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Business Mathematics and Statistics · Ch 6 — Discounting of Bills of Exchange

Banker's Gain and Formula Relationships

5

Banker's Gain and Formula Relationships

The banker's gain (BG) is the extra amount the banker makes by charging discount on the full face value instead of on the present value. It is the difference between the banker's discount and the true discount:

BG=BD−TDBG = BD - TD

Because BDBD is the interest on FF and TDTD is the interest on PVPV, the banker's gain is exactly the simple interest on the true discount for the same period:

BG=TD×r×t100=TD2PV.BG = \dfrac{TD \times r \times t}{100} = \dfrac{TD^{2}}{PV}.

These give a compact family of relationships, all worth memorising:

QuantityFormula
Banker's discountBD=Frt100BD = \dfrac{Frt}{100}
True discountTD=Frt100+rt=F−PVTD = \dfrac{Frt}{100+rt} = F - PV
Present valuePV=100F100+rt=F−TDPV = \dfrac{100F}{100+rt} = F - TD
Banker's gainBG=BD−TD=TD2PV=TD rt100BG = BD - TD = \dfrac{TD^{2}}{PV} = \dfrac{TD\,rt}{100}
Face value from BD, TDF=BD×TDBD−TDF = \dfrac{BD \times TD}{BD - TD}
Banker's discount from TDBD=F×TDPVBD = \dfrac{F \times TD}{PV}
Definition 1Banker's Gain (BG)

The excess of banker's discount over true discount; equals the interest on the true discount, $BG = B …

Definition 2Key relation

Face value from the two discounts: F=BD×TDBD−TDF = \dfrac{BD \times TD}{BD - TD}; and $ …

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