A (slight) improvement on a terrible savings idea…lottery syndicates


Shark attack + lightning = MegaMillions? Source: http://www.onewhale.com via Google Images.

The season finale of Last Week Tonight with John Oliver had a great main story: the many fallacies associated with lotteries.  Among them:

  • The probability of winning the ‘big one’ in the US, MegaMillions, is roughly the same as being stuck by lightning…while being eaten by a shark.  I’m not quite sure this is right, as the odds I see on the internet are the following:
  • Lotteries may not be so good for charity/education as one might think.  The program focuses on the replacement of education funds by lottery money; sometimes the shifted state funds are used elsewhere, and sometimes they’re just wasted.

OK, back on point.  The program got me thinking about the humble lottery syndicate: suppose a group of 5 or so co-workers pool $1 each, on a weekly basis, to buy MegaMillions tickets.  The jackpot would be shared equally.  I’m sure many folks just do this as a way to build camaraderie, but there are indeed many people who honestly believe the lottery will help fund retirement. So let’s discuss a very slight improvement for this group of folks.

  • Objective: a better payout ratio than MegaMillions, while keeping similar ideas of a lottery (e.g. small investment with big potential payout).  So I would like a higher expectancy, which is defined as
    • Prob(win) * Win$ – Prob(not win) * Cost$
    • For MegaMillions, the average jackpot is $105 million.  So the expectancy is:
      • (1/259 million) * ($105 million) – (1- 1/259 million) * ($1) = $(0.60).  I expect to lose $0.60 of my $1 invested each week, on average.
      • NB: I don’t include the other prizes, to simplify analysis.  Take a look here for more rigorous computations.
  • Strategy:  purchase of deep out-of-the-money put options on the S&P500.  Similar to lottery tickets, they will almost always expire worthless, but will sometimes have some value and very infrequently have a very large value.
  • Method: choose a near-dated, highest-strike put option on the S&P500 (SPX) which can be purchased for $5 – the ‘nickel option’.  To give perspective, here are the best value nickel puts on the SPX (current value = 2040) this morning:
    • 2 days to expiry (DTE): 1795 strike.  The S&P would need to ‘crash’ 12% over the next couple days for this to close in the money.  I choose this one.
    • 8 DTE: 1605 strike.  A 20% crash needed.
    • 16 DTE: 1450 strike.  A 29% crash needed.
  • Scenarios:
    • Under most conditions, the 2 DTE 1795 put will expire worthless.  The syndicate has lost its $5 this week.
    • The probability of reaching 1795 in two days can be very roughly modelled as the probability of a 12% move given the S&P’s current 13% annualised volatility.  That is a roughly 11-standard deviation move, meaning 0% probability given a normal distribution of returns.  So conventional stats don’t really help us.
    • Going to our friend Quandl, we can calculate all the 2-day (overlapping) returns for the S&P 500 going back to 1950.  It turns out our 12% OTM Put would have ‘come good’ a few times in that period:
      • 20 Nov 2008 (-12.4%)
      • 20 Oct 1987 (-16.2%)
      • 19 Oct 1987 (-24.6%)
    • So: out of 16,318 overlapping observations, we’ve found 3 that would result in a win for the nickel option.  That’s a historical probability of 3/16318, or around 1 in 5,500.
    • What do you win?  That’s a good question.  With only 3 observations where this has occurred, the expected move given the 12% loss is difficult to estimate.  Just averaging the 3 gives -17.7%, or today’s S&P crashing to around 1678.  Our option would then be worth $11,700 at expiration.
      • Extension: what about volatility? If we are indeed down 17% on the S&P, we can’t expect implied volatility to remain 13% annualised.  Though the VIX wasn’t around in 1987, estimates of implied volatility around Black Monday were about 170% annualised.  Suppose we assumed we bought the nickel option today, then the crash happened; suppose further the volatility expanded to 170% annualised.  Our option would now be worth about $17,500 with the extra vol.  If we’re playing for this scenario, we might actually prefer the options with greater DTE…but I’m getting too complicated.
    • Our expectancy, then: (1/5500) * ($11,700) – (1 – (1/5500)) * ($5) = $(2.87), or $(0.57) for each $1 invested.  I expect to lose slightly less than the lottery.

In sum, this fun little exercise has taught me a few things:

  1. Lotteries are still terrible ways to make money.  The expectation is horrible.
  2. Buying deep OTM options is also a pretty terrible way to make money.  The expectation is nearly as bad.
  3. Statistics are fun when the numbers involved are ludicrous.  I reckon almost every one of the stats mentioned above is dubious in one way or another.  So fun for a thought experiment, but never for creating an actual investment strategy.

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