Bills Discounting

## Bills Discounting – Meaning

Bills Discounting, or Discounting of Bill of Exchange means trading of the Bill (or cashing it at Present Value) before its maturity date (at less than its Face Value). The cash realised at Present Value is computed from unexpired period until maturity at specified Rate of Interest.

Bills Discounting – Example 1: Let us explain with an example. Assume that a merchant A purchases goods worth of ₹5000 from another merchant B at a credit of 3 months. A accepts a Bill of Exchange for Rs 5000 payable to B after 3 months (called Bill Due Date or Maturity Date).

Let us Assume that B needs this money immediately (before the Bill Due Date), for his business working fund. He approaches a banker (or some other financier who discounts bills) to pay Discounting the Bill of Exchange. Now the Banker pays B deducting Interest on face value of Bill (called Bankers Discount). A will pay the Banker the face value of the bill on maturity date.

So, in this arrangement, both seller and buyer gets credit. The Buyer gets credit offered by the Seller. Buyer pays to the Bank on maturity of the Bill. The seller gets credit from Banker, by getting money instantly, before maturity date.

So, Bill Discounting is a mechanism for availing Trade Credit.

## Bills Discounting – Terminology

• Face Value of Bill: (F)= Amount be paid on due date, as stated on the Bill.
• Present Value or Present Worth : (PW) = Amount to become equal to face value, maturing after time T, at interest rate of R.
• Banker’s Discount :  (BD) = Simple interest on face value for unexpired time.
• True Discount :  (TD) = Simple interest on present value for unexpired time.
• Banker’s Gain : (BG) = Difference between banker’s discount and true discount

Bills Discounting – Example 2

Let us explain these terms by example.

Suppose Ram (Debtor) purchase goods worth Rs 1,00,000 from Shyam (Creditor). Ram (who does not have ready money) is ready to pay Interest for 9 months @6% pa, allowing him to pay after 9 months.

Interest on 1,00,000 for 9 months @ 6% pa =1,00,000× $\left( {\frac{6}{{100}}} \right)x\left( {\frac{9}{{12}}} \right)=$ Rs 4500. So, Ram accepts a Bill of Exchange in favour of Shyam, due after 9 months for 100000 (Value of Goods + Interest 4500)=1,04,500.

Now, Shyam would get 1.04,500 from the Bill, after 9 months. But Shyam wants money now. So, Shyam goes to a baker to discount the Bill. The Banker Discounts the Bill charging Bankers Discount @6% pa for 9 months on Rs 1.04.500 (full Bill Value), i.e $104500\times \left( {\frac{6}{{100}}} \right)x\left( {\frac{9}{{12}}} \right)=$4702.50. So, the Banker would pay 104500-4702.50=99797.50. Thus although the present worth is 1,00,000, Shyam gets less i.e (1,04,500 – 4702) = 99,797.50.

So Shyam gets 202.50 (which is Banker’s gain) lesser than present worth i.e. 100,000. Bankers Gain is the Simple Interest (i.e@ 6%) for Discount period (i.e 9 months) on Bankers Discount (i.e. 4500) $BG=4500\times \left( {\frac{9}{{12}}} \right)\times ~\left( {\frac{6}{{100}}} \right)$ = 202.50

## Bill Discounting – Short Term Trade Credit

Through these two examples, we have explained purpose of Bill Discounting as mechanism of availing short term Trade Credit.

• In first example, we have explained the concept of Bill Discounting In this example, the Supplier allows credit to Buyer. Supplier bears the Interest, as Bank deducts Banker’s Discount while discounting the Bill from Supplier.
• In second example, Supplier adds the Interest amount in the Bill. So, the Buyer pays the Principal as well as Interest amount.

## Bankers Gain through Bill Discounting

So, in summary, Banker gets extra gain because Banker compute Banker Discount on future Maturity Value (not the present worth), but pays money at Present Worth (which is less than Maturity Value). This extra gain, called Bankers gain, is the Interest difference of these two amounts i.e. (Interest on Face Value – Interest on Present Worth).

In both way, banker gain by paying preset value of Bill, charging Banker Discount on Face Value of Bill. The Interest on Present Value of the Bill (what the Banker pays while discounting the Bill) is called True Discount. So, Banker’s Gain = Banker Discount – True Discount.

## Bill Discounting Basic Formulas

As per definition explained above, we derive the following formulas

BD=Face Value (F) × Interest rate (R) ×Unexpired Period (T)= F×R×T

Present Worth = PW = $\frac{F}{{\left[ {1+\left( {\frac{{R\times T}}{{100}}} \right)} \right]}}$

TD = Present Worth (PW) × Interest rate (R) ×Unexpired Period (T)= PW×R×T

BG=Bankers’ Gain = TD-BD

## Bill Discounting – Important Formulas

With the above basic formula, we deduce other often used derived formula, for various mathematical calculation related to Bill Discounting, listed below at one place for easy reference, for solving problems.

(Note: In the formula shown R is considered Rate in % (like 7%, so value of R is 7), for ease of computation in practical problems. So, in the formula, Rate is divided by 100 because, in the problems the Rate is provided as %. If you use the absolute rate (like .07 for 7%), there is no need to divide by 100. Both methods are explained in the examples.

PW = $\frac{F}{{\left[ {1+\left( {\frac{{R\times T}}{{100}}} \right)} \right]}}$
BD= $\frac{{F\times R\times T}}{{100}}$ TD= $PW\times \frac{{R\times T}}{{100}}$ BG=BD−TD
PW+TD=F
TD $\left( {1+\frac{{R\times T}}{{100}}} \right)=$BD
TD= $\frac{{F\times R\times T}}{{100+\left( {R\times T} \right)}}$

F= $\frac{{BD\times TD}}{{BD-TD}}$   BG= $TD\times \frac{{R\times T}}{{100}}$
BG= (TD)2 / PW $\frac{{BD}}{{TD}}$= $\frac{F}{{PW}}$

Bill Discounting – Example 3

Find the Present Value of Rs. 1800 due in 73 days hence, at 7.5% p.a. (1 year = 365 days)
Given F,T & R. Find PW
Here, F = 1800,  years=  years=.2 years, R = = .075, PW=?
PW = $\frac{F}{{\left[ {1+\left( {\frac{{R\times T}}{{100}}} \right)} \right]}}$
PW= $\frac{{1800}}{{\left( {1+\left[ {\left( {\frac{{7.5}}{{100}}} \right)\times \left( {\frac{1}{5}} \right)} \right)} \right]}}$ = $\frac{{1800}}{{1+\left( {.075\times .2} \right)}}=$ $\frac{{1800}}{{1+.015}}$= $\frac{{1800}}{{1.015}}$. or PW=1773.40
PW= $\frac{{1800}}{{1+.075\times .2}}$ = $\frac{{1800}}{{1+.015}}$= $\frac{{1800}}{{1.015}}$. or PW=1773.40

Bill Discounting – Video in Hindi

Bill Discounting – Example 4

Find the present value, true discount, banker’s discount and banker’s gain on a bill of ₹104500 due in 9 months at 6% per annum?

Given F,T & R. Find PW,BD,TD,BG

Given, F =₹104500, T =9 months = $\frac{9}{{12}}$ years = $\frac{3}{4}$ years (or 0.75 years), R =6% pa. (or $\frac{6}{{100}}$= .06),

Banker’s Discount, BD = $\frac{{F\times R\times T}}{{100}}$= $\frac{{104500\times 6\times \frac{3}{4}}}{{100}}$=4702.50 [or 104500×.06x.75= 4702.50]

Present value, PW
= $\frac{F}{{1+\frac{{RT}}{{100}}}}=\frac{{104500}}{{1+6\times \frac{3}{4}\times \frac{1}{{100}}}}$= $104500/(1+\frac{9}{{200}}$)= $(104500/\left( {\frac{{209}}{{200}}} \right)$= $\frac{{209000}}{{200}}$=100000

[or $\frac{{104500}}{{1+\left( {.06\times .75} \right)}}$ = $\frac{{104500}}{{1+.045}}$= $\frac{{104500}}{{1.045}}$=Rs.100000]

True Discount, TD
= $\frac{{\left( {PW} \right)\times \left( {R\times T} \right)}}{{100}}$= $\frac{{100000\times 6\times \frac{3}{4}}}{{100}}$=4500

[or 100000×.06×.75=4500]

Banker’s Gain, BG = BD – TD =4702.50−4500=₹202.50

Bill Discounting – Example 5

The True Discount of a Bill due after 6 months @ 8% p.a is Rs 40, Find the Amount of the Bill

Given T=6 months = $\frac{6}{{12}}$ years = $\frac{1}{2}$ years, TD=40 (Given), F =?

TD= $\frac{{F\times R\times T}}{{1+R\times T}}$ , or 40= $\frac{{F\times \frac{1}{2}~\times \frac{8}{{100}}}}{{1+\frac{1}{2}\times \frac{8}{{100}}}}$ , or $~F\times \frac{1}{2}~\times \frac{8}{{100}}$ = 40 ( $1+\frac{1}{2}\times \frac{8}{{100}})$,

or, .04F= 40+1.6=41.6 or F= 41.6/.04= =1040

Bill Discounting – Example 6

Find the time after which the due amount will become 1060 @ 6% pa, when TD is Rs.60

Here F=1060, R=6%= $\frac{6}{{100}}$, T=?

TD= $\frac{{1060\times \text{T}\times \frac{6}{{100}}}}{{1+\left( {\text{T}\times \frac{6}{{100}}} \right)}}$, So, 60= $\frac{{1060\times \text{T}\times \frac{6}{{100}}}}{{1+\left( {\text{T}\times \frac{6}{{100}}} \right)}}$, So, 63.6×T=60+3.6T, or 60T=60, or T=1 year

Let us verify the result, TD = $\frac{{1060\times 1\times \frac{6}{{100}}}}{{1+\left( {1\times \frac{6}{{100}}} \right)}}$ = 63.6/1.06=60 (verified)

Bill Discounting – Example 7

If the TD on 11000 due for 15 month is 1000, Find rate of Interest

Here=11000, T= $\frac{{15}}{{12}}$ years= $\frac{5}{4}$.  years, TD=1000, R=?

TD= $\frac{{F\times R\times T}}{{100+\left( {R\times T} \right)}}$
TD= $\frac{{11000\times \frac{5}{4}\times R}}{{1+\frac{5}{4}\times R}}$, or 1000= $\frac{{13750R}}{{1+1.25R}}$, or R=13750R=1000+1250R,
Or, 13750×R-1250×R=1000, or 12500×R=1000, or R=1000/12500=.08 or 8% pa

Bill Discounting – Example 8

Find the TD. on a sum of Rs. 1750 due in 18 months and 6% p.a.

In this problem, F (Amount Due), Period (T) and Rate of Interest (R), are given. We have to find out value of TD.

Here : F =1750. T= $\frac{{18}}{{12}}$ years = 1.5 years. R= $\frac{6}{{100}}$ = .06.

So, TD = $\frac{{F\times R\times T}}{{1+\left( {R\times T} \right)}}$= $\frac{{1750\times \left( {1.5} \right)\times \left( {.06} \right)}}{{1+\left[ {\left( {1.5} \right)\times \left( {.06} \right)} \right]}}$ = $\frac{{1750\times \left( {.09} \right)}}{{\left( {1+.09} \right)}}$ = $\frac{{157.5}}{{1.09}}$ = 144.50 (appx). Ans

Bill Discounting – Example 9

A bill for Rs. 1224 is due in 6 months. Find the difference between true discount and banker’s Discount (Interest @4% p.a).

Here F-1224, T=6 months = 0.5 years. R= $\frac{4}{{100}}$= .04, We have to find out T.D, B.D, and then compute B.G

TD= $\frac{{F\times R\times T}}{{1+R\times T}}$ = $\frac{{1224\times \left( {.05} \right)\times .04)}}{{1+\left[ {\left( {.05} \right)\times \left( {.04} \right)} \right]}}$= $\frac{{24.48}}{{1.02}}$ = 24. BD= F×R×T=1224×(.05)×(.04)= 24.48. So. BG=BD- TD= 24.48-24=0.48

We may cross verify B.G =  Interest on  T.D = 24×(.04)×(.05) = 0.48

Bill Discounting Problems and Solutions – Video in Hindi

Bill Discounting – Example 10

TD. and BG. on a certain bill of exchange due after a certain time is respectively Rs 50 and

Re. 0.50. Find the face value of the bill.

Here TD =50. BG=.50, F=? . To get the value of F, we have to first find out the value of R (rate of Interest).

BG. = Interest on TD. or, 0.50 = 50 × R. or  R= $\frac{{0.50}}{{50}}$ = .01 . BD= TD+ BG = 50 + 0.50 = 50.50.

Again BD = Interest on A.

Now 50.50 = A×(.01). So, A= $\frac{{50.50}}{{.01}}$ = 5050 (Face Value of the Bill).

Bill Discounting – Example 11

A bill of exchange drawn on 5th  Jan for Rs 2,000 payable at 3 months was accepted on the

same date and discounted on 14th Jan, at 4% p.a. Find out amount of Discount.

Here the Bill Value (A), period (n), Rate of Interest (i) are given. We need to compute B.D

Unexpired number of days 14 Jan (Date of Acceptance) to 8 April (legal Maturity date) = 17 (Jan, excluding 14th Jan) + 28 (Feb, assuming not leap year) + 31 (Mar) + 8 (Apr) = 84 days

So, n= $\frac{{84}}{{365}}$ years. i=4%= $\frac{4}{{100}}.~$B.D= 2000 × $\frac{{84}}{{365}}$ × $\frac{4}{{100}}$ = 18.41 (appx).

Bill Discounting – Example 12

BD on a Bill due 4 months hence, @15% pa, is Rs. 420. Find TD.

TD = $~\frac{{B.D}}{{1+rt}}$ = $\frac{{420}}{{1+\left[ {\left( {.15} \right)\times \left( {\frac{1}{3}} \right)} \right]}}$ = $\frac{{420}}{{\left( {1+.05} \right)}}$ = $\frac{{420}}{{1.05}}$ = 400.

Ex. The banker’s discount on a bill due 4 months hence at 15% is Rs. 420. What is the true discount?

Here BD=420, T= $\frac{4}{{12}}$= $\frac{1}{3}$, R=15, TD=?

TD= $BD\times \frac{{100}}{{100+\left( {T\times R} \right)}}$ = $\frac{{420\times 100}}{{100+\frac{1}{3}\times 15}}$ = $420\times \frac{{100}}{{100+5}}$ = 420 $\times \frac{{100}}{{105}}$ = $\frac{{42000}}{{105}}$=400. So, True Discount is Rs 400

Bill Discounting – Example 13

The present worth of a sum due sometimes hence is Rs.5760 and the banker’s gain is Rs.10. What is the true discount? Here PW=

T.D =  =√(5760×10) =√57600= Rs. 240

Bill Discounting – Example 14

A bill for Rs. 3000 is drawn on 14th July at 5 months. It is discounted on 5th October at 10% pa. What is the Banker’s Discount? Here F = Rs. 3000, R = 10% , bill drawn on 14th July at 5 months. Nominal Due Date = 14th Dec, Legal Due Date = 14th Dec + 3 days = 17th Dec. Date of Discounting = 5th October

Unexpired Time = 26 (6th to 31st of Oct) + 30 (Nov) + 17 (Dec) = 26 + 30 + 17 = 73 Days =( $\frac{{73}}{{365}}$) years = $\left( {\frac{1}{5}} \right)$ years.

B.D = Simple Interest on face value of bill for unexpired time =F × T × R= $\frac{{3000\times \frac{1}{5}\times 10}}{{100}}$= 30 × $\frac{1}{5}$ ×10 = 60

Bill Discounting – Example 15

The bankers discount and the true discount of a sum at 10% per annum simple interest for the same time are Rs.100 and Rs.80 respectively. What is the sum and the time?

B.D = Rs.100, T.D = Rs.80, R = 10%=  = 0.1

B.D = Rs.100, T.D = Rs.80, R = 10%= $\frac{{10}}{{100}}$ = 0.1

F= $\frac{{BD~\times TD}}{{\left( {BD-TD} \right)}}$ = $\frac{{100~\times 80}}{{100-80}}$ = $\frac{{100~\times 80}}{{20}}$ = 400.

So, 400 × T ×0.1= 100, or 40×T=100, or T=2.5 years. So, Sum is Rs.400 and Time is 2.5 years.

Bill Discounting – Example 16

The banker’s discount of a certain sum of money is Rs. 36 and the true discount on the same sum for the same time is Rs. 30. What is the sum due?

F= $\frac{{BD~\times TD}}{{\left( {BD-TD} \right)}}$ = $\frac{{36~\times 30~}}{{36-30}}$ = $\frac{{36~\times 30}}{6}$ = Rs 180.

Bill Discounting – Example 17

A bill is discounted at 10% per annum. If banker’s discount is allowed, at what rate percent should the proceeds be invested so that nothing will be lost?

Let the amount = Rs.100. R is the annual rate of Interest in %

BD= Simple Interest on face value of the bill for 1 year= 100 $\times ~\frac{{10}}{{100}}=Rs.10$

Hence, Proceeds = Rs.100 – Rs.10 = Rs.90
Hence we should get Rs.10 as the interest of Rs.90 for 1 year, so that nothing will be lost.

So, 90 x 1 x $\frac{R}{{100}}$= 10. Or, R= $\frac{{10\times ~100}}{{90}}$ = $\frac{{100}}{9}$ =11 $\frac{1}{9}~$% (or 11.11%).

Bill Discounting – Example 18

The banker’s gain on a bill due 1 year hence at 10% per annum is Rs. 20. What is the true discount?

Here R=10%= $\frac{{10}}{{100}}$ = 0.1

TD= $\frac{{BG}}{{T\times R}}$ = $\frac{{20~}}{{1\times 0.1}}$ = Rs. 200

Bill Discounting –  True Discount Bankers Discount – Hindi Video

## One Comment

1. Rohon says:

Nice one