Solvency is No Free Lunch
By Arnold Kling
Suppose for example that you’ve got $100,000 in assets. You know with virtual certainty that the market will eventually fall from its present level of 100 down to its fundamental value of 50. The catch: You don’t know when it will fall. Every year, the market has a 50% of going up 20%, and a 50% chance of plummeting down to 50. So if you sell $100,000 short with contracts that resolve after a year, you’ll lose your shirt if the market goes up five years in a row.
His solution is sell short only $10,000 each year.That does increase your chances of staying solvent. However, it also increases your chances of missing out on a big payoff. If the stock drops to 50 in the first year, you make $5,000. You could have made $50,000 if you had sold short $100,000.
What if we compare the strategies in terms of expected value?Suppose we set a five-year time frame, not ten years, and make the initial assets $50,000. Also, make the upward movement of the stock each year 25 percent if it goes up, so we lose our shirt if it goes up four years in a row. Then, if we bet $10,000 each year (and cumulate our bets), we have
win first year (p = .5) $5000
win second year (p = .25) $10,000
win third year (p = .125) $15,000
win fourth year (p = ..0625) $20,000
win fifth year (p = .03125) $25,000
never win (p = .03125) negative $50,000
Compare this with betting the ranch the first year:
win first year (p = .5) $25,000
win second year (p = .25) $18,750
win third year (p = .125) $14,062
win fourth year (p = .0625) $10,547
never win (p = .0625) negative $50,000
Do you still like the idea of stretching out your bets?