Insurance companies of any sort make money in much the same way. You, the insurance buyer, bet that something is going to happen. The insurance provider, on the other hand, bets that it won’t. For example, when you insure your home contents, you are betting that something of yours will be stolen or damaged. If something is stolen or damaged, you receive a pay-out and you win: the insurance company has given you more money than you have given it. If nothing is stolen and the insurance company isn’t required to make a pay-out, then the insurance company wins, and they get to keep all the lovely money you’ve given them without having to pay you a single penny.

This is of course, a very simplistic and reductionist way of looking at insurance, but for the sakes of what we will discuss, it will do.

So, how do the insurance companies make money? Well the obvious answer is that they win the bet more times than the customer. Whilst insurance companies inevitably have to make ludicrous pay-outs at certain times, over the large number of customers they have, more often than not, a customer will pay more money to them than the insurance company will give to the customer.

But how does the insurance company stack the odds in their favour? How do they ensure that, more often than not, they’ll receive more money than they give out? In reality, there are hundreds of methods that insurance companies employ to make sure they stay in the black, but we’ll only focus on a few, using life insurance as an example. Now, there are many types of life insurance policy, but for this example, we will use a “whole-life” policy. With a whole-life policy, customers pay a monthly or annual fee for the remainder of their lives, to receive a predetermined lump sum upon their death.

__The basics__

Life insurance policies, at their very base, are quite simple. When you take out a life insurance policy, you are betting the insurance company that you are going to die before they estimate you will. If you do, you receive a pay-out that is greater than the amount you have paid them over the years. If you die after the company’s estimate, you will have paid more money to the company than you will receive and the company wins. At its very base, a life insurance policy will follow this equation:

Or in other words:

So, if you contacted a life insurance broker to insure someone from the time they are born, the company would take that equation and enter some numbers. They’d estimate your time of death, how much they’d give you, and then they would work out the annual payments from that. For example, let’s imagine they estimate that I’m going to die at 65 and that they’re going to give me £500,000 upon my death. Slotting those numbers into the formula we get…

So my annual payment would be around £7,700.

Of course, people don’t die exactly when the company estimates. So what’s the result when that happens? Well, say that I pay my annual payment from as soon as I’m born (I’d be an extremely rich baby but no matter), but I die at age 50. In this case, the insurance company will still pay me the £500,000, but I’ve only given them 50 x £7,700 = £385,000, so the company would lose money. £115,000 to be precise. Now let’s say that instead, I die at age 90. In this case, the insurance company gives me the £500,000, but I’ve already given them 90 x £7,700 = £693,000, so the company has made a profit of £193,000. So how does the company balance these out to ensure that they stay in business? And is there anything they can do to maximise their profits? Let’s have a look.

*(Now for the purposes of the little bit of maths we’re going to do, we’ll imagine that people take out their life insurance policies as soon as they’re born. We know this isn’t true, but it makes the maths a little bit easier).*

Let’s imagine that we are going to start a company in the world of life insurance. How are we going to make money? Well, according to our equation we’ve been given, we know that every time someone dies before we predicted they would, we lose money. But every time a customer dies after we’ve predicted they would, we make money. So theoretically, to break even, we just have to make sure that the same number of people die before our estimation as those who die after, right? Not quite, but it’s not a bad place to start. For now then, we’ll set that as our goal and then take a look what happens.

So how are we going to make sure that roughly 50% of people are going to die on or after our estimation, and that roughly 50% will die on or before our estimation? Arguably the best way to decide this is via a graph.

This graph shows the number of deaths at any given age as a percentage of the total number of deaths. As you can see (and as you probably would have guessed), the majority of people live to the ages of around 80, with relatively few deaths in the first 40 years of life and fewer individuals (and therefore deaths) after the ages of 90. But the important bit of information we need is the age to which 50% of the population live to. From there we can ensure that half of our customers die before or when we expect them to, and half die after or when we expect them to.

To find this out, we can use the area under the line in our graph. The area under the graph between any two points as a percentage of the total area represents the proportion of people who pass away within that age category. For example, if we looked at the percentage of the area under the line between 55-59 and 70-74, we would find that around 18% of the total area under the graph would lie between those 2 points. That means that 18% of the population die between the ages of 55 and 74. So we just need to find the age where 50% of the population have died and 50% of the population are still alive. Using the previous mathematics, we find that 50% of the area of the graph underneath the line is located before (and including) the age range 80-84, and 50% of the area of the graph underneath the line is located after that range. Therefore, we know that the age we’re looking for lies somewhere between the ages of 80 and 84.

Now, as you may have already realised, what we’re doing is essentially finding out what the life expectancy is of our above sample. Because of where I got the data from, I know that the life expectancy of the sample (which is all of the women in Australia in 2012) is around 81. But, if were to find out where exactly in our age range the life expectancy was, we would just employ exactly the same tactics as before, except we would need information that was not grouped into classes, but was specific for each age. Nonetheless, we’ll pretend that’s what we did and now our brand new company has got its estimation!

So, now we know that 50% of the population die before the age of 81 and 50% die after that age, so what would happen if we used only this information to start our company? Well, in reality, we would end up losing a lot of money.

The reason for this is pretty simple. While, theoretically, having 50% of our customers die before our estimation and 50% die after could lead to us breaking even, this would also require that our customers that die either side of our prediction to die an equal difference in time from each other. For example, if we set our estimation as 81, if one customer dies aged 80, we’d lose 1 year’s payment worth of money. To get that money back (i.e. have a customer pay 1 year’s payment more than we’d give them back) we need another to die at age 81. If one customer dies at age 20, we’d lose 61 years’ worth of payments, so we’d need another customer to die at age 142 to gain that 61 years’ worth of payments back, and so on and so forth. That way, for every amount of money lost by a premature death, the equivalent amount would be regained by someone dying an equal amount of time after our estimation.

But this isn’t what happens. Your chances of dying either side of the average are not equal. While 1% of the population will die before age 1 (80 years **before** our prediction), 1% percentage of the population do not die at age 161 (80 years **after** our prediction), and so we’ll lose a considerable amount of money.

But how big a deficit does this place us in? We’ll find out.

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