Simple Interest and Discount
Role in the exam plan
This is prerequisite revision. The current syllabus places Annuities and Sinking Funds in Week 8 Quiz 2. Simple interest and discount were Quiz 1 topics. Revisit this unit when a short-term cash flow or quoted discount rate is confusing; prioritize Units 3–5 for Quiz 2.
Draw the cash flows before choosing a formula
A simple-interest loan starts with principal P received now and ends with F repaid at maturity. Interest is I = Prt and F = P(1 + rt), with annual r in decimal form and t measured in years. The original principal stays the interest base throughout the term. A simple bank-discount loan instead quotes the face/maturity value F and deducts interest in advance: D = Fdt; proceeds P = F − D = F(1 − dt). The discount rate d applies to face value, not the cash actually received.
| Unknown | Simple interest | Simple discount |
|---|---|---|
| Interest/discount | I = Prt | D = Fdt |
| Cash now | P = F/(1 + rt) | P = F(1 − dt) |
| Cash at maturity | F = P(1 + rt) | F = P/(1 − dt) |
| Annual quoted rate | r = I/(Pt) | d = D/(Ft) |
For simple discount require dt < 1 to obtain positive proceeds. A valid algebraic answer with negative proceeds is not a sensible ordinary loan.
Four day-count combinations
Actual/approximate time concerns the numerator. Exact/ordinary interest concerns the denominator. Actual time counts elapsed calendar days, ordinarily excluding the starting day and including the ending day. Approximate time treats each month as 30 days. Exact interest uses a 365-day year (or the stated leap-year convention); ordinary interest uses 360 days. The problem must specify a convention when the distinction matters.
For 18 January to 6 August 2026, actual time is 200 days; approximate time is 30(8 − 1) + (6 − 18) = 198 days. On PHP 75,000 at 9%, actual/ordinary interest is 75,000(.09)(200/360) = PHP 3,750. Approximate/exact is 75,000(.09)(198/365) = PHP 3,661.64. Do not silently combine a 30-day numerator with a 365-day denominator unless requested. Months stated as a term use months/12 unless the problem asks for dates/day counts.
Worked case: compare what the borrower receives
A 120-day note has face value PHP 100,000 and a 12% bank discount rate using a 360-day year. D = 100,000(.12)(120/360) = PHP 4,000. Proceeds are PHP 96,000, while repayment is PHP 100,000. The borrower pays PHP 4,000 on the PHP 96,000 actually usable.
The effective simple annual borrowing rate is r = D/(Pt) = .12/[1 − .12(120/360)] = 12.5%. This is an annualized simple rate for this one term, not a compounded annual percentage yield. A 12% simple-interest loan with the same term and PHP 100,000 principal supplies more cash initially, so comparing face values alone is misleading.
If a borrower needs exactly PHP 96,000 now, solve F = 96,000/[1 − .12(120/360)] = PHP 100,000. Multiplying 96,000 by 1.04 incorrectly treats the quoted discount as a simple-interest rate.
Worked case: discount an interest-bearing note
An investor holds a PHP 80,000 note earning 10% simple interest for nine months. Its maturity value is F = 80,000[1 + .10(9/12)] = PHP 86,000. The investor sells it with three months remaining at an 8% bank discount rate. The bank discount is 86,000(.08)(3/12) = PHP 1,720; sale proceeds are PHP 84,280. Calculate the note's maturity value using its original contract, then discount that maturity value only for the remaining time. Applying bank discount to original principal gives the wrong proceeds.
Partial payments: the U.S. Rule
Under the stated U.S. Rule, accrue simple interest on the unpaid principal until a payment date. Apply the payment to accrued interest first; only the remainder reduces principal. Do not charge interest on unpaid interest in this model. If a payment is insufficient to cover accrued interest, it does not reduce principal; carry the unpaid interest separately according to the stated contract.
Example: PHP 60,000 is borrowed at 12% simple interest. After three months, PHP 15,000 is paid. Accrued interest = 60,000(.12)(3/12) = PHP 1,800. Principal reduction = 15,000 − 1,800 = PHP 13,200; new principal = PHP 46,800. Three months later, final payment is 46,800[1 + .12(3/12)] = PHP 48,204. Subtracting PHP 15,000 from principal first would understate the debt.
Set up equations of value carefully
When replacing several short-term obligations, pick a focal date and express every cash flow at that date under the stated simple-interest convention. Unlike compound interest, repeated simple-interest transfers can depend on the chosen focal-date convention. State it and follow it consistently. Never add pesos at different dates before moving them to a common date.
Checks and traps
- Convert 8% to .08; convert a 90-day ordinary term to 90/360, not 90/365.
- Maturity value exceeds principal at a positive simple-interest rate. Bank-discount proceeds are below face value.
- For the same principal, positive rate and actual days, ordinary interest exceeds exact interest because 360 is the smaller divisor.
- Keep the annualized borrowing rate separate from the fraction of proceeds paid as interest during the term.
- Round at the final monetary result. Keep a date timeline and units beside every rate.
Independent self-check
Explain why a 10% bank discount rate on a six-month note costs more than 10% simple interest on cash proceeds. Answer: the upfront discount is based on the larger face value; the comparable simple rate is .10/(1 − .10 × .5) = 10.5263%. This reasoning matters more than memorizing two similar formulas.
Source basis
Paraphrased concept map from the course simple-interest/discount teaching files and Slater Chapters 16–17. All worked numbers and assessment cases in this package are independently authored; original Canvas files are retained locally and are not public attachments.