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 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.

What Is Adverse Possession?

The term “adverse possession” refers to a legal principle that grants title to someone who resides on or is in possession of another person’s land. The property’s title is granted to the possessor as long as certain conditions are met including whether they infringe on the rights of the actual owner and whether they are in continuous possession of the property. Adverse possession is sometimes called squatter’s rights, although squatter’s rights are a colloquial reference to the idea rather than a recorded law.

Understanding Adverse Possession

As mentioned above, adverse possession is a legal situation that occurs when one party is granted title to another person’s property by taking possession of it. This can happen intentionally or unintentionally with or without the property owner’s knowledge.

In cases of intentional adverse possession, a trespasser or squatter—someone who occupies another person’s land illegally—knowingly comes on to another person’s land to live on it and/or take it over. In other cases, adverse possession may be unintentional. For example, a homeowner may build a fence separating their yard without realizing they’ve crossed over and encroached on their neighbor’s property line. In either case, the adverse possessor—also referred to as the disseisor—can lay claim to that property. And if the claimant is successful in proving adverse possession, they are not required to pay the owner for the land.

Requirements to Prove Adverse Possession

The requirements to prove adverse possession tend to vary between jurisdictions. In many states, proof of payment for the taxes on a property and a deed is essentially required for the claimant to be successful. Each state has a time period during which the landowner of record can invalidate the claim at any time. For example, if the state threshold is 20 years and the landlord paints or pays for other maintenance on the house in question in the 19th year, then the claimant will have a difficult time proving adverse possession. That said, landowners are advised to remove the possibility of adverse possession as soon as possible by having signed agreements for any use of an owned property.

To successfully claim land under adverse possession, the claimant must demonstrate that his or her occupation of the land meets the following requirements:

• Continuous use: Under this condition, the adverse possessor must show they’ve been in continuous and uninterrupted possession of the property in question.
• Hostile and adverse occupation of the property: Although this doesn’t mean that the disseisor uses force to take the land, they must show there is no existing agreement or license from the landowner such as a written easement, lease, or rent agreement.
• Open and notorious possession: The person seeking adverse possession must occupy a property in a manner that is open, notorious, and obvious. The true owner is not required, however, to be aware of the occupation.
• Actual possession: The possessor must actively possess the property for the state’s predetermined statutory period, which may vary from three to 30 years. Possession may involve maintaining the land and—depending on state law—paying taxes.
• Exclusive use: The property is used solely by the disseisor, excluding any others from using it as well.

How to Prevent Adverse Possession

If you are a landowner, you can prevent a trespasser from gaining property ownership by taking some easy measures:

• Identify and mark your property boundaries. Inspect your land regularly for signs of trespassers. You may want to use “no trespassing” signs and block entrances with gates. Although many states will not find a “no trespassing” sign sufficient to prevent an adverse possession claim, it’s a good way to deter trespassers.
• Offer to rent the property to the trespasser. With a proper rental agreement in place, the trespasser cannot claim adverse possession.
• Grant written permission to someone to use your land, and make sure you get their written acknowledgment.
• Act fast. In the event of trespassing, you must act before the trespasser has been on your land for the period of time detailed by your jurisdiction, in order to make a successful case.

Adverse Possession vs. Homesteading

Adverse possession is similar to homesteading in practice. In homesteading, government-owned land or property with no clear owner on record is granted to new owners provided they are using and improving it. If a homesteader doesn’t use the land, they can lose it. Adverse possession can operate in a similar manner by freeing up land with an unclear title for productive use.

Of course, adverse possession can also be abused in ways homesteading cannot. If there is an informal easement between two farms where one farmer’s fence has an acre of the neighbors’ land in it, for example, the farmer using it can claim adverse possession to essentially bite off that chunk of land if there is no written easement agreement.