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Comparing insurance with expected value

Video transcript
Voiceover:Paul has the option between a high-deductible or a low-deductible health insurance plan. When we talk about the deductible in health insurance, if someone says that they have a plan with $1,000 deductible, that means that the insurance company only pays the medical cost after the first $1,000. So if you have a $1,000 deductible and you, say, incur medical cost of $800, you're going to pay that $800. The insurance company won't pay anything. If you have a deductible of $1,000 and your total medical costs are $1,200, you're going to pay the first $1,000, and then the insurance company will kick in after that. So with that out of the way, let's think about his 2 plans. If Paul chooses the low-deductible plan, he will have to pay the first $1,000 of any ... let me do that in purple, the first $1,000 of any medical cost. The low-deductible plan costs $8,000 for a year. So in this situation, he's going to pay $8,000 to get the insurance. If he has $900 of medical expenses, the insurance company still pays nothing. If he has $2,000 of medical expenses, then he pays the first $1,000, and then the insurance company would pay the next $1,000. If he has $10,000 in medical expenses, he would pay the first $1,000, and then the insurance company would pay the next $9,000. If Paul chooses the high-deductible plan, he will have to pay the first $2,500 of any medical cost. The high-deductible plan costs $7,500 a year. It makes sense that the high-deductible plan costs less than the low-deductible plan because here, the insurance doesn't kick in until he has over $2,500 of medical expenses, while here, it was only $1,000. To help himself choose a plan, Paul found some statistics about common health problems for people similar to him. Assume that the table below, and I put it up here on the right, correctly shows the probabilities in cost of total medical incidents within the next year. So this right over here, it's saying what's the probability he has $0 in medical costs? What's the probability of ... he has a 25% probability of $1,000, 20% probability, $4,000. And this is a simplification, a pretty dramatic simplification from the real world. In the real world, the way this makes it sound is there's only 5 possible medical costs that someone might have, $0, $1,000, $4,000, $7,000, and $15,000. In the real world, you could have $20 of medical cost. You could have $20,000. You could have $999. So in the real world, there is many, many more situations here that you would have to redistribute the probabilities accordingly. But with that said, this isn't a bad approximation. It's just saying, OK, roughly, if you wanted to construct this so it's easier to do the math, which the problem writers have done, say, look, OK, 30% $0, 25% $1,000. This is pretty indicative if you had to group all of the possible costs into some major bucket. So it's probably at least a pretty good method for figuring out which insurance policy someone should use. So they say, including the cost of insurance, what are Paul's expected total medical costs with the low-deductible plan? Round your answer to the nearest cent. Actually, I'll get the calculator out for this. With the low-deductible plan here, low-deductible plan, he's going to have to spend his total cost, he's going to spend $8,000 no matter what. Whoops, what happened to my calculator? He's going to spend $8,000 ... My God, I'm having issues. He's going to spend $8,000 no matter what, so that is $8,000. Then let's see. There is a 30% probability he spends nothing. I could just write that as plus 0.3 times $0, and I will write it just so that you see I'm taking that into account. There's a 25% chance that he has $1,000 in medical costs. In the low-deductible plan, he still has to pay that $1,000, so plus 0.25% chance that he pays $1,000, $1,000. And then you might say, OK, plus 0.2 times $4,000, but remember, if his medical costs are $4,000, he is not going to pay that $4,000. He is only going to pay the first $1,000. So it's really plus 0.2. In this situation, his out-of-pocket costs are only $1,000. The insurance company will pay the next $3,000. So 0.2 times $1,000. And then plus 0.2, 20% chance, even if he has $7,000 in medical costs, he's only going to have to pay the first $1,000, so 0.2 times $1,000 again. And then plus 0.05 times, once again, $1,000. If he has $15,000 in expenses, he is only going to have to pay the first $1,000, times $1,000. And we get $8,700. One way you could have thought about it is OK, he is going to have to pay $8,000 no matter what, and all of the situations where he ends up paying, that's anything, that's these 4 situations right over here, there is a 70% probability of falling into one of these 4 situations. At any one of these, he only has to pay $1,000. The insurance company pays everything after that. So you could say $8,000 plus there is a 70% chance that he is going to pay $1,000. And once again, this table is a pretty big simplification from the real world. There's probably a lot of scenarios where you would have to pay $500 or $600 or whatever it might be. But let's just go with this. That's essentially a simplification. There's a 70% chance that he's going to have to pay $1,000, and so that's $700 expected cost from that, plus the $8,000 from the insurance gets us to $8,700. So let's write that down. So $8,700 ... My pen is really acting up. I don't know what's going on here. I think I have to get a new tablet. You can't even read that. Let me write this. $8,700. Including the cost of insurance, what are Paul's expected total medical costs with the high-deductible plan? Round your answer to the nearest cent. Let's look at the high-deductible plan. He's going to pay $7,500 no matter what, $7,500 no matter what, and then there's a $0 ... We could write $0 times 30% or 30% times $0, but that's just going to be $0. There is a 25% chance he spends $1,000, so plus 0.25 times $1,000. There is a 20% chance, plus 0.2, he's not going to spend $4,000 here. He's going to have to spend the first $2,500. So it's a 20% chance he spends $2,500, so times $2,500. Insurance company will pay the next $1,500. Plus another 20% chance. Even in this situation, he only has to pay $2,500, so times $2,500, $2,500. And then finally, plus there's a 5% chance. Even in this situation, he only has to pay the first $2,500, times $2,500 gets us to $8,875. So $8,875. Once again, you could think about it as, OK, there is a 25% chance that he pays $1,000, and then there is a 45% chance that he pays $2,500. All of these situations, he is paying $2,500. But either way, you would get $8,875. If Paul wants the best payoff in the long run and must buy 1 of the 2 insurance plans, he should purchase the? Well, his expected total cost of insurance, including medical costs, is lower with the low-deductible plan. So this one, he should go with the low, low-deductible. Which, once again, you shouldn't use these videos as insurance advice. This is actually ... But also, it's an interesting way to think about it. It's typically ... Well, it's not always typically the case that the low-deductible plan is going to have a higher long-term ... or the low-deductible plan is going to be a better deal. It's usually the ... Well, I won't make any actuarial statements. But at least in this situation, the low-deductible plan seems like the better deal. He has lower expected total costs given these probabilities.