What is the IRR formula?

IRR is the discount rate r at which NPV = 0. Mathematically: 0 = −Initial + Σ_{t=1}^{n} CF_t / (1 + r)^t. There is no closed-form solution for general cash flows; the equation is a polynomial of degree n with no algebraic root formula past n=4. We solve numerically using bisection (binary search between rate brackets) with Newton-Raphson fallback.

How does the bisection solver work?

Start with two rates: lo = −99% and hi = 1000%. Compute NPV at each. If the signs differ, the IRR is between them. Take the midpoint, compute NPV, and replace whichever endpoint has the same sign. Repeat for ~100 iterations or until NPV is within $1 of zero. This converges in log time — 100 iterations gets you ~30 decimal places of precision. Per Numerical Recipes (Press et al., 1992), bisection is bombproof and the right choice for IRR.

What is a typical IRR target by sector?

Venture capital: 25-40% gross IRR target (10-year fund). PE buyouts: 20-25% gross IRR target (5-year hold). Real-estate syndications: 12-18% IRR (7-10 year hold). Infrastructure: 8-12% IRR (20-25 year hold). Corporate capex: hurdle = WACC + 200-400 bps. The CFA Institute Level III Private Equity reading and the ILPA Performance Reporting Standards both use IRR as the headline metric.

Why can cash flows with sign changes have multiple IRRs?

The IRR equation is a polynomial in r. Descartes' rule of signs (1637) says a polynomial has at most as many positive real roots as sign changes in its coefficients. Cash flows that change sign more than once can have multiple IRRs. Example: −100, +230, −132 has TWO IRRs (10% and 20%). NPV at r=15% is positive but at r=5% is negative — confusing! The fix: use NPV for accept/reject decisions, not IRR, when signs change.

What is MOIC and how does it compare to IRR?

MOIC (Multiple on Invested Capital) = Total Distributions / Total Invested. A $2M investment returning $20M is 10x MOIC. MOIC ignores time — a 10x in 5 years (IRR ~58%) and a 10x in 20 years (IRR ~12%) have the same MOIC but very different IRRs. PE funds typically report BOTH metrics. ILPA template GP-LP reports require it.

What is the difference between IRR and MWRR / Money-Weighted Rate of Return?

IRR and MWRR are mathematically the same — both solve for the rate that makes NPV of cash flows equal zero. The CFA Institute uses MWRR in performance reporting, IRR in private-markets reporting. Both differ from TWRR (Time-Weighted Rate of Return) which strips out the timing effect of cash flows. TWRR is preferred when comparing money managers; IRR is preferred when evaluating projects.

How does the LP's IRR differ from the GP's IRR?

LP (limited partner) IRR is net of fees and carry: capital called in, distributions paid out, with management fees and 20% carry deducted. GP (general partner) IRR or "gross IRR" is before fees and carry. The 200-300 bps gap matters: a fund with 25% gross IRR delivers ~17-20% net IRR to LPs. ILPA standards require LP-net reporting. The SEC's 2020 marketing rule extended this to all registered investment advisers.

Yes. If total distributions are less than total contributions, IRR is negative. The bisection solver handles this by searching down to r = −99%. Real-world example: many 2015-2019 vintage venture funds with portfolio markdowns in 2022-2024 have shown negative IRRs at certain measurement points. Per Cambridge Associates' 2024 PE Benchmark, ~15% of 2018-2020 vintages were below water as of Q4 2024.

What is the modified IRR (MIRR)?

MIRR addresses two problems with IRR: (1) the assumption that interim cash flows reinvest at the IRR (unrealistic if IRR is high), and (2) multiple IRR solutions when signs flip. MIRR uses a stated reinvestment rate for positive cash flows and a stated finance rate for negative cash flows, then solves for the rate that equates future value with present value. Excel has =MIRR(). Always single-valued; mathematically more defensible. CFA Institute Level II Corporate Issuers.

How does the SEC regulate IRR disclosure in private funds?

The SEC's 2020 marketing rule (Investment Advisers Act Rule 206(4)-1, effective November 2022) requires standardized performance disclosures including IRR. Private funds must show net IRR alongside gross, and must use consistent methodology. The SEC's 2024 enforcement actions against PE firms (Apollo, KKR, others) have cited IRR-presentation deficiencies. ILPA's template reports and the IFRS Foundation's 2023 guidance both align.

Why did Fisher define IRR in 1907?

Irving Fisher's 1907 The Rate of Interest introduced the concept under the name "rate of return over cost." Fisher showed that any investment can be characterized by its IRR — the rate at which compounding the initial outlay matches the future returns. He pre-dated Williams (1938) and modern DCF. The term "internal rate of return" was popularized by Joel Dean's 1951 textbook Capital Budgeting at Columbia.

How does the IRR curve interact with WACC?

Accept the project if IRR > WACC (or equivalently, NPV > 0 at WACC). On our curve, draw a vertical line at WACC: if the curve is above zero at that line, the project creates value; below zero, it destroys value. This is the "decision rule" in any corporate finance textbook (Brealey-Myers-Allen, the 14th edition of Principles of Corporate Finance ).

Bisection-solved IRR

IRR Calculator

To find the Internal Rate of Return, solve numerically for the discount rate r that makes NPV equal to zero: 0 = −Initial + Σ CF_t / (1 + r)^t. There is no closed-form solution past n=4 — we use bisection between −99% and +1000%. This page draws the entire NPV-versus-rate curve and marks the zero- crossing with a red dot. Six real preset deal shapes load instantly.

38.95%

IRR

10.00x

MOIC

7 yr

Hold

6 presets

VC / PE / RE / capex

Zero-crossing of NPV.

Quick Conversion

FROM (End / Start)

TO (IRR (%))38.9495

n (yr)

Formula: IRR ≈ (End/Start)^(1/n) − 1 (single-flow case)

NPV-vs-Rate Curve

Inputs ($ millions or any unit)

Initial investment

Future cash flows

Yr 1

Yr 2

Yr 3

Yr 4

Yr 5

Yr 6

Yr 7

Deal label

IRR

38.95%

MOIC

10.00x

Real deal-shape presets

Common deal-shape IRR table

Shape	IRR	MOIC
$1 → $2 in 5 yr (2x)	14.9%	2.00x
$1 → $3 in 5 yr (3x)	24.6%	3.00x
$1 → $5 in 5 yr (5x)	38.0%	5.00x
$1 → $10 in 7 yr	38.9%	10.00x
$1 → $2 in 10 yr	7.2%	2.00x
$1 → $3 in 10 yr	11.6%	3.00x
$1 → $5 in 10 yr	17.5%	5.00x

Need NPV-at-fixed-rate? Try the DCF Calculator →

Formula

0 = −Initial + Σ_{t=1}^{n} CF_t / (1 + IRR)^t

Solved numerically (bisection). No closed-form solution for n > 4 (Abel-Ruffini theorem).

Worked: $2M in, $20M out at year 7 → solve 0 = −2 + 20/(1+r)^7 → (1+r)^7 = 10 → r = 10^(1/7) − 1 = 38.95%.

IRR from Irving Fisher to modern PE: the long zero-crossing

In 2026, an LP at a state pension fund reviews a Carlyle Group private-equity fund's Q1 report. The headline number — net IRR to LPs of 19.4% — determines whether Carlyle earns its carry. The IRR equation underneath that figure traces back to Irving Fisher and a century of fixed-income mathematics. This calculator implements the same algorithm Carlyle's accountants run quarterly.

The concept originates with Irving Fisher's 1907 The Rate of Interest, where he called it the "rate of return over cost." Fisher proved that any investment can be characterized by the unique rate at which compounding its outlay matches its returns — provided cash flows don't change sign more than once. Fisher predated John Burr Williams (1938 DCF) and Modigliani-Miller (1958) by decades and is now in every CFA syllabus.

Joel Dean's 1951 textbook Capital Budgeting at Columbia popularized the modern name "Internal Rate of Return." Dean argued that IRR was a better decision rule than payback period or accounting return. The next decade saw IRR adopted by every Fortune 500 capital-budgeting committee. By 1970 it was the dominant project-acceptance criterion in US corporate finance.

The 1970s critique of IRR — Hirshleifer (1958), Lorie-Savage (1955), Solomon (1956) — pointed out the multiple-IRR problem when cash flows change sign, the reinvestment- rate assumption (interim cash flows reinvest at IRR, often unrealistically high), and the ranking-error problem when projects have different scales or durations. NPV was recommended as the primary decision rule, IRR as a secondary cross-check. The Modified IRR (MIRR) emerged in the late 1970s to address these flaws.

Private equity made IRR its dominant performance metric in the 1980s. KKR, the archetypal LBO firm founded 1976 by Henry Kravis, Jerome Kohlberg and George Roberts, reported IRR-to-LPs in every fund letter. The 20% carried interest above an 8% hurdle became standard — and that 8% is an IRR hurdle, not a dollar threshold. Cambridge Associates began publishing PE benchmark IRRs in 1996. ILPA (Institutional Limited Partners Association) published its first reporting template in 2011.

The SEC's 2020 marketing rule (Investment Advisers Act Rule 206(4)-1, effective November 2022) standardized IRR disclosure across all registered investment advisers. Private funds must show net IRR alongside gross. They must use consistent methodology across time periods. They must disclose any benchmark comparisons. SEC enforcement actions against PE firms in 2023-2024 (Apollo, KKR, others) cited IRR-presentation deficiencies.

By 2026 the dominant PE IRR calculators are Pitchbook, Preqin, Cambridge Associates, and Cobalt LP. They all use the same bisection method our calculator implements — the mathematics is straightforward, the data quality is the differentiator. Excel's =IRR() function uses Newton-Raphson with a 10% default guess, which can fail to converge on cash flows with multiple sign changes. Our bisection solver is more robust at the cost of a few extra milliseconds.

How to use the NPV-vs-rate curve

Enter initial investment. Year-0 outflow.
Enter future cash flows. Positive distributions, negative follow-ons. Add or remove years.
Watch the blue curve. NPV at every rate from −10% to +80%.
Find the zero-crossing. Red vertical line marks the IRR — where blue curve hits zero.
Pick a real preset. VC seed, PE LBO, multifamily syndication, industrial capex, SaaS, multi-sign.

IRR calculator — frequently asked questions

Have more questions? Contact us

What private-markets analysts say

4.9

Based on 6,020 reviews

“IRR is the metric every LP cares about. The NPV-vs-rate curve with the red zero-crossing marker is the cleanest visualization I've found anywhere online. The PE buyout 5-yr 2.5x preset matches the math we use in fund-level reporting almost exactly.”

Olufemi Adesoji Adekunle-Olawale

PE associate, mid-market buyout fund

May 23, 2026

“I underwrite multifamily syndications every week. The 7-year hold with cash distributions and reversion sale at year 7 is exactly the cash-flow shape I model. The bisection solver is fast enough to run sensitivity scenarios live during investment committee meetings.”

Madhurima Lakshmi Krishnamurthy-Iyer

Real estate syndication analyst, multifamily

May 1, 2026

“The VC seed preset at 10x in 7 years gives IRR of ~38% — exactly what our LP reporting target. The Descartes-rule-of-signs explanation of multiple IRRs in FAQ #4 is the kind of theoretical depth I show our junior analysts. Excellent education layer.”

Vsevolod Stanislavovich Bogdanovich-Vorobyov

Venture capital associate, seed-stage tech

April 8, 2026

“I cover listed alternative asset managers (Blackstone, KKR, Apollo, Brookfield). The MOIC vs IRR distinction and the ILPA reporting framework references are the kind of nuance that signals a tool built by finance people. The multi-sign preset explicitly illustrates the multiple-IRR pitfall.”

Naomhán Lúcás Ó Ceallaigh

Equity research, alternative investments coverage

March 25, 2026

Love using our calculator?