# Mathematics of Insurance

This is a guest post by Izzy Woods

Life Is A Gamble: Mathematics Of Expected Value And Insurance Explained

It’s always interesting when someone makes the claim that they don’t gamble. Even this early in the year most people will have made several decisions where the outcome was uncertain and those outcomes were mostly financial – just like a bet. Whether it was deciding to start the year with a new job or take out some kit car insurance on the newly built love of their lives they can’t be sure whether their financial position will be stronger or weaker as a result of their decision.

We’ll first explore the simple mathematics behind these gambles using a gambling example and explain why some decisions you should take each and every time they are offered to you. Interestingly products like insurance rely on our fear of ruin and are not always a fair gamble; we’ll explain why that is and how that leads to profit for insurance companies.

Assessing The Expected Value Of A Decision

When faced with a decision which has primarily financial outcomes it is natural that you would want to choose the option which provides the greatest expected financial return. Assessing this is simple and is shown below:

Expected Value = Probability 1 x Value of Outcome 1 + Probability 2 x Value of Outcome 2…

A simple example is a game of dice. The drunken gambler at the local bar offers you the chance to play a game. If the dice rolls a 4, 5 or 6 he will give you 15. If the dice rolls a 1, 2 or 3 you will give him 10.

EV = (0.5 x 15) + (0.5 x -10)
= 2.5

A normal casino game is always designed so the casino will win in the long run. In this game the player is being offered by the drunk gambler a game where the expected win is 2.5 per game by paying more when a 4, 5 or 6 comes despite the chance of them rolling is exactly the same as a 1, 2 or 3. Since this is a positive expected value the player should sit and play all night. We can calculate the expected value of a thousand games simply by multiplying the expected value of one game by one thousand.

EV(1000 Games) = 1000 x 2.5
= 2,500

The Profits Of Insurance

Insurance has sometimes been called a ‘sucker bet’ because it is perceived to have a negative expected value to the buyer of the policy and the likelihood of claiming is relatively low. The combination of these two factors usually makes it extremely profitable for insurance companies. The annual results of the big life insurance firms make this point very nicely!

“I got an online quote from an insurance company in the UK and was given the figure of £10 a month for a 40-year package of £100,000 cover. So (breaks out calculator) I pay the insurance company £4800 over the course of the policy and when (because it’s not an if) I die they pay my beneficiary £100,000.”
http://boards.straightdope.com/sdmb/showthread.php?t=473242

This question shows a big part of why insurance looks to be better value to most people than it actually is. In this case by giving the insurance company £10 per month the customer is forgoing the ability to invest that money elsewhere. If investment rates were 0% so all we lost by buying the policy was £4,800 we can calculate the break even (EV=0) chance of death at that policy price as below where Pd is the probability of death.

EV = Pd x £100,000 + probability of surviving x -4800

=> 0 = (Pd x £100,000) – £4800(1 – Pd)

=> 0 = £100,000Pd – 4800 + £4800Pd

=> 4800 = 104800Pd

=> Pd = 4.58%

In reality though we lose the total value we could have received for that money. If the £10 per month had been invested and returned £10,000 at the end of the period then the probability of death to break even is 9.09%. This is important because insurance companies will be investing the money and will know what your probability of death is fairly well based on their actuarial calculations. If the applicants actual probability of death is 7% during that term let’s compare the perceived EV(1) seen from the perspective of the person asking the question with the actual EV(2) based on the value of the money if it had been invested.

EV(1) = 0.07 x 100,000 – 0.93 x 4800
= £7,000

EV(2) = 0.07 x 100,000 – 0.93 x 10000
= -£2300

As you can see our applicant perceives a positive EV situation when in fact the insurer is set to make a nice profit over their thousands of customers by calculating their investment returns and probabilities of death correctly.

Fear Of Ruin

The final factor to consider when understanding why consumers will take these negative EV bets by taking insurance is the insurers attitude to risk and their corresponding fear of ruin. If our applicant owns a nice house with a £100,000 mortgage in the above example and wants to protect their family from losing their home should they die the £10 per month is a very small commitment in the event they survive whereas the £100,000 mortgage is a very large commitment for the surviving family members.

Consumers are prepared to pay (through taking a negative EV bet) for this peace of mind and this allows the insurers to make a larger profit than they could simply through better calculating investment returns and risk of death than consumers are able to – although naturally as we saw in the question earlier this better ability to perceive these factors is a significant part of their profitability.

This is a guest post by Izzy Woods.

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#### Vineet Patawari

Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!