Banker's Discount

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Banker's DiscountA. Important Concepts - Banker's DiscountAssume 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.

B. Important Formulas - Banker's DiscountLet 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 = $\frac{\text{FTR}}{100}$

$\text{PW =}\frac{\text{F}}{1+\text{T}\left(\frac{\text{R}}{100}\right)}$

TD = Simple Interest on the present value for unexpired time $=\frac{\text{PW}\times \text{TR}}{100}=\frac{\text{FTR}}{100+\text{TR}}$

TD = $\frac{\text{BD}\times 100}{100+\text{TR}}$

PW = F - TD

F = $\frac{\text{BD}\times \text{TD}}{\text{BD – TD}}$

BG = BD – TD = Simple Interest on TD = $\frac{(\text{TD}{)}^2}{\text{PW}}$

TD = $\sqrt{\text{PW}\times \text{BG}}$

TD = $\frac{\text{BG}\times 100}{\text{TR}}$

C. A Simple Example to understand the Basic Concepts of Banker's DiscountWhat is the present value, true discount, banker's discount and banker's gain on a bill of Rs.104500 due in 9 months at 6% per annum?

F = Rs. 104500

T = 9 months = $\frac{9}{12}$ years = $\frac{3}{4}$ years

R = 6%

Banker's Discount, BD $=\frac{\text{FTR}}{100}$
$=\frac{104500\times \frac{3}{4}\times 6}{100}=1045\times \frac{3}{4}\times 6=\text{Rs.}4702.50$

Present value, PW $=\frac{\text{F}}{1+\text{T}\left(\frac{\text{R}}{100}\right)}$
$=\frac{104500}{1+\left(\frac{3}{4}\right)\left(\frac{6}{100}\right)}=\text{Rs.}100000$

True Discount, TD $=\frac{\text{PW}\times \text{TR}}{100}$
$=\frac{100000\times \frac{3}{4}\times 6}{100}=1000\times \frac{3}{4}\times 6=\text{Rs.}4500$

Banker's Gain, BG = BD – TD = 4702.50 - 4500 = Rs.202.50