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Simple interest

Simple interest keeps the earning base unchanged. Calculate a two-year loan, compare annual compounding, and separate the quoted annual rate from the actual accrual fraction.

Simple interest is interest calculated on an unchanged principal, without adding earlier interest to the amount that earns the next period's interest. For a funding desk, it converts a quoted annual interest rate into a cash amount owed over a specified borrowing interval.

“Simple” describes the calculation basis. It does not establish whether a loan is cheap, safe or certain to be repaid.

The principal stays at USD 1,000

Suppose a borrower receives USD 1,000 at 6% simple interest per year for two years. Assume a fixed principal and rate, an explicit year fraction of exactly two, no fees or tax, and all principal and interest paid at the end. Accrued interest is never capitalised.

The first year adds USD 60. The second year adds another USD 60, because its earning base is still USD 1,000. Recording interest as it accrues does not itself make that interest earn interest.

Accrual intervalPrincipal used to calculate interestNew interestTotal interest accrued
First yearUSD 1,000USD 60USD 60
Second yearUSD 1,000USD 60USD 120
The borrower repays USD 1,120 at maturity. From the lender's perspective, the cash flows are minus USD 1,000 initially and plus USD 1,120 after two years, assuming payment as promised.
Two-year simple-interest ledger: USD 60 accrues each year on an unchanged USD 1,000 principal, giving USD 1,120 due. Annual compounding at the same 6% quote ends at USD 1,123.60.
Two-year simple-interest ledger: USD 60 accrues each year on an unchanged USD 1,000 principal, giving USD 1,120 due. Annual compounding at the same 6% quote ends at USD 1,123.60.
Synthetic two-year comparison at the same 6% annual quote. Simple interest is not capitalised; annual compound interest is. No interim payments, fees or tax.
With compound interest, the first year's USD 60 would join the earning base. Annual compounding would then earn USD 63.60 in year two and produce 1,000(1.06)2=USD 1,123.601{,}000(1.06)^2=\text{USD }1{,}123.60. The additional USD 3.60 is interest on the first year's interest. Keeping the same headline rate while changing the compounding rule changes the cash flows.

Rate × time × principal

For constant principal PP, a simple annual rate rr written as a decimal, and contractual year fraction τ\tau,

I=Prτ,A=P+I=P(1+rτ).I=Pr\tau, \qquad A=P+I=P(1+r\tau).

Here II is the interest and AA is the total due when everything is paid at the end. For the loan above, I=1,000×0.06×2=120I=1{,}000\times0.06\times2=120. A separate loan with the same principal and quote over an explicitly specified half-year fraction would accrue 1,000×0.06×0.5=USD 301{,}000\times0.06\times0.5=\text{USD }30. Its period interest is 3% of principal, despite the 6% annual quote.

The word “half-year” is not enough to settle a dated trade. The day count convention determines the year fraction from the contract's dates. For example, a 90-day simple loan at the same principal and rate gives:
  • On Actual/360, τ=90/360=0.25\tau=90/360=0.25, so interest is USD 15.00.
  • On Actual/365 Fixed, τ=90/365\tau=90/365, so interest is USD 14.79, rounded to cents.
The days and annual percentage are identical; the quote basis differs. Neither calculation says that 90 days is universally one quarter of a calendar year. The ACT's interest-rate explainer develops this distinction between annual quotes and period interest.

Where the shortcut stops

Simple interest is useful for a single accrual interval, including many short-term funding quotes. It is not a rule that every instrument with a short maturity must use simple interest.

If principal is repaid partway through, calculate each interval on its applicable outstanding principal. If interest is paid and then reinvested elsewhere, that reinvestment has its own cash flows and rate. Rolling over successive simple-interest deposits can produce compounding when each new deposit includes earlier interest; the original two-year simple contract does not do that.

The formula also permits a negative rr: interest is then negative for positive principal and positive time. An ordinary positive repayment amount requires 1+rτ>01+r\tau>0. Do not extrapolate a negative simple rate indefinitely past that boundary. Nor does a positive quoted rate guarantee a realised investment return: payment failure, fees and other contractual terms can change what the lender receives.

SelfMadeQuant's Time Value of Money lesson takes the next step from accumulation to present value. Its public page is a preview; full lesson access requires membership.

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