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### Course: Statistics and probability > Unit 9

Lesson 8: More on expected value- Term life insurance and death probability
- Getting data from expected value
- Expected profit from lottery ticket
- Expected value while fishing
- Comparing insurance with expected value
- Expected value with empirical probabilities
- Expected value with calculated probabilities
- Making decisions with expected values
- Law of large numbers

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# Term life insurance and death probability

Understanding an insurance company's sense of my chances of dying. Created by Sal Khan.

## Want to join the conversation?

- What If there is epidemic and many people die.

Wouldnt the company go bankrupt and most of people not receive their money?(64 votes)- Good point. Insurance companies actually purchase their own insurance for events like that from even bigger insurance companies with even more money. For example, maybe Sal's insurance company can afford to pay out $1 million per year so they are ok if only 1 person dies. If more than one person dies, then they can collect some of their insurance money (from the bigger company) so they can afford to pay the policies they sold. And if we new the premium the bigger company charged Sal's insurance company, we could calculate how likely the bigger insurance bigger thinks this is just like Sal did calculated above.(88 votes)

- How do you know the probability of death? I know they look into risk factors but how can a company be sure that they will make a profit from the investment?(19 votes)
- Generally things happen similar to how they have happened in the past. There is an incredible amount of statistics and data in regards to how and when people die. A company can take that data and build models that put the odds in their favor of making a profit.(50 votes)

- At2:53why is it that at most 1 Sals out of 100 may die in order for the insurance company not to loose money? What's the explanation? Sal didn't give any here.(14 votes)
- Because each Sal only pays 1% of the insurance payout over the life of the policy. So in this case 100 Sals each pay $10,000 for a total of $1,000,000. For each Sal that dies, the insurance company needs to pay that Sal's family $1,000,000. So if 2 Sals die the Insurance pay out $2,000,000 but have only collected $1,000,000 so they are losing money.(57 votes)

- If that one person in 100 died they would probably not die at exactly year 20. Given that they would die sometime earlier and stop paying premiums at that point, wouldn't the probability of death need to be <1% for the company to break even?(12 votes)
- I think the best way to look at this "contract" between the buyer and the insurer is that the insurer figures there is some probability p that the user will die during the 20 year interval. A reasonable assumption is that this probability is uniformly distributed over the 20 years, with the expected (mean value) of death being 10 years. At $500/yr, this means the insurer can expect the buyer to have paid on average (in thousands of dollars) $5K before dying. Suppose the buyer does die during the 20 year period. In this case the insurer will be out $995K= $1000K-$5K. If the buyer does not die during the 20 years, then the insurer will have gained $10K. Therefore the expected return for the insurer is -$995Kp + 10K(1-p) with a probability of p that the buyer dies during the 20 years. The insurer wants this expected return to be >=0. Therefore the insurer is estimating that p satisfies -1005Kp + 10K >=0 or that p satisfies p <= 10/1005, which is slightly less than the 1/100 suggested in the video.(14 votes)

- I am having trouble finding a definition of Expected Value? Can somebody give me a definition and also if there is a general equation I can use for the problem sets?(8 votes)
- http://www.artofproblemsolving.com/Videos/index.php?type=counting#chapter11

I found these videos to be a great introduction to Expected Value. In fact, you should definitely give all the rest of their video library a try :)(10 votes)

- What are some of the things an insurance company would look at when estimating one's probability of death in the next say, 20 years?(4 votes)
- Typically the big 3 are Age, Gender, and Health (which includes things like BMI, family history, blood pressure, cholesterol, and most importantly "Are you a Smoker?"). Other things that get accounted for might include DMV records (do you have reckless driving tickets or DUIs), Occupational risks (are you a crab fisherman or skydiver?), and income level (This might not be considered directly, but in general if you purchase a bigger policy the insurance company assumes you will have better access to medical care and thus lower mortality) . Additionally, what country you live in plays a factor.

Source: work in life insurance(12 votes)

- Lets say a person who is in great shape gets a policy of 500/year and his family is guaranteed 1Million Dollars after his death. But he dies in a car accident a few days later. Then What?(3 votes)
- Death is death. If it was an accident, the policy pays off. That's the whole point of insurance. It deals with uncertainty.

(Most life insurance has an exclusion for suicide that takes place within some period of time after the policy is purchased, because if the person is planning suicide, then death is not uncertain).(7 votes)

- Why has Sal not accounted for the time value of money(interest rates) while calculating the net payment for 20 years? Does the insurance company considers the interest when they pay the insurance after 20 years(in case of death)?(4 votes)
- No, if you have a $1000 policy, it pays out $1000 regardless of whether it pays out tomorrow or 20 years from now.

That's how the insurance company makes money. They earn interest on the premiums you pay in, but they don't pay out interest. In place of paying out interest, they pay out claims as they come up.(3 votes)

- Whoever dies is not going to continue paying their premiums, are they? If they do, then fine. But if they don't, then the company is really getting less than $1 mill for the 100 Sal's over 20 years-especially if that Sal dies earlier in the 20 years. So, do the premiums get paid even for the person who dies, and if not then how does this work?(4 votes)
- Those who die do not continue paying premiums. The premium should be slightly higher in order to balance out that effect. You can do the math to figure out exactly what the break-even premium should be.(3 votes)

- what if the insurance company went bankrupt after some years what will happen to my insurance !(5 votes)

## Video transcript

I'm thinking about getting
life insurance because I have a mortgage and
I have a young son and another baby on the way. And so if anything
were to happen to me, I'd want them to at least be
able to pay off the mortgage and then maybe have some
money left over for college and to live, and whatever else. And so I went to the
insurance company, and I said I want to
get a $1 million policy. And what I'm actually
getting a quote on is a term life policy,
which is really-- I just care about
the next 20 years. After those 20 years, hopefully,
I can pay off my mortgage. There'll be money saved up. Hopefully, my kids
would kind of at least have maybe gotten
to college or I would have saved up
enough money for college. So that's why I'm willing
to do a term life policy. The other option is to do
a whole life policy, where you could pay a
certain amount per year for the rest of your life. At any point you die, you
get the million dollars. In a term life, I'm only
going to pay a $500 per year for the next 20 years. If at any point over
those 20 years I die, my family gets a million. At the 21st year, I have
to get a new policy. And since I'm going
to be older and I'd have a higher chance
of dying at that point, then it's probably going
to be more expensive for me to get insurance. But I really am just worried
about the next 20 years. But what I want to
do in this video is think about given
these numbers that have been quoted to me by
the insurance company, what do they think that
my odds of dying are over the next 20 years? So what I want to think
about is the probability of Sal's death in
20 years, based on what the people at
the insurance company are telling me. Or at least, what's
the maximum probability of my death in order
for them to make money? And the way to think about it,
or one way to think about it, kind of a
back-of-the-envelope way, is to think about what's the
total premiums they're getting over the life of this
policy divided by how much they're insuring me for. So they're getting $500
times 20 years is equal to, that's $10,000 over the
life of this policy. And they are insuring
me for $1 million. So they're getting-- let's
see those 0s cancel out, this 0 cancels out--
they're getting, over the life of the
policy, $1 in premiums for every $100 in insurance. Or another way to
think about it. Let's say that there were 100
Sals, 100 34-year-olds looking to get 20-year term
life insurance. And they insured all of them. So if you multiplied
this times 100, they would get $100 in premiums. This is the case where
you have 100 Sals, or 100 people who are
pretty similar to me. 100 Sals. They would get $100 in premium. And the only way that they
could make money is if, at most, one of those Sals-- or really
just break even-- if, at most, 1 of those Sals were to die. So break even if
only 1 Sal dies. I don't like talking about this. It's a little bit morbid. So one way to think
about it, they're getting $1 premium
for $100 insurance. Or if they had 100 Sals, they
would get $100 in premium, and the only way they
would break even, if only 1 of those Sal dies. So what they're really
saying is that the only way they can break even is if
the probability of Sal dying in the next 20 years is less
than or equal to 1 in 100. And this is an
insurance company. They're trying to make money. So they're probably
giving these numbers because they think the
probability of me dying is a good-- maybe it's 1
in 200 or it's 1 in 300. Something lower, so
that they can insure-- one way to think about it--
they could insure more Sals for every $100 in premium
they have to pay out. But either way, it's a
back-of-the-envelope way of thinking about it. And it actually makes me
feel a little bit better because 1 in 100 over the
next 20 years isn't too bad.