What Is the Addition Rule for Probabilities?

The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.

The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.

The Formulas for the Addition Rules for Probabilities Is

Mathematically, the probability of two mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y)+P(Z)$P(Y or Z)=P(Y)+P(Z)

Mathematically, the probability of two non-mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y) + P(Z) – P(Y \text{ and } Z)$P(Y or Z)=P(Y)+P(Z)−P(Y and Z)

What Does the Addition Rule for Probabilities Tell You?

To illustrate the first rule in the addition rule for probabilities, consider a die with six sides and the chances of rolling either a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the chances of rolling a 6 are also 1 in 6, the chance of rolling either a 3 or a 6 is:

1/6 + 1/6 = 2/6 = 1/3

To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the term, 5 girls and 4 boys receive a grade of B. If a student is selected by chance, what are the odds that the student will be either a girl or a B student? Since the chances of selecting a girl are 11 in 20, the chances of selecting a B student are 9 in 20 and the chances of selecting a girl who is a B student are 5/20, the chances of picking a girl or a B student are:

11/20 + 9/20 – 5/20 =15/20 = 3/4

In reality, the two rules simplify to just one rule, the second one. That’s because in the first case, the probability of two mutually exclusive events both happening is 0. In the example with the die, it’s impossible to roll both a 3 and a 6 on one roll of a single die. So the two events are mutually exclusive.

Mutual Exclusivity

Mutually exclusive is a statistical term describing two or more events that cannot coincide. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other.  For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. Furthermore, getting a three on an initial roll has no impact on whether or not a subsequent roll yields a five. All rolls of a die are independent events.

What Is the Accounting Rate of Return (ARR)?

The accounting rate of return (ARR) is a formula that reflects the percentage rate of return expected on an investment or asset, compared to the initial investment’s cost. The ARR formula divides an asset’s average revenue by the company’s initial investment to derive the ratio or return that one may expect over the lifetime of an asset or project. ARR does not consider the time value of money or cash flows, which can be an integral part of maintaining a business.

Understanding the Accounting Rate of Return (ARR)

The accounting rate of return is a capital budgeting metric that’s useful if you want to calculate an investment’s profitability quickly. Businesses use ARR primarily to compare multiple projects to determine the expected rate of return of each project, or to help decide on an investment or an acquisition.

ARR factors in any possible annual expenses, including depreciation, associated with the project. Depreciation is a helpful accounting convention whereby the cost of a fixed asset is spread out, or expensed, annually during the useful life of the asset. This lets the company earn a profit from the asset right away, even in its first year of service.

The Formula for ARR

The formula for the accounting rate of return is as follows:

$ARR = \frac{Average\, Annual\, Profit}{Initial\, Investment}$ARR=InitialInvestmentAverageAnnualProfit​

How to Calculate the Accounting Rate of Return (ARR)

1. Calculate the annual net profit from the investment, which could include revenue minus any annual costs or expenses of implementing the project or investment.
2. If the investment is a fixed asset such as property, plant, and equipment (PP&E), subtract any depreciation expense from the annual revenue to achieve the annual net profit.
3. Divide the annual net profit by the initial cost of the asset or investment. The result of the calculation will yield a decimal. Multiply the result by 100 to show the percentage return as a whole number.

Example of the Accounting Rate of Return (ARR)

As an example, a business is considering a project that has an initial investment of $250,000 and forecasts that it would generate revenue for the next five years. Here’s how the company could calculate the ARR:

• Initial investment: $250,000
• Expected revenue per year: $70,000
• Time frame: 5 years
• ARR calculation: $70,000 (annual revenue) / $250,000 (initial cost)
• ARR = 0.28 or 28%

Accounting Rate of Return vs. Required Rate of Return

The ARR is the annual percentage return from an investment based on its initial outlay of cash. Another accounting tool, the required rate of return (RRR), also known as the hurdle rate, is the minimum return an investor would accept for an investment or project that compensates them for a given level of risk.

The RRR can vary between investors as they each have a different tolerance for risk. For example, a risk-averse investor likely would require a higher rate of return to compensate for any risk from the investment. It’s important to utilize multiple financial metrics including ARR and RRR to determine if an investment would be worthwhile based on your level of risk tolerance.

Advantages and Disadvantages of the Accounting Rate of Return (ARR)

Advantages

The accounting rate of return is a simple calculation that does not require complex math and is helpful in determining a project’s annual percentage rate of return. Through this, it allows managers to easily compare ARR to the minimum required return. For example, if the minimum required return of a project is 12% and ARR is 9%, a manager will know not to proceed with the project.

ARR comes in handy when investors or managers need to quickly compare the return of a project without needing to consider the time frame or payment schedule but rather just the profitability or lack thereof.

Disadvantages

Despite its advantages, ARR has its limitations. It doesn’t consider the time value of money. The time value of money is the concept that money available at the present time is worth more than an identical sum in the future because of its potential earning capacity.

In other words, two investments might yield uneven annual revenue streams. If one project returns more revenue in the early years and the other project returns revenue in the later years, ARR does not assign a higher value to the project that returns profits sooner, which could be reinvested to earn more money.

The accounting rate of return does not consider the increased risk of long-term projects and the increased uncertainty associated with long periods.

Also, ARR does not take into account the impact of cash flow timing. Let’s say an investor is considering a five-year investment with an initial cash outlay of $50,000, but the investment doesn’t yield any revenue until the fourth and fifth years.

In this case, the ARR calculation would not factor in the lack of cash flow in the first three years, while in reality, the investor would need to be able to withstand the first three years without any positive cash flow from the project.

The Bottom Line

The accounting rate of return (ARR) is a simple formula that allows investors and managers to determine the profitability of an asset or project. Because of its ease of use and determination of profitability, it is a handy tool in making decisions. However, the formula does not take into consideration the cash flows of an investment or project, the overall timeline of return, and other costs, which help determine the true value of an investment or project.

What Are Affiliated Companies?

Companies are affiliated when one company is a minority shareholder of another. In most cases, the parent company will own less than a 50% interest in its affiliated company. Two companies may also be affiliated if they are controlled by a separate third party. In the business world, affiliated companies are often simply called affiliates.

The term is sometimes used to refer to companies that are related to each other in some way. For example, Bank of America has many different affiliated companies including Bank of America, U.S. Trust, Landsafe, Balboa, and Merrill Lynch.

Companies may be affiliated with one another to get into a new market, to maintain separate brand identities, to raise capital without affecting the parent or other companies, and to save on taxes. In most cases, affiliates are associates or associated companies, which describes an organization whose parent has a minority stake in it.

Understanding Affiliated Companies

There are several ways companies can become affiliated. A company may decide to buy out or take over another one, or it may decide to spin off a portion of its operations into a new affiliate altogether. In either case, the parent company generally keeps its operations separate from its affiliates. Since the parent company has a minority ownership, its liability is limited, and the two companies keep separate management teams.

Affiliates are a common way for parent businesses to enter foreign markets while keeping a minority interest in a business. This is especially important if the parent wants to shake off its majority stake in the affiliate.

There is no single bright-line test to determine if one company is affiliated with another. In fact, the criteria for affiliation changes from country to country, state to state, and even between regulatory bodies. For instance, companies considered affiliates by the Internal Revenue Service (IRS) may not be considered affiliated by the Securities and Exchange Commission (SEC).

Affiliates Versus Subsidiaries

An affiliate is different from a subsidiary, of which the parent owns more than 50%. In a subsidiary, the parent is a majority shareholder, which gives the parent company’s management and shareholders voting rights. Subsidiary financials may also appear on the parent company’s financial sheets.

But subsidiaries remain separate legal entities from their parents, meaning they are liable for their own taxes, liabilities, and governance. They are also responsible for following the laws and regulations where they are headquartered, especially if they operate in a different jurisdiction from the parent company.

An example of a subsidiary is the relationship between the Walt Disney Corporation and sports network ESPN. Disney owns an 80% interest in ESPN, making it a majority shareholder. ESPN is its subsidiary.

SEC Rules Surrounding Affiliates

Securities markets around the world have rules that concern affiliates of the businesses they regulate. Here again, these are complex rules that need to be analyzed by local experts on a case-by-case basis. Examples of rules enforced by the SEC include:

• Rule 102 of Regulation M prohibits issuers, selling security holders, and their affiliated purchasers from bidding for, purchasing, or attempting to induce any person to bid for or purchase, any security which is the subject of a distribution until after an applicable restricted period has passed.
• Before disclosing nonpublic personal information about a consumer to a nonaffiliated third party, a broker-dealer must first give a consumer an opt-out notice and a reasonable opportunity to opt out of the disclosure.
• Broker-dealers must maintain and preserve certain information regarding those affiliates, subsidiaries, and holding companies whose business activities are reasonably likely to have a material impact on their own finances and operations.

Tax Consequences of Affiliates

In nearly all jurisdictions, there are important tax consequences for affiliated companies. In general, tax credits and deductions are limited to one affiliate in a group, or a ceiling is imposed on the tax benefits that affiliates may reap under certain programs.

Determining whether companies in a group are affiliates, subsidiaries, or associates is done through a case-by-case analysis by local tax experts.