## Friday, December 23, 2016

### The New Year's Game

US tax code creates some… interesting incentives. One of these is the New Year’s Game: a once-a-year opportunity for individuals to make an expected annualized return of 10% to 20% on a \$3000 investment.

Background
In the US, money made from selling any sort of investment instrument (usually financial assets or real estate) held for at least one year, is classified as long-term capital gains. So if I buy a stock or option contract on December 28 this year, and sell it on December 29 next year, then any profit made on trade is classified as long-term capital gains. Long-term capital gains are taxed at a much lower rate than ordinary income: 0% for individual income up to \$37650, 15% up to \$415050, and 20% from there. (Note that this is federal tax only.)

Aside from long-term capital gains tax, the other interesting thing about investment income is what happens when investments generate a loss. Individual investors are allowed to deduct up to \$3000 of capital losses from their taxable income each year.

Opportunity
Together, long-term capital gains tax and capital loss deduction create an asymmetry: gains on investments are taxed at the lower capital gains rate, while losses are deducted from higher-taxed ordinary income. Let’s look at an example to see how it plays out.

Let’s say I pay 15% tax on long-term capital gains, and 25% tax on ordinary income (I wish it were that low...). If I hold a stock for one year, and make \$100 profit off of it, then I pay \$15 in taxes and keep \$85 after taxes. On the other hand, if I lose \$100 on that stock, then I deduct that \$100 loss from my ordinary income and get \$25 back with my tax return, so my effective after-tax loss was only \$75. Thus the asymmetry: even if this investment is a coin flip, I still gain more in the good case than I lose in the bad case.

For simplicity, let’s assume that we’re investing in an efficient market with zero discount rate. That means the price of any asset is its average expected payout: if an asset has 50% chance of paying off \$1.00 and 50% chance of paying off \$0, then it costs 0.5*1.00 + 0.5*0.00 = \$0.50.

In this simplified efficient market, an asset with a 50% chance of gaining \$100 and a 50% chance of losing \$100 costs 0.5*100 + 0.5*(-100) = 0, i.e. the asset is free to buy (we’ll talk later about how to implement this). But after accounting for taxes, this asset would return 0.5*85 + 0.5*(-75) = \$5 on average. So we have an asset which is free to acquire, but has an expected after-tax return of \$5!

Optimization
The idea of the New Year’s Game is to push this idea to the maximum.

Abstractly, the trick outlined above works by turning ordinary income into long-term capital gains. In our simplified efficient market, everything has an expected return of zero, so all expected gains are offset by expected losses. But the expected losses are deducted from ordinary income, while expected gains are long-term capital gains. So for every expected dollar taken away from ordinary income, we add an expected dollar to capital gains. In other words, expected dollars are taken out of ordinary income, and put into long-term capital gains.

The expected gain is just the difference in tax rates times the amount of money moved from ordinary income to long-term capital gains. In the example above, the difference in tax rates was 25% - 15% = 10%, while the number of expected dollars moved was 0.5*100 = 50, for an expected gain of 0.1*50 = \$5.

Since the profit comes from turning ordinary income into long-term capital gains, we maximize profit by turning as much income as possible into capital gains. Counterintuitively, the trick here is to maximize expected losses: because the market is efficient, expected (pre-tax) loss equals expected (pre-tax) gain equals the amount of money transformed from ordinary income to capital gains. The losses are capped at \$3000: that’s the maximum amount we’re allowed to deduct from ordinary income, so that’s the maximum amount we can transform into expected capital gains.

So we want an investment which, in most years, loses exactly \$3000.

Let’s choose an investment which has some small probability p of making a huge windfall return, and probability (1-p) of losing \$3000. The expected loss will be (1-p)*3000, so on average, the investment returns (1-p)*3000 times the difference in tax rates. (The difference in federal tax rates is always 10% - 20%, depending on income.)

Implementation
In practice, this can roughly be done with some number of long call options and the same number of short calls at adjacent strike prices, so the investment pays off some amount if the underlying stock ends up above \$K, and loses \$3000 if the underlying stock ends up below \$(K-1). By adjusting the strike price, we can adjust the probability p that the investment pays off in any given year.

The main tradeoff is the probability p. For example, if p is ½, then the investment pays off every other year on average, but the long-run average return is only (1-p)*3000 = \$1500 times the difference in tax rates. If p is ¼, then the long-run average return improves to (1-p)*3000 = \$2250 times the difference in tax rates, but the investment only pays off once every 4 years on average. As p decreases, the long-run average return approaches its maximum limit of \$3000 times the difference in tax rates, but the investment pays off less and less often (though the windfall when it does pay off gets larger and larger).

Once we pick a value for p, we need to pick a strike price for the long/short option pair. If the gain when it pays off is G, and the loss when it doesn’t is L, then with zero expected return, p*G = (1-p)*L, or L/G = p/(1-p). Pick a p-value, and this formula will tell you what L/G ratio you need. I won’t spell out all the details, but it should be straightforward to compute the payoff for both the gain and loss scenarios with each strike price K, and then pick an option pair close to your target L/G. After that, just divide \$3000 by the loss amount to figure out how many to buy.

The final implementation detail is the one-year constraint: in order to qualify for long-term capital gains treatment, the investment must be held for at least one year. The final trading day of 2017 will be Friday, December 29, so the last chance to enter the New Year’s game for 2017 will be this coming week on Wednesday, December 28.