Risk and Return Part 3: Lottery tickets, roulette wheels and expected value

In Parts 1 and 2 of this series, I made the case at length that there are no low risk/high return investments; that risk and return are basically perfectly positively correlated.  At this point you might wonder:  Why is it important to realize that there are no low risk/high return investments?  The reason it is so important is that once you truly understand that there are effectively no low risk/high return investments you must face  an important realization.

You only have two options: 

1. Continue to take the same level of risk you have been taking and get satisfied with your current return.

2. Take more risk and potentially generate higher return.

Note that taking more risk does not guarantee that you will receive higher return as higher risk means a higher variance of returns.  If you take higher risk, you may receive higher, lower or the same returns.

But not taking higher risk does guarantee that you will continue to receive at most the same level of returns that you are currently receiving.

This concept can be applied not just to investment returns but to many important questions in your life.  Do you currently have as much money as you want?  Are you satisfied with your current job?  In any area of your life where you seek to generate meaningful outcomes, your level of return will be perfectly positively correlated with the level of risk you are taking in that area.

What about…….

I expect that if you still don’t agree that there are no low-risk/high-return investments, you have already thought of several exceptions and would like to ask: “What about…?” 

What about lottery tickets? 

Aren’t lottery tickets low-risk/high return investments? 

For example, in the US, you can buy a Powerball lottery ticket for $2 and if you match all 5 numbers, you win $1,000,000.  $2 is a very low amount to invest to potentially win $1,000,000. 

If you match all 5 numbers plus the Powerball, you can win $345,000,000 today and sometimes even more (the highest Powerball jackpot to date was $2.04 billion in Nov 2022.)

How can a $2 investment that potentially returns $2 billion not be a low risk/high return investment?

The first mistake here is conflating the amount of the investment with the level of risk of the investment.  A $2 investment is not more or less risky than a $2 million investment.  The amount risked is independent of the investment risk.

Of course, most people would be much more willing to risk $2 than $2 million, but willingness to risk is not the same as level of risk.

Remember that we define risk as the variance of returns.  The variance of returns remains the same no matter the amount being risked.  For example, if we bet on the flip of a theoretical fair coin, the level of risk is 50% (half the time you will win and half the time you will lose) no matter if we are betting $2 or $2 million on this flip.  (For real coin flips, the coin lands on the same side that it started on slightly more often, so the level of risk is not exactly 50% for real coin tosses).

In Powerball, your chances of winning $1 million are 1 in 11,688,053, which means that your risk is beyond high.  Your “investment” in a Powerball ticket is actually a guaranteed loss.  High risk/negative return. 

How can a Powerball ticket be a guaranteed loss, even though you still have some chance to win $1 million?  To understand this, we need to know about expected value.

What is expected value?

The expected value of an investment is the return an investor can expect to receive given the probabilities of the different potential outcomes.

For example:

A fair coin flip where you bet $1	Win = $1 Lose = $1	Win = $2 Lose = $1	Win = $1 Lose = $2
How much you win on average	$1 * 50% = 0.50	$2 * 50% = 1	$1 * 50% = 0.50
How much you lose on average	($1) * 50% = -0.50	($1) * 50% = -0.50	($2) * 50% = -1
Add together	(0.50) + 0.50 = 0	(0.50) + 1 = 0.50	0.50 + (1) = -0.50
Expected value	0	$0.50	($0.50)
Invest or not	Indifferent	Always invest	Never invest

A negative expected value investment is a guaranteed loss, and you should never invest in a guaranteed loss.

What is the expected value of a Powerball ticket?

The expected value of all lottery tickets, including Powerball, is negative.  Excluding the jackpot (which varies), the expected value of a $2 Powerball ticket is about $0.32.  Assuming a maximum Jackpot of $1 billion and factoring in taxes and the probability of a split pot, the expected value is about $0.852, still highly negative.

Since lottery ticket buyers are all losing money on every ticket, the lottery ticket seller is guaranteed to make money, which makes running a lottery a low-risk/high-return investment. 

Notice that the state reserves this right to itself by making private lotteries illegal.  Individuals who run their own lotteries face legal risks on top of their financial risks, preserving the no low-risk/high return rule here for anyone other than government.  Insider trading laws limit another kind of low-risk/high return investment; government is quite serious about reserving this right for themselves.

But someone will win the lottery, won’t they?  Won’t that person have made a low-risk/high-return investment?

No, they won’t have.  The lottery ticket’s expected value was still negative when they bought it; they just got lucky this time.  Getting lucky doesn’t retroactively change the risk/return profile of the investment.  And no one can be consistently lucky (then it wouldn’t be luck). 

What about Casino games?

The masters of negative-expected-value investments are the casinos.  Every bet in the casino (except for one) is structured so that it has a negative expected value for the player.  This means that every time you place a bet in a casino, you lose money. 

The proof can be found in the $14.8 billion in gambling revenue that Nevada casinos earned in 2022.  Where does all that money come from?  It comes from the guaranteed losses of gamblers. 

Most lottery ticket buyers probably know that the odds are against them when they buy a ticket.  But casinos are highly skilled at misdirecting players about their true odds of winning (and the negative expected value of each of their bets).

Consider roulette for example.  If you bet $1 on red on the roulette wheel and you win, you get $2 – your original $1 plus $1 winnings.  Since there are only red and black numbers on the wheel, this should be a fair bet, correct? 

It would be, if there were only red and black numbers on the wheel.  In fact on a typical roulette wheel, there are 17 red and 17 black numbers and one green number 0.  It is the addition of this one little green 0 that allows casinos to transfer hundreds of millions of dollars a year from players to themselves. 

With the green 0, the odds of winning are 17/35 or 49/51 instead of 50/50.  This means that the expected value of a $1 bet is 98 cents, which is another way of saying that you lose two cents every time you bet on a color in roulette.  After 50 bets on average you lose your entire $1.  This is called the house “edge.” 

As one casino executive famously said: “Give me a .001 edge and enough time and I will take all your money.” And they do.

The house edge in blackjack is a little less obvious.  Where does the edge come from, even if the player plays just like the dealer?  The dealer has to hit any two cards 16 and below and stand on 17 or above.  If both dealer and player have the same count, the hand is a push.  So if the player plays the dealer’s strategy, can’t they neutralize the house edge? 

No, and all because of one subtle exception to the “same count is a push” rule.  This is not the case if  both have over 21 (a bust).  If the player busts, and then the dealer busts, the player still has to pay.  This is why the dealer always acts last in blackjack, to maintain that all important house edge. 

Card counting in blackjack is one way to reduce or in some situations eliminate the house edge, which is why casinos kick out (or do worse to) card counters.   There are no low risk/high return investments in the casino, just guaranteed losses for players.

Like the government in the lottery, casinos are on the winning side of negative expected value bets, guaranteed to win. Doesn’t this make owning a casino a low risk/high return investment?  Not quite - casinos are subject to many other risks outside of the casino floor, where they don’t control the odds. Competition from other casinos, on-line gambling, other forms of entertainment, economic downturns, etc., leave casinos as medium risk/medium return investments.

What is the difference between gambling and high risk/high return investments?

What exactly is the difference between gambling and high risk/high return investments?  Some people don’t see any and lump the two together.

Chat GPT-4 defines gambling as “the wagering of money or something of value on an event with an uncertain outcome.”

Britannica defines it as “betting or staking of something of value, with consciousness of risk and hope of gain, on the outcome of a game, a contest, or an uncertain event.”

These definitions do not seem to make a meaningful distinction between gambling and investing.  Chat GPT-4 even adds that “some consider activities like stock or commodities trading to be a form of gambling, although this is a topic of debate.”

A definition that groups investment in the same category as buying a lottery ticket is meaningless.  A much more useful definition of gambling is an investment with a known negative expected value.  This would clearly separate gambling from investment, including high-risk/high-return investment.

I mentioned above that one bet in the casino is not negative expected value:  there is no house edge in poker.  The casino takes a rake on each hand (a fixed fee), but the players compete against one another, not against the house.  When we use a meaningful definition of gambling, poker is not considered gambling.

Why are they not taking more risk?

Now that we have dispensed with some of the most common exception cases, we should be able to agree that risk and return are highly correlated.  In Part 1 of this series, I said that although most people will have heard “high risk/high return” before, they act as if they don’t understand what it really means.  If they did understand what it means, and they were not satisfied with their current level of returns, they would already be taking more risk to try to increase their returns, rather than wasting time searching for low risk/high return investments.  But they are not. 

Why aren’t they? 

Maybe they are actually satisfied with their current returns, but the vast majority of people I ask this question of say they are not.  The world seems to be full of people chasing illusory low risk/high return investment opportunities, which they would have no reason to do if they were satisfied with their current returns.

So what is stopping them from taking more risk?

Maybe they want to take more risk, but they don’t know how.

Maybe they are afraid to take more risk.

Maybe both. 

If they don’t know how, it’s understandable they are not doing it, but why are they afraid?