Bank discount: find proceeds and compare interest
See why the amount received differs from the amount repaid, and why a discount rate is not the rate on cash received.
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Proceeds = F − D
Face value minus the upfront discount
Bank discount is calculated from the amount due at maturity. The borrower receives that amount minus the discount.
Start with the cash timeline
For a simple discount note, let F be the face value due at maturity, d the annual discount rate, and t the term in years. The lender deducts D = F × d × t at the start. The cash received, called proceeds, is F − D.
In a simple interest loan, interest is calculated on the principal received and added to the maturity payment. In a discount loan, the quoted discount is calculated on the face amount and withheld upfront. Identify which amount the problem gives you before selecting a formula.
These examples use a 360-day year and assume no extra fees. The quoted rates are hypothetical classroom values.
Example: a 100-day discount note
A borrower signs a $12,500 face-value note at a 7.2% annual bank discount rate for 100 days.
- Convert time: t = 100/360.
- Find discount: D = 12,500 × 0.072 × (100/360) = $250.00.
- Find proceeds: 12,500 − 250 = $12,250.00.
| When | Borrower’s cash flow |
|---|---|
| Today | Receives $12,250.00 |
| At maturity | Repays $12,500.00 |
The cost is $250, but the borrower did not receive $12,500 to use. That is why comparing the discount rate directly with a simple interest rate can mislead.
Compare rates on the same cash received
For a simple annualized comparison on the same 360-day basis, divide the $250 charge by the $12,250 proceeds, then divide by the fraction of a year:
r = (250 ÷ 12,250) ÷ (100/360) = 0.073469…
That is about 7.35%, higher than the quoted 7.2% discount rate. This calculation is a simple annualized rate on proceeds, not a compound annual yield or a regulatory APR calculation.
For comparison, a simple interest loan that actually advances $12,250 at 7.2% for the same 100 days would charge:
I = 12,250 × 0.072 × (100/360) = $245.00.
Its maturity payment would be $12,495.00. Both examples advance $12,250; the discount note costs $5 more under these assumptions. Comparing equal face values instead would hide the difference in cash received.
Work backward from the amount needed
Suppose the borrower needs exactly $9,800 in proceeds for 120 days at a 6% annual discount rate. The desired cash is not the face amount.
From proceeds = F(1 − dt), rearrange:
F = proceeds ÷ (1 − dt)
F = 9,800 ÷ [1 − 0.06 × (120/360)] = 9,800 ÷ 0.98 = $10,000.00.
Check the answer: discount = 10,000 × 0.06 × (120/360) = $200.00. Subtracting $200 from $10,000 leaves the required $9,800.
The factor 1 − dt must be positive for positive proceeds. Keep full precision until the final money amount, and verify the face value by substituting it into the proceeds formula.
Try it yourself
An $18,000 face-value note is discounted at 9% for 80 days on a 360-day basis. How much cash does the borrower receive?
Show the worked answer
D = 18,000 × 0.09 × (80/360) = $360.00. Proceeds = 18,000 − 360 = $17,640.00. The $18,000 face value is still due at maturity.
Keep practicing
Notes & discounting · Chapter 17References
- Slater, supplied Chapter 17, pp. 450–457
Promissory notes, bank discount, proceeds, and solving for face value. Page ranges follow the references in StudySoda’s supplied reviewer.
- U.S. Treasury: Understanding Pricing and Interest Rates
The Bills section gives the face-value discount-price formula on a 360-day basis. Treasury bills are an illustration of discount pricing, not a claim that every loan uses the same convention.
Original examples for learning. Check the conventions and rounding required by your course.
