Should you buy a lottery ticket? A look into the $656M MegaMillion Jackpot

Would you pass an opportunity to become rich instantly, with all your dreams and wishes fulfilled for just over a buck. That's the promise modern lotteries offer and it's quite an interesting affair.
Singapore has its own Toto, and the most recent jackpot is worth ~$650k SGD ($520k USD). I often see people queuing outside the Singapore Pools, eager to pick their lucky numbers.

However, it wasn't the pools that inspired me to dedicate a post entirely on the rationale of lottery buying. Thousands of miles away, something interesting happened on the 30th of March. The MegaMillion Jackpot hit $640 Million, sending US into a lotto buying frenzy. Nearly as many dollars were spent on tickets on the last day alone than what ended up being given away. And the game is pretty simple. You pick 5  numbers between 1-56 and a mega number between 1-46. You win the jackpot when you match all six numbers.
After 18 consecutive draws with no winners, the jackpot had accumulated to an all time high of $656 Million, and finally a trio of people picked the right numbers 2, 4, 23, 38, 46 with a Mega Ball of 23, and walked home with around $218 Million (pretax and annuity) each.
That's pretty incredible right. Now before you rush out to try your luck in the next lotto,  let's evaluate what your chances are, and should you waste your hard earned money on it. We will use the MegaMillion as an example.

The odds of picking all 6 right numbers are a cool  1 in a 175,711,536
Math: Probability of winning = p(Winning) = 1/(56 C 5) * 1/ (46 C 1)

The incredibly small odds also mean that your chances improve ever so slightly  even if you spend 1000s of dollars on picking different numbers.

Another thing that people often forget is the final payout. In US, you typically have a 35% federal tax and a state specific tax. In addition, the Jackpot payout is either an annuity over the next 26 years or an upfront cash payment. We know that a $1 is a worth a lot less 26 years from now. The Present value is ~63% of the Jackpot, almost same as the upfront cash payout that Mega Million offers, if you chose to accept it over the annuity.

Finally, the pot is split among all the winners, thus your individual share reduces with more winners.

Even in a state where there are no state taxes, our 3 winners will eventually take home only $89M (~218 * 63% * 65%) of the initial $218M today.

There obviously are other smaller prizes if you match just a few numbers, but I will ignore them in our discussion here.

Question is, How do you evaluate whether you should buy into a lottery.

Idea is pretty simple. We compare the expected value of a lottery with it's price. If the EVA is equal or greater than the price, we buy it. Below is short video tutorial on what Expected Value is.

Expected Value = p(Loss)*0 + p(Winning) * Your Winning Amount
EVA = 0 + 1/176M * Winning Amount* 63% * (1 - 35%)

where p stands for probability

Assuming If there is only 1 winner of the jackpot, solving the above for EV = 1, will give us a break even Winning Amount of $430 Million Prize Money before you should even consider investing in the lottery.

When the jackpot touched $650 Million, the expected value was a cool $1.5. But wait, before you rush out, did I mention there can be more than 1 winner. With each additional winner the pot shrinks and so does the Expected  value.

Expected Value with Multiple Winners

To estimate the number of possible winners, let's turn to Poisson distribution.
λ = np
Where n is the total number of tickets sold ~652 Million
p is the  probability of winning, since each ticket independently has a 1 in 176 Million chance

Based on the distribution, there is a 97.5% chance there will be a winner. 9% chance of just 1 winner, 17% chance of 2 winners, 21% chance of 3 winners (actual result), 19% chance of 4 winners and 14% chance of 5 winners.

Now using this, let us calculate the new expected value when there is a tie (multiple winners). We use another tool called Bayes' Theorem, which will help us estimate, what are the chances of more than 1, 2, 3 or more winners, trying with me (i.e when I have already won)

EVA = 0 + 1/176M *63% * (1 - 35%) * Winning Amount * [p(Sole Winner) + p(tie with 1)/2 + p(tie with 2)/3..... ]

Solving for the above, gives an EVA before tax for annuity at $1.33. Discounting it for the Present Value and taxes gives us just around $0.54, which is clearly less than the price of the ticket ($1). The details of my working are on the google spreadsheet. The EVA of all the other prizes for Mega Million Lottery gives you just an additional ~$0.11. Thus Mega Millions is never a rational decision for most people.

You could argue that when you buy a ticket, a few parameters are unknown. For instance, the eventual number of sold tickets. But you could look at the historical data, from places like lottoreport to estimate the future sales.

Concluding remarks
I have never bought a lottery ticket in my life. For me, the ticket price outweighs its expected value and can never be a profitable decision. And it's not just Mega Million. No matter where you are, or which Lotteries you buy, the odds are generally always stacked against you. But people still continue to queue, waste their time and hard earned money, week after week, with a hope that it might just be their lucky day. Rationality says NO but decisions are generally driven by emotions. And till then, people will continue buying lottery tickets.


Why do we buy Insurance? Expected Outcomes, Risk pooling and Risk preferences  

In life a lot of events and outcomes are uncertain. Economic risk is the possibility of losing economic security to satisfy an individual’s needs/desires for food, shelter, clothes and other amenities, due to unforeseen events in future.

Thus uncertainty leads to risk. And insurance is just a means to manage this risk. As we will see later, insurance enables you to trade an uncertain outcome/risk (and potential loss) with a known amount (Premium). Behavioral economists have concluded that we humans, buy too less insurance. We don’t anticipate fringe (low probability) outcomes, which may eventually result in a huge payouts (loss).

Let’s take a simple example.
Consider a risk averse car owner has a chance of less than 20% of being in a single accident in a year. And no chance of being in more than one accident. 50% of Accidents are minor and involve repairs worth $500. 40% require repairs worth $5000. And 10% may require replacement of car, which is worth $20,000.

Expected Loss and Standard Deviation

Car Owner’s expected loss is the mean of the above loss distribution
E[X] = 80% * 0 + 20% * (50% * $500 + 40% * $5000 + 10% * $20000)
E[X] = $850

Thus, on an average, car owner spends $850 each year on repair but, the possibility of having to fork out $5000 or $20,000 in medium to severe accidents could create potential concern.

To measure the variability of outcomes, we should also look at the standard deviation of the loss distribution.
Var[X] = 80% * (0 – 850)^2 + 20% * (50% * (500 – 850) ^ 2+ 40% * (5000 – 850) ^ 2 + 10% * (20000 – 850)^2 )
Var[X] = $ 9,302,500
SD[X] = Sqrt (Var[X]) = $3050
CV = Coefficient of Variation of distribution = SD/Mean = 3.6

Thus the variance of outcome (dispersion of possible loss) is pretty high.

Insurance Policies – Issuers and Holders

Now let us see how insurance instruments can be used to reduce/manage individual risk. Insurance is an agreement where, for a stipulated payment (policy premium), one party (the policy issuer) agrees to pay  the policy holder (car owner), a defined amount (policy claim) upon the occurrence of a specific loss (repairs from accident).
The issuer issues the policy to a pool of policy holders, assesses the Expected Total Loss and the Variation and charges a Policy Premium to cover all projected loss claims in the pool. The premium is thus the policy holder’s share of the total premium of the risk pool.

The idea is quite simple. Insurance is all about risk pooling. By subscribing to an insurance policy, the car owners can transfer the risk to the policy issuer. In a given year, not all car owners will be involved in accidents. The issuer pays  these unfortunate few using the premium collected from all the holders. Thus each policy holder exchanges their uncertain losses for a known premium. Also, as seen above, greater the standard deviation, greater the risk, and thus will require higher Premiums (will see again later).

The issuer could restrict the potential loss scenarios by defining or exempting certain “perils” in the contract. For instance, damages from natural disasters like Floods and Earthquakes could be exempted.
Now, extending the earlier example. Assuming, 100 car owners subscribe to an insurance policy which guarantees against all potential losses.
Expected Loss for the insurance pool now is,
E[nX] = n E[X] = 100 * 850 = $85,000

Variance = Var[nX] = n * Var[X]
SD = Sqrt (Var[nX] ) = Sqrt(n) * SD[X] = 10 * 3050 = 30,500
CV = SD/Mean = 30,500/85,000 = 0.36

Had, none of them subscribed to insurance, Expected loss for the group of 100 car owners would still have been nE[X] = $85,000
But, the Variance would have been the sum of their individual variances
n Var[X] = $305,000

Coefficient of Variation is an useful tool to measure variability between positive distributions of different mean. General form of the CV of the group is
CV = Sqrt(n) * SD[X] / n E[X] = SD[X]/ (E[X] * Sqrt (n)
Thus, as number of policy subscribers increases and becomes really large, CV tends to zero. In simple terms, it becomes easier for the issuer to predict expected losses.

Pricing Risk Premium

It should be clear that the existence of the insurance provider does not change the chances of the loss (Ignoring Moral Hazard for now). Since the issuer expects to pay out at least $85,000, the insurance premium would be priced above $85,000/1000 = $850. Generally, it would include some markups for administrative overheads, processing fees, some reserves for uncertainty and a small Profit. In a perfectly competitive market, the Profit margin is generally not that high. Assuming the issuer adds a 30% markup to cover all these, the policy holder ends up paying a premium of $1105 for the policy.
Net Premium = $850
Additional expenses and margins = $255
Gross Premium = $1105

Do note, some risk averse car owners are still willing to pay this amount, even though it is higher than their expected loss ($850) since they can exchange an unmanageable outcome (due to uncertainty) with a non-zero variance for a known amount (Premium) with zero variance. The number of eventual policy subscribers would depend on their risk preference. Had the Gross Premium per holder been $850 (equal to the Expected Loss), the insurance would have been actuarially fair. And all risk averse car owners would have immediately signed up for the insurance.

Issues of Moral Hazard and Adverse Selection

Earlier we mentioned that the insurance provider does not change the chances of the loss. However, studies have found that individuals seem to be more risk-seeking when they know they are insured. For instance, car owners might start driving more recklessly, increasing the chances of accidents and insurance pay outs. This is know as the problem of Moral Hazard.

Adverse Selection occurs when buyers and sellers have asymmetric information. It describes a situation where an individual's demand for insurance (either the propensity to buy insurance, or the quantity purchased, or both) is positively correlated with the individual's risk of loss. Thus, more smokers opt in for health insurance, and more reckless drivers opt in for car insurance. To manage adverse selection, Insurance providers have a differential premium structure where high-risk users (identified from background checks and profile questionnaires) are charged higher Premium than the rest.

Some interesting variations

Modern insurance instruments are really complex. Two simple variations are Deductions and Benefit Limits (Ceilings). In Deductions, the issuer pays out after a minimum threshold. In our earlier example, if the minimum deduction is $500, and if the repairs are worth $5000, the issuer pays out only $4500. For any repair below $500, the owner has to pay from his own pocket. As expected, the probability of claim drops from 20% to 10% and so does the risk premium. Some stand-out advantages are –Lower Premiums, A check against Moral Hazard (since owners will be more careful due to the first $500 out-of-pocket payment) and Lesser overheads –Admin and Processing Fee (since small claims are not processed).

In Benefit Limits, the issuer sets an upper bound on the claim pay-out. Policy holders can subscribe to additional coverage by paying incremental Premium. It also provides a safety net for the issuer and reduces the risk burden. Though a thing to note is that the additional Premium per holder is generally not that high since, as seen earlier, the risk pool is shared among a large number of subscribers.

Well, that’s the basics. The model discussed above was between an issuer and a policy holder. Insurance portfolios are also traded by the issuer in the financial sector (Securitization).  There is a lot of interesting literature out there on premium/insurance pricing. We briefly touched on the risk preference of the consumer. What we did not discuss was the optimal amount of insurance to buy. Intuitively, a consumer would want to maximize his Utility under uncertainty, that is, maximize his Expected utility -with and without the probability of loss, inclusive of his premium payouts and claims. Thus, the coverage or Premium that maximizes the solution will rely on individual utility and risk preferences. Let us leave it for a later post.

Risk and Insurance: Society of Actuaries -
Premium Calculation and Insurance Pricing -
Applications of Expected Utility Theory -
An Introduction to Risk-Aversion -