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 Aptitude Test?

An aptitude test is an exam used to determine an individual’s skill or propensity to succeed in a given activity. Aptitude tests assume that individuals have inherent strengths and weaknesses and have a natural inclination toward success or failure in specific areas based on their innate characteristics.

Aptitude tests are generally used for job placement, college program entry, and to help people to get an idea of where their interests and aptitudes can take them regarding careers.

Understanding Aptitude Tests

Aptitude tests can be used to determine your capabilities in a variety of subjects. For example, individuals may take an aptitude test to determine the careers that are a good match for their skills and interests. Similarly, high school students may take an aptitude test when thinking about what would be an appropriate college major or whether college is the best choice for them.

In general, aptitude tests measure one’s competence in logical reasoning, numerical skills, or verbal ability; competency can be evaluated through problem-solving tasks and testing one’s critical thinking across various contexts.

When You Might Take an Aptitude Test

Some schools administer aptitude tests to students beginning in elementary school. Combined with intelligence and achievement tests that measure student mastery of academic content, aptitude tests may be used to determine placement in gifted and talented programs or other specific educational tracks.

Language Aptitudes

For example, the Modern Language Aptitude Test (MLAT) measures a student’s potential for successfully mastering foreign languages. Aptitude tests can also help determine if a student needs special education services.

Older Students

For older students, Differential Aptitude Tests (DATs) assess a range of aptitudes from spatial relations to language usage. The results can help administrators make curricular recommendations.

Curriculum Guidance

Counselors might use high scores in mechanical reasoning tests to guide students toward courses that prepare them for engineering or designing studies in college. For instance, students who score well in tests measuring speed, accuracy, and problem-solving might choose coursework in computer science, finance, or other fields requiring attention to detail.

Hiring Decisions

Some companies use aptitude tests to help them make hiring decisions. These tests, called career assessment tests, help human resources (HR) personnel learn more about a prospective employee’s strengths and weaknesses. Career assessment tests may also be used within a company to make promotion decisions.

Situational judgment tests are a specific kind of career aptitude test that can help predict how an employee might respond to specific situations in the workplace. They are also used to assess an employee’s communication style and ability to work within a team.

Some situational judgment tests focus specifically on customer service careers. These tests analyze personality traits predictive of successful public interaction, such as empathy, diplomacy, and patience. Situational judgment tests may also be used to predict future competency in sales-related careers.

Types of Aptitude Tests

Many types of aptitude tests are taken for educational or career discovery and ability purposes. The most common aptitude tests are:

• Mechanical reasoning: These types test your knowledge of physical concepts and are generally used to evaluate you for technical positions.
• Situational judgment: These tests gauge your reactions to situations and your decision-making.
• Diagrammatic/spatial reasoning: These tests see how well you can reach a conclusion based on processes contained in diagrams.
• Abstract reasoning: These measure problem-solving abilities and identify relationships between abstract arrays.
• Numerical reasoning: In these tests, your abilities with numbers, math, and data are tested.
• Verbal reasoning: Your language, reading comprehension, and vocabulary are tested in these types.
• Inductive reasoning: Under the pressure of time, these tests see how well you analyze patterns and data.
• Logical reasoning: Logical reasoning tests measure how well you recognize patterns and sequences and identify relationships between objects.
• Clerical aptitude: Clerical tests measure how fast and accurate you can be, seeing how well you can concentrate.

How to Pass an Aptitude Test

You might have heard that you can’t prepare for an aptitude test, but nothing could be further from the truth. Study guides and practice aptitude tests are readily available online and in community or high school libraries.

Conduct an online search for the type of test you’ll be taking. You’ll find many guides that offer tips on subjects you can read about and practice tests. Once you’ve found the practice tests and some study material:

• Practice the test daily: Practicing every day gives you continuous exposure to the questions and answers and helps you identify weak spots.
• Learn the test format: Remembering the format helps you move through the test. It also allows you to identify areas you need to work on.
• Carefully read the instructions: When you take the test, don’t dive into it. Instead, read all instructions and ensure you understand them, so you don’t work through a section giving wrong answers.
• Manage your testing time: Answer the questions you know first. Then, go back and work on the more challenging questions.

The Bottom Line

An aptitude test is a measure of your ability to learn or perform required tasks and succeed in the environment you’re in. They are not a measure of intelligence—these tests look for your strengths and weaknesses, giving you and the evaluator an idea of your abilities.

Most aptitude tests can indicate success in education and a career, but they are not always 100% accurate predictions. For example, it is possible to score very low on an aptitude test for a specific job and be successful with hard work and dedication—the test may have merely taken a snapshot of your knowledge and abilities at a point where you didn’t know much about the field.

Remember that preparation can help you pass an aptitude test even if you have no prior knowledge in an area—if you think of it as learning something new and interesting, you may surprise yourself with what you can do.

What Is an Autoregressive Integrated Moving Average (ARIMA)?

An autoregressive integrated moving average, or ARIMA, is a statistical analysis model that uses time series data to either better understand the data set or to predict future trends. 

A statistical model is autoregressive if it predicts future values based on past values. For example, an ARIMA model might seek to predict a stock’s future prices based on its past performance or forecast a company’s earnings based on past periods.

Understanding Autoregressive Integrated Moving Average (ARIMA)

An autoregressive integrated moving average model is a form of regression analysis that gauges the strength of one dependent variable relative to other changing variables. The model’s goal is to predict future securities or financial market moves by examining the differences between values in the series instead of through actual values.

An ARIMA model can be understood by outlining each of its components as follows:

• Autoregression (AR): refers to a model that shows a changing variable that regresses on its own lagged, or prior, values.
• Integrated (I): represents the differencing of raw observations to allow the time series to become stationary (i.e., data values are replaced by the difference between the data values and the previous values).
• Moving average (MA):  incorporates the dependency between an observation and a residual error from a moving average model applied to lagged observations.

ARIMA Parameters

Each component in ARIMA functions as a parameter with a standard notation. For ARIMA models, a standard notation would be ARIMA with p, d, and q, where integer values substitute for the parameters to indicate the type of ARIMA model used. The parameters can be defined as:

• p: the number of lag observations in the model, also known as the lag order.
• d: the number of times the raw observations are differenced; also known as the degree of differencing.
• q: the size of the moving average window, also known as the order of the moving average.

For example, a linear regression model includes the number and type of terms. A value of zero (0), which can be used as a parameter, would mean that particular component should not be used in the model. This way, the ARIMA model can be constructed to perform the function of an ARMA model, or even simple AR, I, or MA models.

ARIMA and Stationary Data

In an autoregressive integrated moving average model, the data are differenced in order to make it stationary. A model that shows stationarity is one that shows there is constancy to the data over time. Most economic and market data show trends, so the purpose of differencing is to remove any trends or seasonal structures. 

Seasonality, or when data show regular and predictable patterns that repeat over a calendar year, could negatively affect the regression model. If a trend appears and stationarity is not evident, many of the computations throughout the process cannot be made and produce the intended results.

How to Build an ARIMA Model

To begin building an ARIMA model for an investment, you download as much of the price data as you can. Once you’ve identified the trends for the data, you identify the lowest order of differencing (d) by observing the autocorrelations. If the lag-1 autocorrelation is zero or negative, the series is already differenced. You may need to difference the series more if the lag-1 is higher than zero.

Next, determine the order of regression (p) and order of moving average (q) by comparing autocorrelations and partial autocorrelations. Once you have the information you need, you can choose the model you’ll use.

Pros and Cons of ARIMA

ARIMA models have strong points and are good at forecasting based on past circumstances, but there are more reasons to be cautious when using ARIMA. In stark contrast to investing disclaimers that state “past performance is not an indicator of future performance…,” ARIMA models assume that past values have some residual effect on current or future values and use data from the past to forecast future events.

The following table lists other ARIMA traits that demonstrate good and bad characteristics.

The Bottom Line

The ARIMA model is used as a forecasting tool to predict how something will act in the future based on past performance. It is used in technical analysis to predict an asset’s future performance.

ARIMA modeling is generally inadequate for long-term forecastings, such as more than six months ahead, because it uses past data and parameters that are influenced by human thinking. For this reason, it is best used with other technical analysis tools to get a clearer picture of an asset’s performance.

What Is Accrued Liability?

The term “accrued liability” refers to an expense incurred but not yet paid for by a business. These are costs for goods and services already delivered to a company for which it must pay in the future. A company can accrue liabilities for any number of obligations and are recorded on the company’s balance sheet. They are normally listed on the balance sheet as current liabilities and are adjusted at the end of an accounting period.

Understanding Accrued Liability

An accrued liability is a financial obligation that a company incurs during a given accounting period. Although the goods and services may already be delivered, the company has not yet paid for them in that period. They are also not recorded in the company’s general ledger. Although the cash flow has yet to occur, the company must still pay for the benefit received.

Accrued liabilities, which are also called accrued expenses, only exist when using an accrual method of accounting. The concept of an accrued liability relates to timing and the matching principle. Under accrual accounting, all expenses are to be recorded in financial statements in the period in which they are incurred, which may differ from the period in which they are paid.

The expenses are recorded in the same period when related revenues are reported to provide financial statement users with accurate information regarding the costs required to generate revenue.

The cash basis or cash method is an alternative way to record expenses. But it doesn’t accrue liabilities. Accrued liabilities are entered into the financial records during one period and are typically reversed in the next when paid. This allows for the actual expense to be recorded at the accurate dollar amount when payment is made in full.

Types of Accrued Liabilities

There are two types of accrued liabilities that companies must account for, including routine and recurring. We’ve listed some of the most important details about each below.

Routine Accrued Liabilities

This kind of accrued liability is also referred to as a recurring liability. As such, these expenses normally occur as part of a company’s day-to-day operations. For instance, accrued interest payable to a creditor for a financial obligation, such as a loan, is considered a routine or recurring liability. The company may be charged interest but won’t pay for it until the next accounting period.

Non-Routine Accrued Liabilities

Non-routine accrued liabilities are expenses that don’t occur regularly. This is why they’re also called infrequent accrued liabilities. They aren’t part of a company’s normal operating activities. A non-routine liability may, therefore, be an unexpected expense that a company may be billed for but won’t have to pay until the next accounting period.

Journal Entry for an Accrued Liability

Accounting for an accrued liability requires a journal entry. An accountant usually marks a debit and a credit to their expense accounts and accrued liability accounts respectively.

This is then reversed when the next accounting period begins and the payment is made. The accounting department debits the accrued liability account and credits the expense account, which reverses out the original transaction.

When Do Accrued Liabilities Occur?

Accrued liabilities arise for a number of reasons or when events occur during the normal course of business. For instance:

• A company that purchases goods or services on a deferred payment plan accrues liabilities because the obligation to pay in the future exists.
• Employees may perform work for which they haven’t received wages.
• Interest on loans may be accrued if interest fees were incurred since the previous loan payment.
• Taxes owed to governments may be accrued because they are not due until the next tax reporting period.

At the end of a calendar year, employee salaries and benefits must be recorded in the appropriate year, regardless of when the pay period ends and when paychecks are distributed. For example, a two-week pay period may extend from December 25 to January 7.

Although they aren’t distributed until January, there is still one full week of expenses for December. The salaries, benefits, and taxes incurred from Dec. 25 to Dec. 31 are deemed accrued liabilities. These expenses are debited to reflect an increase in the expenses. Meanwhile, various liabilities will be credited to report the increase in obligations at the end of the year.

Accrued Liability vs. Accounts Payable (AP)

Accrued liabilities and accounts payable (AP) are both types of liabilities that companies need to pay. But there is a difference between the two. Accrued liabilities are for expenses that have not yet been billed, either because they are a regular expense that doesn’t require a bill (i.e., payroll) or because the company hasn’t yet received a bill from the vendor (i.e., a utility bill).

As such, accounts payable (or payables) are generally short-term obligations and must be paid within a certain amount of time. Creditors send invoices or bills, which are documented by the receiving company’s AP department. The department then issues the payment for the total amount by the due date. Paying off these expenses during the specified time helps companies avoid default.

Examples of Accrued Liability

As noted above, companies can accrue liabilities for many different reasons. As such, there are many different kinds of expenses that fall under this category. The following are some of the most common examples:

• Wage expenses: This is for work already performed by employees. The work is paid for in the next accounting period. This is common with employers who pay their employees bi-weekly, because a pay period may extend into the following accounting month or year.
• Goods and services: Some companies place orders and receive goods and services from their suppliers without paying for them immediately. As an accrued expense, the receiving company pays for these goods and services at a later date.
• Interest: A company may have an outstanding loan for which the interest isn’t yet due. The lender may require this expense.