What is the missing TVM value?

How the TVM calculator works

The five values. N is the number of payment periods, I/Y the nominal yearly interest rate in percent, PV the amount at the start, PMT the payment each period and FV the amount after the last period. P/Y is the number of payments a year and C/Y the number of times a year interest compounds (whole numbers from 1 to 365). Compute picks the value to work out; the other four are typed. An empty PV, PMT or FV counts as 0; N and I/Y must be typed when they are not computed.

Signs. Money you receive is positive and money you pay is negative, so a deposit is negative and the balance you get back is positive.

Rate per period. i = (1 + I/Y ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1, as a decimal.

The equation. With t = 0 for payments at the end of each period and t = 1 for payments at the start:

PV × (1 + i)^N + PMT × (1 + i × t) × ((1 + i)^N − 1) ÷ i + FV = 0

At i = 0 it is PV + PMT × N + FV = 0.

Each value, with g = (1 + i)^N and a = (1 + i × t) × (g − 1) ÷ i (a = N at i = 0):

• FV = −(PV × g + PMT × a)
• PV = −(FV + PMT × a) ÷ g
• PMT = −(PV × g + FV) ÷ a
• N = ln(g) ÷ ln(1 + i) with g = (PMT × (1 + i × t) − FV × i) ÷ (PMT × (1 + i × t) + PV × i); at i = 0, N = −(PV + FV) ÷ PMT.
• I/Y: every yearly rate above −100% and at most 1,000% where the left side of the equation changes sign, each refined by bisection. The search checks every 0.01% from −20% to 50%, every 0.1% from −99.9% to −20% and from 50% to 1,000%, and −99.95%, −99.99%, −99.999%, −99.9999%, −99.99999% and −99.999999%; two rates closer together than those steps can be missed. When two rates fit (the cash flows change sign twice), both are shown.

Other results. Rate per period = 100 × i. Effective yearly rate = 100 × ((1 + i)^(P/Y) − 1). Total of payments = N × PMT. Net interest = PV + N × PMT + FV (positive: interest earned; negative: interest paid).

Precision and overflow. The rate search reads the sign of the equation from the form above when (1 + i)^N is at most 1, and from the same equation divided by (1 + i)^N when it is above 1, so no step overflows; a rate at which (1 + i)^N is not a finite number above 0 in 64-bit floating point is dropped. With PV, PMT and FV all 0 (or empty) the equation holds at every rate, so there is no rate. The other computed values use the same divided equation when (1 + i)^N is above 1, so an answer inside the limits never fails on a larger number on the way. The total of payments and the net interest are exact decimal arithmetic on the typed amounts and the computed value, each read as the decimal it prints as, rounded once at the end. The computed value, the rates and the powers are 64-bit floating point, so a value that is exactly half a cent in decimals can land just below it and show rounded down.

Limits and no answer. N is above 0 and at most 12,000; I/Y is above −100% and at most 1,000%; PV, PMT and FV are within ±$1 trillion. A computed value outside those limits is no answer. So is a (1 + i)^N that is not a finite number above 0 in 64-bit floating point, N at 0% with no payment, N when g ≤ 0 or g = 1, and I/Y when no rate fits.

Display. Money shows to the cent, rounded half up from the value worked out; N at most 2 decimals; I/Y and the effective rate at most 4; the rate per period at most 6.

Assumptions

• The rate is the same in every period and every payment is the same size.
• No fees or taxes are included. This is an estimate for planning, not financial advice.

Worked examples by hand

Savings plan (compute FV). N = 120, I/Y = 7, PV = −10,000, PMT = −500, P/Y = C/Y = 12, END. i = 0.07 ÷ 12 = 0.0058333. g = 1.0058333^120 = 2.009661. a = (g − 1) ÷ i = 173.0848. FV = 10,000 × 2.009661 + 500 × 173.0848 = $106,639.02.

Excel's FV example 2 (compute FV). N = 12, I/Y = 12, PV = 0, PMT = −1,000, monthly, END. i = 0.01; g = 1.01^12 = 1.126825; a = 12.682503. FV = 1,000 × 12.682503 = $12,682.50. The payments total −$12,000, so the net interest is $682.50.

Excel's FV example 3 (compute FV). N = 35, I/Y = 11, PMT = −2,000, monthly, BGN. i = 0.11 ÷ 12 = 0.0091667; g = 1.0091667^35 = 1.376263; a = 1.0091667 × 0.376263 ÷ 0.0091667 = 41.423123. FV = 2,000 × 41.423123 = $82,846.25.

Excel's PV example (compute PV). N = 240, I/Y = 8, PMT = 500, FV = 0, monthly, END. i = 0.0066667; g = 1.0066667^240 = 4.926803; a = 589.0204. PV = −500 × 589.0204 ÷ 4.926803 = −$59,777.15: to receive $500 a month for 20 years you pay $59,777.15 today.

Excel's PMT example 3 (compute PMT). N = 216, I/Y = 6, PV = 0, FV = 50,000, monthly, END. i = 0.005; g = 1.005^216 = 2.936766; a = 387.3532. PMT = −50,000 ÷ 387.3532 = −$129.08 a month.

No interest (compute FV). N = 10, I/Y = 0, PV = −1,000, PMT = −100. FV = −(PV + PMT × N) = 1,000 + 1,000 = $2,000, with no interest.