An implicit interest rate is a rate which is not explicitly stated. For example, a client may offer to pay in multiple installments instead of up-front, but not have the sophistication, or need, to explicate the interest rate implied by the offer.
Implicit interest can be easily made explicit using a spreadsheet or financial calculator, especially if the periods of payment are evenly spaced.
An informality to the transaction (such as a loan between friends), a desire to hide the interest rate from an unsophisticated borrower, or even just an inability to calculate it are all reasons that an interest rate might not be stated.
It may be unstated in an offer to pay in installments, in lieu of an up-front payment. For example, a client may owe $1,000, but offers to pay in 3 installments of $350 each. The seller may wish to know the explicit interest rate to determine if he is getting a fair deal.
Calculation by calculator
An inexpensive financial calculator can be used to make an implicit interest rate explicit, if the payments are evenly spaced.
For an option to pay $100,000 up-front or pay $16,275 per period for 10 periods, enter -100000 into PV (present value), 16275 into PMT (payment), and 10 into N (number of periods). Then press these buttons in sequence: CMPT i, for "compute interest." The result will be 10%.
The names on the buttons may differ from calculator to calculator, but should not be hard to decode.
Note that since future value (FV) is not entered in, the calculator will presume it to be zero. Therefore, the problem is treated as a single loan or debt to be paid off, as opposed to a comparison of deals, so the payment and present value must be of opposite signs to indicate the difference in direction of cash flow. Entering 100000 and -16275 should also return the answer of 10%.
In Microsoft Excel, the RATE function calculates the implied interest of evenly spaced payments. As with a financial calculator, a sign change must be entered to indicate a difference the direction of cash flow. Therefore, for a payment that can be made either $100,000 up-front or in installments of $16,275 for 10 payments, the proper syntax is
Either of these two syntaxes return an implicit interest rate of 10%.
The fourth argument, future value, is left blank here, which, as in a financial calculator, defaults its value to zero.
Using the "present value table for a series of future payments" link at the U.S. Chamber of Commerce reference below, the above example can be repeated to calculate an interest rate of 10%.
Because of the size constraint of such tables, normalization is used, with the payment amount constrained to $1. To convert the above problem for table use, the corresponding comparison of payments to present value is
$16,275 / PV(actual) = $1 / PV(table)
$16,275 / $100,000 = $1 / PV(table) .
This produces PV(table) = 6.1444, which closely matches the 6.144567 in the table under 10% interest for 10 periods of payments. The interest rate per period is therefore 10%.
The interest can also be solved for by numerical analysis. The relevant equating of the present-value with the payments with interest-earned backed out is
PV = PMT * [ 1/(1+i) + 1/(1+i)^2 + ... + 1/(1+i)^n)]
PV - PMT [1 - 1/(1+i)^n]/[i] = 0
In Microsoft Excel, the left-hand side is typed in, with i pulled from another cell that holds an initial estimate of the interest rate. Opening Solver from the Tools menu, the target cell is the formula cell, "By Changing Cells" points to the interest cell, and "Equal to" is set to zero, since the object is to find i such that the left-hand expression equals zero. Clicking "Solve" changes the interest cell to the solution of the zero-equality equation.
The zero-equality equation can also be used in code, using methods that find the zeros of an equation, such as the Newton-Raphson method.
Using Solver affords the opportunity to allow uneven payment periods. The exponents in the first PV equation would be modified before entering the equation in total into a cell for Solver to reference.