Actuarial Mathematics for Modelling · Equation of value

Solving for Unknown Interest Rate and Payments

Updated 11 October 2026 · Fact-checked

Solving for an unknown rate or payment means writing the equation of value, with present value of money in equal to present value of money out at a chosen date, then solving for the unknown. Payments are found by direct algebra. Rates usually need linear interpolation or iteration, such as Newton-Raphson.

Understand Solving for Unknown Interest Rate and Payments

The equation of value says that, at a chosen date and a chosen interest rate, the value of all money paid in equals the value of all money paid out. Every question in this topic starts from that one statement.

There is always one unknown. It can be a payment, a time, or the interest rate. If the unknown is a payment, the equation is linear in that payment. You collect terms and divide. This is the easy case.

If the unknown is the interest rate, the equation is a polynomial in v = 1 ÷ (1 + i). Apart from simple cases, it has no neat formula. The rate that makes the equation balance is called the yield, or the internal rate of return (IRR). You find it by trial: compute the value at two rates, then use linear interpolation to estimate the rate that gives the target. Where more accuracy is needed, you repeat the process or use Newton-Raphson iteration.

If the unknown is a time, you usually solve an annuity equation for n using logarithms. Where the time is not a whole number, state your assumption clearly.

The choice of comparison date does not change the answer when the rate is fixed. Pick the date that gives the simplest algebra, often the date of the unknown payment. For a yield, the answer is the same whatever date you pick, but some dates make the working shorter.

Key rules to remember

Equation of value: Σ (payments in) × v^t = Σ (payments out) × v^t, where v = 1 ÷ (1 + i); Use any single comparison date, but value every cash flow at that same date and at the same rate.
Level annuity-immediate: a_n = (1 − v^n) ÷ i; Payments of 1 at the end of each period for n periods. For payment X, the present value is X × a_n.
Linear interpolation for the rate: i ≈ i1 + (i2 − i1) × (T − f(i1)) ÷ (f(i2) − f(i1)); f(i) is the value at rate i and T is the target value. Choose i1 and i2 close together with T between f(i1) and f(i2).
Newton-Raphson iteration: i_(k+1) = i_k − f(i_k) ÷ f′(i_k); Here f(i) is the equation written as f(i) = 0. Repeat until two successive values agree to the required accuracy.
Unknown term of a level annuity: n = −ln(1 − i × A ÷ P) ÷ ln(1 + i); From A = P × a_n, where A is the present value, P the payment and i the rate per period. Check the answer is sensible.

How to solve Solving for Unknown Interest Rate and Payments questions

Use this method for any question that asks for an unknown rate, payment or time.

1Draw a timeline. Mark every payment in and out, with its amount and its time. Mark the unknown.
2Choose a comparison date. Pick the one that makes the algebra simplest, usually the time of the unknown payment or time zero.
3Write the equation of value: present value of money in = present value of money out, using the rate and date you chose.
4If the unknown is a payment, collect its terms on one side, then divide to find it.
5If the unknown is a time, isolate the annuity or power term, then take logarithms.
6If the unknown is a rate, define f(i) as the value of inflows minus the value of outflows. Try two rates that give values of opposite sign, or that straddle the target.
7Apply linear interpolation to find the rate. Then check it by substituting back, or repeat the interpolation with closer rates.
8State the answer with units, such as per annum effective, and round only at the end.

Quickest way: Bracket and interpolate

When to use it: Use this when you need a yield and have a calculator but no solver, as in most written questions.

Rearrange so that one side is a known number and the other is a function of i, for example a_2 = 5 ÷ 3.
Make a rough guess using total return over time, then test it.
Test one rate above and one below. Keep them 1% apart so interpolation is accurate.
Interpolate using the formula, and give the answer to the accuracy asked.
If the question says to use the interpolation, do not iterate further. Give the interpolated answer.

Common mistakes in Solving for Unknown Interest Rate and Payments

Valuing different cash flows at different dates
Students discount some payments to time 0 and others to the date of the unknown, without a plan.
Fix: Fix one comparison date before you write anything. Apply that date to every term.
Interpolating with values that do not straddle the target
Students pick two trial rates quickly and do not check the signs.
Fix: Check that the target lies between f(i1) and f(i2). If not, you are extrapolating, so try a better pair of rates.
Mixing up the direction of the sign in f(i)
The value of inflows and outflows is subtracted in the wrong order, or the sign changes partway through.
Fix: Define f(i) once, as inflows minus outflows. Keep the same definition for both trial rates.
Using a nominal rate as the effective rate
The problem gives i(12) or a per-month rate, and the annuity formula is applied with the wrong period.
Fix: Convert to the effective rate per payment period before using a_n, or solve in the payment period and convert at the end.
Rounding trial values too early
Rounding v or a_n to 2 or 3 decimals changes the interpolated rate noticeably.
Fix: Keep at least 5 or 6 significant figures during the working. Round only the final answer.
Counting the wrong number of payments or wrong timing
Annuity-immediate and annuity-due are confused, or a deferred first payment is mistimed.
Fix: Write the times of the first and last payment on the timeline. Then choose a_n, ä_n or a deferred form to match.

Worked examples

Example 1

An investor pays ₹1,00,000 now and receives ₹60,000 at the end of each of the next 2 years. Find the effective annual yield, using linear interpolation between 13% and 14%.

Show the solution

Equation of value at time 0: 1,00,000 = 60,000 × (v + v²).
Divide by 60,000: v + v² = 5 ÷ 3 = 1.666667. Define f(i) = v + v².
At i = 13%: v = 0.884956, v² = 0.783147, so f(13%) = 1.668103.
At i = 14%: v = 0.877193, v² = 0.769468, so f(14%) = 1.646661.
The target 1.666667 lies between 1.668103 and 1.646661, so interpolation is valid.
i ≈ 13% + 1% × (1.668103 − 1.666667) ÷ (1.668103 − 1.646661) = 13% + 1% × 0.001436 ÷ 0.021442.
0.001436 ÷ 0.021442 = 0.0670, so i ≈ 13.07%.
Check: the exact root of v² + v − 1.666667 = 0 gives v = 0.88444, so i = 13.07%.

Answer: The yield is about 13.07% per annum effective.

Example 2

A loan of ₹5,00,000 is repaid by 5 equal payments at the end of each year. The effective annual rate of interest is 8%. Find the annual payment.

Show the solution

Let the payment be X. Compare at time 0.
Equation of value: 5,00,000 = X × a_5 at 8%.
v^5 = 1.08^(−5) = 0.680583, since 1.08^5 = 1.469328.
a_5 = (1 − 0.680583) ÷ 0.08 = 0.319417 ÷ 0.08 = 3.992710.
X = 5,00,000 ÷ 3.992710 = 1,25,228 to the nearest rupee.

Answer: The annual payment is about ₹1,25,228.

Exam tips

Always write the equation of value in words or symbols before you substitute numbers. Marks are given for the setup.
For IRR questions, state f(i), the trial rates, the values, and the interpolation formula. Examiners look for each.
In computer-based papers, you can use a solver or a function such as an IRR routine. Still state the equation and the assumptions, such as annual cash flows.
Check your answer by substituting it back into the equation. This takes less than a minute and often catches a slip.
In multiple-choice questions, estimate the yield first. Wrong options are often near values from common slips such as the wrong period.

Practice questions from Equation of value

Solving for Unknown Interest Rate and Payments: frequently asked questions

How do I find the yield from an equation of value?

Write the equation so that the value of inflows minus outflows is zero, then solve for i. Try two rates, one above and one below the root. Interpolate linearly, and check by substituting the result back.

Is the internal rate of return the same as the yield?

Yes. The IRR is the rate of interest at which the net present value of the cash flows is zero. It is the same as the yield from the equation of value.

How accurate is linear interpolation?

It is accurate when the two trial rates are close and the function is smooth. The wider the gap between the rates, the larger the error. Use rates 1% apart for a good estimate.

Can the equation of value have more than one solution for i?

Yes, if the cash flows change sign more than once, there can be more than one positive root. In exam questions the cash flows usually change sign once, so there is one sensible yield.