Simple Interest

The Formula — I Equals PRT

The simple interest formula has three variables — principal, rate, and time — whose product produces the total interest amount.

Simple interest equals principal multiplied by rate multiplied by time.

I equals P multiplied by R multiplied by T.

Principal — P — is the original amount of money lent, borrowed, or invested. It is the base amount on which interest is calculated for every period of the arrangement and remains constant throughout the life of the simple interest calculation. Under compound interest, the principal grows each period as accrued interest is added to it — under simple interest, the principal never changes.

Rate — R — is the interest rate expressed as a decimal or percentage per unit of time, almost always expressed as an annual rate. A five percent annual rate is expressed as zero point zero five in the formula. It is critical that the rate and the time period be expressed in the same time unit — if the rate is annual, the time must be expressed in years or fractions of years. If the rate is monthly, the time must be expressed in months.

Time — T — is the number of periods for which the interest calculation applies, expressed in the same unit as the rate. One year at an annual rate produces T equals one. Six months at an annual rate produces T equals zero point five or six-twelfths. Ninety days at an annual rate using the money market Actual/360 day count convention produces T equals ninety divided by three hundred and sixty.

The total amount owed or received at the end of the period — principal plus interest — equals the principal plus the simple interest, or equivalently the principal multiplied by the quantity one plus the rate multiplied by time.

Total amount equals principal multiplied by one plus rate multiplied by time.

Total amount equals P multiplied by the quantity one plus R times T.

Worked Examples — From Simple to Applied

Three worked examples illustrate the formula across progressively more applied contexts.

Example one — the basic calculation. An investor deposits ten thousand dollars in a savings certificate at a five percent annual simple interest rate for three years. Simple interest equals ten thousand multiplied by zero point zero five multiplied by three — equalling one thousand five hundred dollars. Total value at maturity equals ten thousand plus one thousand five hundred — equalling eleven thousand five hundred dollars. The investor earns five hundred dollars of interest in each of the three years — interest in year one, year two, and year three is identical because the calculation never incorporates prior interest into the base.

Example two — the fractional year calculation. A commercial paper note with a face value of one million dollars is issued at a discount rate equivalent to an annual simple interest rate of four and a half percent with a term of ninety days. Using the money market Actual/360 day count convention, T equals ninety divided by three hundred and sixty, equalling zero point two five. Simple interest equals one million multiplied by zero point zero four five multiplied by zero point two five — equalling eleven thousand two hundred and fifty dollars. The note pays eleven thousand two hundred and fifty dollars of interest for its ninety-day term.

Example three — solving for the rate. A three-month Treasury bill is purchased for ninety-eight thousand dollars and matures at face value of one hundred thousand dollars in ninety-one days. 

The interest earned is two thousand dollars. Using the Actual/360 convention, the simple interest rate equals interest divided by principal divided by time — two thousand divided by ninety-eight thousand divided by ninety-one divided by three hundred and sixty — equalling approximately eight point zero eight percent annualised. This calculation demonstrates how the simple interest formula is inverted to solve for the unknown rate — the technique used to compute the bond equivalent yield on Treasury bills.

Simple Interest and the Money Market Day Count Conventions

The money market — the market for short-term debt instruments with original maturities of one year or less — calculates interest using simple interest rather than compound interest, because the short-term nature of the instruments makes compounding within the term immaterial and because market convention favours the transparency and calculational simplicity of linear interest accrual.

United States dollar money market instruments — Treasury bills, commercial paper, bankers acceptances, federal funds, and repurchase agreements — use the Actual/360 day count convention. Under this convention, the number of calendar days in the actual period is counted precisely, and that actual count is divided by a 360-day year rather than the actual 365 or 366-day calendar year. The Actual/360 convention slightly overstates the annualised rate relative to an Actual/365 calculation — a ninety-day period represents ninety divided by three hundred and sixty, equalling twenty-five percent of the year under Actual/360, compared to only ninety divided by three hundred and sixty-five, equalling twenty-four point six six percent under Actual/365. This difference is small but meaningful for precise comparison of returns across instruments using different day count conventions.

United States Treasury notes and bonds — longer-term instruments — use the Actual/Actual day count convention for interest accrual calculations, counting the exact number of calendar days and dividing by the exact number of days in the coupon period. Corporate bonds in the United States traditionally use the Thirty/Three Sixty day count convention — also called the Thirty over Three Sixty convention — which assumes every month has exactly thirty days and every year has exactly three hundred and sixty days, producing predictable and uniform coupon payment amounts.

Understanding day count conventions is important for Series 7 examination candidates because the accrued interest calculation on bonds — the interest that has accrued since the last coupon payment and that must be paid by the buyer to the seller in a secondary market bond transaction — depends directly on which day count convention the bond uses.

Simple Interest and Coupon-Paying Bonds — The Direct Application

The most practically important application of simple interest in the securities markets is the coupon payment structure of fixed income securities. Every coupon payment on a bond is a simple interest calculation — the coupon rate multiplied by the face value of the bond equals the annual interest payment, and each semiannual payment equals half of the annual coupon amount regardless of how long the investor has held the bond.

A ten thousand dollar face value corporate bond with a six percent annual coupon rate pays three hundred dollars every six months — six percent of ten thousand is six hundred dollars per year, divided by two for the semiannual payment frequency, equalling three hundred dollars per payment. This three hundred dollar payment is identical at every coupon date throughout the bond's life — it does not increase as each coupon payment is received, because coupons are simple interest payments calculated on the unchanged face value rather than compound interest that accumulates on a growing base.

This simple interest structure of coupon payments is what gives rise to reinvestment risk — the risk discussed in the Reinvestment Risk entry of this dictionary — because the yield to maturity calculation assumes that each coupon received can be reinvested at the same yield to maturity rate. If actual reinvestment rates differ from the yield to maturity, the investor's realised total return differs from the promised yield, and that difference is entirely attributable to the compounding assumption embedded in the yield calculation rather than to the simple interest coupon payments themselves.

Simple Interest Versus Compound Interest — The Critical Distinction

The distinction between simple interest and compound interest is among the most foundational and most tested concepts in the SIE and Series 7 examination framework — understanding precisely how the two differ and in which circumstances each applies is essential for both examination success and practical financial analysis.

Simple interest calculates interest exclusively on the original principal for every period — the interest earned in period two is identical to the interest earned in period one because neither period's interest is incorporated into the base for subsequent calculations. The total interest over any number of periods is a linear function of time — doubling the time period exactly doubles the total interest. This linear relationship is the defining mathematical characteristic of simple interest.

Compound interest calculates interest on the principal plus all previously accrued interest — each period's interest becomes part of the base for the following period's calculation, creating an exponentially growing interest amount. The total interest over a given period under compounding is always greater than under simple interest for the same principal, rate, and time, because compound interest includes interest on interest. The longer the time period and the higher the compounding frequency, the greater the difference between compound and simple interest for the same nominal rate.

A comparison using a ten thousand dollar principal at six percent annual interest makes the distinction precise. Under simple interest over five years, total interest equals ten thousand multiplied by zero point zero six multiplied by five — equalling three thousand dollars. Under annual compounding over five years, total interest equals ten thousand multiplied by the quantity one plus zero point zero six raised to the fifth power, minus ten thousand — equalling ten thousand multiplied by one point three three eight two three, minus ten thousand — equalling three thousand three hundred and eighty-two dollars and twenty-six cents. The compound interest produces three hundred and eighty-two dollars and twenty-six cents more over five years — the dollar amount of interest earned on previously accrued interest.

The practical significance of this distinction for securities professionals is that almost all investment returns over periods longer than one year compound — dividends reinvested, coupon payments reinvested, and capital appreciation that compounds on a growing base all follow compound rather than simple interest mechanics — while the quoted yields on money market instruments and the stated coupon rates on bonds follow simple interest conventions for their periodic payment calculations.

The Annual Percentage Rate Versus the Annual Percentage Yield

The distinction between the annual percentage rate and the annual percentage yield — sometimes called the effective annual rate — directly reflects the distinction between simple and compound interest and is tested in the context of consumer financial disclosures.

The annual percentage rate — APR — is the simple interest expression of the cost of credit, calculated as the periodic rate multiplied by the number of periods in a year without compounding. A credit card that charges one and a half percent monthly has an APR of eighteen percent — one and a half percent multiplied by twelve. This APR represents the simple interest annualised rate without accounting for the compounding that occurs when unpaid interest is added to the balance each month.

The annual percentage yield — APY — or effective annual rate — is the compound interest expression that accounts for the frequency of compounding — how many times per year interest is added to the principal and begins earning interest on itself. The APY for the same one and a half percent monthly rate equals the quantity one plus zero point zero one five raised to the twelfth power, minus one — equalling approximately nineteen point five six percent — materially higher than the eighteen percent APR because the APY captures the effect of monthly compounding on the annualised cost.

The Truth in Lending Act — Regulation Z at 12 CFR Part 1026 — requires creditors to disclose the APR in consumer credit transactions, allowing borrowers to compare the cost of credit across different lenders and product types on a standardised basis. The APY is required to be disclosed for deposit accounts under the Truth in Savings Act — Regulation DD — allowing depositors to compare the return on savings products that compound at different frequencies.

Simple Interest in Regulatory and Legal Contexts

Simple interest appears in regulatory and legal contexts beyond its role in investment calculations — particularly in the context of disgorgement and prejudgment interest in SEC enforcement proceedings and in the calculation of interest on defaulted obligations.

When the SEC brings an enforcement action and obtains disgorgement of a defendant's ill-gotten gains, it typically also seeks prejudgment interest on the disgorged amount — calculated from the date of the violation through the date of the court order at the Internal Revenue Service underpayment rate under IRC Section 6621. This prejudgment interest is calculated as simple interest rather than compound interest in most SEC enforcement cases — applied to the principal disgorgement amount at the applicable quarterly IRS rate for each quarter from the violation date through judgment.

In contract law, the default rate of interest on overdue obligations — when no specific rate has been agreed in the contract — is governed by applicable state law, which typically specifies a simple interest rate. The Uniform Commercial Code's provisions on interest on defaulted payment obligations generally adopt simple interest as the default calculation method in the absence of contractual specification.

The Time Value of Money — Where Simple Interest Sits in the Broader Framework

Simple interest is the foundational concept from which the time value of money framework — the core analytical infrastructure of finance — is built, but the full time value framework almost always uses compound interest rather than simple interest for multi-period calculations because compound interest more accurately reflects the economic reality of money's ability to earn returns on returns.

The present value and future value relationships for single cash flows — discussed in the Present Value and Future Value entries of this dictionary — use compound interest. The present value of a future amount equals the future amount divided by the quantity one plus the discount rate raised to the power of the number of periods. The future value of a present amount equals the present amount multiplied by one plus the interest rate raised to the number of periods. Both formulas reflect compounding — the exponential relationship between present and future value that makes compound interest the appropriate tool for multi-period financial analysis.

Simple interest future value — where the future amount equals the principal multiplied by one plus rate multiplied by time — is a linear rather than exponential function and is appropriate only for single-period or very short-term calculations where the approximation error from ignoring compounding within the period is immaterial. The money market conventions discussed above — Actual/360 for Treasury bills and commercial paper — use simple interest precisely because the short maturities of money market instruments make compounding within the term economically immaterial.

Examination Relevance and Key Takeaways

Simple interest is tested on the SIE and Series 7 examinations in the context of the interest calculation framework, the time value of money, the distinction from compound interest, and the money market conventions for short-term instruments.

The key points to retain are these.

Simple interest equals principal multiplied by rate multiplied by time — I equals P times R times T. The principal never changes in a simple interest calculation — interest is always computed on the original amount borrowed or invested, not on a growing balance that includes previously accrued interest. Total amount at maturity equals principal plus interest — or equivalently principal multiplied by one plus rate multiplied by time — P multiplied by the quantity one plus R times T.

The rate and time must be expressed in the same time unit — an annual rate requires time expressed in years or fractions of years. Money market instruments in the United States use the Actual/360 day count convention — dividing the actual number of days in the period by three hundred and sixty to express the time fraction. Corporate bonds use the Thirty/Three Sixty day count convention. Treasury notes and bonds use the Actual/Actual convention. Coupon payments on bonds are simple interest calculations — the stated coupon rate multiplied by the face value produces the annual interest regardless of whether the investor reinvests those coupons.

Simple interest produces a linear relationship between time and interest — doubling the time exactly doubles the interest. Compound interest produces an exponential relationship — interest earned on previously accrued interest causes total interest to grow faster than linearly with time. For the same principal, rate, and time period, compound interest always produces more total interest than simple interest — the difference equals the interest earned on prior interest. The annual percentage rate is the simple interest expression of credit cost — required by the Truth in Lending Act and Regulation Z at 12 CFR Part 1026 for consumer credit disclosures. The annual percentage yield is the compound interest expression capturing the effect of compounding frequency — required for deposit account disclosures. Prejudgment interest in SEC enforcement proceedings is typically calculated as simple interest at the IRS underpayment rate under IRC Section 6621 applied to the disgorged principal amount from the date of violation through the date of judgment.