## How Life Insurance Premiums Are Determined

The example will be a five year term insurance premium for \$100,000 of coverage on a 45 year old man. In practice, five years is a short term, but it will help to keep the sample calculations more manageable, and a longer term wouldn’t help clarify the concepts any further. For simplicity, we’ll assume that premiums are paid annually at the beginning of the year, and death benefits occur at the end of the year.

The basic principle behind a life insurance premium is that the present value of the premiums should be equal to the present value of the death benefits, policy expenses, and company profits. This might not make sense at the moment, but the sections below will expand on all of this.

## Mortality

Mortality is probably the component of life insurance that people are most familiar with. It refers to the likelihood of dying and is used to calculate the amount of death benefits expected to be paid. For this example, we’ll use these mortality rates:

These are the same sample mortality rates that I used in this article on calculating life expectancy, which might not be a bad idea to read if you want a little more information. The letter “q” is standard for mortality rates, and these rates represent the probability of dying for someone who has attained the given age. In other words, someone currently 47 years old has a .096% chance of dying in the next year, but someone aged 45 today does not have a .096% chance of dying at age 47. They are conditional probabilities, so some people might find this a little confusing.

In order to calculate death benefits that are expected to be paid, we need to know the actual, or unconditional, probability of dying at each age. This is easy enough, as we can figure out the likelihood of living to a given age and then multiply it by the conditional probability of dying at that age (which we have in the table above).

The “p” column in the table above is the probability that someone with the given age survives to the end of the year. It is just the complement of the probability of death from the earlier table (i.e. 1 minus the probability of death), so it is still a conditional probability. The rightmost column is the probability that a 45 year old survives to the end of a given year. You can calculate this column by successively multiplying the values in the “p” column. For example, the value in the age 47 row can be found by (.99932)*(.99918)*(.99904). Since the numbers are small, it is important to keep a lot of decimal places.

Now that we have both the probabilities of a 45 year old living to a given age and the probabilities of death, we can multiply them to get the probabilities of the 45 year old dying at each age:

It is important to keep a lot of decimals for these calculations. If you round, you could mistake the value at age 47 (for example) as .00096, which is the mortality rate from the first table (which is not rounded). If you add up the probabilities in the table above, there is only a .479% chance of dying at some point in the next five years, so we should expect that the premium won’t be very large.

It is now easy to calculate the amount of death benefits that are expected to be paid at the end of each year. Multiply the death benefit (\$100,000 in this example) by the probability of death using the numbers from the table above. (For display purposes these are rounded to two decimals, but all the calculations throughout the example continue to use the actual values).

Year 1: \$68.00

Year 2: \$81.94

Year 3: \$95.86

Year 4: \$108.73

Year 5: \$124.56

We also need to take into account the mortality rates for calculating the premium. Since some people die each year, there will be fewer people to pay premiums, so this has to be taken into account. But all we need to do is use the probabilities of surviving to each age that we already have from one of the tables above.

Since premiums are paid at the beginning of the year, everyone pays a premium in the first year. In the second year, 99.32% of people pay, 99.85% in the third year, and so on. For the five year period, we can add up the expected death benefits and divide by the number of expected premiums to get a level premium that would be due each year.

This is \$479.09 divided by 4.9918, or \$95.97 per year if we only consider mortality and nothing else. So, like we expected, the premium in this example is low. It would be great if we could stop here, but this premium does not take into account all of the other factors that we need to discuss, which will affect the premium.

Of course, mortality rates will vary by age, gender, health status and history, etc. Insurance companies have huge amounts of mortality data and are able to classify people into different risk categories.

## Interest

When calculating an insurance premium, interest also needs to be taken into account. Insurance companies invest the premiums that they receive, which helps bring down premiums. Related to interest, it is important to understand the concepts the time value of money and present/future value.

In economics, the term “time preference” refers to the relative value a person places on a good at different points in time. Take an apple as a simple example. Most people would place more value on having an apple today than six months from today. Apples can go rotten and people need calories to survive. A rotten apple does no good, especially if you’re not around to eat it.

People can be categorized as having either a high time preference or a low time preference. A person with a high time preference is generally more concerned with his well-being in the present and near future, while someone with low time preference is more concerned with his well-being in the future.

It’s easy to see that most people will have a high time preference for apples, but how about with money? Time preference can also be used here. Suppose that your friend asks if he can borrow \$500 and that he’ll pay you back in six months. Even ignoring inflation or the risk that he might not repay you, you probably will want him to give you more than \$500 back. You could’ve used that \$500 to buy something for yourself, invested it, or any number of other things. Because you have a preference for \$500 now versus \$500 six months from now, you might ask him for \$515 to compensate for this preference.

The point is simple and one that most people intuitively understand: a dollar today does not have the same value as a dollar in the future. Money has a “time value.”

Present value and future value are intertwined with the time value of money. The present value is the worth, in today’s dollars, of some amount in the future. The future value, as its name implies, is the worth, in future dollars, of some amount today.

Using the example from earlier, \$500 is the present value of \$515 to be received six months from today. Conversely, \$515 is the future value in six months of \$500 to be paid today.

So, how does one calculate present and future values? In the above example we just pulled a number out of thin air because it seemed like a reasonable amount to satisfy a time preference. In reality, present and future values need to account for time preference, uncertainty, inflation, etc. in a more explicit way. The easiest and most common way to do this is through interest.

Two common ways to account for interest are with simple interest and compound interest. With simple interest, the amount of interest earned during each period is constant. For example, if you borrow \$100 at 6% simple interest per year and make no payments, you will owe \$6 of interest after one year, \$12 after two years, \$18 after three years, etc. This leads to the following formula that relates present and future values:

Future Value = (Present Value)*(1 + i*t)

Here, i and t are interest rate and time, respectively. Make sure the units of time are consistent with the interest rate.

Example: at what simple rate of interest will \$100 accumulate to \$122 in four years?

Answer: \$122 = (100)*(1 + 4*i); i = .055 or 5.5%

With compound interest, the rate of interest earned during each period is constant. For example, if you borrow \$100 at 6% compound interest per year and make no payments, you will owe \$6 of interest after one year, \$12.36 after two years, and so on. In other words, the interest accrues interest on itself. This is a much more realistic treatment of interest and also much more common (and what we’ll use) in real life. The formula relating present and future values using compound interest is as follows:

Future Value = (Present Value)*(1 + i)t

Here, i and t are interest rate and time, respectively. Make sure the units of time are consistent with the interest rate. In this case, you will usually need to use a calculator to solve problems.

One final thing to note is that you don’t have to find the present value of only one payment. For example, you might wonder what the present value of \$10 one year from today and \$25 three years from today is. You can simply add up the individual present values (assuming 6% compound interest):

Present Value = 10/(1.06) + 25/(1.06)3 = 9.43 + 20.99 = 30.42.

With that long aside out of the way, it should be clear why interest is important when determining a life insurance premium. Both the premiums and the death benefits are paid at different points in the future. They both need to be evaluated at a common time so that we can get them on an equal basis. One difference between this and the above method of finding present values is that we don’t know if/when the cash inflows and outflows will occur for life insurance. So, we refer to it as an actuarial present value, which is just a fancy way of saying that the present value involves cash flows that are contingent on the occurrence some underlying event(s), and that the likelihood of the event(s) needs to be taken into account.

We already determined the expected premiums and death benefits in the section on mortality. This time around, before adding up the expected premiums and death benefits, we need find the present value of each (when we did this in the mortality section, we were essentially doing this step with a 0% interest rate).

If we use an interest rate of 4.5%, the table below shows the values.

Because we are now including interest in the calculation, the premium went from \$95.97 when only taking into account mortality, to \$90.66. At 6% interest, the premium would be \$89.00, and at 8% \$86.86, which doesn’t seem like much of a change. But on a longer term policy the differences would be larger. In practice, it is much more complicated than this and companies can’t just pick an interest rate out of thin air, but for our purposes it works fine.

## Expenses

There are various expenses associated with operating an insurance company and issuing policies, and premiums need to account for these. Some of these expenses include underwriting expenses, policy fees, and administrative/overhead expenses, and commissions.

For policies that undergo medical underwriting, insurance companies cover the cost of sending someone out to perform a medical assessment and interview, laboratory fees, etc. And for policies with larger a death benefit, even more extensive and thorough medical underwriting may take place. Additionally, not all people who go through underwriting become policyholders. Some people may be declined coverage and others may ultimately decide not to purchase a policy (even if they are approved, perhaps at a different rate than they expected) for various reasons.

Many companies, especially for term insurance, have explicit policy fees (e.g. \$60 a year). In this case it is simple to determine how it affects your premium since it is stated up front, and it tends to be fixed regardless of the size of the policy. So if a company includes a policy fee, it will make up a larger percentage of the premium for policies with a small death benefit. Some companies have ways to avoid paying policy fees, such as having more than one type of policy with the company (e.g. a term and a permanent policy). Of course, you shouldn’t buy multiple policies for the sole purpose of avoiding a policy fee.

Administrative and overhead expenses are things you would normally think of when running a company, such as employee salaries, benefits, advertising costs, servicing clients, etc. These aren’t really attributable to specific policies, but in aggregate premiums need to be sufficient to cover them.

And finally, commissions. If you’ve ever heard an agent mention that commissions don’t affect how much you’re paying since the company pays him the commission and not you, it’s not really accurate. It is true that the agent isn’t setting the commission rate and he does not determine your premium. (And if he did try to reduce your premium, such as by giving you a portion of the commission, that would be illegal). But the money to pay commissions must come from somewhere, even if it is the company paying it. One thing that is true is that if you buy a policy (identical policy, company, etc.) from Agent Smith or Agent Jones, online or anywhere else, you’ll pay the same price.

Since this is an expense directly attribute to a specific policy, I’ll include it in the example calculation. For this policy, I’ll assume that the commission is 50% of the first year premium, paid when the first premium is paid. In this case it will bring the premium to \$101.76, or \$11.11 greater than the premium without the commission. You can see this is the case because the commission is \$50.88 (50% of the premium), and its present value is \$50.88 since it is paid at issue. From the interest section, we know that the actuarial present value of \$1 is \$4.580375, so the annual premium necessary to cover the commission is 50.88/4.580375, or \$11.11.

## Profits

The last item we’ll consider is the profit margin. It should come as no surprise that insurance companies, like all businesses expect to make a profit. There are many ways that profit could be measured or priced in, but in this example we’ll use a simple percentage of premium. We’ll assume that the profit will be equal to 5% of the premiums (each year).

This brings the premium to \$95.43, an increase of \$4.77 over the policy with no profit priced in. Note that we are only considering the premium increase from profits in isolation and not including commissions, which we will do later. To solve for this, you can divide the present value of the death benefits (\$415.24) by 95% of the present value of \$1 (.95*\$4.58) (since 95% of the premium is to cover death benefits and the other 5% is for profit). As a percentage, the premium increase is 5.26%, which is equal to 1/(1 – profit margin).

## Other/Miscellaneous

One major thing that we did not discuss is the fact that not everyone keeps their policy forever. We assumed that everyone who survives each year pays their premium. In real life, policies lapse (for reasons such as no longer being able to afford it, no longer needing it, or finding a cheaper policy), people convert them to permanent policies, etc. If we were to reflect this, it would mean that we would expect fewer premiums to be collected (and obviously those who lapse would no longer be eligible to receive a death benefit). This is an important consideration, but is beyond the scope of this article.

Another thing to consider is that each company operates differently. Certain companies may try to be competitive or more aggressive in some market segments that they want to target. On the other hand, they may not want to be involved in other markets, so they could forgo it altogether or essentially price themselves out of it. This is one reason it is important to shop around or work with someone who is familiar with underwriting standards and practices at various companies. The best rate for one person might be from company A, while the best rate for another person could be from company B.

These are just two other things that might be considered, but in reality insurance companies consider many, many things, and pricing an insurance product is a complex process.

## Putting It All Together

To wrap up our example, you might think you can add each piece of the premium together (death benefit, commissions, and profits), but this is not quite the case. In the previous sections we considered commissions and profits independently, but when both are included they will affect one other: if you increase the premium to account for profits, the increase premium will increase commissions; and conversely, if you increase the premium to account for commissions, this will also increase the profits (since both are defined as a percentage of premium).

If you include both at the same time the premium would be \$107.82, an increase of \$17.16 (a little bit more than just adding up \$11.11 for commissions and \$4.77 for profits). You could calculate this by dividing the present value of the death benefits (\$415.24) by (.95*\$4.58 – .5). This second term is the portion of the present value of the premiums available to fund the death benefit.

If we use the actuarial present value of \$1 that we calculated (\$4.58), we can lay things out simply:

Actuarial present value of premiums: \$4.58*\$107.82 = \$493.84

Actuarial present value of death benefit: \$415.24 (given in a table above)

Actuarial present value of commissions: \$53.91 (50% of the first year premium)

Actuarial present value of profits: .05*\$493.84 = \$24.69 (5% of present value of premiums)

If you add everything up, the present value of the premiums is equal to the present value of all the other components, which was our initial goal. (If you’re following along with calculations, you might get slightly different answers; for display purposes I rounded many numbers, but kept all precision when performing calculations).

## Conclusion

If you got through all of that, congratulations! I hope that this article gave you at least a little bit of insight into some of the things that determine a life insurance premium. It’s certainly more complicated than this in practice, especially with all of the complex product designs that exist today.

top