Batch 3 - Class 186 - Probability in Insurance (2)

Pre-Class Exercise

MartinShColl 2.4 - Three points are selected at random on a sphere's surface. What is the probability that they fall within the same hemisphere?

Attendance Kushagra, Aashvi, Muskaan, Anshi, Siddhant, Damini, Rhea, Rehaan, Arjun, Smiti, Mahi

Class puzzles

Review of Basic Probability (Independent Events, Expected Value)

Bank Savings Example (Single Person and Mutual)

Term Life Insurance

What if I have an inverse problem. Lets say the person above now has a family, and wants to make sure that in event of his death, the family gets Rs 1 million to take care of their expenses. He is willing to pay Rs 5000 per year as a "premium" to get this guarantee, and is really only worried for next 20 years (after which he believes the family can take care of themselves). How would you think about the risks here? Let us ignore interest costs for now.

Total premium = 5000 * 20 = 100,000
Total payment in event of death = 1,000,000
So just to breakeven, only one in 10 people can die. Which is 10% probability over next 20 years.

Note that insurance typically works where risk of an event is small, but the downside is high. So you essentially get everyone to pay for the risk, and give the downside to where it actually materializes

Interest costs - which way does it swing the economics - does it reduce or increase the term insurance premium?

Lets do a simple example with 5% interest rate to see what effect it has on premium

If the person died in 5th year and insurance company is paying out the million rupees, what happens to the premium for remaining 15 years - doesn't come through. This means that the insurance company will actually earn lower than 100,000 on an average case.

So now, you don't just need the probability of a person dying in the twenty year period, but where exactly they died. In the above example, let us assume that 1000 people buy the insurance, and 5 people die each year (premium is collected at beginning of year). Can you find out what the premium should be for breakeven?

What is the effect of combining the two factors? Note that for someone who dies early, the money stops coming, but the payout has to be made immediately as well.

Actuarial tables will quantify risk of death at each age, and use that probability to offer insurance at each age

Another variation is unit linked, where if the person does not die before end of term, they can get an investment return

Differential Pricing

Suppose an insurance company determines that their expected payout for 1000 people is 6,000 each (because they make a million rupee payout to each one who dies). It charges 6,000 from each client. (All numbers net of interest etc).
However, now a smart analyst discovers that their expected payout on 500 of their female clients is 5,000 and on male clients 7,000. What are the pricing choices?

They can continue with old pricing and still breakeven
What will happen if another company comes up which starts to offer female clients 5,000 pricing and male clients 7,000

Adverse selection. All male clients will go to first firm and all female clients will choose the new firm
As a result, the new firm will breakeven, but the old firm will lose money on each client

Suppose we take another factor - smoking, and realize that smoker loss rate is 7,000 whereas non smoker is 5,000. How will we think about composing these two factors?

We could treat those independently. For example, base rate is 5,000. If you are a male you pay 2,000 extra and if you smoke, you pay another 2,000 extra.

There could be correlation between the two factors. For example, more males smoke. In the example above, what seems like the correlation?

Annuity Contracts

Let us now consider a different problem. A person has a million rupees and he doesn't want to spend it now. He wants to make sure that he can get a monthly income out of it for rest of his life. What are the options?
Option 1: The person can invest this in mutual funds, and keep taking out 5000 Rs a month from it. What are the pros and cons of this approach?

If he needs more or less at some point, he can adjust
However, he may live longer than the money stream
Or, he can land up with a balance amount when he dies, which is lost (remember, the inefficiency in our first example)

Option 2: An insurance company "pools" several of these individuals, and offers an annuity basis expected mortality predictions

Lets do an example. Rs 1 million at Rs 5000 a month is a 200 month stream (assuming no interest). Now suppose 4 people do it, and probabilistically, we know that one of them will die after 100 months, another after 150 months, another after 200 months and the last after 250 months.

Total months to be served are 700, so now the annuity amount can be 1million/175, i.e. around Rs 5700
Moreover, if the person outlives 200 months, he still continues to get paid

Homework Problem:

References:

The Colossal Book of Short Puzzles and Problems, by Martin Gardner