IRR‑Rechner

This IRR calculator calculates an annualized rate-of-return plus profit (loss).

• Supports dated cash flows
• Easy bulk data entry
• Save entries to a file for later recall
• Know your rate of return across multiple accounts and investments

Answers the question: “How am I doing?”

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The Internal Rate of Return (IRR) is the annualized rate of return on an investment. It is calculated from the amounts and dates of the cash flows. It does not require an externally specified interest rate. For this reason, it is called “internal.” This calculator uses the Newton–Raphson method to compute the IRR.

An Internal Rate of Return (IRR) calculator computes the investment outcome. The results let you compare two or more investment options on a consistent basis.

This calculator determines the IRR for a complex series of cash flows. It also reports the total invested amount, the total returned amount, and the profit (or loss). The calculator supports both irregular time periods and exact date data entry.

The frequency option defines regular cash flows, such as daily, monthly, or quarterly payments. There are 11 frequency choices.

Review the usage tips below (click to scroll). …

The Internal Rate of Return (IRR) converts uneven project cash flows into a single annualized rate of return. It lets investors compare opportunities on a consistent basis. Because the IRR reflects both the amounts and the timing of cash flows, it standardizes results across investments that have different patterns of outlays and receipts.

For example, consider two rental properties for sale. The asking prices are approximately the same, and the projected rents are also approximately the same. One property requires a higher initial renovation cost. The other has higher property taxes. How can an investor determine which purchase is the better investment?

An investor can use an IRR calculator to make this comparison.

Caution: Do not compare internal rates of return calculated with different calculators.

Why is this important?

Two different calculators may compute results slightly differently, and neither calculator is necessarily incorrect. For example, Microsoft Excel includes two IRR functions that may return different results for the same cash flows. Users do not need to focus on this point, but they should be aware of it when interpreting results.

For the record, this calculator determines the IRR by using the Newton–Raphson method and counting days (some calculators instead count periods).

To try a calculator that applies a different IRR algorithm, use this site’s Annual Percentage Rate (APR) calculator. The APR calculator follows the method specified in the Truth-in-Lending Act for calculating APR, which is a form of IRR.

Variablendefinitionen

r

Periodic rate of return. For example, per year when cash flows are annual.

IRR

Nominale annualisierte Rendite, berechnet als IRR = r × f.

f

Frequency (the number of periods per year). For annual spacing, f = 1.

PMT

Cash flow at period index t. By convention, outflows are negative and inflows are positive. Values may differ across periods.

n

Gesamtzahl der Perioden nach t = 0. Die Summation von t = 0 bis t = n umfasst sowohl den anfänglichen Cashflow bei t = 0 als auch den finalen Cashflow bei t = n.

t

Period index. An integer with t = 0, 1, …, n, measured in equal time steps. (The calculator does not require cash flows to be equally spaced.)

How do you calculate the IRR?

To calculate the Internal Rate of Return (IRR), solve for the interest rate that makes the Net Present Value (NPV) of a series of cash flows equal to zero. Because the IRR equation is nonlinear, it is usually solved using an iterative method such as Newton–Raphson.

Detailed Explanation

The IRR equation is nonlinear and cannot be solved algebraically. To find the rate r that makes NPV equal to zero, treat the problem as a root-finding problem. This means solving the following equation:

We want the value of r such that f(r) = 0. This calculator applies the Newton–Raphson method to find this value. The method starts with an initial estimate and refines that estimate using both the value and the slope (derivative) of the function at that point.

The slope is the derivative of f(r), denoted f’(r), which shows how sensitive the NPV is to changes in r. It is computed as:

The Newton–Raphson update formula is:

Each iteration produces a value that is closer to the IRR. This process is illustrated in Fig. 2, which shows the calculation using sample cash flows.

Calculation Steps Explained—Fig. 2.

What is the IRR for the cash flows −50,000 (investment), −10,000, −12,000, +90,000 (returned), with each cash flow one year apart?

Solve for the periodic IRR by setting the Net Present Value (NPV) to zero and defining f(r) as the sum of discounted cash flows and f’(r) as its derivative. Then apply Newton–Raphson updates (Equation (6)) until f(r) converges to zero.

1. Define the NPV function f(r) from Equation (2):
   f(r) = −50,000 − 10,000 ÷ (1 + r)^1 − 12,000 ÷ (1 + r)^2 + 90,000 ÷ (1 + r)^3
2. Apply the general derivative rule (Equation (5)) to compute f’(r):
   f’(r) = 10,000 ÷ (1 + r)^2 + 24,000 ÷ (1 + r)^3 − 270,000 ÷ (1 + r)^4(Each term follows the pattern −t × PMT_t ÷ (1 + r)^(t+1).)
3. Choose an initial guess for the periodic rate: r₀ = 0.10.
4. Compute discount factors at r₀ (first iteration shown in full):
   (1 + r₀) = 1.10 (1 + r₀)^−1 = 1 ÷ 1.10 ≈ 0.90909091 (1 + r₀)^−2 = 1 ÷ (1.10)^2 ≈ 0.82644628 (1 + r₀)^−3 = 1 ÷ (1.10)^3 ≈ 0.75131480 (1 + r₀)^−4 = 1 ÷ (1.10)^4 ≈ 0.68301346
5. Evaluate f(r₀) using Equation (2):
   f(r₀) = −50,000 + [−10,000 × 0.90909091] + [−12,000 × 0.82644628] + [90,000 × 0.75131480] ≈ −50,000 − 9,090.90910 − 9,917.35536 + 67,618.33200 ≈ −1,389.93238167

   Result: f(r₀) ≈ −1,389.93238167

6. Evaluate f’(r₀) using Equation (5) (term by term):
   
   1. t = 1, PMT₁ = −10,000:
      −1 × (−10,000) ÷ (1 + r₀)^2 = +10,000 × (1 + r₀)^−2 ≈ 10,000 × 0.82644628
   2. t = 2, PMT₂ = −12,000:
      −2 × (−12,000) ÷ (1 + r₀)^3 = +24,000 × (1 + r₀)^−3 ≈ 24,000 × 0.75131480
   3. t = 3, PMT₃ = +90,000:
      −3 × (+90,000) ÷ (1 + r₀)^4 = −270,000 × (1 + r₀)^−4 ≈ −270,000 × 0.68301346
   Sum: 10,000×0.82644628 + 24,000×0.75131480 − 270,000×0.68301346 ≈ −158,117.61491701

   Result: f’(r₀) ≈ −158,117.61491701

7. Apply the Newton–Raphson update (Equation (6)):
   r₁ = r₀ − f(r₀) ÷ f’(r₀) = 0.10 − (−1,389.93238167) ÷ (−158,117.61491701) = 0.10 − 0.00879049676 ≈ 0.09120950
8. Discount factors at r₁ (results only):
   (1 + r₁)^−1 ≈ 0.91641431
(1 + r₁)^−2 ≈ 0.83981518
(1 + r₁)^−3 ≈ 0.76961865
(1 + r₁)^−4 ≈ 0.70528954
9. Evaluate at r₁ (results only):
   f(r₁) ≈ 23.75294757
f’(r₁) ≈ −163,559.17595169
10. Update (results only):
    r₂ = r₁ − f(r₁) ÷ f’(r₁) ≈ 0.09135473
11. Discount factors at r₂ (results only):
    (1 + r₂)^−1 ≈ 0.91629236
(1 + r₂)^−2 ≈ 0.83959169
(1 + r₂)^−3 ≈ 0.76931145
(1 + r₂)^−4 ≈ 0.70491420
12. Evaluate at r₂ (results only):
    f(r₂) ≈ 0.00666170
f’(r₂) ≈ −163,467.44351228
13. Update (results only):
    r₃ = r₂ − f(r₂) ÷ f’(r₂) ≈ 0.09135477
14. Discount factors at r₃ (results only):
    (1 + r₃)^−1 ≈ 0.91629233
(1 + r₃)^−2 ≈ 0.83959163
(1 + r₃)^−3 ≈ 0.76931136
(1 + r₃)^−4 ≈ 0.70491410
15. Final convergence (results only):
    f(r₃) ≈ 0.00000000
f’(r₃) ≈ −163,467.41777956
r ≈ r₃ − f(r₃) ÷ f’(r₃) ≈ 0.09135477
16. Annualize using frequency f = 1:
    IRR = r × f ≈ 0.09135477
IRR ≈ 9.135477%

Thus, the periodic IRR is r ≈ 0.09135477, and, with annual spacing (f = 1), the internal rate of return is R ≈ 9.135477%.

Endergebnis

The final answer (IRR) is approximately 9.135%.

Validate the calculator. Three-year internal rate of return calculation.

Calculated result: