Sunday, January 22, 2006

Expected Value: the Lottery and Insurance

At dinner the other night my little brother said that he was going to start buying one lottery ticket a week. I told him this was a bad idea, that he should put $1 into a sharebuilder account each week and once a year buy $48 worth of an emerging market index (china this year, india next, malasia in three, etc.).

I explained to him that emerging market stocks are psychologically very similar to the lottery - some weeks you're up double others you're down by half (extreme example, not how much they actually fluctuate on a regular basis), and that you have a reasonable expectation that your money is going to increase over time. More on this expectation and emerging markets in a future post.

I further explained that with the lottery, every ticket you buy has about a $.50 expected value for every dollar spent.

He looked at me like I was crazy and said "an what value?" I realized he had never been taught expected values, and without that information the chance of winning the lottery seemed good - or at least worth a shot.

Expected value is the sum of (the value of each possible outcome * the probability of getting that outcome). But that is likely not clear at first.

Expected value is best explained using an example:

If you and five buddies put $1 each into a pot and then draw straws to take home all of it, what is your expected value of this game?

The good outcome:
What is the probability you'll get the pot assuming the game is not fixed and completely random? Answer: 1 in 5
What will you get if you win the pot? Answer: $5

The bad outcome:
What is the probability you'll get nothing? Answer: 4 in 5
What will you get if you get nothing? Answer $0
The expected value is then: (.2 * $5) + (.8 * $0) = $1. You get $1 because nobody is taking a profit from this game.

With the lottery there are thousands of possible outcomes, but the approximate result is that you can expect on the average $.5 for every dollar spent.

If people are only getting $.5 back for every dollar spent, then they have to be getting something else as well for that dollar. You can then say that the fantasy of winning is worth at least $.5 to someone playing the lottery. And, in all likelyhood that fantasy is worth more than $.5. (this has to be true because people only trade $1 for something if they believe that they are getting at least $1 in value back) When I play the lottery (which I do to the tune of about $5 a year), I consider all of the money to be an entertainment expense. If I win, that's fun, but if not, oh well. That's why a lottery ticket almost always has a kind of a game as you scratch it.

With the lottery you are buying: Expected value + (Entertainment or a Fantasy) >= $1.

Then the conversation turned to the fact that I had bought an extended warrany on my new car. I explained that I was on the average going to loose money there as well. (I recognize that being a personal financial blogger, this is a sin unparrallell - I've entered into a transaction where I expect to loose money. Let me explain)

If an insurance company is offering you something, You are loosing money on the average. That's how insurance companies make money. If you haven't already, get your mind around that - it will help you accurately anylize insurance.

In spite of this fact, Insurance is often a very good deal!

Just like in the lottery, for every dollar you spend on insurance your expected value is less than $1. I don't know the numbers exactly, they could be computed easy enough, but I don't have the time right now. So let's say that the expected value of an insurance policy is $.75 per dollar spent.

If you're going to be loosing money on the average, how is insurance a good deal? Insurance companies act like risk brokers. They buy risk from you and millions of people like you for more than the actual cost of that risk. Put another way, they sell you security for more than the average cost that security is going to have for them.

With Insurance you are buying: Expected value + value of Security >= $1.

For the Insurance company: Expected value of risk < $1.

Thus on the average they will make money, and they are doing thousands or millions of these transactions a year so they enjoy the statistics of large numbers - namely regression towards the mean.

Back to my extended warranty. I had originally budgeted $62.5 a month to put into a "car repairs & maintainance" fund so that if something goes wrong I have a reserve. The cost of an 80,000 Mile, 5 year waranty ended up being $24 a month for the life of my loan (that is 80,000 miles and 5 years from today). I will still have maintainance like oil changes, tires, etc. But I don't have to worry about the cost of many other problems. I bought a Nissan and all I have to do if something that is warantied breaks is take it to a nissan dealer and they'll take care of the rest. That security is worth much more than $6 a month ($24 * .25 gap from expected value to the price I paid), the security alone is worth $24 a month to me.

I did, however, have a small victory when I was deciding to purchase the warranty. The salesman at the dealership was saying "If you use it once for something like the air conditioning, that's $2000 (on $1,200 total), and it pays for itself. It is a good investment." I looked him straight in the eye and said "No it's not. If the insurance company is selling it to me, it is a bad investment. I am going to loose money on the average if I buy this thing." He looked at me, blinking, as I continued "What I'm trying to decide is if the security is worth the cost." He then promptly calculated the cost and changed his sales tactic to "For only $24 a month you don't have to worry about any major repairs. That's not a lot of money for that assuredness." Car salesmen have an amazing ability to turn sales tactics on a dime.

Upshot: Realize that in the lottery and insurance you're going to be losing money on the average. Knowing that, ask yourself if the entertainment/fantasy in the lottery or security in insurance is worth the money you're going to lose.


Anonymous said...

Emerging market equity risk (standard dev) runs 10-25% pa - currently 10% while US stocks run 8-20%. you are overestimating crapshoot characteristics of EME. Yuu have stumbled on a mispricing which has been been repricing for the lat five years.

franky said...

I did an extensive lotto expected value post for the carnival of pf last week. Hopefully you saw it? Maybe you should let your brother take a gander.

Amit said...

Great post. I loved your example of how the salesman was able to change his extended warranty pitch to satisfy your concerns.