What is Common Risk, Independent Risk, and Diversification. Overview and Explanation

Let's say that you moved to a small town on the Mississippi River you're right up against the river you got a nice little home in the town of Sulphur Springs and let us say in Sulphur Springs there are a total of 100 homes altogether and when you buy this home the bank requires that you get insurance. So you go over to Tom's insurance and you get two types of insurance you get flood insurance in case the mississippi river floods and damages your home and you also get title insurance.

So let's just say that the risk of each of these catastrophes of happening, let's say there's a 2% chance that there will be a flood that damages your home and a 2% chance that you will have some kind of problem with your title. For those of you who haven't owned a home title insurance is basically insurance that after you've bought title to the home that somebody doesn't come later and say "Hey wait I actually have titled that home, my great-grandfather gave it to me in his will." So these two types of risks are different and that flood insurance is a common risk. This is common and when we say that something is a common risk what we mean is that it's perfectly correlated and you might be thinking correlated with what? Well, remember that you have 1 home here but there are 99 other homes and sulfur springs. Let's just assume that Tom's insurance has ensured all 100 of these homes, so from the perspective of Tom's insurance the risk of a loss due to flood is basically perfectly correlated because if you have flood damage probably the other 99 homes do as well because you all live in a tiny tight little corner of town. Let's say all the homes are pretty much right up on top of each other, you're right against the river and so if the Mississippi River floods all 100 homes are probably going to be flooded as well and so it's a common risk and it's perfectly correlated. If there's one payout that the insurance company has to make due to this flood it has to pay out all 100 homes.

Now title insurance is different, so title insurance is an independent risk it's not a common risk and that means that it's not correlated with other losses. So if you have some kind of problem with the title of your home that probably has nothing to do with the other 99 homes. It's just probably something unique to your home there was some kind of issue so that's an independent risk. This is going to become relevant as we talk about diversification because ultimately what diversification does is that averages out independent risks but it can't do anything about common risk and so let's get into an example it'll make it a little less abstract.

So let's say if we want to calculate the standard deviation of the chance of a claim from the homeowners perspective, so from your perspective as a homeowner, you can either have assumed you have flood insurance or title insurance you can either have nothing happen which we'll just categorize as 0 or you could have a claim which will collect categorize as 1. So it's just a binary option and so basically one of these things is going to happen and so now we know that there's a 98% chance that you won't have a claim and there's a 2% chance that you will have a claim. So now we can just calculate the standard deviation of a claimed weight.

The standard deviation is just the square root of the variance. So all we're gonna do is just plug in some numbers to calculate the standard deviation and so I'm just gonna go through this we've got 0.98 so this is the 98% chance right that you don't have any claim at all which would be if there's no claim minus 0.02 which is our expected value our mean. We expect that 2% of the time we'll have a claim. Then we square that, that's just part of the formula for a variance. Then we're gonna add to that the 2% chance that we do have a claim which would be the 1, if we have a claim it's 1 and then we subtract the average expected value of 0.02 and then we square that. So we add all that together and then we take the square root and that's going to give us 0.14 or 14%. So that's the standard deviation of our claim is 14% from the perspective of the homeowner and this is the same for the flood or title insurance either way if we're going we've calculated exactly the same result will come.

Now let's take it from the perspective of the insurance company. So from Tom's insurance perspective, the standard deviation with the flood claim is going to be exactly the same as from the homeowner's perspective, it's gonna be this 14%. So it's gonna be calculated the exact same way but it's going to be different when we come to the title insurance. The title insurance instead of calculating the standard deviation we're gonna take something called the standard error. The standard error is basically the standard deviation divided by the sample size and the square root of that sample size.

So we've got a hundred homes, so we're going to take in the denominator the square root of a hundred. So that's gonna be 10 and then in the numerator, we've got the standard deviation. You might be wondering why are we doing this? Why are we taking this standard error? Well, in this case, the risk is of the title loss like somebody coming in and saying "Hey wait I actually have title to that house." and then they sue you and they get the house and so then the insurance company Tom's insurance has to pay you, they have to reimburse you for that. So when those risks are identical which I mean obviously a title loss for one home would be the same as another and independent, independent again we're getting back to the idea that's not correlated. So it's not like if you have some kind of problem with the title on your home that the other 99 homes are also going to have some kind of problem. So the risks that are gonna be identical and independent then we take the standard error. So basically all we're doing is take the standard deviation and we're scaling it by the root of the sample size. So then that's gonna give us 14% divided by the root of 100 which is 10. It's gonna give us 1.4%.

Now if you see 1.4% is a lot lower than 14%, so there's gonna be less risk with the title insurance than there is with the flood from the perspective of the insurance company. So less risk for insurers. Now if it seems a little confusing to you, this is the value of diversification and the idea is that all the independent risks kind of cancel out. So like some years, it might be the case that there was a title claim and the insurance company lost some money but 99 out of the other homes didn't have any problem at all or 98 didn't have any problem at all something like that. So the insurance company can kind of predict and say "Well look you know we're not going to have all kinds of problems in any given year with the title, there would be like one or two every year or something like that." but with the flood when the flood happens all 100 homes get hit. So it's like all or nothing, so there is no averaging out. If you think about it basically diversification is the averaging out of the independent risks over time.

Now here's another way to think about it if this isn't intuitive enough for you you can think about it like this, from the insurers perspective the amount of money that they would have to keep on hand to satisfy all its customers, in terms of the reserves insurance company have reserved this is cash or resources they keep on hand in order to make a payout. If there's a bunch of insurance claims in a given year they need to make a payout they're gonna have to keep more reserves on hand for the flood insurance than they are for the title claim. Why? because if a flood happens, on the off chance on that 2% chance there is a flood they need to be able to reimburse 100 people. They need to reimburse all 100 because the flood hits everybody so they got to keep a lot of reserves on hand to be able to satisfy a hundred claims. But with the title claims it's like okay there's gonna be one, two, or three of them every year it's not like all 100 are gonna hit us at once. So you don't have to keep as many reserves on hand but in any case, the way to think about it is just if you can remember that diversification is the averaging out of the independent risk. They don't all happen at the same time and so they average out and it's actually less risky for the insurer to provide title insurance than it is for it to provide flooding.

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