What’s an Actuarial Life Table?

An actuarial life table is a table or spreadsheet that shows the probability of a person at a certain age dying before their next birthday. It’s often used by life insurance companies to calculate the remaining life expectancy for people at different ages and stages, and the probability of surviving a particular year of age.

Because men and women have different mortality rates, an actuarial life table is computed separately for men and women. An actuarial life table is also called a mortality table, life table, or actuarial table.

How an Actuarial Life Table Works

Insurance companies utilize actuarial life tables to help price products and project future insured events. Mathematically and statistically based actuarial life tables assist life insurance companies by showing event probabilities, such as death, sickness, and disability.

An actuarial life table can also include factors to differentiate variable risks such as smoking, occupation, socio-economic status, and even gambling, and debt load. Computerized predictive modeling allows actuaries the ability to calculate for a wide variety of circumstances and probable outcomes.

Actuarial Science

Actuarial science uses primarily two types of life tables. First, the period life table is used to determine mortality rates for a specific time period of a certain population. The other type of actuarial life table is called the cohort life table, also referred to as a generation life table. It is used to represent the overall mortality rates of a certain population’s entire lifetime.

The population selection must be born during the same specific time interval. A cohort life table is more commonly used because it attempts to predict any expected change in mortality rates of a population in the future.

A cohort table also analyzes observable mortality patterns over time. Both types of actuarial life tables are based on actual populations of the present and educated predictions of a population’s near future. Other types of life tables may be based on historical records. These types of life tables often undercount infants and understate infant mortality.

Insurance companies use actuarial life tables to primarily make two types of predictions: the probability of surviving any particular year of age and the remaining life expectancy for people of different ages.

Other Uses of Actuarial Life Tables

Actuarial life tables also play an important role in the sciences of biology and epidemiology. In addition, the Social Security Administration in the United States uses actuarial life tables to examine the mortality rates of people who have Social Security in order to inform certain policy decisions or actions.

Actuarial life tables are also important in product life cycle management and for pension calculations.

What Is an Annuity Table?

An annuity table is a tool for determining the present value of an annuity or other structured series of payments. Such a tool, used by accountants, actuaries, and other insurance personnel, takes into account how much money has been placed into an annuity and how long it has been there to determine how much money would be due to an annuity buyer or annuitant.

Figuring the present value of any future amount of an annuity may also be performed using a financial calculator or software built for such a purpose.

How an Annuity Table Works

An annuity table provides a factor, based on time, and a discount rate (interest rate) by which an annuity payment can be multiplied to determine its present value. For example, an annuity table could be used to calculate the present value of an annuity that paid $10,000 a year for 15 years if the interest rate is expected to be 3%.

According to the concept of the time value of money, receiving a lump sum payment in the present is worth more than receiving the same sum in the future. As such, having $10,000 today is better than being given $1,000 per year for the next 10 years because the sum could be invested and earn interest over that decade. At the end of the 10-year period, the $10,000 lump sum would be worth more than the sum of the annual payments, even if invested at the same interest rate.

Annuity Table and the Present Value of an Annuity

Present Value of an Annuity Formulas

The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows:


$\begin{aligned}&\text{P} =\text{PMT}\times\frac{ 1 – (1 + r) ^ -n}{r}\\&\textbf{where:}\\&\text{P} = \text{Present value of an annuity stream}\\&\text{PMT} =\text{Dollar amount of each annuity payment}\\&r = \text{Interest rate (also known as the discount rate)}\\&n = \text{Number of periods in which payments will be made}\end{aligned}$​P=PMT×r1−(1+r)−n​where:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as the discount rate)​

Assume an individual has an opportunity to receive an annuity that pays $50,000 per year for the next 25 years, with a discount rate of 6%, or a lump sum payment of $650,000. He needs to determine the more rational option. Using the above formula, the present value of this annuity is:


$\begin{aligned}&\text{PVA} = \$50,000 \times \frac{1 – (1 + 0.06) ^ -25}{0.06} = \$639,168\\&\textbf{where:}\\&\text{PVA}=\text{Present value of annuity}\end{aligned}$​PVA=$50,000×0.061−(1+0.06)−25​=$639,168where:​

Given this information, the annuity is worth $10,832 less on a time-adjusted basis, and the individual should choose the lump sum payment over the annuity.

Note, this formula is for an ordinary annuity where payments are made at the end of the period in question. In the above example, each $50,000 payment would occur at the end of the year, each year, for 25 years. With an annuity due, the payments are made at the beginning of the period in question. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):


$\begin{aligned}&\text{P} = \text{PMT} \times\left(\frac{1 – (1 + r) ^ -n}{r}\right) \times (1 + r)\end{aligned}$​P=PMT×(r1−(1+r)−n​)×(1+r)​

If the above example of an annuity due, its value would be:


$\begin{aligned}&\text{P}= \$50,000\\&\quad \times\left( \frac{1 – (1 + 0.06) ^ -25}{0.06}\right)\times (1 + 0.06) = \$677,518\end{aligned}$​P=$50,000​

In this case, the individual should choose the annuity due, because it is worth $27,518 more than the lump sum payment.

Present Value of an Annuity Table

Rather than working through the formulas above, you could alternatively use an annuity table. An annuity table simplifies the math by automatically giving you a factor for the second half of the formula above. For example, the present value of an ordinary annuity table would give you one number (referred to as a factor) that is pre-calculated for the (1 – (1 + r) ^ – n) / r) portion of the formula.

The factor is determined by the interest rate (r in the formula) and the number of periods in which payments will be made (n in the formula). In an annuity table, the number of periods is commonly depicted down the left column. The interest rate is commonly depicted across the top row. Simply select the correct interest rate and number of periods to find your factor in the intersecting cell. That factor is then multiplied by the dollar amount of the annuity payment to arrive at the present value of the ordinary annuity.

Below is an example of a present value of an ordinary annuity table:

What Is an Amortized Loan?

An amortized loan is a type of loan with scheduled, periodic payments that are applied to both the loan’s principal amount and the interest accrued. An amortized loan payment first pays off the relevant interest expense for the period, after which the remainder of the payment is put toward reducing the principal amount. Common amortized loans include auto loans, home loans, and personal loans from a bank for small projects or debt consolidation.

How an Amortized Loan Works

The interest on an amortized loan is calculated based on the most recent ending balance of the loan; the interest amount owed decreases as payments are made. This is because any payment in excess of the interest amount reduces the principal, which in turn, reduces the balance on which the interest is calculated. As the interest portion of an amortized loan decreases, the principal portion of the payment increases. Therefore, interest and principal have an inverse relationship within the payments over the life of the amortized loan.

An amortized loan is the result of a series of calculations. First, the current balance of the loan is multiplied by the interest rate attributable to the current period to find the interest due for the period. (Annual interest rates may be divided by 12 to find a monthly rate.) Subtracting the interest due for the period from the total monthly payment results in the dollar amount of principal paid in the period.

The amount of principal paid in the period is applied to the outstanding balance of the loan. Therefore, the current balance of the loan, minus the amount of principal paid in the period, results in the new outstanding balance of the loan. This new outstanding balance is used to calculate the interest for the next period.

Amortized Loans vs. Balloon Loans vs. Revolving Debt (Credit Cards)

While amortized loans, balloon loans, and revolving debt–specifically credit cards–are similar, they have important distinctions that consumers should be aware of before signing up for one.

Amortized Loans

Amortized loans are generally paid off over an extended period of time, with equal amounts paid for each payment period. However, there is always the option to pay more, and thus, further reduce the principal owed.

Balloon Loans

Balloon loans typically have a relatively short term, and only a portion of the loan’s principal balance is amortized over that term. At the end of the term, the remaining balance is due as a final repayment, which is generally large (at least double the amount of previous payments).

Revolving Debt (Credit Cards) 

Credit cards are the most well-known type of revolving debt. With revolving debt, you borrow against an established credit limit. As long as you haven’t reached your credit limit, you can keep borrowing. Credit cards are different than amortized loans because they don’t have set payment amounts or a fixed loan amount.

Example of an Amortization Loan Table

The calculations of an amortized loan may be displayed in an amortization table. The table lists relevant balances and dollar amounts for each period. In the example below, each period is a row in the table. The columns include the payment date, principal portion of the payment, interest portion of the payment, total interest paid to date, and ending outstanding balance. The following table excerpt is for the first year of a 30-year mortgage in the amount of $165,000 with an annual interest rate of 4.5%

What Is an Amortization Schedule?

Amortizing loans feature level payment amounts over the life of the loan, but with varying proportions of interest and principal making up each payment. A traditional mortgage is a prime example of such a loan.

A loan amortization schedule represents the complete table of periodic loan payments, showing the amount of principal and interest that comprise each level payment until the loan is paid off at the end of its term. Early in the schedule, the majority of each payment goes toward interest; later in the schedule, the majority of each payment begins to cover the loan’s remaining principal.

Understanding an Amortization Schedule

If you are taking out a mortgage or auto loan, your lender should provide you with a copy of your loan amortization schedule so you can see at a glance what the loan will cost and how the principal and interest will be broken down over its life.

In a loan amortization schedule, the percentage of each payment that goes toward interest diminishes a bit with each payment and the percentage that goes toward principal increases. Take, for example, a loan amortization schedule for a $165,000, 30-year fixed-rate mortgage with a 4.5% interest rate:

Amortization schedules can be customized based on your loan and your personal circumstances. With more sophisticated amortization calculators, like the templates you can find in Excel you can compare how making accelerated payments can accelerate your amortization. If for example, you are expecting an inheritance, or you get a set yearly bonus, you can use these tools to compare how applying that windfall to your debt can affect your loan’s maturity date and your interest cost over the life of the loan.

In addition to mortgages, car loans and personal loans are also amortizing for a term set in advance, at a fixed interest rate with a set monthly payment. The terms vary depending on the asset. Most conventional home loans are 15- or 30-year terms. Car owners often get an auto loan that will be repaid over five years or less. For personal loans, three years is a common term.

Formulas Used in Amortization Schedules

Borrowers and lenders use amortization schedules for installment loans that have payoff dates that are known at the time the loan is taken out, such as a mortgage or a car loan. There are specific formulas that are used to develop a loan amortization schedule. These formulas may be built into the software you are using, or you may need to set up your amortization schedule from scratch.

If you know the term of a loan and the total periodic payment amount, there is an easy way to calculate a loan amortization schedule without resorting to the use of an online amortization schedule or calculator. The formula to calculate the monthly principal due on an amortized loan is as follows:

Principal Payment = Total Monthly Payment – [Outstanding Loan Balance x (Interest Rate / 12 Months)]

To illustrate, imagine a loan has a 30-year term, a 4.5% interest rate, and a monthly payment of $1,266.71. Starting in month one, multiply the loan balance ($250,000) by the periodic interest rate. The periodic interest rate is one-twelfth of 4.5% (or 0.00375), so the resulting equation is $250,000 x 0.00375 = $937.50. The result is the first month’s interest payment. Subtract that amount from the periodic payment ($1,266.71 – $937.50) to calculate the portion of the loan payment allocated to the principal of the loan’s balance ($329.21).

To calculate the next month’s interest and principal payments, subtract the principal payment made in month one ($329.21) from the loan balance ($250,000) to get the new loan balance ($249,670.79), and then repeat the steps above to calculate which portion of the second payment is allocated to interest and which is allocated to the principal. You can repeat these steps until you have created an amortization schedule for the full life of the loan.

An Easier Way to Calculate an Amortization Schedule

Calculating an amortization schedule is as simple as entering the principal, interest rate, and loan term into a loan amortization calculator. But you can also calculate it by hand if you know the rate on the loan, the principal amount borrowed, and the loan term.

How to Calculate the Total Monthly Payment

Typically, the total monthly payment is specified by your lender when you take out a loan. However, if you are attempting to estimate or compare monthly payments based on a given set of factors, such as loan amount and interest rate, you may need to calculate the monthly payment as well.

If you need to calculate the total monthly payment for any reason, the formula is as follows:

Total Monthly Payment = Loan Amount [ i (1+i) ^ n / ((1+i) ^ n) – 1) ]

where:

• i = monthly interest rate. You’ll need to divide your annual interest rate by 12. For example, if your annual interest rate is 6%, your monthly interest rate will be .005 (.06 annual interest rate / 12 months).
• n = number of payments over the loan’s lifetime. Multiply the number of years in your loan term by 12. For example, a 30-year mortgage loan would have 360 payments (30 years x 12 months).

Using the same example from above, we will calculate the monthly payment on a $250,000 loan with a 30-year term and a 4.5% interest rate. The equation gives us $250,000 [(0.00375 (1.00375) ^ 360) / ((1.00375) ^ 360) – 1) ] = $1,266.71. The result is the total monthly payment due on the loan, including both principal and interest charges.

30-Year vs. 15-Year Amortization Table

If a borrower chooses a shorter amortization period for their mortgage—for example, 15 years—they will save considerably on interest over the life of the loan, and they will own the house sooner. That’s because they’ll make fewer payments for which interest will be amortized. Additionally, interest rates on shorter-term loans are often at a discount compared to longer-term loans.

There is a tradeoff, however. A shorter amortization window increases the monthly payment due on the loan. Short amortization mortgages are good options for borrowers who can handle higher monthly payments without hardship; they still involve making 180 sequential payments (15 years x 12 months).

It’s important to consider whether or not you can maintain that level of payment based on your current income and budget.

Using an amortization calculator can help you compare loan payments against potential interest savings for a shorter amortization to decide which option suits you best. Here’s what a $500,000 loan with a 6% interest rate would look like, with a hypothetical 30-year and 15-year schedule to compare: