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 the 79 365 fraction in our example is equal to the 2 164 10 000 fraction ignoring rounding variances pro rata is therefore simply taking one fraction and finding its equal given a specific denominator 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 each of you would receive 50 of the pizza if the slices were shared equally this pro rata example is trying to determine which fraction with a denominator of four is equal to 1 2 each of you gets two slices because 2 4 1 2 examples of pro rataeach investor is paid according to their holdings when a company pays dividends to its shareholders the total amount of dividends paid would be 200 if a company has 100 shares outstanding and issues a dividend of 2 per share the total dividend payments can t exceed this limit no matter how many shareholders there are the pro rata calculation must be used to determine the appropriate portion of that 200 whole due to each shareholder assume there are only four shareholders they 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 is therefore 50 100 x 200 100 this makes sense because the shareholder owns half the shares and receives half the total dividends the remaining shareholders get 50 30 and 20 respectively be mindful of situations where weight isn t given to individuals every american citizen eligible to vote can cast a tally equal to every other citizen regardless of their age income or any other factor the pro rata value of the vote is equal to one divided by the total population that votes in this example 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 the insurance company must prorate the annual premium to determine what s owed if a policy is needed for a shorter term 2 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 to do this assume an auto policy that typically covers a full year carries a premium of 1 000 the company must reduce the premium accordingly if the insured only requires the policy for 270 days the pro rata premium due for this period is 1 000 365 x 270 739 73 pro rata calculations are also used to determine the amount of interest that will be earned on an investment 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 if an investment earns an annual interest rate the amount of interest earned in two months on an investment that yields 10 interest each year is 10 12 x 2 1 67 payment on accrued interest on bonds is calculated on a pro rata basis accrued interest is the total interest that has accumulated on a bond since its last coupon payment the bondholder is still entitled to the interest that accrues up until the time the bond is sold if they sell the bond before the next coupon date 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 ai face value of bond coupon rate time factorwhere ai accrued interestcoupon rate annual coupon ratenumber of periods per yeartime factor days lapsed since last paymentdays in payment period 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 3consider a bondholder who sells their corporate bond on june 30 the bond has a face value of 1 000 and a 5 coupon rate that pays semiannually on march 1 and sept 1 the buyer of the bond will pay the seller 1 000 5 2 122184 16 58 begin aligned 1 000 times frac 5 2 times frac 122 184 16 58 end aligned 1 000 25 184122 16 58 