Cost of Probability
In a previous post on Insured Probabilities I posed a few questions regarding the value of insurance: what is that value, how can we calculate it, and how do we know when it’s worth it without just guessing? Then I ignored them and went on. I’m sorry, but I’m here now. Don’t get too excited though, I still don’t have any exact answers.
I do, however, want to provide some interesting ways of thinking about the value of insurance that may aid in determining whether it makes sense to pay for it. Consider car insurance for example. Most reasonable people buy insurance on a new car to cover any damages that may occur either to the car or the driver. The insurance company makes money though because they have the odds on their side.
On the other side, the expected value of that insurance is almost always less than what people pay for it, but still almost everyone buys it! I find that most people just dislike the risk of not carrying insurance, but why? Like the horse racing tickets, it becomes apparent if we look at the bigger picture.
Consider a simple example of car insurance in which I have a 1% probability of causing a $10,000 accident per month. Over time, that’s going to average $100 per month and the insurance will cost $110 per month, but it may still make sense for me to take the insurance!
The only way it makes sense is if the average dollar costs me more than normal when I have to pay a large amount at once. This is generally the case since the less money I have, the more each dollar is worth. If I have $1 of spendable income, it will cost me $1 if I lose it, but if I have $0 of spendable income, losing $1 will cost me more since I will have to pay interest that I wouldn’t normally have had to pay.
Assuming I can put the $10,000 debt on my credit card that charges 10% per year in interest and it takes me one year to pay it off, I will end up paying something like $10,400 for the accident. Calculating the expected value from that figure is much more accurate, but $104/month is still less than $110/month. However, if I didn’t have the cash flow to pay it off in one year or my interest rate on the debt was above 10%, it could easily end up costing more than $11,000. The cost really comes in though if I don’t have the credit available to me at all, then I have to start selling things to make up for it. Since I would need the cash immediately, I’m probably going to have to sell for a discount rate higher than the insurance premium of 10%. If I don’t have enough to sell I’m really going to be hurting since bankruptcy is going to cost way more than the $1,000/year the insurance company charges.
It’s not that the expected value of the policy is incomplete in determining whether or not it’s worth buying, it just has to account for the added cost of money I can’t afford. If I can afford to lose the full amount without paying the 10% premium, insurance just isn’t worth it, at least not the reduced risk part.
The same sort of thing applies to guaranteed investment returns (paying off loans) and volatile investment returns (stock market). Despite averaging over 10% annualized returns, withdrawing just 8% per year would almost surely leave nothing left over a 30-year period since withdrawals during low performance years would likely end up taking much more than their fair share. The volatility, though averaging to 10%, cannot be worth 10% every year! The very fact that the 10% is only an average of probabilities rather than guaranteed costs money. Consider that before buying into the stock market instead of paying off a debt just because it has a higher average return!
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