Author: Sophia Tutorial

This lesson will introduce the concept of risk.

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What's Covered

This tutorial is going to explain to you the concept of risk from a probability standpoint by focusing on:

  1. Risk


Risk is essentially the negative average value, the expected value, you would incur by losing something.

  • Diamonds are very valuable. But consider diamonds in a safe. This is not very high risk. Even though the diamonds themselves are valuable and losing them would mean a huge loss of money, the probability of losing them is so small that it offsets their large value and makes it not very risky.
  • Firefighters-- their lives obviously are very, very important to them. Economists would say that your life is a valuable asset. However, a firefighter's profession carries with it a fairly high probability of losing his life, so a high probability of losing something that's very valuable. And that high value combined with a high probability of loss is what makes fighting fires defined as being a high risk occupation.
  • A paper clip in your pocket. Almost certainly, you'll use it or lose it by the end of the day. So the probability of loss is very high. However, it's still not considered risky. Because the paper clip isn't a very valuable asset to you.

So what are we seeing here? We're seeing that the risk associated with having an item is equal to the value that you would incur by losing it times the probability of losing it. And this is what makes it an expected value.

Term to Know

    • Risk
    • The negative expected value of losing something. Something becomes less risky if it is not worth much, or if the probability of loss is very small.

A house that costs $300,000 has approximately one in 5,000 chance of burning down, a total loss. So what people will do is they will take that negative $60 of risk and buy insurance. They'll pay the insurance company a little bit more than $60 to assume that $300,000 risk.

So buying insurance protects against loss by agreeing to pay you back for if you do lose it. So since they're assuming the risk, you have to compensate them for that with the $60 by paying premiums.

Now risk is a very unpredictable issue here. So when you're talking about the expected value, you would think about it in terms of that negative $60 mean.

Think about 5,000 homeowners with $300,000 homes. 4,999 homeowners don't lose anything at all. And one of them loses everything. What would that happen to be like? The standard deviation of the data set here would be 4,999 zeroes and a negative $300,000. And the standard deviation of that data set is over $4,200.

Compare that with the fact that almost all the values are the same, that standard deviation is awfully high. It's a very unpredictable quantity with a lot of variability to it. Because of the high variability, it's not beneficial to have only a few policies out there.

ExampleSuppose that you are an upstart insurance company and you only have five policies. It would be possible that one of your five policies would be one of the ones that burned down. You wouldn't have enough money coming in to offset what you would have to pay to that homeowner.So the more policies an insurance company issues, the more predictable the gains and losses become.

The more policies they have out there, the more of a guarantee it is that they'll have a few large payouts to pay. But the more policies they have out there, the more premiums they'll be able to collect. They'll be able to collect relatively small amounts from people who don't need them to pay them. So they're going to more than offset their large payouts by collecting many, many small premiums. So it becomes less risky for them, because it becomes so much more predictable with more policies out there.


Risk is the product of the value of something times the probability of losing it. So it's an expected value. It's a negative expected value. And to offset that negative expected value and protect against loss, insurance companies assume the risk for multiple people and their homes. So whoever purchases insurance policies agrees to pay premiums to the insurance company. And in exchange, insurance company assumes the risk and will pay you in the case of your loss. People will pay good money for that kind of peace of mind.

So we talked about the idea of risk from a probability standpoint.

Good luck!

Source: this work is adapted by sophia author jonathan osters.

Terms to Know

The negative expected value of losing something. Something becomes less risky if it is not worth much, or if the probability of loss is very small.