When to self insure for things like cars? (prefer mathematical model)

Non-investing personal finance issues including insurance, credit, real estate, taxes, employment and legal issues such as trusts and wills
Topic Author
adherenceEnergy
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by adherenceEnergy » Thu Jul 23, 2020 5:41 pm

shess wrote:
Thu Jul 23, 2020 12:24 am
If you cut your comprehensive and collision coverage out, how long will it take for that savings to double your $100M?
Irrelevant. Why take a losing insurance bet when you won't notice a loss? I was offered a warranty for $8 on a $12 microphone I purchase at best buy. $8 meant nothing to me, but so did $12. I passed. Paying for insurance would have almost no effect on my net worth growth, but neither would shelling out another $12 if the mic happened to break. Insuring a 15k car for a guy with 100 million is a similar situation as insuring a $12 mic for us peasants. Yeah insurance cost is meaningless, but so is the car.

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Thu Jul 23, 2020 5:55 pm

adherenceEnergy wrote:
Thu Jul 23, 2020 5:41 pm
shess wrote:
Thu Jul 23, 2020 12:24 am
If you cut your comprehensive and collision coverage out, how long will it take for that savings to double your $100M?
Irrelevant. Why take a losing insurance bet when you won't notice a loss? I was offered a warranty for $8 on a $12 microphone I purchase at best buy. $8 meant nothing to me, but so did $12. I passed. Paying for insurance would have almost no effect on my net worth growth, but neither would shelling out another $12 if the mic happened to break. Insuring a 15k car for a guy with 100 million is a similar situation as insuring a $12 mic for us peasants. Yeah insurance cost is meaningless, but so is the car.
On average, I expect richer people to drive nicer cars. If we define anything less than 0.1% of your net worth as noise, then on a $75k vehicle, then a $75M net worth might be the tipping point where the cost of the vehicle crosses into the noise territory.

inbox788
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by inbox788 » Thu Jul 23, 2020 6:16 pm

adherenceEnergy wrote:
Thu Jul 23, 2020 5:41 pm
shess wrote:
Thu Jul 23, 2020 12:24 am
If you cut your comprehensive and collision coverage out, how long will it take for that savings to double your $100M?
Irrelevant. Why take a losing insurance bet when you won't notice a loss? I was offered a warranty for $8 on a $12 microphone I purchase at best buy. $8 meant nothing to me, but so did $12. I passed. Paying for insurance would have almost no effect on my net worth growth, but neither would shelling out another $12 if the mic happened to break. Insuring a 15k car for a guy with 100 million is a similar situation as insuring a $12 mic for us peasants. Yeah insurance cost is meaningless, but so is the car.
Would you have paid $0.25 or $0.02 for the warranty? It's still not going affect your net worth, and likelihood is the gas you use to come back to Best Buy cost more. Let P be the probability that function that the microphone fails during the normal warranty and P' the probability that it fails during the extended time added by the additional cost labeled C. Now Q and Q' are the probabilities that you get your behind off the chair to go to Best Buy to claim the warranty that is covered. The optimal solution space for the utility function of the warranty U(P, P', Q, Q', C) has a maximum that can be solved. I leave the solution to maximizing the function as an exerciser for the reader.

As far a Collision and Comprehensive insurance, if there is a U() that is maximized at a cost C, but I'm willing to pay an additional delta (how does one enter Greek symbols in this interface if we're doing mathematical modeling? oh, never mind) for an additional U'() function for the convenience of having a default repair process and known cost so I don't have to waste time trying to figure out if I should fix it or where to fix it.

Short answer is while I can probably afford to self-insure a $10k or $15k car, I can also afford and it's worth paying a few hundred a year just in case (and there might be a rental car/borrowed car coverage benefit; potential extra peace of mind). I use a 5 or 10 year model for deductible coverage too, going from $500 to $1000 deductible if the 10 year savings is $50-100 or more, and I might pay for the lower deductible if it's less than $50. For $1 or $2, sure. Realistically, I assume I'm paying about twice what it's worth statistically (hence that $50 extra for lower deductible takes 1 accident in 20 years to pay off; pay $1000 total, get back $500). How many accidents does the average and/or median drive get into in their lifetime (and how many years of driving lifetime)? Found 1 in every 18 years on my first and only google search. YMMV https://cederberglaw.com/how-many-car-a ... -lifetime/

If you're only responsible (financially) for half those accidents, maybe adjust appropriate numbers by 50%.

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michaeljc70
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by michaeljc70 » Fri Jul 24, 2020 7:34 am

megabad wrote:
Thu Jul 23, 2020 4:19 pm
Since it doesn’t seem like a bunch of folks are considering this, be aware that every time you have a claim it gets noted in your file and has a very high chance of affecting your premiums. If you hold comprehensive thinking you can just use it to replace your windshield a couple times and get your moneys worth, you may end up paying that back many times over for many years with increased premiums.
If a rock hits your windshield and you file a claim it is unlikely your premium will go up just from that. Even if it happens a couple of times. There are chargeable claims and non-chargeable claims.

eigenperson
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by eigenperson » Fri Jul 24, 2020 8:04 am

No one has actually given you a mathematical model, so I'll give it a go. Here's a rather simplistic model based on the Kelly criterion. The model is for someone who has just bought insurance and is considering canceling it for a full refund, because that makes the formulas simpler. If you don't yet have insurance, but you still want to use the model, you need to pretend that you have already bought the insurance; in particular, you have to subtract the premium you would pay from your net worth before doing the calculations. (Obviously this doesn't matter much unless the premium is quite large.)

The model implicitly assumes logarithmic utility. It only works in the case where there is a single, known loss insured against and where the loss can only happen once during the insured period. It ignores the possibility that a claim results in surcharges in the future. Look, I did say it was a rather simplistic model.

Definitions:

L = loss insured against as a fraction of net worth
p = probability of loss during the insured period
M = ratio of actual premium charged by the insurance company to the "fair" premium

The "fair" premium is the one that a insurer would charge if they modeled your individual risk perfectly, had infinite capital, had no expenses other than paying claims, and wanted to make no profit on average. Thus it takes into account any money the insurer makes on the float, but does not consider any money they have to spend on sales, claims adjustment, or whatever, and does not consider their opportunity cost. In practice, you can ignore the float (interest rates are really low anyway) and just equate the "fair" premium with the expected loss, p * (loss insured against).

Then the model recommends the following decision:

* If M < 1, keep the insurance; it has a positive expected value and also reduces variance.
* If pM > 1, drop the insurance; you are paying more than the expected loss, and therefore always lose money.
* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)

Let's see how the model works in practice.

As a special case, if p is very small, keep the insurance if L > (M-1)/M. For example, for home insurance, the industry average appears to be something like M = 1.3. (Note: most of the rest goes to expenses, not to profit.) Thus the model says you should keep insuring your house against total loss (a very rare event, so we estimate p -> 0) if your equity is more than 23% of your net worth.

What about a case where p is not all that small? Suppose you bought insurance to replace your $1000 phone if you drop it in the toilet. You're a total klutz and there is a 20% chance you will drop it in the toilet each year. The insurance costs $210 per year. M = 1.05 and p = 0.2, so the insurance should be canceled if the phone is less than 6% of your net worth. The same company insures your $1,000 wedding ring. This time the chance of loss is only 0.2%, and the cost is only $2.10/year. Again M = 1.05, but this time p is lower, so the insurance should be canceled if the ring is less than 4.8% of your net worth. As your net worth grows, the ring insurance will be kept longer than the phone insurance. Thus, the model shows that insurance against lower-probability events is a better buy than insurance against higher-probability events, even if the insured amount and the insurance company's profit margins are the same.

What about a case where the model looks bad? Consider title insurance. For title insurance, M = 20 or something ridiculous like that. The model tells you to buy title insurance against total loss only if the risk is 95% of your net worth. This probably strikes you as wrong. In practice, a lot of people are more risk-averse than Kelly.

Anyway, there's a mathematical model. Whether it is a useful one, I don't know.

JackoC
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by JackoC » Fri Jul 24, 2020 9:04 am

michaeljc70 wrote:
Fri Jul 24, 2020 7:34 am
megabad wrote:
Thu Jul 23, 2020 4:19 pm
Since it doesn’t seem like a bunch of folks are considering this, be aware that every time you have a claim it gets noted in your file and has a very high chance of affecting your premiums. If you hold comprehensive thinking you can just use it to replace your windshield a couple times and get your moneys worth, you may end up paying that back many times over for many years with increased premiums.
If a rock hits your windshield and you file a claim it is unlikely your premium will go up just from that. Even if it happens a couple of times. There are chargeable claims and non-chargeable claims.
I both replaced and repaired a windshield on last BMW and already repaired once on current one, three incidents in three very long road trips over 3 yrs (large truck threw up a rock in eastern PA, pick up did in Shelby MT, we seem to have kicked up a rock which somehow ricocheted and hit the windshield in Absolute-Middle-of-Nowhere, NM). The replacement was low cost and two repairs were free with Safelite via Geico. Insurance premium has been tending down, maybe it would have gone down more if not for those incidents, but no obvious impact.

On the math, seems a tendency toward a 0/0 expression as both car and the *net* cost of insurance (not the premium but the difference between the insurance 'fair value' and the premium) tend toward insignificance relative to NW. Again I'd say the key piece of info would be more quantification of fair value v premium. Yes obviously in total insurance co's must charge more than fair value to cover non-claim costs plus return commensurate with their owners' risk. However profitability might vary significantly by type of insurance, and also it's not totally inconceivable a given car owner would have knowledge they are a higher risk than insurance co thinks, though again overall in general that can't be true or the ins co would disappear.

talzara
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by talzara » Fri Jul 24, 2020 10:46 am

eigenperson wrote:
Fri Jul 24, 2020 8:04 am
Here's a rather simplistic model based on the Kelly criterion. ...

* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)
Please post a derivation using standard notation for the Kelly Criterion, which is defined in terms of f, p, and b.
eigenperson wrote:
Fri Jul 24, 2020 8:04 am
What about a case where the model looks bad? Consider title insurance. For title insurance, M = 20 or something ridiculous like that. The model tells you to buy title insurance against total loss only if the risk is 95% of your net worth. This probably strikes you as wrong. In practice, a lot of people are more risk-averse than Kelly.
It's worse than that. What the title insurance industry call a loss ratio is actually a loss and LAE ratio. Title insurance has a loss and LAE ratio of 5%, but the loss ratio is only 0.5%. Most claims are for $0 of loss, and they only require some legal fees to fix some faulty paperwork.

Since you define M to exclude loss adjustment expenses, M = 200.

orangesherbet
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by orangesherbet » Fri Jul 24, 2020 11:14 am

We have a hard time letting go of our paid-off cars. So I have a couple of cars that are 20 years old and we don't really need them, but we still have them.

My not very mathematical formula is:

How much does it cost to insure it for comprehensive & collision
vs
How much would I care if it got damaged and I had to junk it with no insurance compensation

The good news is the premiums got lower and lower as the car's value went down.
At some point it still crossed a tipping point and I no longer thought it worthwhile to pay even the small premium. For me that point has been at around when the car was worth $3k.

mnnice
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by mnnice » Fri Jul 24, 2020 5:13 pm

oldfort wrote:
Wed Jul 22, 2020 11:28 pm
mnnice wrote:
Wed Jul 22, 2020 10:42 pm
How much the comp and collision costs is pretty germane. If the comp and collision cost $1000 per year and the car is worth $10k I would be way more likely to skip it than if it was $100.

Theoretically, not having it could go a long way towards the purchase of something different. My kid totaled my car in 2019. I had bought it new in 2004. I had full coverage til circa 2011. By 2019 I had probably saved enough by skipping comp and collision to buy a replacement vehicle.
It also depends on who the driver is. A 17-year-old teen boy driver is going to be a lot higher risk on average than a 50 year old driver.
Hmm the exact ages in my scenario.

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Fri Jul 24, 2020 6:04 pm

mnnice wrote:
Fri Jul 24, 2020 5:13 pm
oldfort wrote:
Wed Jul 22, 2020 11:28 pm
mnnice wrote:
Wed Jul 22, 2020 10:42 pm
How much the comp and collision costs is pretty germane. If the comp and collision cost $1000 per year and the car is worth $10k I would be way more likely to skip it than if it was $100.

Theoretically, not having it could go a long way towards the purchase of something different. My kid totaled my car in 2019. I had bought it new in 2004. I had full coverage til circa 2011. By 2019 I had probably saved enough by skipping comp and collision to buy a replacement vehicle.
It also depends on who the driver is. A 17-year-old teen boy driver is going to be a lot higher risk on average than a 50 year old driver.
Hmm the exact ages in my scenario.
I could easily believe a 17 year old boy has a 10% chance of totaling a vehicle within the first year.

inbox788
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by inbox788 » Fri Jul 24, 2020 6:25 pm

mnnice wrote:
Fri Jul 24, 2020 5:13 pm
oldfort wrote:
Wed Jul 22, 2020 11:28 pm
mnnice wrote:
Wed Jul 22, 2020 10:42 pm
How much the comp and collision costs is pretty germane. If the comp and collision cost $1000 per year and the car is worth $10k I would be way more likely to skip it than if it was $100.

Theoretically, not having it could go a long way towards the purchase of something different. My kid totaled my car in 2019. I had bought it new in 2004. I had full coverage til circa 2011. By 2019 I had probably saved enough by skipping comp and collision to buy a replacement vehicle.
It also depends on who the driver is. A 17-year-old teen boy driver is going to be a lot higher risk on average than a 50 year old driver.
Hmm the exact ages in my scenario.
Premium might be $100 for the 50 year old and $1000 for the 17 year old.

Isn't that priced into the premium already? Or does that adjustment play a factor in the model taking into account NW?

Also, there's the average loss vs total loss to consider. A total loss on a new $30k car may be far higher than an average loss of $5k, but 3-5 years later, the risks haven't changed much but the total loss is only $15k vs average loss still around $5k.

Insurance premium pricing is probably more related to higher probability average loss vs. risk aversion of owners/drivers makes them more concerned about low probability/high loss events.

https://www.sawayalaw.com/blog/breaking-cost-accident/

FWIW, last couple of times I upgraded cars, replacing decade old ones, the added cost of insurance was surprisingly low, increasing about $200/year for probably 5X max loss potential. I guess from that the risk of a total loss is less than 1%/year.

MathWizard
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by MathWizard » Sat Jul 25, 2020 10:45 am

I've only had comprehensive on the one occasion that I didn't have enough money to buy the car outright.

I've always believed the you should almost never own a car that you could not afford to replace, so only liability almost all the time.


The exception I had was to buy a reliable car to get to interviews after my BS. That was probably not an optimal move, but I did not want to lose out on an interview due to car trouble.

I have only been in one fender bender that was my fault in 45 years of driving. My car only had minor damage to rear quarter panel , which I never fixed.

novemberrain
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by novemberrain » Sat Jul 25, 2020 10:58 am

I would put the threshold at like 50x or more. That is, if you have a $10k car, you can "self insure" it if your NW is more than $500k.

Of course , still carry the max liability.

novemberrain
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by novemberrain » Sat Jul 25, 2020 11:05 am

Watty wrote:
Wed Jul 22, 2020 11:38 pm

That could work if the accident was your fault but if someone else is at fault and their insurance is not paying for the damage, or they were uninsured, then having your insurance would be useful since your insurance company would deal with collecting from the other driver.
Isn’t that a separate type of insurance? The “uninsured motorist coverage”?

novemberrain
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by novemberrain » Sat Jul 25, 2020 11:15 am

tiburblium wrote:
Thu Jul 23, 2020 4:03 pm
I assume whoever hits me will have no/ minimal insurance, have no assets and immune to lawsuits, and will inflict $,$$$,$$$ in medical costs
Is My hospital expenses covered if I have my own medical insurance? I am wondering what is the need to buy medical coverage from car insurance company

TheJoelfather
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by TheJoelfather » Sat Jul 25, 2020 11:38 am

My mental model:
  1. What am I considering insuring?
  2. You're considering insuring?! That means you maybe shouldn't afford it. Can you do with a less expensive version of that thing?
  3. If yes, you're right, buy the lesser thing.
  4. If no, buy the insurnce with a deductible as large as you can stomach. Adjust your emergency fund based on the deductible.
So if you can stomach a hit $10k at a time, my model would output the following as a sample and your emergency fund should account for a $10k hit at any time:
  • No cell phone insurance ($1k)
  • No collectibles insurance ($5k)
  • Liability-only coverage on older car ($7k)
  • Comprehensive & collision on newer car ($20k) with the highest allowable deductible
  • Homeowners insurance ($500k) with a deductible of $10k

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Sat Jul 25, 2020 1:18 pm

novemberrain wrote:
Sat Jul 25, 2020 11:05 am
Watty wrote:
Wed Jul 22, 2020 11:38 pm

That could work if the accident was your fault but if someone else is at fault and their insurance is not paying for the damage, or they were uninsured, then having your insurance would be useful since your insurance company would deal with collecting from the other driver.
Isn’t that a separate type of insurance? The “uninsured motorist coverage”?
No, uninsured motorist coverage is for if the other driver doesn't have insurance. That's different than your insurance company trying to collect from the other driver's insurance.

Topic Author
adherenceEnergy
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by adherenceEnergy » Mon Jul 27, 2020 12:20 pm

eigenperson wrote:
Fri Jul 24, 2020 8:04 am
* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)
So I plugged in some numbers for car insurance, and I think I understand your model. I used some simplistic numbers based on googling. 6% was just any claim in a year, not total loss (0.3% was total loss on a newish car, fwiw, but the end result wasn't significantly different). Insurance net premium was $100 collected to $73 paid out (with $27 to underwriting expenses, as you talked about wrt all the excess going to underwriting expenses).

x >(100/73-1)/((100/73) - 0.06(100/73)^2)
x>29.4%

Meaning on a 20k car, and a >68k net worth, you shouldn't insure it according to the model.

This is a really cool model, imo. Thanks for taking the time to post about it. I agree Kelly is usually much more aggressive than most people are comfortable with. This leads me to believe my 10x rule of thumb is probably in the ballpark that corresponds to my risk tolerance, as in other scenarios, I'm usually about 3x more conservative than Kelly would recommend, except in really high EV bets, where it recommends very very large % of your bankroll.

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Mon Jul 27, 2020 12:47 pm

adherenceEnergy wrote:
Mon Jul 27, 2020 12:20 pm
eigenperson wrote:
Fri Jul 24, 2020 8:04 am
* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)
So I plugged in some numbers for car insurance, and I think I understand your model. I used some simplistic numbers based on googling. 6% was just any claim in a year, not total loss (0.3% was total loss on a newish car, fwiw, but the end result wasn't significantly different). Insurance net premium was $100 collected to $73 paid out (with $27 to underwriting expenses, as you talked about wrt all the excess going to underwriting expenses).

x >(100/73-1)/((100/73) - 0.06(100/73)^2)
x>29.4%

Meaning on a 20k car, and a >68k net worth, you shouldn't insure it according to the model.

This is a really cool model, imo. Thanks for taking the time to post about it. I agree Kelly is usually much more aggressive than most people are comfortable with. This leads me to believe my 10x rule of thumb is probably in the ballpark that corresponds to my risk tolerance, as in other scenarios, I'm usually about 3x more conservative than Kelly would recommend, except in really high EV bets, where it recommends very very large % of your bankroll.
This isn't the Kelly criterion. The Kelly criterion says you should never take a negative expected value bet. In other words, never buy insurance regardless of your net worth.

fyre4ce
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by fyre4ce » Mon Jul 27, 2020 3:47 pm

eigenperson wrote:
Fri Jul 24, 2020 8:04 am
No one has actually given you a mathematical model, so I'll give it a go. Here's a rather simplistic model based on the Kelly criterion. The model is for someone who has just bought insurance and is considering canceling it for a full refund, because that makes the formulas simpler. If you don't yet have insurance, but you still want to use the model, you need to pretend that you have already bought the insurance; in particular, you have to subtract the premium you would pay from your net worth before doing the calculations. (Obviously this doesn't matter much unless the premium is quite large.)

The model implicitly assumes logarithmic utility. It only works in the case where there is a single, known loss insured against and where the loss can only happen once during the insured period. It ignores the possibility that a claim results in surcharges in the future. Look, I did say it was a rather simplistic model.

Definitions:

L = loss insured against as a fraction of net worth
p = probability of loss during the insured period
M = ratio of actual premium charged by the insurance company to the "fair" premium

The "fair" premium is the one that a insurer would charge if they modeled your individual risk perfectly, had infinite capital, had no expenses other than paying claims, and wanted to make no profit on average. Thus it takes into account any money the insurer makes on the float, but does not consider any money they have to spend on sales, claims adjustment, or whatever, and does not consider their opportunity cost. In practice, you can ignore the float (interest rates are really low anyway) and just equate the "fair" premium with the expected loss, p * (loss insured against).

Then the model recommends the following decision:

* If M < 1, keep the insurance; it has a positive expected value and also reduces variance.
* If pM > 1, drop the insurance; you are paying more than the expected loss, and therefore always lose money.
* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)

Let's see how the model works in practice.

As a special case, if p is very small, keep the insurance if L > (M-1)/M. For example, for home insurance, the industry average appears to be something like M = 1.3. (Note: most of the rest goes to expenses, not to profit.) Thus the model says you should keep insuring your house against total loss (a very rare event, so we estimate p -> 0) if your equity is more than 23% of your net worth.

What about a case where p is not all that small? Suppose you bought insurance to replace your $1000 phone if you drop it in the toilet. You're a total klutz and there is a 20% chance you will drop it in the toilet each year. The insurance costs $210 per year. M = 1.05 and p = 0.2, so the insurance should be canceled if the phone is less than 6% of your net worth. The same company insures your $1,000 wedding ring. This time the chance of loss is only 0.2%, and the cost is only $2.10/year. Again M = 1.05, but this time p is lower, so the insurance should be canceled if the ring is less than 4.8% of your net worth. As your net worth grows, the ring insurance will be kept longer than the phone insurance. Thus, the model shows that insurance against lower-probability events is a better buy than insurance against higher-probability events, even if the insured amount and the insurance company's profit margins are the same.

What about a case where the model looks bad? Consider title insurance. For title insurance, M = 20 or something ridiculous like that. The model tells you to buy title insurance against total loss only if the risk is 95% of your net worth. This probably strikes you as wrong. In practice, a lot of people are more risk-averse than Kelly.

Anyway, there's a mathematical model. Whether it is a useful one, I don't know.
I'd be interested in seeing a derivation too. I thought it would be obvious when looking at the Taylor series expansions of ln(1+Pr) and ln(1+Pr-L), but I'm not seeing how things line up. Here Pr is the premium = p*L*M.

It'd also be nice to have a model that includes a deductible. That could be useful for trading off different sets of {deductible, premium} options.

investing engineer
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by investing engineer » Mon Jul 27, 2020 11:07 pm

oldfort wrote:
Mon Jul 27, 2020 12:47 pm
adherenceEnergy wrote:
Mon Jul 27, 2020 12:20 pm
eigenperson wrote:
Fri Jul 24, 2020 8:04 am
* Otherwise, keep the insurance if L > (M-1)/(M - pM^2)
So I plugged in some numbers for car insurance, and I think I understand your model. I used some simplistic numbers based on googling. 6% was just any claim in a year, not total loss (0.3% was total loss on a newish car, fwiw, but the end result wasn't significantly different). Insurance net premium was $100 collected to $73 paid out (with $27 to underwriting expenses, as you talked about wrt all the excess going to underwriting expenses).

x >(100/73-1)/((100/73) - 0.06(100/73)^2)
x>29.4%

Meaning on a 20k car, and a >68k net worth, you shouldn't insure it according to the model.

This is a really cool model, imo. Thanks for taking the time to post about it. I agree Kelly is usually much more aggressive than most people are comfortable with. This leads me to believe my 10x rule of thumb is probably in the ballpark that corresponds to my risk tolerance, as in other scenarios, I'm usually about 3x more conservative than Kelly would recommend, except in really high EV bets, where it recommends very very large % of your bankroll.
This isn't the Kelly criterion. The Kelly criterion says you should never take a negative expected value bet. In other words, never buy insurance regardless of your net worth.
The model here has already changed the frame of reference to having already purchased the insurance. The question is whether to drop the insurance.

talzara
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by talzara » Tue Jul 28, 2020 2:25 pm

fyre4ce wrote:
Mon Jul 27, 2020 3:47 pm
It'd also be nice to have a model that includes a deductible. That could be useful for trading off different sets of {deductible, premium} options.
Just model the deductible as a separate insurance policy, then subtract it out.

High deductibles make insurance a better deal. The deductible takes care of the high-frequency, low-severity losses that you can self-insure. What remains are the low-frequency, high-severity losses that you actually want to buy insurance for.

For example, most insurers offer up to 5% deductibles on homeowners policies. At a 5% deductible, almost all of the premium goes to paying for fire, liability, and catastrophes. All the other coverage becomes very cheap: theft, hail, lightning, etc.

Homeowners insurance doesn't allow you to drop coverage, but high deductibles allow you to avoid paying for most of it by covering only the long-tail risks. A $100 theft premium could drop to $5, because most thieves won't steal 5% of the value of your house. However, if a well-organized gang shows up with a moving truck and cleans out your house, then you're still insured.

jbmitt
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by jbmitt » Tue Jul 28, 2020 5:15 pm

reln wrote:
Wed Jul 22, 2020 11:15 pm
jbmitt wrote:
Wed Jul 22, 2020 10:32 pm
reln wrote:
Wed Jul 22, 2020 8:38 pm
adherenceEnergy wrote:
Wed Jul 22, 2020 7:37 pm
Let me give you an example. Let's say you had 10k to your name and your had a 15k car, buying comprehensive and collision insurance should be a no brainer. Now let's say you had 100 million, obviously you wouldn't need insurance on a 15k car, you'd just write a check without blinking. What is the threshold on when you should go without insurance?

My gut instinct is that if your net worth is 10x the value of what you're insuring, you should self insure, but it's not really based on anything. Does anyone know of any mathematical models of when to self insure for items based on your current net worth?
"Self insurance" is an oxymoron.

Not really. There are several states that allow you to place a surety bond in lieu of liability insurance. It’s not a good solution for most people. It is an option for some businesses.
Which isn't insurance.
Since you’re a contrarian. You could also insure via a captive arrangement where you develop your own insurance for your risk. It’s a fancy way that may companies ‘self insure’.

Starfish
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by Starfish » Tue Jul 28, 2020 5:51 pm

sk2101 wrote:
Thu Jul 23, 2020 1:22 pm
michaeljc70 wrote:
Wed Jul 22, 2020 8:06 pm
Unless you have tens of millions, I wouldn't consider self insuring (normal things-not counting getting collision on a $3k car).
People worth tens of millions are the ones who most need the highest coverage they can get. Lawyers are salivating over the opportunity to sue them. There was a guy in Florida that settled for almost $50 million in a car accident that resulted in death.

Auto insurance covers 300k or so. How would it help?
And also the discussion is about collision, not liability.

reln
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by reln » Tue Jul 28, 2020 11:43 pm

jbmitt wrote:
Tue Jul 28, 2020 5:15 pm
reln wrote:
Wed Jul 22, 2020 11:15 pm
jbmitt wrote:
Wed Jul 22, 2020 10:32 pm
reln wrote:
Wed Jul 22, 2020 8:38 pm
adherenceEnergy wrote:
Wed Jul 22, 2020 7:37 pm
Let me give you an example. Let's say you had 10k to your name and your had a 15k car, buying comprehensive and collision insurance should be a no brainer. Now let's say you had 100 million, obviously you wouldn't need insurance on a 15k car, you'd just write a check without blinking. What is the threshold on when you should go without insurance?

My gut instinct is that if your net worth is 10x the value of what you're insuring, you should self insure, but it's not really based on anything. Does anyone know of any mathematical models of when to self insure for items based on your current net worth?
"Self insurance" is an oxymoron.

Not really. There are several states that allow you to place a surety bond in lieu of liability insurance. It’s not a good solution for most people. It is an option for some businesses.
Which isn't insurance.
Since you’re a contrarian. You could also insure via a captive arrangement where you develop your own insurance for your risk. It’s a fancy way that may companies ‘self insure’.
This reminded me of so many fun memories of facultative reinsurance deals. Those were the good old days. It was also fun to figure out good ways to separate risks into different legal entities. Though it is not company 'self insurance.' It is the opposite. It is legal separation of risks by insinuating liabilities.

I'm not a contrarian, though. The term 'self insurance' is a contradiction onto itself. Insurance implies spreading of risks such as risk pooling (life insurance, car insurance, health insurance, longevity insurance etc). Self insurance on the other hand does the opposite thing. It concentrates risk.

Not saying anything about what you should or shouldn't do. Just pointing out that the option to self insure is the same as the option to not insure. Nothing wrong with that. I choose to not insure some risks. I just don't fool myself into thinking I self insured something.

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Re: When to self insure for things like cars? (prefer mathematical model)

Post by MathIsMyWayr » Wed Jul 29, 2020 12:46 am

I wouldn't rely on an amateurish flimsy math based on a questionable logic for making a decision. Math does not improve the underlying logic, but provides a structural flow to the logic. I assume insurance companies come ahead moneywise. The deciding factors are 1) can you afford a loss 2) can you deal with the inconvenience of taking care of the loss on your own?

azianbob
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by azianbob » Wed Jul 29, 2020 12:28 pm

I believe the OP is talking about self insuring vs collision & comprehensive. I do not think anywhere they are suggesting to drop the liability and medical portion. That would be quite stupid imo.

I generally keep C&C until the car is worth less than $7,500 (this happens around 100k miles). That's generally the point where I feel the cost of the C&C is not worth the payout. If I were to get in a major accident, the repairs have a good chance to be greater than the car is worth and they would just pay me out anyways. I would want C&C for the cases where most accidents would result in me being able to repair the car and keep driving it.

afan
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by afan » Wed Jul 29, 2020 12:37 pm

It also depends on how small a claim one might actually file. If the value of a claim, after deductible, is small, then it may not be worth the potential impact on premiums to file at all. Total of an expensive new car- sure you would file. $4,000 of damage to a car worth more than that? Would you file a claim with a deductible of $3,000, thus get $1,000? Would you file a claim if the deductible were $2,000?
The cost of the insurance vs amount and likelihood of a loss are not the only factors. One would need to know the impact of the claim on future premiums to help decide whether it is worth filing at all.
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JackoC
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by JackoC » Wed Jul 29, 2020 1:19 pm

MathIsMyWayr wrote:
Wed Jul 29, 2020 12:46 am
I wouldn't rely on an amateurish flimsy math based on a questionable logic for making a decision. Math does not improve the underlying logic, but provides a structural flow to the logic. I assume insurance companies come ahead moneywise. The deciding factors are 1) can you afford a loss 2) can you deal with the inconvenience of taking care of the loss on your own?
I agree generally, but would just say that besides some lack of certainty on your points 1 and 2 as they apply to me, the key missing piece is my inability to analytically determine *how much* more than fair value the ins so is charging me for comp/coll in a good deal available in the market. A model is not what I need to refine my estimate it's better inputs.

Eg. I get competing quotes every once and in a while and nobody touches Geico with a 10' pole in overall premium for us, high limit liability for late middle age couple plus one adult kid as additional driver, two vehicles only one with comp/coll. I'm listed as primary driver of the 2018 BMW M2 ~$527/yr for comp/coll w/ $2k deductible. 'Geico has to make money, in general, to exist', but it's very difficult IMO to nail down whether the net cost of that insurance is $50, $300, -$50? per yr, which would make a big difference in any analytical model. My gut feel is that since I push the car fairly hard on winding roads for most of its ~8k miles/yr, though I'm quite conservative in traffic and the car is garaged ~90% of the time pretty safe from theft or vandalism, the net cost of that insurance isn't that much. In $'s and cents 1) total loss without compensation would be entirely affordable but a bummer, 2) dealing inside the insurance system is a lot easier. I will probably drop it at some point. Again I don't think there's enough solid information to where a model becomes the key tool to process it.

CT-Scott
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by CT-Scott » Wed Jul 29, 2020 8:00 pm

azianbob wrote:
Wed Jul 29, 2020 12:28 pm
I generally keep C&C until the car is worth less than $7,500 (this happens around 100k miles). That's generally the point where I feel the cost of the C&C is not worth the payout. If I were to get in a major accident, the repairs have a good chance to be greater than the car is worth and they would just pay me out anyways. I would want C&C for the cases where most accidents would result in me being able to repair the car and keep driving it.
Wait...so is C&C only for accidents that get repaired? If I have a car worth $4000 and don't have C&C and the car gets "totaled" the auto insurance will pay me the car's current KBB value? What coverage covers that, if not C&C?

MathIsMyWayr
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by MathIsMyWayr » Thu Jul 30, 2020 12:07 am

CT-Scott wrote:
Wed Jul 29, 2020 8:00 pm
azianbob wrote:
Wed Jul 29, 2020 12:28 pm
I generally keep C&C until the car is worth less than $7,500 (this happens around 100k miles). That's generally the point where I feel the cost of the C&C is not worth the payout. If I were to get in a major accident, the repairs have a good chance to be greater than the car is worth and they would just pay me out anyways. I would want C&C for the cases where most accidents would result in me being able to repair the car and keep driving it.
Wait...so is C&C only for accidents that get repaired? If I have a car worth $4000 and don't have C&C and the car gets "totaled" the auto insurance will pay me the car's current KBB value? What coverage covers that, if not C&C?
The CT-Scott's statement has no bearing on the underlined statement of yours, just out of the blue. The CT-Scott's reasoning has much more value than many lines of equations.

azianbob
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by azianbob » Thu Jul 30, 2020 1:49 am

CT-Scott wrote:
Wed Jul 29, 2020 8:00 pm
Wait...so is C&C only for accidents that get repaired? If I have a car worth $4000 and don't have C&C and the car gets "totaled" the auto insurance will pay me the car's current KBB value? What coverage covers that, if not C&C?
If you have a car and do not have C&C and your car gets totaled, if you are not at fault, you will get the value of your car from the other driver's insurance. Since it's their fault, your insurance does not matter.

On the other hand, if the accident was your fault, you will get $0 since you did not have C&C.

If you have C&C and it is your fault, then they will pay up to the value of your car minus the deductible for repairs. If you have a $4000 car and the repair costs $3000 and your deductible is $1000 then they will pay you $2000. If the repairs costs $7000 then they will just pay you the $3000 (value of your car $4000 minus the deductible of $1000). If your car is totaled, they will pay you out $3000. So that's why I'm saying at a certain point, I feel it's not really worth paying for C&C.

But I do carry 250/500k liability.

MathIsMyWayr
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by MathIsMyWayr » Thu Jul 30, 2020 7:18 am

azianbob wrote:
Thu Jul 30, 2020 1:49 am
CT-Scott wrote:
Wed Jul 29, 2020 8:00 pm
Wait...so is C&C only for accidents that get repaired? If I have a car worth $4000 and don't have C&C and the car gets "totaled" the auto insurance will pay me the car's current KBB value? What coverage covers that, if not C&C?
If you have a car and do not have C&C and your car gets totaled, if you are not at fault, you will get the value of your car from the other driver's insurance. Since it's their fault, your insurance does not matter.

On the other hand, if the accident was your fault, you will get $0 since you did not have C&C.

If you have C&C and it is your fault, then they will pay up to the value of your car minus the deductible for repairs. If you have a $4000 car and the repair costs $3000 and your deductible is $1000 then they will pay you $2000. If the repairs costs $7000 then they will just pay you the $3000 (value of your car $4000 minus the deductible of $1000). If your car is totaled, they will pay you out $3000. So that's why I'm saying at a certain point, I feel it's not really worth paying for C&C.

But I do carry 250/500k liability.
There is a little twist with C&C. If you have C&C and your car is damaged through the 100% fault of the other driver, your insurance company may withhold your deductible until a subjugation.

eigenperson
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by eigenperson » Thu Jul 30, 2020 7:24 am

talzara wrote:
Fri Jul 24, 2020 10:46 am

Please post a derivation using standard notation for the Kelly Criterion, which is defined in terms of f, p, and b.
fyre4ce wrote:
Mon Jul 27, 2020 3:47 pm
I'd be interested in seeing a derivation too. I thought it would be obvious when looking at the Taylor series expansions of ln(1+Pr) and ln(1+Pr-L), but I'm not seeing how things line up. Here Pr is the premium = p*L*M.

It'd also be nice to have a model that includes a deductible. That could be useful for trading off different sets of {deductible, premium} options.
The standard formulation of the Kelly criterion is:

f = p - (1-p)/b

where p is the probability of winning the bet, b is (upside / downside), and f is the fraction of net worth to bet. In this model, you win the bet if you don't have a loss. It is more convenient to talk in terms of 1 - p or the probability of loss -- let's call it q (in my previous post I called it p, but obviously can't use that now).

Before handling the Kelly cases let's handle the non-Kelly ones:

* If M < 1 then dropping insurance both has a negative expected value and increases variance, so you should never do it (always insure).
* If qM >= 1 then you end up paying more than the possible loss in a single premium. Thus, dropping insurance has a positive (or 0) EV and reduces variance, so you should always do it (never insure).

For the rest we can therefore assume M >= 1 and qM < 1.

Then the Kelly bet size is:

f = 1 - q - q/b

Now for b = upside / downside. The upside of canceling insurance, if you win, is that you save the premium, which in this model is equal to qLM. The downside is that you lose the loss L, but in this case at least you still saved the premium, so the overall downside is L - qLM. (Of course the loss is actually L * net worth rather than L... but we're dividing and net worth will cancel.) Thus we can write the entire RHS of the equation in terms of the model parameters:

f = 1 - q - q(L-qLM)/(qLM) = 1 - q - (1 - qM)/M = 1 - q - 1/M + q = 1 - 1/M = (M-1)/M.

For the LHS need to know how much we are betting, as a fraction of net worth. That is L - qLM, as previously calculated. We don't actually get to choose the bet size in this model. Suppose we don't want to exceed the Kelly bet. Then we will bet (i.e. *not* insure) if:

L - qLM < (M-1)/M

Now we just divide by (1-qM). We can restrict to the case where qM < 1 as noted above. Thus:

L < (M-1)/(M(1-qM)) = (M-1)/(M - qM^2).

This is the condition for *not* insuring. The condition for insuring is therefore:

L > (M-1)/(M-qM^2).

Except for the fact that I replaced "p" with "q" to avoid a notation conflict, this is the same formula.
oldfort wrote:
Mon Jul 27, 2020 12:47 pm
This isn't the Kelly criterion. The Kelly criterion says you should never take a negative expected value bet. In other words, never buy insurance regardless of your net worth.
As someone else already noted, this model tells you when to cancel insurance, not when to buy it.

Purchasing insurance should not be viewed as a bet in the context of the Kelly criterion. You can try, but then you're ignoring that your "bet" is perfectly negatively correlated with an exogenous loss. Under such circumstances, the Kelly criterion doesn't maximize log utility anymore.

CT-Scott
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by CT-Scott » Thu Jul 30, 2020 7:42 am

azianbob wrote:
Thu Jul 30, 2020 1:49 am
CT-Scott wrote:
Wed Jul 29, 2020 8:00 pm
Wait...so is C&C only for accidents that get repaired? If I have a car worth $4000 and don't have C&C and the car gets "totaled" the auto insurance will pay me the car's current KBB value? What coverage covers that, if not C&C?
If you have a car and do not have C&C and your car gets totaled, if you are not at fault, you will get the value of your car from the other driver's insurance. Since it's their fault, your insurance does not matter.

On the other hand, if the accident was your fault, you will get $0 since you did not have C&C.

If you have C&C and it is your fault, then they will pay up to the value of your car minus the deductible for repairs. If you have a $4000 car and the repair costs $3000 and your deductible is $1000 then they will pay you $2000. If the repairs costs $7000 then they will just pay you the $3000 (value of your car $4000 minus the deductible of $1000). If your car is totaled, they will pay you out $3000. So that's why I'm saying at a certain point, I feel it's not really worth paying for C&C.
Thanks for the reply. But if the 2007 Honda CR-V (say 150k miles) that my still-inexperienced 21-year-old daughter is driving is worth maybe $3500-4000, and collision ($500 deduct) and comp ($100 deduct) is costing me $239/year, isn't it still worth paying that? If she gets in an accident that is her fault and it gets totaled, I get $3000-3500 (after accounting for the $500 deductible).

Also, can anyone suggest the best source to use for determining a car's value, that would be closest to what an insurance company would likely value it? The KBB site asks you to give a subjective rating of the car being in Fair, Good, Very Good, or Excellent condition, and then gives you a trade-in value "range", so the low-end value for an older car in "Fair" condition vs the high-end value if rated as "Excellent" condition can be pretty wide, and if the car is demolished in an accident, how would they know what condition it was really in before the accident?

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Thu Jul 30, 2020 7:57 am

deleted
Last edited by oldfort on Thu Jul 30, 2020 9:50 pm, edited 1 time in total.

eigenperson
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by eigenperson » Thu Jul 30, 2020 10:18 am

oldfort wrote:
Thu Jul 30, 2020 7:57 am
This is all wrong. You're using the Kelly Criterion for even money bets. Insurance is not an even money. A more general form for the Kelly Criterion
is
{f=p/a-q/b.}
No, I'm not. The formulation for even-money bets is f = p - q = 2p - 1. I used f = p - q/b, which is one of many equivalent formulations for non-even-money bets.

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Thu Jul 30, 2020 10:20 pm

eigenperson wrote:
Thu Jul 30, 2020 10:18 am
oldfort wrote:
Thu Jul 30, 2020 7:57 am
This is all wrong. You're using the Kelly Criterion for even money bets. Insurance is not an even money. A more general form for the Kelly Criterion
is
{f=p/a-q/b.}
No, I'm not. The formulation for even-money bets is f = p - q = 2p - 1. I used f = p - q/b, which is one of many equivalent formulations for non-even-money bets.
You're right on this point, but there's other issues with using the Kelly Criterion in this context. It assumes you can bet an arbitrary percentage of your bankroll vs. in insurance only being allowed to bet a constant dollar amount. If you're trying to maximize the expected value of the log of wealth, your terminal wealth is guaranteed to be 0 after N = Bankroll/Premiums number of years, assuming no other sources of income and you always buy insurance.

eigenperson
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by eigenperson » Fri Jul 31, 2020 6:59 am

oldfort wrote:
Thu Jul 30, 2020 10:20 pm
eigenperson wrote:
Thu Jul 30, 2020 10:18 am
oldfort wrote:
Thu Jul 30, 2020 7:57 am
This is all wrong. You're using the Kelly Criterion for even money bets. Insurance is not an even money. A more general form for the Kelly Criterion
is
{f=p/a-q/b.}
No, I'm not. The formulation for even-money bets is f = p - q = 2p - 1. I used f = p - q/b, which is one of many equivalent formulations for non-even-money bets.
You're right on this point, but there's other issues with using the Kelly Criterion in this context. It assumes you can bet an arbitrary percentage of your bankroll vs. in insurance only being allowed to bet a constant dollar amount. If you're trying to maximize the expected value of the log of wealth, your terminal wealth is guaranteed to be 0 after N = Bankroll/Premiums number of years, assuming no other sources of income and you always buy insurance.
Unfortunately, in this model, your net worth goes to zero no matter what you do. Your only choice is whether it goes there via premiums or losses. Using the Kelly criterion ensures that you get there later rather than (possibly) sooner. Note that once your net worth gets low enough that the loss places you into bankruptcy, the actual loss is no longer equal to the nominal loss. You should adjust L accordingly. If you do this, the model will eventually tell you to stop buying insurance.

The other critique (you can't vary your bet size) is accurate. This model tells you not to make a super-Kelly bet, but in reality a super-Kelly bet might be better than no bet, if those are your only two options. In practice, though, I don't think this is a serious weakness, because this model generally instructs the user to drop insurance earlier than they are comfortable doing.

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Re: When to self insure for things like cars? (prefer mathematical model)

Post by MikeG62 » Fri Jul 31, 2020 7:54 am

We could easily afford to self-insure (on the collision and comprehensive sides), but we don't. For full disclosure, we do drive newer higher-end luxury cars (not 10+ year old high mileage vehicles as is typical among many Bogleheads). We do carry a high(er) deductible ($1,000). Also, we now lease and I think the finance companies require full insurance coverage (so it's a moot point).
Real Knowledge Comes Only From Experience

oldfort
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by oldfort » Fri Jul 31, 2020 4:27 pm

eigenperson wrote:
Fri Jul 31, 2020 6:59 am
Unfortunately, in this model, your net worth goes to zero no matter what you do. Your only choice is whether it goes there via premiums or losses. Using the Kelly criterion ensures that you get there later rather than (possibly) sooner. Note that once your net worth gets low enough that the loss places you into bankruptcy, the actual loss is no longer equal to the nominal loss. You should adjust L accordingly. If you do this, the model will eventually tell you to stop buying insurance.
I'm not sure your strategy gets you there later rather than sooner. If you forego insurance, there's a possibility your wealth will last longer than (N/premium) years, versus being guaranteed to go bust. L might be capped at one, but it will always be greater than (M-1)/M, telling you to buy insurance.

prairieman
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Re: When to self insure for things like cars? (prefer mathematical model)

Post by prairieman » Fri Jul 31, 2020 5:53 pm

What is the value of hassles? I can afford to self insure and I still buy collusion and comprehensive insurance, even on my 2006 Honda van.

Two life experiences:

I was hit by a person who lied to their company and claimed I was at fault. I had just stopped paying collision insurance on the car, but was not going to let them get away with it. I eventually had to go to small claims court to get vindication. I carefully prepared a case and won but never want to do that again. Man, I was angry, too.

My wife was hit hard in the rear by an uninsured motorist. The damage was $6000, close to the value of the car, but this time we had collision insurance. The other driver, who was hauled away by ambulance, said it wasn’t his car. The current “owner” who he borrowed the car from had not transferred the title after buying the car, either. The previous owner was still title holder and, therefore, liable for everything. I called him and he refused to pay, so what a mess he was in! My insurance company handled everything for my car right away. Then, months after I had repaired the car, the man called and said he wanted to pay the damages afterall - so I suspect the police had gotten involved.

Anyway, I will carry collision and comp insurance because I do not want to hassle with courts if someone hits my car again.

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