes 15% of that. What if our interest rate is just higher than inflation - say, 4.3% compared to inflation at 4%? Here I'll zoom in the y-axis so the detail is a little more clear:


You actually lose purchasing power at first before it stabilizes and starts to climb. How much does your interest rate need to be to start growing your purchasing power right away? Differentiate f and you get:


Set that equal to zero, let t = 0 and solve for r:


For inflation of 4% and tax of 15%, you need to ear just over 4.7% interest to start actually making a post-tax/post-inflation profit right away. However, if you take a close look at f(t) you'll see that eventually you will make a real profit as long as your interest does exceed inflation even a tiny bit. Under non-ideal conditions it might take a while; for instance 4% inflation with 4.1% interest and a 90% tax rate would take 2300ish years to break even again. Presumably the numbers involved in most real investments are better.

On the other hand, even in this model it's always and everywhere better to be invested at any (non-negative) rate than to just let your money sit in a mattress and get eroded by inflation. These days it's the non-negative rate that's the tricky part. Good luck!

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I read your blog every now and then and I'm proud to say I FINALLY understand one of your posts! Go figure it would be one about capital gains tax.
Keep blogging, oh brillant one, and Geaux Tigers!


But of course, the inflation rate hasn't been 4% or more since September of 2008, and is currently hovering at a sliver above 1%. On the other hand, good luck finding an investment with a 4.1% rate of return...

Great post. Way to break it down.

Nobody has a crystal ball when it comes to the market, but consider this: income gets taxed at almost 3x the rate of capital gains, and the yield on 30-year treasuries is now under 4%. Inflation -- right now -- is probably less than 4%, but not much, and the long-term average of inflation is pretty close to 4%, and often much higher after long-periods of low rates. And the debt market is huge relative to the equity market, at least 10x.

Where is all the money in long-term treasuries going to go when rates start to notch up?

Not a big surprise. If a nation offshores all it's wealth production, leaving only finance, real estate, and Mc Jobs, then wealth becomes a zero-sum game. Sure, the currency it's denominated in inflates, but once you adjust for that, investments don't earn. And neither should they. Why should you be paid for sitting on your ass watching numbers in an account increment?

Shouldn't T be a scaled heaviside function, since The Man doesn't (to first oder) give you money back when you lose money?

@Joseph: Actually, he does. You are allowed to use capital losses to offset capital gains, and if your losses exceed your gains in a given year you can apply up to $3000 against your regular income (if your net capital loss is larger you can carry the rest forward). I took advantage of this provision last year, since I had a net capital loss of $2xxx in 2009.

Real world calculations are somewhat more difficult because the special rate(s) on capital gains only apply to long-term capital gains (the definition of which I would have to look up since I know it has changed recently). If you haven't held the investment for the prescribed minimum time, your capital gain is treated as regular income. The idea behind this rule (and I agree with the concept) is to give this advantage only to bona fide investment and not to short-term speculation.

By Eric Lund (not verified) on 27 Sep 2010 #permalink

"And neither should they. Why should you be paid for sitting on your ass watching numbers in an account increment?"

Paul: If you would leave your mailing address in the comments section of this post, then I will happily send you a pre-paid envelope. You can then write me a check for all the money you have accumulated "sitting on your ass" while your personal savings account at your local bank has accumulated interest.

By Max Fagin (not verified) on 27 Sep 2010 #permalink

Matt, I'm sure this is simple for you, but it isn't for me :-) I want to know the probability of the results in a School Board election held in our parish yesterday. Just thought it would be an interesting topic for a few of our local political blogs.

Kay Padgett Byrd - 556 Votes
Martin Bourgeois - 338 Votes
Mike Tucker- 218 votes

Love you son! Mom

"You actually lose purchasing power at first before it stabilizes and starts to climb."

I seem to recall that the original case for capital gains taxes was precisely that. People who made a habit of blaming financial crises on short-term investment proposed capital gains taxes as a way of enforcing patience. The participants in that initial debate who differed on the question "Should capital gains be taxed?" really differed on the question "Is the overall effect of speculation on the financial system destabilizing?" Perhaps that still is the true structure of the debate now.

The idea behind this rule (and I agree with the concept) is to give this advantage only to bona fide investment and not to short-term speculation.

"The idea behind this rule (and I agree with the concept) is to give this advantage only to bona fide investment and not to short-term speculation."

Why exactly would one want to discourage short term speculation? Short term speculation IS bona fide investment. It's an attempt to grow one's wealth, differing only in the size of the potential profits/losses and the risk the assumed by the investor. How does the incidental fact that an investment is short term imply that it should be discouraged?

By Max Fagin (not verified) on 13 Oct 2010 #permalink