Bret D. Whissel’s paper derives formulas for the periodic payment x and interest rate i of an amortizing loan given the original principal P, total number of payments N, the number of payments made r, and the remaining principal R. First, Whissel uses the standard annuity formula for the payment (Equation 6* in his paper), which is the well-known amortization formula:
equivalent to the present-value-of-annuity formula . He also uses a formula for the remaining balance after r payments (Equation 12), which can be written as:
By rearranging the balance formula to solve for x in terms of R and equating it to the payment formula, Whissel eliminates x and arrives at a single equation in i. This results in Equation (16), which relates the fraction of principal repaid to the exponential terms:
This can be recognized as a compact form of the amortization relationship; indeed, . So (16) is equivalent to:
Whissel notes that due to the typically high power of in this equation, one cannot solve it for i in closed-form and “finding i will require numerical methods”. Plugging the solved i back into either earlier formula yields the periodic payment x.
Whissel’s approach is mathematically sound and aligns with standard financial theory. Equation (16) is derived by correctly combining two fundamental relationships: the payment formula and the remaining balance formula. The steps leading up to (16) involve basic algebraic manipulations (dividing through by , isolating terms, etc.) that are done carefully with no apparent errors. We can verify Equation (16) independently by recognizing that it essentially equates the fraction of principal remaining to the fraction of the annuity factor remaining. In fact, from standard annuity theory, the remaining balance after r payments in a fully amortizing loan is
so dividing both sides by P yields exactly Whissel’s Equation (16). This is consistent with textbooks and finance references on loan amortization. It reflects the intuitive idea that after r payments, the portion of principal repaid relative to P equals the ratio of the sum of a geometric series of length r to that of length N. There is no contradiction with accepted financial theory; on the contrary, Whissel’s result is a restatement of the well-known relationship between loan balance and the annuity formula.
No major oversights are evident in the derivation itself under the assumptions stated. (Whissel assumes a conventional fully amortizing loan with no balloon payment and no interest-free periods.) The algebraic steps are all reversible provided . The case of zero interest rate is a special-case (the formula would degenerate since makes the payment simply ); but in practice one would handle that separately. Apart from that edge case, the derivation holds for any positive interest rate. It’s worth noting that the formula also assumes a fixed payment x each period – which is the standard scenario for an amortizing loan. Loans with unconventional structures (graduated payments, negative amortization, etc.) would not directly satisfy these formulas without modification. Indeed, general finance sources caution that the “remaining balance” formula used here applies only to standard fixed payment loans; exotic loans (negative amortization, graduated payments) require special calculations. Whissel’s paper, however, confines itself to the standard amortization context, so this assumption is appropriate.
In terms of the use of exponential terms and rearrangements, Whissel’s handling is appropriate. He combines the equations in a way that isolates the terms involving . The resulting Equation (16) is a transcendental equation in – it cannot be simplified into a polynomial in simple form because appears inside the exponential expressions. Whissel stops at this point and rightly concludes that numerical root-finding is needed. This is aligned with the consensus in financial mathematics: solving for the interest rate in an annuity or loan formula is not analytically trivial. The equation essentially becomes a high-degree polynomial if expanded (in fact, if one cleared denominators and expanded, it would yield a polynomial of degree N in ). For large N (e.g. a 30-year mortgage with monthly payments gives ), this polynomial is of very high degree. Even for moderate N, there is no general algebraic solution. (By the Abel–Ruffini theorem, polynomial equations of degree 5 or higher have no solution in radicals.) Financial math texts emphasize that there is no closed-form algebraic way to isolate i in the annuity equation and that one must resort to iteration. Whissel’s derivation fully acknowledges this, and his result is consistent with that understanding.
In summary, the approach is mathematically correct. Equation (16) is derived without error and it encapsulates the same relationship one could obtain from standard formulas for present value and remaining balance. It does not violate any accepted financial principle; rather, it is a direct consequence of them. The only “insight” it relies on is that two equations (for payment and remaining balance) can be solved together to back out i, which is a standard strategy. There is no sign of an oversight in the mathematical steps. The use of exponential terms is proper – Whissel carefully moved terms and factored differences like in a way that maintained algebraic equality. Each step (such as dividing through by , which assumes , or isolating the bracketed terms) is justified. Therefore, the derivation stands on solid ground and aligns perfectly with established theory of loan amortization.
Whissel’s method of solving for the interest rate and payment is essentially the same approach used by standard amortization calculators and spreadsheet financial functions – the difference is mostly in form, not substance. In practice, tools like Excel’s PMT and RATE functions implement these very formulas:
Periodic Payment (PMT): Excel’s PMT (for a fully amortizing loan) is built on the formula . This is exactly Whissel’s Equation 6*. In other words, Whissel’s payment derivation reproduces the standard payment formula that one finds in textbooks or Excel’s documentation. There is no discrepancy in the payment amount; if one plugs the same P, N, and i into Whissel’s formula or Excel’s PMT, the result for x will be identical (save for rounding). This indicates Whissel’s approach is fully consistent with common financial calculations for loan payments.
Interest Rate (RATE): Excel’s RATE function is used to find the interest rate that satisfies a present value equation given NPER, PMT, PV, and FV. In the context of a loan, one would input the total number of periods, the payment (as a negative cash flow), the present value (loan principal), and the future value (remaining balance, or 0 if fully paid). Internally, Excel’s RATE does not have a closed-form formula; it uses an iterative numerical method (Newton–Raphson or a variant of the secant method) to converge on the interest rate. This is fundamentally the same task as solving Whissel’s Equation (16). In fact, if we set up Excel’s RATE with nper = r, pv = P, pmt = –x, and fv = R (with appropriate sign conventions), Excel is effectively solving the same equation for i. There is no analytical shortcut inside Excel; it’s performing root-finding just as Whissel suggests doing. The accuracy of both methods will depend on the numeric precision and convergence criteria, but for practical purposes both yield the same result for i. In other words, Whissel’s equation must be solved iteratively, and that is exactly what Excel or a financial calculator does when finding an unknown interest rate.
Crucially, Whissel’s final equation is equivalent to the typical internal rate of return (IRR) formulation of the problem. If one were to treat the loan as a series of cash flows – P received at time 0, x paid each period for r periods, and with R still owed at time r (which could be thought of as an outgoing cash flow if the loan were settled at that point) – the interest rate i is the IRR of that cash flow stream. Standard finance theory states that IRR is found by solving a polynomial equation (the net present value set to zero) which generally requires numerical methods. This is exactly what Equation (16) represents. Thus, from a theoretical standpoint, Whissel’s method is not introducing a new formula but rather recapitulating the IRR/annuity equation in a convenient form. It fully aligns with widely accepted computational methods.
In comparison to other academic sources, there is agreement that solving for the interest rate of an annuity cannot be done by simple algebraic inversion. For example, educational texts note that the interest rate appears both as an exponent and as a multiplier in the annuity formula, making isolation impossible by elementary means. The common recommendation is to use trial-and-error, financial calculators, or tools like Excel’s RATE function. Whissel’s paper explicitly acknowledges this need for numerical solution and thus is in line with standard teaching. There is no indication that Whissel proposes an alternative analytical solution; he explicitly suggests using a computer program or spreadsheet to find i. This matches typical practice.
Practicality: In practice, using Whissel’s formula (16) is as practical as using any other formulation of the problem. One could plug the equation into a spreadsheet and use Goal Seek or Solver to find i, just as one might do with the present value formula. The level of difficulty is comparable. Excel’s built-in RATE function is arguably more convenient (since it’s pre-packaged and optimized), but under the hood it’s solving the same problem. Both approaches will converge to the correct interest rate given a reasonable initial guess. For typical loan parameters, convergence is usually fast and unproblematic. Thus, Whissel’s method is neither more nor less accurate than the standard methods – it is essentially the same computation. Any differences in practice would come down to numerical tolerances or initial guesses rather than the formula itself.
Computational efficiency: From a computational standpoint, solving Equation (16) requires evaluating functions like , which, for large N, might raise concerns about potential overflow or precision loss. However, in realistic scenarios (e.g. up to a few hundred or even a few thousand), modern computing environments handle this easily in double precision. The iterative root-finding process typically converges in only a few iterations (Excel’s RATE, for instance, usually converges within 20 iterations or fewer). The cost of computing and at each iteration is trivial for a computer. Standard financial calculators also solve these using efficient iterative techniques almost instantaneously. Therefore, Whissel’s method is computationally efficient enough for practical use – on par with other approaches. It does not offer a speed advantage over Excel’s built-in algorithms (which are also very fast), but it is not meaningfully slower either. Both will find the solution in milliseconds for typical inputs.
One area to consider is robustness. Excel’s RATE function and similar routines use Newton–Raphson or secant methods, which require an initial guess and can occasionally fail to converge if the guess is poor or if the function behaves pathologically. In well-behaved loan scenarios (positive interest, standard amortization), the function is continuous and monotonic in i, so convergence is usually reliable. Excel by default uses 10% as a starting guess, which works in most cases, but if the true interest rate is very low or very high, the user might need to provide a better guess to ensure convergence. This is not a flaw of the formula but of the iterative method. Whissel’s paper doesn’t delve into the solution technique details, but any implementation of (16) would face the same need. In comparison, financial calculators typically implement a safe combination of bisection and Newton methods to guarantee a result. In essence, all methods must deal with the numerical nature of the solution. The formula by itself is accurate; it’s the solving technique that must be chosen carefully to be robust. A benefit of Whissel’s explicit equation is that one can plug it into a solver directly. The formula’s structure (a difference of exponentials divided by another difference) is well-behaved for positive i and poses no extraordinary stability problems in the domain of typical interest rates.
In summary, Whissel’s method is equivalent in practicality and accuracy to using Excel’s financial functions or other standard calculators. It does not provide a closed-form answer (nor does any standard method), and it requires numerical iteration just like any other approach. The results it produces for i and x will match those from standard tools, since all are rooted in the same fundamental equations. This equivalence is supported by the financial literature, which notes that such interest rate calculations “typically require numerical methods or financial calculators to solve for the interest rate directly”. Whissel’s work essentially confirms the standard practice with a clear derivation.
Equation (16) itself is a well-behaved equation for numerical solving, and it does not introduce significant instability beyond what is inherent in the problem. The equation can be rewritten as . The function is continuous for (since required for real-valued powers) and, in the context of loans, we only consider (non-negative interest rates). Over , is monotonic: when i = 0, ; when i grows large, the fraction tends to 1, so tends to . Typically , so and will go from negative to positive (or vice versa) exactly once, ensuring a unique root for i. This guarantees that simple bracketing and bisection can always find the root in a known interval (for instance, , where could be a very high rate that definitely overshoots the needed interest). In practice one might bracket between 0% and, say, 100% (or some high value) and then refine. Because is smooth and unimodal in typical cases, numerical root-finding is stable. Newton’s method converges quadratically when near the solution, and in this case the derivative can be computed or approximated without much difficulty if needed. The derivative would involve terms like , etc., which are large in magnitude for big N, but that is manageable in double precision arithmetic. Excel and other software handle this by their built-in algorithms; for a custom implementation, one could also use the bisection method (which is guaranteed to converge albeit more slowly) as a fail-safe.
One potential numerical issue is when i is extremely small. If the interest rate is near 0, the terms and are both very small, and their ratio tends toward (the 0% interest case). Directly computing these differences for very tiny i could lead to floating-point cancellation error. However, this is easily managed by using higher precision or series expansion if needed. For example, one can use the Taylor expansion for tiny i to get an initial sense of the solution. In practice, interest rates are not astronomically small in loan scenarios (if i is essentially zero, one can handle that as a special case). Similarly, if N is very large (say, thousands of periods) and i is moderately large, might overflow a standard float. But in realistic financial contexts, N and i remain in ranges that are safe (even a 100-year monthly loan with a moderate rate is computable). Nonetheless, a robust implementation might incorporate checks: for example, if is extremely close to 0, use a linear approximation; if is very high, consider scaling or using logarithms to evaluate (since can be computed without overflow). These are common precautions in writing financial algorithms but are not specific to Whissel’s formula – they would apply to any formulation of the problem.
Another consideration is the initial guess for iteration. Whissel’s paper doesn’t provide a starting guess, but a good practice is to start with something like as a rough estimate (this scales the fraction of principal paid vs. expected fraction if interest were zero). Financial calculators often start at 10% or use an interpolation based on the ratio . Excel uses 0.1 (10%) by default. If the actual rate is far from the guess, Newton’s method might diverge on the first step (e.g. a guess that yields a negative denominator or overshoot). In such cases, switching to bisection or bounding the step can ensure convergence. The MrExcel forum discussion, for instance, points out that one must solve by iteration (no closed form) and suggests using bisection or Newton–Raphson as the means to get the result. Modern solver implementations often combine these: they bracket the root and then use Newton’s method for fast convergence, which is a good strategy for Equation (16) as well.
It’s also worth noting that Equation (16) is nicely scaled between 0 and 1 on both sides (being a ratio of differences). This tends to keep the values moderate. For example, will always yield a number between 0 and 1 for positive i. The left side is also between 0 and 1. This means during iteration, we aren’t dealing with extremely large or small values; the function is bounded between –1 and 1 initially. This mitigates numerical extremes. Contrast this with solving a high-degree polynomial expanded out – coefficients might be large, causing intermediate blow-ups or root-finding difficulties. Whissel’s arranged form avoids that, which is a positive from a numerical stability perspective. In fact, it’s often recommended to solve such equations in their rational form rather than expanding into a raw polynomial, to avoid coefficient pathologies. Whissel’s form is suitable for stable computation.
In conclusion, the numerical aspects of Whissel’s equation are well-behaved. The need for iteration is inherent and not a drawback of his derivation but a fact of the problem. With standard root-finding techniques, one can solve (16) accurately and efficiently. The method shares the same stability considerations as any IRR or interest rate calculation. In practical spreadsheet or software implementation, one would either use a built-in solver or implement a reliable algorithm (e.g. bracket and then Newton). These are well-understood techniques, and Whissel’s formula poses no novel difficulties in this regard.
While Whissel’s formulation is correct and functional, we can discuss a few improvements or alternatives in the context of solving for i and x:
Direct Use of Financial Functions: The most straightforward “alternative” is not algebraic but using existing tools. As mentioned, Excel’s RATE function or financial calculator routines are optimized for this exact problem. They often implement a combination of Newton’s method and bisection with good initial estimates. In a spreadsheet model, one could simply use =RATE(N, -x, P, R) (if x were known) or use a Goal Seek to adjust i until the remaining balance formula matches R. These approaches don’t change the mathematics, but they are highly practical. If implementing this in software, one could borrow algorithms from open-source libraries or references. For example, the LibreOffice documentation explicitly states: “RATE uses the Newton–Raphson method for root finding (there is no general theoretical alternative to this iterative approach)”. Ensuring that the implementation can handle edge cases (very low or high rates, or non-convergence with a bad guess) is an improvement over a naive implementation of Whissel’s equation. In summary, leveraging tried-and-true financial solver functions is an improvement in terms of usability and reliability, even though it doesn’t yield a different analytical result.
Bisection Method for Robustness: If one is writing a custom solution (say in a Python script or custom calculator), using a bisection algorithm on could be a highly robust alternative. Bisection will always converge to a root as long as you can find an interval where changes sign (which we can, since is typically negative and is positive in the usual case of ). Bisection doesn’t diverge and doesn’t require a derivative. Its convergence is slower than Newton’s, but given the simplicity of the function evaluation, it’s still very fast on modern hardware. This method might be preferable in a spreadsheet if one were to implement it via a simple macro, because it does not require carefully chosen initial guesses and is insensitive to function slope. The trade-off is that Newton’s method (or Excel’s built-in approach) will find the answer in a handful of iterations, whereas pure bisection might take more iterations to reach the same precision. However, since even 50 iterations is inconsequential computationally, robustness might be worth that cost. This is essentially an implementation detail – using bisection vs. Newton doesn’t change the equation being solved, but it improves reliability.
Hybrid or Secant Methods: Many implementations use a secant method (which approximates the derivative by finite difference) or a hybrid (for example, Brent’s method which combines bisection and secant). These could be considered improvements if one is concerned about ensuring convergence. Excel’s solver likely uses a secant variant. In academic literature, algorithms like Brent’s are recommended for solving financial equations since they guarantee convergence and are reasonably fast. Adopting such an algorithm in any custom implementation would make Whissel’s equation solution more robust without any significant performance penalty.
Alternate Algebraic Forms: One might attempt to rearrange Equation (16) into forms that could be more convenient for certain cases. For instance, Whissel’s equation can be manipulated into a polynomial in or . One alternative formulation comes from writing the remaining balance in present value terms: since , one could derive a polynomial by clearing denominators. Another approach is to set . Then . Equation (16) becomes . Cross-multiplying gives . Multiply through by (to avoid fractional exponents): . This simplifies to . Now substitute back : . Notably, this is just the original equation again (it’s essentially a tautology). The point here is that there’s no obvious algebraic simplification that avoids the iterative solution for i. The form we have is already about as simple as it gets. However, one might consider solving for instead of i to avoid dealing with the singularity at . Let . Then equation (16) becomes . This is a polynomial equation in : . Expanding, . Rearranging, This is a polynomial of degree N in the variable . For example, if and , it’s a 360th-degree polynomial (with most intermediate coefficients zero except the and terms). There is no formula to solve that explicitly for general N. Using this polynomial form for numerical solution is possible (one could apply a polynomial root-finding algorithm), but it’s not necessarily more efficient or stable than dealing with the rational form directly. In fact, directly finding roots of a high-degree polynomial can be numerically unstable, whereas using the structured form of (16) is stable. Thus, while this is an “alternative formulation,” it’s not an improvement for computation – it’s essentially the same challenge in a different guise.
Lambert W function: Occasionally, when faced with exponentials, one might consider if the Lambert W could solve it. Lambert W solves equations of the form . Our equation in i (or y) is not reducible to a single exponential times variable; it’s a sum of two exponentials. For certain special cases (e.g., if r or N-r were very small), one could solve explicitly. For instance, if , the equation becomes . With some manipulation, that can reduce to a simpler form: . Cross-multiplying and simplifying leads to a quadratic in in that particular case, which can be solved. But for general r, no closed-form emerges. So Lambert W doesn’t really help except in trivial cases. An “improvement” here is simply recognizing that a closed-form solution would require solving a transcendental equation that doesn’t fit known closed-form inverses. So focusing on robust numeric solution is the sensible path.
Initial guess improvements: While not a different formula, one practical enhancement is to choose a good initial guess for i when using iterative methods. A possible heuristic: compute the interest that would be implied if the loan were interest-only for r periods and then suddenly paid down to R. For example, one could guess . This guess comes from the idea that if no principal were paid at all in r periods, . This overestimates the interest because in reality some principal is paid, but it provides an upper bound. Another guess could be based on linear amortization: and as above provide a bracket. A mid value could be a reasonable start for Newton. Excel’s fixed 0.1 works decently for many typical loans, but in an automated solver one could tailor the guess using such formulas to possibly reduce iterations. This is a minor computational improvement.
Solving for x without i: In some scenarios, one might wonder if we can determine the payment x directly from given P, R, N, r without first finding i. Generally, i is needed because x depends on i. However, if one only needs x (and not i explicitly), one could attempt a two-dimensional root find: solve the system of two equations (payment formula and remaining balance formula) in two unknowns simultaneously. This is essentially what Whissel did analytically by eliminating x. Alternatively, one could eliminate i instead: for example, from the payment formula express , substitute into the balance formula, and try to solve for x. That turns out to be just as hard, since i reappears inside . So there is no direct closed form for x either unless i is known. Thus, solving for i first (via (16)) and then computing x = \frac{iP}{1 - (1+i)^{-N}} is the sensible approach. This is exactly what Whissel recommends: find i with (16), then plug into Eq. 6* or 14 for x. There’s no simpler method to get x given R other than going through i. Any improvement here would be algorithmic (e.g., one could use a multi-dimensional solver like Newton’s method in 2D, but that’s overkill when one equation in one unknown suffices).
Generality: One could extend Whissel’s approach to more general situations as an improvement. For example, if a loan had a balloon payment at the end or irregular payments, one could derive analogous equations. Those would also likely require numeric solutions. An improvement in generality would be to set up a framework where any four of the five variables can be solved for given the other four. Financial calculators do this: for instance, if i is known and you want to find R after r payments, you plug values into the remaining balance formula directly. If R is known and i known, find how many payments r have been made (this would also require solving a logarithmic equation for r, which actually can be solved explicitly with log because r appears only in exponents, not both inside and outside). The hardest unknown to solve analytically is always i, because it appears non-linearly. In that sense, Whissel’s equation (16) is the canonical hard case. It doesn’t simplify further, so the improvement is in solving techniques as discussed. For completeness, one might note that if x were known instead of R, solving for i would involve a similar equation derived from . That leads to , which again is solved by iteration (Excel’s RATE essentially solves that when fv=0). Whissel’s case is a bit more general (nonzero R), but the difficulty is the same. Thus, our discussion of methods applies broadly: there’s no closed form for interest in an annuity equation except via numeric methods or, rarely, special-case simplifications.
In summary, suggested improvements revolve around how to solve Whissel’s equation rather than deriving a fundamentally different equation. The derivation itself is sound and minimal. To recap actionable improvements:
Use reliable numerical algorithms (Newton with a fallback to bisection) to solve Equation (16) for i. This ensures convergence and accuracy in all cases.
Provide a good initial guess or bracket for i to speed up convergence (e.g., bracket i=0 and a high i that would make nearly 1, then bisect).
Once i is found, compute x using the stable formula rather than Whissel’s rearranged Eq. 14 (they are algebraically identical, but the former is the standard implementation).
If implementing in a spreadsheet without the RATE function, consider using a VBA macro to implement the iteration, since doing it by hand can be tedious. Alternatively, use Excel’s Solver/Goal Seek on either Whissel’s equation or directly on the remaining balance formula. This is a practical improvement for users who may not be comfortable coding their own root-finding.
Acknowledge edge cases: if is extremely small (loan hardly paid down) then i will be very high – ensure the solver can handle high-interest guesses; if is exactly 0 (loan fully paid) then the formula simplifies and one could directly use the standard full-term formulas (this is essentially the case of solving , which is the usual full payoff scenario).
For teaching or communication, one might present Equation (16) in the more intuitive form . This clearly shows “fraction of principal paid = fraction of annuity factor used.” This is just a rearrangement of Whissel’s (16) but can help users understand the meaning. Sometimes understanding the formula helps in choosing better guesses for solving it.
Overall, Whissel’s method is already optimized in form for the problem it addresses. The primary “improvement” one can suggest is to delegate the solving to robust numerical routines or to follow best practices in implementing those routines. The problem of finding interest rates given partial amortization data is a classic one with no closed-form answer, so all improvements are about making the numerical solution more user-friendly and fail-safe. In an academic sense, one could also cite that this equation leads to a polynomial of high degree which “requires numerical methods”, as multiple sources confirm. This theoretical acknowledgment supports why our focus is on computational approach improvements.
In conclusion, Bret Whissel’s derivation leading to Equation (16) is mathematically sound, aligns with standard financial formulas, and correctly encapsulates the relationship needed to solve for the interest rate. The equation is essentially a reformulation of the amortization problem that any standard method (Excel’s RATE, financial calculators, IRR calculations) would also have to solve. There are no evident errors in his algebraic manipulation; the use of exponential terms is appropriate and unavoidable in this context. The need to resort to numerical methods is expected and is explicitly noted by Whissel, reflecting the consensus that such equations “cannot be solved algebraically” in general.
When comparing to standard tools, Whissel’s method yields the same results and requires the same computational effort – it is as practical and accurate as the widely used approaches. It does not magically circumvent the iterative nature of the problem (nor does it claim to). Thus, its value is in providing a clear derivation and a single equation that one can plug into a solver. Standard calculators and spreadsheet functions have the advantage of convenience and built-in algorithms (e.g., using Newton–Raphson, which Excel’s RATE does), but fundamentally they are solving the same transcendental equation.
In terms of computational implementation, the equation is stable and well-behaved for numeric solving. Potential numerical issues (small interest rates causing subtractive cancellation, large exponents, etc.) can be managed with careful programming or simply using high precision if needed. The monotonic nature of the function ensures that simple methods like bisection will find the root reliably. Newton’s method can be applied for fast convergence, with caution around initial guesses and extreme values.
Potential improvements lie in using robust numerical algorithms and leveraging known functions to carry out the solution. These include using bracketing methods to guarantee convergence and employing hybrid root-finding routines for efficiency and stability. Additionally, presenting the formula in intuitive terms or providing a reasonable initial guess can be helpful practical enhancements. Another “alternative” approach is to frame the problem as an IRR calculation of a cash flow series, which is exactly equivalent to Whissel’s equation and would be solved by the IRR functions available in many software packages.
Finally, it’s worth emphasizing that Whissel’s approach is not only sound but also comprehensive within its scope – it addresses solving for both unknowns (i and x) given the other parameters. This dual solution capability is useful, as many textbook treatments only show how to compute x given i, and leave the inverse problem (finding i) to numerical techniques without deriving an explicit equation. Whissel filled that gap by showing the equation you need to solve. In doing so, he remained consistent with accepted financial theory and computational practices. Any practitioner or student using Equation (16) can trust its correctness and focus on applying appropriate computational tools to obtain the interest rate. In summary, Whissel’s derivation stands up to scrutiny, and while it doesn’t eliminate the hard work of iteration (which no analytical method can), it provides a clear roadmap to find the solution, fully in line with standard amortization theory and methods.
References:
Whissel, B. D., Solution for Interest Rate and Payment Amount, equation derivations.
Standard annuity formula for present value and payment.
LibreTexts Business Math: on impossibility of algebraically solving for annuity interest rate.
LibreOffice Calc Documentation: Excel/Calc RATE uses Newton’s method (iterative).
Studocu/Academic QA: interest rate solution requires numerical methods or financial calculator (Excel RATE).
MrExcel Forum discussion: solving RATE leads to high-degree polynomial, no general formula, use Newton or bisection.
Mathematics Stack Exchange: example of deriving a quintic for interest rate, confirming need for numerical root-finding.
FinanceFormulas.net: notes on remaining balance formula applicability (fixed payment loans only).