Pro Rata: What It Means and the Formula To Calculate It

What Is Pro Rata?

Pro rata is a Latin term used to describe a proportionate allocation. It essentially translates to "in proportion," which means a process where whatever is being allocated will be distributed in equal portions.

If something is given out to people on a pro rata basis, it means assigning an amount to one person according to their share of the whole. While a pro rata calculation can be used to determine the appropriate portions of any given whole, it is often used in business finance.

Understanding Pro Rata

Pro rata typically means that each party or person receives their fair share in proportion to the whole. Pro rata calculations can be used in many areas, including determining dividend payments, which are cash payments by corporations paid to shareholders.

In insurance, pro rata is used to determine the amount of premium due for a policy that only covers a partial term. Allocating the appropriate portion of an annual interest rate to a shorter time frame can also be done via pro rata.

Pro rata is also used to determine how much of a distribution from a qualified retirement account—such as an IRA, SEP, or 401(k)—is taxable when the account contains before and after-tax dollars. For example, an account holder has a 401(k) funded with 20% pre-tax dollars and 80% post-tax dollars. As a result, withdrawals will consist of 20% taxable and 80% non-taxable money.

How To Calculate Pro Rata

Pro rata is calculated based on three items:

1. The number of items true, owned, or having been incurred.
2. The total quantity of that item (i.e. maximum quantity possible).
3. The quantity of a related item that will be assigned in the second bullet above.

Pro Rata Share = (Number of "True" Items / Maximum Quantity Possible)

Pro Rata Distribution = Pro Rata Share * Quantity of Related Item

For example, imagine an employee set to receive a $10,000 bonus this year. If the employee leaves at any point, they will receive a pro rata share of the bonus. The employee leaves on March 20th. The agreement awards the bonus based on the number of days worked (inclusive of the last day).

To calculate the pro rata share, divide the number of true items by the maximum quantity possible. In this example, the number of true items is the number of days worked. Assuming a non-leap year, there are 79 days (inclusive) between January 1 and March 20. In addition, the maximum number of days possible is 365.

Pro rata share = 79 / 365 = 21.64%

To calculate the pro rata distribution, multiply the pro rata share by the related item. In this example, the item we want to pro rate is the annual bonus.

Pro rata distribution = 21.64% * $10,000 = $2,164

Therefore, in this example, by leaving on March 20, the employee will receive a pro rata distribution of $2,164 based on the prorated number of days worked of the year.

Why Pro Rata Works

The mathematical concept of pro rata works because the proportion of one good is imposed on another. Pro rata entails taking a fraction of one item and conveying the same fraction on another base.

The root of pro rata is grounded in proportionally equal fractions with different denominators. In the example above, the fraction 79/365 is equal to the fraction 2,164/10,000 (ignoring rounding variances). Therefore, pro rata is simply taking one fraction and finding its equal given a specific denominator. Stated another way, pro rata attempts to solve for the numerator to make two fractions equal given specific criteria.

Consider another example where you and a friend want to proportionally share four pieces of pizza. If the slices are shared equally, each of you would receive 50% of the pizza. Therefore, this pro rata example is trying to determine which fraction with a denominator of 4 is equal to 1/2. Since 2/4 = 1/2, each of you gets two slices.

Examples of Pro Rata

Pro Rata and Dividends per Shareholder

When a company pays dividends to its shareholders, each investor is paid according to its holdings. If a company has 100 shares outstanding, for example, and issues a dividend of $2 per share, the total amount of dividends paid will be $200. No matter how many shareholders there are, the total dividend payments cannot exceed this limit. In this case, $200 is the whole, and the pro rata calculation must be used to determine the appropriate portion of that whole due to each shareholder.

Assume there are only four shareholders who hold 50, 25, 15, and 10 shares, respectively. The amount due to each shareholder is their pro rata share. This is calculated by dividing the ownership of each person by the total number of shares and then multiplying the resulting fraction by the total amount of the dividend payment.

The majority shareholder's portion, therefore, is (50 / 100) x $200 = $100. This makes sense because the shareholder owns half of the shares and receives half of the total dividends. The remaining shareholders get $50, $30, and $20, respectively.

Pro Rata for Insurance Premiums

Another common use is to determine the amount due for a partial insurance policy term. Most insurance policies are based on a 12-month period, so if a policy is needed for a shorter term, the insurance company must prorate the annual premium to determine what is owed. To do this, divide the total premium by the number of days in a standard term, and multiply by the number of days covered by the truncated policy.

For example, assume an auto policy that typically covers a full year carries a premium of $1,000. If the insured only requires the policy for 270 days, then the company must reduce the premium accordingly. The pro rata premium due for this period is ($1,000 / 365) x 270 = $739.73.

Pro Rata for Interest Rates

Pro rata calculations are also used to determine the amount of interest that will be earned on an investment. If an investment earns an annual interest rate, then the pro rata amount earned for a shorter period is calculated by dividing the total amount of interest by the number of months in a year and multiplying by the number of months in the truncated period. The amount of interest earned in two months on an investment that yields 10% interest each year is (10% / 12) x 2 = 1.67%.

When it comes to bonds, payment on accrued interest is calculated on a pro rata basis. Accrued interest is the total interest that has accumulated on a bond since its last coupon payment. When the bondholder sells the bond before the next coupon date, they're still entitled to the interest that accrues up until the time the bond is sold. The bond buyer, not the issuer, is responsible for paying the bond seller the accrued interest, which is added to the market price.

The formula for accrued interest is as follows:

$\begin{aligned} &\text{AI} = \text{Face Value of Bond} \times \text{Coupon Rate} \times \text{Time Factor} \\ &\textbf{where:} \\ &\text{AI} = \text{Accrued Interest} \\ &\text{Coupon Rate} = \frac{ \text{Annual Coupon Rate} }{ \text{Number of Periods Per Year} } \\ &\text{Time Factor} = \frac{ \text{Days Lapsed Since Last Payment} }{ \text{Days in Payment Period} } \\ \end{aligned}$​AI=Face Value of Bond×Coupon Rate×Time Factorwhere:AI=Accrued InterestCoupon Rate=Number of Periods Per YearAnnual Coupon Rate​Time Factor=Days in Payment PeriodDays Lapsed Since Last Payment​​

The factor is calculated by dividing the length of time the bond was held after the last coupon payment by the time from one coupon payment to the next.

For example, consider a bondholder who sells their corporate bond on June 30. The bond has a face value of $1,000 and a 5% coupon rate, which pays semiannually on March 1 and Sept. 1. The buyer of the bond will pay the seller:

$\begin{aligned}&\$1,000 \times \frac { 5\% } { 2 } \times \frac { 122 }{ 184 } = \$16.58 \\\end{aligned}$​$1,000×25%​×184122​=$16.58​