Mathematics and Statistics · Ch 9 — Commission, Brokerage and Discount
True Discount, Banker's Discount and Banker's Gain
True Discount, Banker's Discount and Banker's Gain
When a bill is discounted before maturity, the holder receives less than the face value, because money received now is worth more than the same money later. There are two ways to measure the deduction, and the difference between them is a small profit for the bank.
The present worth (PW) of a bill is the sum which, invested now at the given rate of simple interest for the unexpired period, would grow to the face value on the legally-due date. The true discount (TD) is the genuine interest on this present worth — the honest cost of receiving the money early.
Present worth and true discount
For a bill of face value , unexpired period years and rate per annum, write . Then
The true discount is simple interest on the present worth: .
Banks, however, find it simpler to charge interest on the whole face value. The banker's discount (BD) is the simple interest on the face value for the unexpired period, and the amount the bank actually hands over is the cash value .
Banker's discount, cash value and banker's gain
Because the bank charges on but the fair charge is only on the smaller PW, the bank keeps a small extra amount, the banker's gain:
Useful relationships …
with ; the sum which, invested now at for the unexpired period , grows …
; the genuine interest on the present worth, i.e. simp …
; the simple interest charged by a bank on the whole face value for the …
; the amount a bank actually pays the holder on discoun …
; the bank's small profit, equal to simple interest …