Arnold Kling  

Solvency is No Free Lunch

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Bryan writes,

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?

Comments and Sharing


COMMENTS (6 to date)
Joe Marier writes:

You could double up if you win early.

JohnW writes:

Repeatedly betting a fraction of one's available bankroll with favorable odds is a well-studied probability problem with an optimal solution having been identified by J. L. Kelly in 1956.

The Kelly criterion states that you should bet a fraction of your money

f = p - (1 - p) / b

where p is the probability of winning, and b is odds paid, b=1 for an even money bet.

Now this does not apply perfectly to shorting stock, since you could potentially lose more than you "bet" and you also can get hit with margin calls, but it can at least give us a ballpark answer. Or you could buy put options, in which case Kelly applies more precisely.

Anyway, with your example, each year, for every $100 shorted, you either lose $25 or win $50. We can look at that as betting $25 with 2 to 1 odds, 50% chance of winning. So Kelly says to bet 0.5 - 0.5 / 2 = 25% of your funds each year, which would correspond to shorting 100% each year. In reality, you would want to decrease that number due to margin calls (and margin regulations), but I think it is clear that you should be shorting at least 50% of your funds with such a favorable bet.

Alternatively, consider put options on SPX. June 2010 options with 1050 strike (near the money) are about $112. If SPX goes up 25%, the options expire worthless, but if it goes down 50%, they are worth $525. So p = 0.5, and b = 525/112 = 4.6875, so the Kelly fraction is 0.39. So if you had $100K, you'd buy $39K of put options the first year. If you lost, you'd have $61K left and buy 39% of $61K, or about $24K the second year.

Ken in Canada writes:

So what ever happened to the idea that the stock market was a place to invest in companies and become a participating owner of the business?

When did we turn it into a casino?

After the recent global financial calamity why is it still okay that it's a casino?


Winton Bates writes:

It seems to me that there is something wrong with the example. If I had "virtual certainty" that the market would eventually fall to its "fundamental value" of 50 then I would expect the probability that it will fall in any particular year to rise over time until it does fall. It seem to me that the longer the price stays above "fundamental value" the higher the probability that other market participants will realize that the market is over-valued, and thus act in ways that will cause the price to fall.

mgunn writes:

This is a well studied problem, and I'm glad someone FINALLY mentioned Kelly! Kelly betting is also equivalent to maximizing expected utility where utility = log (money).

phineas writes:

You can find an elementary, very good explanation of the Kelly Betting System in section 2.7 of the introductory probability book "Understanding Probability" by Jan Tijms and the whole book is freely downloadable at

BTW, I presume that's a LEGAL free book because it's listed at
though in truth I don't understand what legal status such downloads have.

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