Toying around with the Kelly criterion I get that the amount I should spend on insurance increases with my income though my intuition says that the higher your income is the less you should insure. Can someone less confused about the Kelly criterion provide some kind of calculation?
For anyone asking, I wondered if, given income and savings rate how much should be invested in bonds, stocks, etc. and how much should be put into insurance, e.g. health, fire, car, etc. from a purely monetary perspective.
The Kelly criterion returns a fraction of your bankroll; it follows that for any (positive-expected-value) bet whatsoever, it will advise you to increase your bet linearly in your income. Could this be the problem, or have you already taken that into account?
That aside, I’m slightly confused about how you can use the Kelly criterion in this case. Insurance must necessarily have negative expected value for the buyer, or the insurer makes no profit. So Kelly should be advising you not to buy any. How are you setting up the problem?
The Kelly criterion returns a fraction of your bankroll; it follows that for any (positive-expected-value) bet whatsoever, it will advise you to increase your bet linearly in your income. Could this be the problem, or have you already taken that into account?
Well that is exactly the point. It confuses me that the richer I am the more insurance I should buy, though the richer I am the more I am able to compensate the risk in not buying any insurance.
That aside, I’m slightly confused about how you can use the Kelly criterion in this case. Insurance must necessarily have negative expected value for the buyer, or the insurer makes no profit.
Yes and no. The insurer makes only a profit if the total cost of insurance is lower than the expected value of the case with no insurance. What you pay the insurer for is that the insurer takes on a risk you yourself are not able to survive (financially), that is catastrophically high costs of medical procedures, liabilities or similar. It is easily possible for the average Joe to foot the bill if he breaks a $5 mug but it would be catastrophic for him if he runs into an oil tank and has to foot the $10,000,000 bill to clean up the environment. (This example is not made up but actually happened around here.)
It is here where my intuition says that the richer you are, the less insurance you need. I could also argue that if it was the other way around, that you should insure more the richer you are, insurance couldn’t exist, seeing as the insurer is the one who should buy insurance from the poor!
You can use the Kelly criterion in any case, either negative or positive expected value. In the case of negative value it just tells you to take the other side of the bet or to pay to avoid the bet. The latter is exactly what insurance is.
So Kelly should be advising you not to buy any. How are you setting up the problem?
I model insurance from the point of view of the buyer. In any given time frame, I can avoid the insurance case with probability q, saving the cost of insurance b. Or I could lose and have to pay a with the probability p = 1-q. This is the case of not buying insurance, though it is available. So if f = p/a—q/a is negative I should insure, if f is positive, I should take the risk. This follows my intuition insofar that catastrophic but improbable risk (very high a, very low p) should be insured but not probable and cheap liabilities (high p, low a).
The trick is now that f is actually the fraction of my bankroll I have to invest. So the richer I am the more I should insure absolutely but my intuition says I should by less insurance. I know I have ignored something fundamental in my model. Is it the cost of insurance? Is it some hidden assumption in the formulation of the Kelly criterion as applied to bets? Did I accidentally assume that someone knows something the other party doesn’t? Did I ignore fixed costs? This eats me up.
Edit: Maybe the results have to be interpreted differently? Of course if I don’t pay the insurance, Kelly still says to invest the money somehow, maybe in having a small amount always at hand as a form of personally organized insurance. Intuition again says that this pool should grow with my wealth, effectively increasing the amount of insurance I buy, though not from an insurer but in opportunity cost.
I know I have ignored something fundamental in my model.
The Kelly formula assumes that you can bet any amount you like, but there are only so many things worth insuring against. Once those are covered, there is no opportunity to spend more, even if you’re still below what the formula says.
In addition, what is a catastrophic loss, hence worth insuring against, varies with wealth. If the risks that you actually face scale linearly with your wealth, then so should your expenditure on insurance. But if having ten times the wealth, your taste were only to live in twice as expensive a house, drive twice as expensive a car, etc. then this will not be the case. You will run out of insurance opportunities even faster than when you were poorer. At the Jobs or Gates level of wealth, there are essentially no insurable catastrophes. Anything big enough to wipe out your fortune would also wipe out the insurance company.
Your reply provides part of the missing piece. Given that I am over some kind of absolute measure of poverty, empirically having twice as much disposable income won’t translate into twice as much insurable assets. This limits the portion of bankroll that can be spent for insurance. Also, Kelly assumed unlitimited offer of bets which is not that far from the truth. Theoretically I can ask the insurer to give twice the payout for twice the cost of insurance.
And still, your answer doesn’t quite answer my original question. I asked for given (monthly) income, savings rate and maybe wealth, what is a optimal allocation of insurance and investments, e.g. bonds or equity? And even if assuming that I keep my current assets but double my income and wealth, Kelly still says to buy insurance, though you admit that anything Gates would want to insure against would ruin the insurer, but my intuition still says that Gates does not insure anything that I would, like a car, house or health costs.
Perhaps the problem lies in the dichotomy “buy insurance” versus “do not buy”. It seems to me that you have, in fact, got three, not two, options:
a) Buy insurance from someone else
b) Spend the money
c) Save the money, in effect buying insurance from yourself.
I think option c) is showing up in your analysis as “do not buy insurance”, which should be reserved for b). You are no doubt correct that Gates does not buy car insurance (unless perhaps he is forced to by law), but that does not mean he is not insured. In effect he is acting as his own insurer, pocketing the profit.
It seems to me, then, that Kelly is telling you that the richer you are, the more you should set aside for emergencies, which seems to make sense; but it cannot distinguish between self-insurance and buying an insurance policy.
So you say that if Kelly says to buy insurance for $300 if the insurance costs $100 I should not buy the police but set aside $300 in case of emergency?
Insurance whose payout is only three times the policy cost should rather be classified as a scam. More generally, I think the strategy would be thus: Kelly tells you to take some amount of money and spend it on insurance. If that amount is enough to cover the payout of the insurance policy, then you should not pay the premium; instead you should put the money in savings and enjoy the interest payments. Only if the amount Kelly assigns to insurance is too small to cover the payout should you consider paying the premium.
Ok, in that case it’s rather the Xbox that is the scam, but I stand by the use of the word. If that sort of insurance is a good deal, you’re being screwed over somewhere. :)
in that case it’s rather the Xbox that is the scam
I wouldn’t say that; as always, the question is whether the good is +EV and the best marginal use of your money. If the console costs $3 and insurance costs $1 and there’s a >33% chance the console will break and you’ll use the insurance, given how much fun you can have with an Xbox is that really a scam? I wouldn’t say so.
If that sort of insurance is a good deal, you’re being screwed over somewhere.
In practice the insurance that you can buy is too limited and the odds too bad to actually make it a good deal; I did some basic analysis of the issue at http://www.gwern.net/Console%20Insurance and you’re better off self-insuring, at least with post-second-generation Xbox 360s (the numbers look really bad for the first-generation but hard sources are hard to come by).
Maybe in that case. In the case of life insurance I have practically unlimited options. Whether I insure for $1M or $1k in case of my death is up to me.
Term life insurance, which pays out nothing if you live beyond the term, is like other insurance: it is only worth buying to protect against some specific risk (e.g. mortgage payments) consequent on an early death.
Life assurance is different, in that the event insured against is certain to happen. That is why it is called assurance: assuredly, you will die. As such, it is primarily an investment, together with an insurance component that guarantees a payout even if you die prematurely. As an investment, you can put as much as you like into it, but if your heirs will not be financially stricken if you die early, you—or rather, they—do not need the insurance part.
You have it backwards. The bet you need to look at is the risk you’re insuring against, not the insurance transaction.
Every day you’re betting that your house won’t burn down today. You’re very likely to win but you’re not making much of a profit when you do. What fraction of your bankroll is your house worth, how likely is it to survive the day and how much will you make when it does? That’s what you need to apply the Kelly criterion to.
Toying around with the Kelly criterion I get that the amount I should spend on insurance increases with my income though my intuition says that the higher your income is the less you should insure. Can someone less confused about the Kelly criterion provide some kind of calculation?
For anyone asking, I wondered if, given income and savings rate how much should be invested in bonds, stocks, etc. and how much should be put into insurance, e.g. health, fire, car, etc. from a purely monetary perspective.
The Kelly criterion returns a fraction of your bankroll; it follows that for any (positive-expected-value) bet whatsoever, it will advise you to increase your bet linearly in your income. Could this be the problem, or have you already taken that into account?
That aside, I’m slightly confused about how you can use the Kelly criterion in this case. Insurance must necessarily have negative expected value for the buyer, or the insurer makes no profit. So Kelly should be advising you not to buy any. How are you setting up the problem?
Well that is exactly the point. It confuses me that the richer I am the more insurance I should buy, though the richer I am the more I am able to compensate the risk in not buying any insurance.
Yes and no. The insurer makes only a profit if the total cost of insurance is lower than the expected value of the case with no insurance. What you pay the insurer for is that the insurer takes on a risk you yourself are not able to survive (financially), that is catastrophically high costs of medical procedures, liabilities or similar. It is easily possible for the average Joe to foot the bill if he breaks a $5 mug but it would be catastrophic for him if he runs into an oil tank and has to foot the $10,000,000 bill to clean up the environment. (This example is not made up but actually happened around here.)
It is here where my intuition says that the richer you are, the less insurance you need. I could also argue that if it was the other way around, that you should insure more the richer you are, insurance couldn’t exist, seeing as the insurer is the one who should buy insurance from the poor!
You can use the Kelly criterion in any case, either negative or positive expected value. In the case of negative value it just tells you to take the other side of the bet or to pay to avoid the bet. The latter is exactly what insurance is.
I model insurance from the point of view of the buyer. In any given time frame, I can avoid the insurance case with probability q, saving the cost of insurance b. Or I could lose and have to pay a with the probability p = 1-q. This is the case of not buying insurance, though it is available. So if f = p/a—q/a is negative I should insure, if f is positive, I should take the risk. This follows my intuition insofar that catastrophic but improbable risk (very high a, very low p) should be insured but not probable and cheap liabilities (high p, low a).
The trick is now that f is actually the fraction of my bankroll I have to invest. So the richer I am the more I should insure absolutely but my intuition says I should by less insurance. I know I have ignored something fundamental in my model. Is it the cost of insurance? Is it some hidden assumption in the formulation of the Kelly criterion as applied to bets? Did I accidentally assume that someone knows something the other party doesn’t? Did I ignore fixed costs? This eats me up.
Edit: Maybe the results have to be interpreted differently? Of course if I don’t pay the insurance, Kelly still says to invest the money somehow, maybe in having a small amount always at hand as a form of personally organized insurance. Intuition again says that this pool should grow with my wealth, effectively increasing the amount of insurance I buy, though not from an insurer but in opportunity cost.
The Kelly formula assumes that you can bet any amount you like, but there are only so many things worth insuring against. Once those are covered, there is no opportunity to spend more, even if you’re still below what the formula says.
In addition, what is a catastrophic loss, hence worth insuring against, varies with wealth. If the risks that you actually face scale linearly with your wealth, then so should your expenditure on insurance. But if having ten times the wealth, your taste were only to live in twice as expensive a house, drive twice as expensive a car, etc. then this will not be the case. You will run out of insurance opportunities even faster than when you were poorer. At the Jobs or Gates level of wealth, there are essentially no insurable catastrophes. Anything big enough to wipe out your fortune would also wipe out the insurance company.
Your reply provides part of the missing piece. Given that I am over some kind of absolute measure of poverty, empirically having twice as much disposable income won’t translate into twice as much insurable assets. This limits the portion of bankroll that can be spent for insurance. Also, Kelly assumed unlitimited offer of bets which is not that far from the truth. Theoretically I can ask the insurer to give twice the payout for twice the cost of insurance.
And still, your answer doesn’t quite answer my original question. I asked for given (monthly) income, savings rate and maybe wealth, what is a optimal allocation of insurance and investments, e.g. bonds or equity? And even if assuming that I keep my current assets but double my income and wealth, Kelly still says to buy insurance, though you admit that anything Gates would want to insure against would ruin the insurer, but my intuition still says that Gates does not insure anything that I would, like a car, house or health costs.
Perhaps the problem lies in the dichotomy “buy insurance” versus “do not buy”. It seems to me that you have, in fact, got three, not two, options:
a) Buy insurance from someone else
b) Spend the money
c) Save the money, in effect buying insurance from yourself.
I think option c) is showing up in your analysis as “do not buy insurance”, which should be reserved for b). You are no doubt correct that Gates does not buy car insurance (unless perhaps he is forced to by law), but that does not mean he is not insured. In effect he is acting as his own insurer, pocketing the profit.
It seems to me, then, that Kelly is telling you that the richer you are, the more you should set aside for emergencies, which seems to make sense; but it cannot distinguish between self-insurance and buying an insurance policy.
So you say that if Kelly says to buy insurance for $300 if the insurance costs $100 I should not buy the police but set aside $300 in case of emergency?
Insurance whose payout is only three times the policy cost should rather be classified as a scam. More generally, I think the strategy would be thus: Kelly tells you to take some amount of money and spend it on insurance. If that amount is enough to cover the payout of the insurance policy, then you should not pay the premium; instead you should put the money in savings and enjoy the interest payments. Only if the amount Kelly assigns to insurance is too small to cover the payout should you consider paying the premium.
Depends on the cost of the risk, no? For a first generation XBox 360, paying half the price for a new replacement is not obviously a bad deal...
Ok, in that case it’s rather the Xbox that is the scam, but I stand by the use of the word. If that sort of insurance is a good deal, you’re being screwed over somewhere. :)
I wouldn’t say that; as always, the question is whether the good is +EV and the best marginal use of your money. If the console costs $3 and insurance costs $1 and there’s a >33% chance the console will break and you’ll use the insurance, given how much fun you can have with an Xbox is that really a scam? I wouldn’t say so.
In practice the insurance that you can buy is too limited and the odds too bad to actually make it a good deal; I did some basic analysis of the issue at http://www.gwern.net/Console%20Insurance and you’re better off self-insuring, at least with post-second-generation Xbox 360s (the numbers look really bad for the first-generation but hard sources are hard to come by).
You can ask, but your insurer will decline. You can only insure your house for what it’s worth.
Maybe in that case. In the case of life insurance I have practically unlimited options. Whether I insure for $1M or $1k in case of my death is up to me.
Term life insurance, which pays out nothing if you live beyond the term, is like other insurance: it is only worth buying to protect against some specific risk (e.g. mortgage payments) consequent on an early death.
Life assurance is different, in that the event insured against is certain to happen. That is why it is called assurance: assuredly, you will die. As such, it is primarily an investment, together with an insurance component that guarantees a payout even if you die prematurely. As an investment, you can put as much as you like into it, but if your heirs will not be financially stricken if you die early, you—or rather, they—do not need the insurance part.
You have it backwards. The bet you need to look at is the risk you’re insuring against, not the insurance transaction.
Every day you’re betting that your house won’t burn down today. You’re very likely to win but you’re not making much of a profit when you do. What fraction of your bankroll is your house worth, how likely is it to survive the day and how much will you make when it does? That’s what you need to apply the Kelly criterion to.
Have you read my reply to RichardKennaway? I explicitly look at the case you mention.