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April 15, 2018

How Much Is a Lottery Ticket Worth?


Ask most people how much a lottery ticket is worth and the most probable answers will be the price of the ticket, the value of the jackpot or simply: “Is it a winning ticket?” The value of a ticket for a forthcoming drawing is none of these. The reason for this is related to a concept of risk and reward many people often fail to grasp, aren’t aware of or ignore at their own peril.
The concept is expected value. This is the projected value based on the probability of a given set of outcomes occurring. If it is possible to assign odds to a set of monetary outcomes, an expected value can be calculated. Expected value is a concept that works in a variety of situations. Investors, for instance, can use historical data to determine the likelihood of being able to fund retirement with a certain savings rate or allocation strategy.
The two multi-state jackpot lottery games provide great examples of how the concept of expected value works. Both Mega Millions and Powerball have specific and published odds for the possible winning outcomes. They also have published prize amounts. This allows anybody with a calculator—though a spreadsheet makes the calculations easier—to determine when the expected value of a lottery ticket exceeds the $2.00 cost.
The calculation is simply each potential payout times the odds of having a ticket with the right combination of numbers. For example, the odds of having a single matching number and the Powerball number correct on a single ticket is 1 in 92. One divided by 92 equals approximately 1.09%. Multiplying 1.09% by the fixed $4.00 payout results in an expected value of $0.04. Put another way, if you adjust the value of the prize by the probability of actually winning the prize, you end up with an expected value of $0.04. Running this math for all of the fixed payouts gives us cumulative expected values of $0.25 for a Mega Millions ticket and $0.32 for a Powerball ticket.
The big prize in both games is, of course, the jackpots. Both the sheer size and the variable nature of the jackpot give it great influence on the expected value of a lottery ticket. Not surprisingly, the larger the jackpot, the higher the expected value. Consider the March 30, 2018, Mega Millions drawing, which awarded one lucky person a $521 million prize. The jackpot single-handedly added $1.72 to the expected value of each ticket purchased for the drawing. The math was $521 million times the 0.0000003% odds of matching all five primary numbers plus the sixth Mega Ball number.
Those of you who are quick with arithmetic have already summed up the total expected value of the Mega Millions ticket: $1.97. The expected value for a ticket bought before the drawing was the $0.25 for the non-jackpot prizes and $1.72 for the jackpot. Even with a $521 million jackpot, the expected value of a Mega Millions ticket is less than the cost of a ticket. Such an outcome suggests that those who bought tickets lacked the opportunity to win a jackpot large enough to justify the risk.
The expected value for a Mega Millions ticket does not equal the $2.00 cost until the jackpot reaches $530.5 million. Powerball reaches the $2.00 expected value mark sooner with a jackpot of $490.7 million because the expected value for its fixed payouts is higher. Jackpots below these levels do not justify the purchase of the stock from a weighted-odds standpoint.
The impact that the size of the jackpot has should be very apparent. This has significance for investing as well. The riskier the investment is, the higher the required return to justify the excess risk must be. And the higher the required return is, the greater the likelihood of not realizing that return is. This is why so-called “lottery stocks” and similar types of securities are risky. For every Facebook, Apple, Netflix and Google (now Alphabet), there is a very large number of companies that flounder. While it may be hard to assign exact odds, a person can use historical data to judge when a projected rate of growth is unusually high. Adding basic industry and economic research can add to the scenario analysis (e.g., is realistic for the company to sell $X dollars of its product or service). Even when you lack the data necessary to calculate an expected value, you should be able to gather enough information to place some odds on whether an “everything goes right” scenario will or will not occur. For returns large enough to justify the risks, the “everything goes right” scenario is often necessary.
When doing such analysis, be aware of the element of luck. Just as every lottery jackpot has an eventual winner, the odds of losing everything spent on a lottery ticket always remain steady at 96%. The same principle applies to investing in high-risk stocks and securities: While a few will realize huge returns, most will end up losing most of their value.

The Week Ahead
Taxes must be filed by Tuesday, April 17. (Monday will be a holiday in the District of Columbia.) Tuesday is also the deadline for making 2017 contributions to individual retirement accounts (IRAs). Those of you who have yet to file may find our Tax Guide to be helpful.
First-quarter earnings season will start to pick up steam with 61 members of the S&P 500—mostly mega-cap companies—scheduled to report. Included in this group are seven Dow components: UnitedHealth Group Inc. (UNH), Goldman Sachs Group Inc. (GS), International Business Machines (IBM) and Johnson & Johnson (JNJ) on Tuesday; American Express Co. (AXP) on Wednesday; and General Electric Co. (GE) and Procter & Gamble Co. (PG) on Friday.
The week’s first economic reports will be March retail sales, the April Empire State Manufacturing Survey, February business inventories and the April housing market index, all of which will be released on Monday. Tuesday will feature March housing starts and building permits and March industrial production. The Federal Reserve’s periodic Beige Book will be released on Wednesday. The Philadelphia Federal Reserve’s April Business Outlook Survey will be released on Thursday.
Seven Federal Reserve officials will make public appearances: Atlanta president Raphael Bostic on Monday; San Francisco president John Williams, vice-chair Randal Quarles, Philadelphia president Patrick Harker and Chicago president Charles Evans on Tuesday; Cleveland president Loretta Mester on Thursday; outgoing New York president William Dudley on Wednesday; and Evans and Williams on Friday.
The Treasury Department will auction $16 billion of five-year treasury inflation-protected securities (TIPS) on Thursday. 
Courtesy of Charles Rotblut, CFA is the VP and Editor for American Association of Individual Investors (AAII). Charles is also the author of Better Good than Lucky. (EconMatters author archive here)  
The views and opinions expressed herein are the author's own, and do not necessarily reflect those of EconMatters.

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