quite long text incoming, sorry for that: While reading a corporate finance textbook, i came across a section describing the effect of diversification as well as the systematic and unsystematic risk. The book provides an example in order to show the effect of diversification:
A city where there is a 1% chance of a house being robbed and a 1% chance of the city being hit by an earthquake. An insurance company takes out 100,000 policies of theft insurance and 100,000 policies of earthquake insurance. It is to be expected that 0.01⋅100.000=1.000 houses per year will be robbed.
The textbook quantifies the difference in diversifiability using the standard deviation of the percentage of losses (assuming a 1% probability of a house being robbed and a 1% probability of an earthquake occurring). At the end of the year the damage either occurred (100%) or did not occur (0%).
First, the standard deviation of a claim for a homeowner is calculated:
Standard Deviation_Homeowner (Theft Insurance) = √(0,99*(0-0,01)^2+0,01*(1-0.01)^2) =9,95%
Standard Deviation_Home Owner (Earthquake Insurance) = √(0,99*(0-0,01)^2+0,01*(1-0.01)^2) =9,95%
I find it difficult to interpret the standard deviation in this context I would go with: The probability that my house will be robbed or hit by an earthquake is 1%. So on average, out of 100 robberies/earthquakres my house be affected once, right? The SD tells me that, on average, the probability that my house will be robbed or hit by an earthquake fluctuates around 9.95%. But this interpretation makes no sense for me, since the probability 1%-9.95%=-8.95% would be negative.
In a next step, the risk for the insurance company is calculated.
The book states: In the case of earthquake insurance, because the risk is common, the percentage of claims is either 100% or 0%, just as it was for the homeowner.
However, to calculate the standard deviation, e.g. the risk for theft insurance, it is written that "When the risks are independent and identical, the standard deviation of the average is called the standard error". I don't understand why in the case of theft insurance from an insurance perspective, the standard error is suddenly calculated to quantify the risk of the insurance company.
TL;DR: Why is the standard error used to show the diversification effect for unsystematic risk?
Thanks a lot