IRR Formula and the Reinvestment-Assumption Question: A Practitioner's Guide to IRR, MIRR, and XIRR

Key Takeaways

When Your IRR Is 22% and Treasuries Yield 4%

Most IRR explainers stop after writing out the NPV = 0 identity. This one starts there — and then asks the question every honest investment-committee memo eventually has to answer: when your underwritten IRR is 22% but the risk-free reinvestment rate is 4%, what return should you actually expect?

The gap between those two numbers is not an academic question in 2026. With the 10-year Treasury yielding around 4.20% and money-market funds paying 4.00–4.50%, the spread between IRR and MIRR — the modified IRR that explicitly assumes interim cash flows compound at a chosen reinvestment rate rather than at the project's IRR — is now wide enough on most value-add CRE deals to change how the deal gets presented. A 22% IRR with a 5% risk-free MIRR of 14.5% is a different story than the 22% IRR alone. Both numbers are correct. They answer different questions.

This article covers the IRR formula, the reinvestment-assumption controversy (including the modern academic position from Magni and Martin in The Engineering Economist), MIRR mechanics with a CRE worked example, when XIRR is the right call, and why all of this matters more in 2026 than it has in fifteen years. The audience is institutional CRE practitioners building IC memos, not first-year MBA students — though the formula derivation is here for completeness.

The IRR Identity, From NPV

The IRR is defined as the discount rate that makes the net present value of a series of cash flows equal to zero. For cash flows CF₀, CF₁, ..., CF_n occurring at the end of periods 0 through n, the IRR is the value r that satisfies:

Notice what the formula does not say. It says nothing about what happens to CF₁ after it is received. It says nothing about whether the analyst plans to reinvest interim distributions. It is a mathematical identity about the relationship between a stream of cash flows and a discount rate — nothing more. The practitioner intuition that “IRR assumes reinvestment at the IRR” is a true statement about reconciling IRR to a multi-period compound annual growth rate, but it is not a statement about the IRR formula itself. We return to this distinction below.

Why There's No Closed-Form Solution

For cash flows beyond two periods, the IRR equation is a polynomial in r with no general algebraic solution. A deal with seven years of cash flows produces a seventh-degree polynomial; the Abel-Ruffini theorem guarantees that polynomials of degree five or higher have no closed-form solution in radicals. IRR must be solved numerically.

In practice, every IRR calculation — Excel's IRR() function, Google Sheets, financial calculators — uses iterative root-finding. Excel uses a Newton-Raphson method seeded with an initial guess (you can pass one as the second argument; the default is 0.1, or 10%). For most well-behaved CRE cash flows with one sign change (negative outflow at acquisition, positive inflows thereafter), the iteration converges in a handful of steps to a unique IRR.

Cash flows with multiple sign changes — a deal with a recap or major capex spend mid-hold that briefly turns cumulative cash flow negative again — can produce multiple IRRs. Mathematically, Descartes' rule of signs gives an upper bound: the number of positive real roots equals the number of sign changes (or that number minus an even integer). A four-sign-change cash flow may have one, two, three, or four positive IRRs. Excel's IRR() returns the first one it finds and gives no warning. This is one of the practical arguments for MIRR, which resolves the multiple-root problem by construction.

The Reinvestment-Assumption Controversy

This is the article's intellectual core. Almost every IRR explainer on the open web repeats the same line: IRR assumes that interim cash flows are reinvested at the IRR itself. The Investopedia entry, the Corporate Finance Institute tutorial, Wall Street Prep, and the academic textbooks that trained the analysts writing those pages all repeat this claim, with slight variations.

It is also, strictly, false. Or at least: it requires an asterisk that almost no competitor includes.

The Magni-Martin position

Carlo Alberto Magni and Andrea Marchioni Martin published the modern academic resolution of this question across a series of papers culminating in “The Two Sides of the Reinvestment Assumption Fallacy in IRR and NPV,” The Engineering Economist Vol. 70 No. 3 (2025). Their argument: the IRR formula contains no reinvestment assumption. The formula is an algebraic identity relating a stream of cash flows to a single discount rate. There is no statement — explicit or implicit — about what happens to interim cash flows after they leave the project. Reinvestment is a separate question, answered by a separate model.

When practitioners say “IRR assumes reinvestment at the IRR,” what they actually mean is something more specific: if you want to reconcile the IRR to a single-period CAGR computed against only the original equity outlay and the final disposition value, you have to assume the interim cash flows compounded at the IRR for the arithmetic to work out. That statement is true. But it is a statement about a particular reconciliation exercise — not about the IRR formula itself. The reinvestment assumption is imposed exogenously by the analyst who wants the CAGR reconciliation to hold; the formula does not require it.

Resolving the practitioner intuition

Why does the practitioner intuition persist if it is technically wrong? Because the reconciliation exercise is common enough that the assumption feels load-bearing. An investment-committee member who looks at a deal with a quoted IRR of 17% and asks “so if I invest $10M, I'll have $10M × 1.17⁵ = $21.9M at exit, right?” is implicitly assuming the interim distributions reinvested at the IRR. If they didn't — if the distributions sat in cash at 4% — the analyst will have less than $21.9M at exit, and the realized multi-year CAGR on the original $10M outlay will be lower than 17%.

Both statements are correct. The IRR formula does not require a reinvestment assumption. The reconciliation of IRR to a CAGR on the original capital does. The practitioner intuition is a true statement about the second exercise misattributed to the first. Once you separate the two, the controversy collapses.

MIRR: The Formula

MIRR — the modified internal rate of return — reframes the question. Instead of asking “what discount rate zeros NPV,” MIRR asks: if interim cash inflows compound at one rate (the reinvestment rate) and the initial outflow is financed at another (the finance rate), what single annualized return reconciles the initial investment to the future value of all the proceeds?

Two practical choices determine MIRR: the reinvestment rate and the finance rate. The reinvestment rate is the yield the analyst can realistically earn on distributed cash — typically the risk-free rate (10-year Treasury or money-market yields), the LP's stated opportunity cost, or a weighted average of the two. The finance rate is the cost of capital for the negative cash flows — usually the project's debt cost or the firm's WACC. For most CRE deals with a single negative cash flow at acquisition, the finance rate distinction is academic: the initial outlay is already at time zero, so no PV discounting is needed and the finance rate doesn't change the answer.

By construction, MIRR resolves the multiple-IRR problem. There is only one MIRR for any cash flow stream because the formula collapses all positive cash flows into a single future value and all negative cash flows into a single present value before taking the geometric average. The polynomial that might have multiple roots in IRR has exactly one solution in MIRR.

Worked Example: A 7.5-Point Spread on a Value-Add Multifamily Deal

Consider a stylized value-add multifamily acquisition: $10M of equity at close, $2M/year of distributions during a seven-year hold (front-loaded relative to a stabilizing core deal because the property cash-flows aggressively after renovation), a $3M refi event in year 3 that returns capital to LPs, and an $8M equity disposition in year 7. The cash flow stream:

The IRR — the discount rate that zeros the NPV of the seven-year stream — is approximately 22.0%. The MIRR with a 5% reinvestment rate (representative of money-market or short-Treasury yields in 2026) is approximately 14.5%. The 7.5-point spread is the practical consequence of the reinvestment-assumption distinction: $2M of distributed cash compounding at 5% per year produces less wealth at exit than $2M compounding at 22% per year, and MIRR makes that gap explicit.

In an IC presentation, this deal should be quoted with both numbers. The IRR communicates the project's intrinsic return as a discount rate. The MIRR communicates the realized annualized return the LP should expect if interim distributions are reinvested at the risk-free rate — which they typically are, because most LPs don't have another 22% deal sitting in front of them.

XIRR for Irregular Timing

A short sidebar before continuing: the standard IRR formula assumes cash flows occur at equal intervals (end of each period). Most institutional CRE models work in monthly cash flows, and many models include irregular events — refi closings, mid-quarter capex, sale proceeds on a non-standard date. XIRR — Excel's extended IRR function — takes a date for each cash flow and computes the annualized return assuming actual day-count between flows. In modern institutional practice, “IRR” in a model almost always means XIRR.

XIRR does not change the reinvestment-assumption discussion. It changes the discounting precision. The choice between IRR and XIRR is a choice about cash-flow timing granularity; the choice between IRR and MIRR is a choice about how to handle interim distributions. They are independent decisions.

The Complete Return-Metric Stack

No single return metric tells the whole story of a deal. Institutional underwriting practice quotes a stack of metrics, each answering a different question. A reasonable default for a CRE acquisition:

The right IC-memo lead depends on the LP. Open-end core funds underwrite on income (cash-on-cash and stabilized yield) with IRR as the disposition cross-check. Value-add and opportunistic funds lead with IRR but increasingly cite MIRR alongside in the current rate environment. Family offices and direct investors care more about equity multiple than time-value-weighted IRR because their hold periods are flexible. Fund-level LPs care about TVPI and DPI more than deal-level IRR. Knowing which metric to put first is part of the job.

Why This Matters in 2026

For roughly fifteen years — 2009 through 2022 — the U.S. risk-free rate was near zero. Money-market funds yielded less than 1%. The reinvestment-assumption gap between IRR and MIRR-at-cash was small enough to be operationally irrelevant. A 15% IRR and a 13% MIRR-at-1% were close enough that nobody bothered computing the MIRR. IRR alone was the working metric.

That environment is gone. As of May 2026, the 10-year Treasury yields approximately 4.20%. SOFR is around 4.40%. Money-market funds are paying 4.00–4.50% on cash. The IRR-MIRR spread on a typical value-add multifamily deal has widened from 1–2 points to 6–8 points. The 7.5-point spread in the worked example above is not an edge case — it is roughly typical for a multifamily value-add underwritten at a 20% IRR with a 7-year hold.

A second piece of context: the 2024–2025 vintage of value-add deals is producing realized IRRs that systematically underperform underwritten IRRs. Real Capital Analytics and MSCI returns data are documenting this gap across multiple asset classes. Some of the underperformance is from extended hold periods (refi gaps from the debt environment), some from rents coming in below underwriting, and some — meaningfully — from the reinvestment-assumption gap that was always there but invisible at zero rates. Distributions that returned to LPs in 2023–2025 didn't compound at the project IRR; they sat in money market funds at 4–5%, and the multi-year CAGR on the original equity outlay is correspondingly below the underwritten IRR.

Several institutional LPs are now requesting MIRR alongside IRR in offering documents and quarterly reporting. NCREIF and PREA practitioner surveys reflect this shift. The market is converging toward dual-quoting both metrics as the rate environment makes the gap impossible to ignore.

Five Mistakes Practitioners Make

• Quoting IRR alone in a high-rate environment. When the risk-free rate is 4–5%, IRR alone overstates the realized annualized return on the original capital. Quote MIRR alongside, especially for long-hold value-add deals.

• Using IRR() on a cash flow with multiple sign changes. Recap events, major capex spends, and refi-to-loss events can produce multi-sign cash flows. Excel returns one root with no warning. Use MIRR for these deals, or solve the polynomial graphically to see all the roots.

• Choosing a reinvestment rate that's not achievable. Setting MIRR's reinvestment rate equal to the IRR (a default in some calculators) makes MIRR mathematically identical to IRR — the entire point of the calculation is lost. Use the actual achievable reinvestment rate: the risk-free yield, the LP's stated opportunity cost, or a documented blended rate. Don't pick a number that flatters the deal.

• Treating IRR as a forecast of terminal wealth. IRR is a discount rate, not a compound growth rate on the original equity. If you tell an LP “a 17% IRR on your $10M means $21.9M in five years,” you are implicitly assuming reinvestment at 17%. That is not what IRR says. Compute the realized CAGR separately if that's the question being asked.

• Using IRR on monthly cash flows without converting to XIRR. Excel's IRR() on monthly periods returns a monthly rate that has to be annualized. Most analysts forget to annualize or annualize incorrectly ((1+r)¹²−1 vs 12r). XIRR with actual dates removes the ambiguity. In institutional practice, default to XIRR.

Do It in Apers

Related Articles

The IRR methodology pillar links into the broader returns-analysis cluster:

• IRR Calculator and Formula for Real Estate — the calculator pillar with an interactive widget for running IRR on your own cash flows.
• Equity Multiple and MOIC: When the Multiple Matters More Than IRR — the time-blind complement to IRR.
• Cash-on-Cash Return: Levered vs. Unlevered — the current-yield metric.
• IRR Sensitivity Analysis and Stress Testing — how to test IRR under downside scenarios.
• TVPI, DPI, and RVPI: Fund-Level Return Metrics for LPs — the fund-level analog of IRR and equity multiple.
• Cap Rate Calculator and Formula — the entry-yield metric every IRR conversation eventually references.
• Preferred Return: Simple vs Compounding vs Accruing — how IRR interacts with promote structures.

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