Discount

Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The discount, or charge, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

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Assume that a merchant A purchases goods worth, say Rs.1000 from another merchant B at a credit of say 4 months.
Then B prepares a bill called bill of exchange (also called Hundi). On receipts of goods, A gives an agreement by signing on the bill allowing B to withdraw the money from A’s bank exactly after 4 months of the date of the bill.
The date exactly after 4 months is known as nominally due date. Three more days (called grace days) are added to this date to get a date known as legally due date.
The amount given on the bill is called the Face Value (F) which is Rs.1000 in this case.
Assume that B needs this money before the legally due date. He can approach a banker or broker who pays him the money against the bill, but somewhat less than the face value.
The banker deducts the simple interest on the face value for the unexpired time. This deduction is known as Bankers Discount (BD). In another words, Bank Discount (BD) is the simple interest on the face value for the period from the date on which the bill was discounted and the legally due date.
The present value is the amount which, if placed at a particular rate for a specified period will amount to that sum of money at the end of the specified period. The interest on the present value is called the True Discount (TD). If the banker deducts the true discount on the face value for the unexpired time, he will not gain anything.
Banker’s Gain (BG) is the difference between banker’s discount and the true discount for the unexpired time.

Note: When the date of bill is not given, grace days are not to be added.

Important Formulas – Banker’s Discount

Let F = Face Value of the Bill, TD = True Discount, BD = Bankers Discount, BG = Banker’s Gain, R = Rate of Interest, PW = True Present Worth and T = Time in Years

BD = Simple Interest on the face value of the bill for unexpired time = FTR/100
PW = F/[1+T(R100)]
TD = Simple Interest on the present value for unexpired time $= PW \times TR 100 = FTR 100 + TR “> =(\frac{PW \times TR/}{100)} =(\frac{FTR/}{100 + TR)}$
TD = ( $BD \times 100 100 + TR “> \frac{BD \times 100)/(}{100 + TR)}$
PW = F – TD
F = ( $BD \times TD BD - TD “> \frac{BD \times TD)/(}{BD - TD)}$
BG = BD – TD = Simple Interest on TD = $(TD) 2 PW “> \frac{(TD {)^}^{2/}}{PW}$
TD = $PW \times BG “> \sqrt{PW \times BG)}$
TD = ( $BG \times 100 TR “> \frac{BG \times 100)/}{TR}$

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