Journal: Risk Value Analysis and Health Insurance 2
The recent political developments have given us much cause to think about health insurance and its value. Unfortunately, all public discourse on these subjects is based on fear and emotional appeal rather than arguments containing concrete and comparable values that ought to appeal to a Slashdot nerd. So, after some thought, I have endeavored to discover a method of assigning a real dollar value to an insurance premium. While insurance companies undoubtedly have calculated exactly how much selling a policy is worth to them, I have not seen any numerical analysis of how much buying a policy is supposed to be worth to the consumer. And here is my attempt at doing exactly that.
Computing the value of a risk reduction
First we must state that health insurance by itself does not save any lives, but rather allows customers to purchase treatments that they could not otherwise afford, which may then extend their lives by curing whatever illness that endangers them. Each illness has a statistical risk associated with it, with definite numerical values collected by the census bureau and other similar organizations. By curing the illness, a treatment reduces your risk of death from it by the statistical amount of its incidence. For example, if I have a 2%/year chance to have a heart attack, but have health insurance and am able to purchase treatments that enable me to survive it 50% of the time, then my chance of death due to a heart attack becomes 1%/year, twice lower than if I were uninsured and could not afford any treatment.
Now to ask the core question: what is the value of reducing my risk of death by 1% this year? If the risk were higher, like 100%, you might be able to say that you would give everything you have to reduce it, and be right. But would you do so for a 10% risk? 1%? How about 0.015%, which is your chance of dying in a car accident this year? Nobody seems to worry much about that. Or how about the 0.002% chance of getting killed while crossing a street? This shows that we really care about 100% risks and do not care at all about risks less than 0.01%, but says little about the ones in between or about how to place a specific value on a particular increase or decrease.
What we want is a formula that takes in the risk difference and outputs a monetary value for it. Why focus on money? Everyone says that money can not buy happiness, but it certainly can buy an insurance premium, which is what we are trying price here. Would you agree to give up all your income and receive only the basic necessities of life in exchange for a guaranteed lifespan of 100 years? I thought not. It would be a long life, but it sure would not be any fun to live. This sets bounds for any insurance premium you pay to reduce your risk of death to somewhere between zero and your total income. Additionally, this way we can get some objective value for the value of your remaining life. We can all agree that with all other things being equal, a ten year life is worth more than a five year life. If we add the above criterium of higher income in general leading to a higher quality of life, a ten year life on $10k/year is not quite as good as a five year life at $40k/year. I am going to apply a flat value rate of years*income. If you wish, for example, to value more years higher than more money or vice versa, you are welcome to redo the calculations with those additional factors. The important thing is to have some definite way of measuring life value and argue about methods and numbers instead of trying to compare things by handwaving over vague generalities.
Slashdot journals can not have any pictures, so fire up your imagination and draw a graph of population over time assuming a constant death rate of 1%/year. I am going to use some arbitrary population N, even though for the purpose of buying insurance I am only interested in my own chances, because like most people, I have difficulty thinking of myself as 0.8 of a person. So I am going to assume a population N and then divide the results by it to reach a meaningful personal metric. In the graph you have so far imagined, at the end of the first year 1% of the population will be dead. If N=100, then 1 man will die the first year.
In the second step, imagine another group of people with a death rate of 2%, where out of N=100, 2 men will die. Plotted on the same graph, the 2% line will slope more steeply and intersect zero at 50 years, while the first line will intersect it at 100, giving average life expectancies of 25 and 50 years respectively. Now, let's suppose you happen to be one of the poor shmucks in the second group. How much would you be willing to pay to join the first group for one year?
To visualize this choice, draw another 2% line from the value of the 1% line at 1 year. This will be your line if you buy membership in the 1% group, follow their line for one year, and then stop paying and go back to your original group. The area between the original 2% line and your new detour is your life benefit in person*years. Multiplied by your annual income, it will give the monetary value you receive from the membership. That value is equal to the difference between the rates, dr, mulitplied by N and your life expectancy at the time of purchase. This is the group value, so divide by N to calculate your share, equal to dr*t. Observe how the older you are, the less benefit you can expect from the risk reduction, because you would not have quite as much time left to live anyway. In this example, dr is 1% and t is 25, giving the maximum membership fee of 25% of your income. This is the breakeven value, where the fee absorbs all the benefits you incur from the switch, making the transaction economically unfavorable.
V = dr * M * t
The monetary value of a risk reduction is equal to its numeric value multiplied by your annual income and by your life expectancy at the higher risk.
Estimating my own risk
Now I would like to see how the above formula applies to the value of health insurance. Or, in other words, I shall try to compute the maximum premium I should be willing to pay to reduce my health risks. To do this, I first need to first see what those risks are. Then I will need to compute the actual expected risk reduction, keeping in mind that the risk will not be reduced to zero (insured people die too). I am going to assume a free market health care system, where if you can't pay, you don't get treated. (In the US, the ERs are legally obligated to treat you for free, effectively giving everyone insurance for life-threatening use, and making this whole calculation pointless) I am also going to assume that the purchased insurance is comprehensive, paying for absolutely everything. (Computing the optimal coverage amount for variable-cost insurance is left as an exercise for the reader) I'm also only going to look at treatments too expensive to purchase with my own money, which basically includes any kind of surgery. The cheaper treatments, like antibiotics and other common drugs, can be easily purchased without insurance.
I'm also going to have to make guesses as to how each particular risk affects me, because there are no statistics for "% that would have died if they had no health insurance". Remember that not every injury that you go to the hospital for would kill you if you treated it yourself to the best of your ability or not at all. Deep cuts, for example, require stitches to heal cleanly, but will generally not kill you without them. If you clean them out and stop the bleeding (both tasks you can do yourself for free) it will almost always heal, though it may leave an ugly scar (which is the reason you want the stitches). Broken bones will usually mend if immobilized and you could, with the help of a friend, realign the bones, mix the plaster, and make the cast yourself. Again, the outcome may be worse than what you'd get at a hospital, but you'll live and probably not end up with a physical disability. After all, people have been fixing broken bones since the stone age; it isn't rocket science. See "Where there is no doctor" book for more examples of health problems you can treat yourself.
Finally, for the purposes of this article, many of the guesstimates below are to be considered specific to my situation. Feel free to calculate the value for yourself. The numbers here indicate what I personally belive my risks to be, and it is these personal beliefs that determine what I believe health insurance is worth to me, which is the number I am trying to semi-objectively calculate.
What are the major possible risks I need to be concerned with? The CDC publishes mortality data for the US population. According to the Table 9 of the 2006 data, the overall death rate for my age (white male, 33) is 106/100000, or 0.11%/year. Most people have insurance, so this defines the lower limit on my risk of death. Before buying insurance it will be higher. From table 7, my current life expectancy is 45 years. Table 9 further lists my top causes of death as accidents at 37, suicide at 12, homicide at 11, cancer at 8, and heart disease at 9, accounting for 71% of all causes. What are the benefits of health insurance for each of these?
Accident rate of 0.037%/year includes 0.015% motor vehicle accidents. Being a careful driver, I assume that the only car crashes I'll be in, are the ones caused by somebody else, in which case my health care, if any, would be paid for by the car insurance of the driver at fault, so having my own would probably not be beneficial. Regarding the remaining 0.022%, I have no idea how to guess which injuries would be fatal without treatment. Having never had an injury requiring a hospital visit, I can't even imagine what it might be. Lacking data I'm going to assume a 50% probability of survival (which really seems rather high) and the resulting increase in the accident rate by 0.022%.
Suicide obviously receives no benefit from health insurance. For homicide, I would say that if somebody wanted to kill me, he would very likely succeed, rendering any medical assistance pointless. Both of these numbers are extremely unlikely in my neighborhood; I don't think we've ever had a murder here. Very few people know I exist, and nobody at all hates me enough to kill me. Killing myself is also something that I am not going to do. So, I'm going to put up zero risk gain from no health insurance here.
Cancer statistics are available from the American Cancer Society. 1.4 million people get cancer every year, and 0.56 million die, so treatment increases survival chances by a factor of 2.5. However, this likely is too high for me because one of the most common types of cancer is breast cancer, and it is fully treatable most of the time. It is also not one I'm likely to have, and the rest are usually fatal whether treated or not. So I'm going to lower my risk increase to 2x, an absolute value of 0.008%. By the way, it is interesting to note that cancer mortality rates have not changed at all since 1940 (see the historical rates table at the above reference), which seems to indicate that all these modern treatments are not making much difference after all and leaves one to wonder what's really curing all those cancers.
Final entry is for heart disease. The relevant statistics can be found here. When people die from heart disease, they are dying from some type of heart attack. A million people have heart attacks every year; half of them die regardless of treatment. Half of those who die, die in the first hour, before reaching the hospital. Of the survivors, an additional 16% will die within a month. Hence, with treatment, only about a third survive. I'll increase my risk by 50%, increasing the absolute risk by 0.004%.
Hence, my total risk increases by 0.034% without health insurance, nearly all of it due to accidents and my arbitrary guess of how many of them are likely to kill me. The insurance value is 0.034%*45 = 1.53% of my income. Being unemployed, I'm going to use the average wage data from the BLS, giving average income of $45k/year. Subtracting 30% taxes (federal+state), leaves $31k/year disposable income. 1.53% of that is $482/year, which is the zero-gain insurance premium value. Average insurance payment in the US is currently $6400/year. Interesting, isn't it? $6400/year is 20% of my disposable income, requiring this insurance to reduce my yearly risk by 0.45% (a factor of 5 above the 0.11% total insured death risk stated above) before it becomes economical to buy.
Summary: my total risk of death would be 0.14%/year, and the zero-gain insurance premium would be $482/year.
What about later?
Ok, so I really am in the low-risk category, and my belief that nothing is likely to happen to me is justified. Obama may sarcastically call me and my kind "invincibles", but we sure got the numbers to prove it. But what will happen as I get older? How much would it be worth paying for health insurance when I'm 50 and all those bad things start happening? Let's redo the numbers.
Using the same CDC data tables as above I would have my life expectancy decrease to 29 years. The rate of death from all causes increases to 0.43%/year, of which 0.116% is cancer, 0.088% heart disease, 0.045% accidents, 0.017% suicide, 0.017% liver disease, 0.015% stroke, 0.013% diabetes, and 0.014% pneumonia and other respiratory infections, all of which account for 76% of all causes.
Following the same rationale as before, I'm going to adjust for lack of insurance by increasing death risk by 0.116% from cancer, 0.044% from heart disease, 0 from suicide, and 0.03% from accidents.
I'll increase the risk of death from stroke by 25% (absolute increase of 0.004%) because I would largely consider that the same as dying. Whether you are treated or not, you are likely to enter a state of permanent disability, failing both the quality of life and earnings potential criteria. I'll allow for a small increase in positive outcome, but would consider any stroke practically unsurvivable.
Most cases of diabetes are treated by controlling the diet. According to CDC statistics, only 27% of diabetics take insulin or other medication. The remainder, manage their disease by lifestyle changes. I'll assume a 27% increase in mortality from not being able to pay for treatment, giving an absolute risk increase of 0.004%.
Respiratory infections are treated with antibiotics, which we uninsured people can cheaply order online from overseas. So there would be no significant difference here from buying them with health insurance. Liver disease can not be cured, but some kinds of further damage may be prevented by lifestyle changes.
The resultant risk increase is 0.20%, for a total uninsured mortality risk of 0.63%/year. Value of insurance is 0.20% * 29 = 5.8% of my income. With the same disposable income of $33.5k/year, that comes to $1827, which makes US insurance still too expensive to be worth buying. At this age, cancer is the main problem, followed by heart disease.
Summary: at 50, my risk of death would be 0.63%/year, and the zero-gain insurance premium would be $1827/year.
What about other factors?
So far I have shown how much insurance decreases my risk of fatal health problems, and that it is not worth buying at current prices. But what about other things you can do to reduce the risk? What's the risk reduction that can be achieved by quitting smoking, not drinking too much, and maintaining a normal weight? And what is the monetary value of this reduction?
Googling for risk factors for heart disease and cancer turns up many admonishments to stop smoking and lose weight, but there aren't many actual numbers. The only number consistently quoted is 2-4x increase in risk for smokers. Specifically, lung cancer is almost entirely caused by smoking. (Lung cancer usually happens later than age 50, so it did not affect the risk calculations above) The other factors are stated generally, without statistics. So let's make a guess. Let's say that quitting smoking will reduce my chance of cancer and heart disease by a factor of 3 (no, I don't smoke, or drink, or overeat, but previous risk calculations did not account for that) That maintaining a normal weight will reduce both risks by a factor of 2. And that stopping heavy drinking will reduce them by 30% (drinking affects cancer more than heart disease; alcohol apparently causes a lot of damage to the body. But for simplicity let's guess that the effect is the same.) The total is a ninefold reduction of risk.
The cancer mortality rate for the uninsured 50 year old me was calculated above at 0.232% (the same as incidence). 0.232% reduced ninefold is 0.026%, for the absolute value reduction of 0.206%. For heart disease the rate is 0.132%; reduces ninefold to 0.015%, by 0.117%. Reducing drinking should also completely eliminate liver disease, for an additional reduction by 0.017%. The total reduction is 0.34%, which gives monetary value of 0.34% * 29 = 9.86%, * $31.5k = $3106. This is nearly twice as much benefit as I would get from buying insurance!