{ "site": "m-a.wiki", "section": "LBO returns math", "url": "https://m-a.wiki/lbo/", "reviewed": "2026-08-27", "license": "CC BY 4.0", "changes": "https://m-a.wiki/changes.json", "tables": [ { "title": "Sources and uses at entry", "intro": "Base case. EBITDA_0 = 100.0, x_entry = 10.0, so EV_0 = 1,000.0. Fees are 3.0 percent of EV_0. Uses must equal sources; the sponsor equity check is the plug.", "columns": [ "Uses", "Amount", "Sources", "Amount" ], "rows": [ [ "Purchase of enterprise value (10.0x x 100.0)", "1,000.0", "New term debt (5.0x EBITDA_0)", "500.0" ], [ "Transaction and financing fees (3.0% of EV_0)", "30.0", "Sponsor equity (plug)", "530.0" ], [ "Total uses", "1,030.0", "Total sources", "1,030.0" ] ] }, { "title": "Debt schedule and cash sweep", "intro": "Unlevered free cash flow is 60 percent of EBITDA. Cash interest is 6.0 percent of the beginning debt balance. All cash after interest sweeps to debt. Figures are rounded for display; the totals below use unrounded values.", "columns": [ "Year", "Beginning debt", "EBITDA", "Unlevered FCF", "Cash interest", "Debt paydown", "Ending debt" ], "rows": [ [ "1", "500.00", "105.00", "63.00", "30.00", "33.00", "467.00" ], [ "2", "467.00", "110.25", "66.15", "28.02", "38.13", "428.87" ], [ "3", "428.87", "115.76", "69.46", "25.73", "43.73", "385.14" ], [ "4", "385.14", "121.55", "72.93", "23.11", "49.82", "335.32" ], [ "5", "335.32", "127.63", "76.58", "20.12", "56.46", "278.87" ], [ "Cumulative", "-", "580.19", "348.11", "126.98", "221.13", "278.87" ] ] }, { "title": "Value-creation attribution across exit multiples", "intro": "The four-way bridge from sponsor equity in to sponsor equity out. EBITDA growth is (EBITDA_t - EBITDA_0) x x_entry. Multiple expansion is (x_exit - x_entry) x EBITDA_t. Debt paydown is D_0 - D_t. Fees are the entry fees, which are a permanent leakage from equity value. The four components sum exactly to E_t - E_0 in every row. E_0 = 530.0, EBITDA_t = 127.6282, D_t = 278.8655.", "columns": [ "x_exit", "EBITDA growth", "Multiple expansion", "Debt paydown", "Entry fees", "Sum = E_t - E_0", "E_t", "MOIC", "IRR" ], "rows": [ [ "8.0x", "+276.3", "-255.3", "+221.1", "-30.0", "+212.2", "742.2", "1.400x", "6.97%" ], [ "9.0x", "+276.3", "-127.6", "+221.1", "-30.0", "+339.8", "869.8", "1.641x", "10.41%" ], [ "10.0x", "+276.3", "0.0", "+221.1", "-30.0", "+467.4", "997.4", "1.882x", "13.48%" ], [ "11.0x", "+276.3", "+127.6", "+221.1", "-30.0", "+595.0", "1,125.0", "2.123x", "16.25%" ], [ "12.0x", "+276.3", "+255.3", "+221.1", "-30.0", "+722.7", "1,252.7", "2.364x", "18.77%" ] ] }, { "title": "MOIC to IRR, single outflow and single inflow", "intro": "IRR = MOIC^(1/t) - 1. Valid only when there is one cash out at t = 0 and one cash in at t. Any interim distribution breaks the identity. Figures are IRR in percent.", "columns": [ "MOIC", "3 years", "4 years", "5 years", "6 years", "7 years" ], "rows": [ [ "1.5x", "14.47", "10.67", "8.45", "6.99", "5.96" ], [ "2.0x", "25.99", "18.92", "14.87", "12.25", "10.41" ], [ "2.5x", "35.72", "25.74", "20.11", "16.50", "13.99" ], [ "3.0x", "44.22", "31.61", "24.57", "20.09", "16.99" ] ] }, { "title": "Leverage sensitivity", "intro": "Base case with x_exit held at 10.0x. Only D_0 changes; the equity check absorbs the difference, and interest changes with the balance, so the sweep amount changes too. EV_0 = 1,000.0, fees 30.0, EBITDA_t = 127.6282.", "columns": [ "D_0 (x EBITDA_0)", "Sponsor equity E_0", "Cumulative paydown", "D_t", "E_t", "MOIC", "IRR" ], "rows": [ [ "3.0x (300.0)", "730.0", "288.8", "11.2", "1,265.1", "1.733x", "11.62%" ], [ "4.0x (400.0)", "630.0", "255.0", "145.0", "1,131.2", "1.796x", "12.42%" ], [ "5.0x (500.0)", "530.0", "221.1", "278.9", "997.4", "1.882x", "13.48%" ], [ "6.0x (600.0)", "430.0", "187.3", "412.7", "863.6", "2.008x", "14.97%" ] ] }, { "title": "EBITDA growth sensitivity", "intro": "Base case with x_exit held at 10.0x and D_0 at 500.0. Growth affects both the exit enterprise value and the cash available to sweep, so it moves the answer twice.", "columns": [ "g", "EBITDA_t", "Cumulative paydown", "D_t", "E_t", "MOIC", "IRR" ], "rows": [ [ "0.0%", "100.00", "169.1", "330.9", "669.1", "1.262x", "4.77%" ], [ "3.0%", "115.93", "199.5", "300.5", "858.8", "1.620x", "10.13%" ], [ "5.0%", "127.63", "221.1", "278.9", "997.4", "1.882x", "13.48%" ], [ "8.0%", "146.93", "255.7", "244.3", "1,225.0", "2.311x", "18.24%" ] ] } ], "entries": [ { "id": "sources-and-uses", "term": "Sources and uses", "definition": "The funding identity of an acquisition. Total uses - the purchase price plus every fee, refinancing, and cash injection required at close - must equal total sources. In a sponsor deal the equity check is solved as the residual, which is why the equity number moves whenever any other line moves.", "formula": "E_0 = EV_0 + fees + refinanced debt + cash to balance sheet - new debt - rollover equity - assumed liabilities", "fields": [ { "label": "Worked", "value": "EV_0 = 10.0 x 100.0 = 1,000.0; fees = 0.03 x 1,000.0 = 30.0; new debt = 500.0. E_0 = 1,000.0 + 30.0 - 500.0 = 530.0", "mono": true }, { "label": "Check", "value": "Uses 1,030.0 = Sources 500.0 + 530.0", "mono": true } ], "notes": [ "Fees are the line most often left out of a quick model, and they are pure equity leakage: they buy no asset and never come back. At 3 percent of EV on a 5.0x-levered deal they consume 5.7 percent of the equity check before day one.", "If the target's existing debt is being refinanced rather than assumed, the refinancing appears in uses at its payoff amount including any make-whole or call premium, not at book value. Reading the payoff figure off the balance sheet understates uses.", "Rollover equity from a selling management team is a source, not a discount to price. It reduces the sponsor's cheque without reducing the enterprise value paid, so it lowers the sponsor's absolute dollars at risk while leaving the entry multiple unchanged." ] }, { "id": "entry-enterprise-value", "term": "Entry enterprise value and the entry multiple", "definition": "The entry multiple is enterprise value divided by an EBITDA measure. Which EBITDA measure is used determines what the multiple means, and the two conventions are not interchangeable.", "formula": "EV_0 = x_entry * EBITDA_0", "fields": [ { "label": "Worked", "value": "x_entry = 10.0, EBITDA_0 = 100.0, EV_0 = 1,000.0", "mono": true }, { "label": "LTM basis", "value": "EBITDA for the last twelve months actually reported", "mono": false }, { "label": "Adjusted or run-rate basis", "value": "LTM EBITDA plus agreed add-backs and annualised effects", "mono": false } ], "notes": [ "A 10.0x multiple on adjusted EBITDA of 100.0 and an 11.5x multiple on reported EBITDA of 87.0 are the same enterprise value. Comparing an entry multiple against anything requires knowing which denominator each side used.", "Add-backs matter twice, not once: they set the entry multiple and they usually set the leverage covenant denominator in the credit agreement. An add-back that is generous at signing is a covenant cushion for the life of the loan.", "The exit multiple in a model is almost always applied to a differently-constructed EBITDA than the entry multiple, because five years later there is no add-back story. That asymmetry, not multiple contraction in the market, is the most common source of a model that flatters entry and punishes exit." ] }, { "id": "moic", "term": "MOIC", "definition": "Multiple on invested capital. Total value returned to the sponsor divided by total capital invested, ignoring time. A pure measure of dollars made per dollar risked.", "formula": "MOIC = (sum of distributions + residual value) / sum of contributions", "fields": [ { "label": "Worked", "value": "E_t = 997.4161, E_0 = 530.0, MOIC = 997.4161 / 530.0 = 1.8819x", "mono": true }, { "label": "Gross vs net", "value": "Gross MOIC is measured at the deal; net MOIC is after fund management fees and carried interest", "mono": false } ], "notes": [ "MOIC is the only returns metric that cannot be manipulated by timing. That is its entire value in a diligence conversation and the reason it is quoted alongside IRR rather than instead of it.", "MOIC is blind to the cost of the hold. A 2.0x over three years and a 2.0x over eight years are the same MOIC and very different investments.", "Where a deal has multiple equity injections, MOIC uses the sum of contributions undiscounted. Two 100.0 injections four years apart count as 200.0 of invested capital even though they are not economically equivalent." ] }, { "id": "irr-from-moic", "term": "IRR from MOIC, and when the identity holds", "definition": "For a single cash outflow at inception and a single inflow at exit, IRR is the geometric annualisation of MOIC. The identity fails the moment there is an interim cash flow.", "formula": "IRR = MOIC^(1/t) - 1; equivalently MOIC = (1 + IRR)^t", "fields": [ { "label": "Worked", "value": "MOIC = 1.8819x over t = 5: IRR = 1.8819^(1/5) - 1 = 0.1348 = 13.48%", "mono": true }, { "label": "Inverse", "value": "Required MOIC for a 20% IRR over 5 years = 1.20^5 = 2.4883x", "mono": true }, { "label": "General case", "value": "IRR solves sum over j of CF_j / (1 + IRR)^t_j = 0", "mono": true } ], "notes": [ "The 2.5x-in-five-years shorthand for a 20 percent IRR is exact to two decimals: 1.20^5 = 2.4883.", "IRR is a rate, not an amount. A short, small, high-IRR deal can be worth less in dollars than a long, large, lower-IRR one, which is why fund-level dollar metrics exist alongside it.", "IRR implicitly assumes interim distributions are reinvested at the IRR itself. In a fund that receives a distribution and holds it in cash, the realised compound return is lower than the reported IRR. This is the reason MOIC is quoted next to it, not because MOIC is a better metric." ] }, { "id": "value-creation-attribution", "term": "Four-way value-creation attribution", "definition": "The standard decomposition of the change in sponsor equity value into EBITDA growth, multiple expansion, debt paydown, and a reconciling item for fees and other leakage. Each component answers a different question about whether the return was earned or bought.", "formula": "E_t - E_0 = (EBITDA_t - EBITDA_0)*x_entry + (x_exit - x_entry)*EBITDA_t + (D_0 - D_t) - fees", "fields": [ { "label": "Worked, x_exit = 10.0x", "value": "(127.6282 - 100.0)*10.0 = +276.2816; (10.0 - 10.0)*127.6282 = 0.0; 500.0 - 278.8655 = +221.1345; fees -30.0. Sum = +467.4161", "mono": true }, { "label": "Check", "value": "E_t - E_0 = 997.4161 - 530.0 = 467.4161. Ties exactly.", "mono": true }, { "label": "Worked, x_exit = 12.0x", "value": "+276.2816 + (2.0*127.6282 = +255.2563) + 221.1345 - 30.0 = +722.6724 = 1,252.6724 - 530.0", "mono": true } ], "notes": [ "The cross-term between growth and multiple change has to be assigned to one component or the other, and the convention above assigns it to multiple expansion by valuing the multiple change on exit EBITDA. Valuing growth at the exit multiple instead and multiple change at entry EBITDA also ties, and produces a different-looking split of the same total. Ask which convention a bridge uses before comparing two of them.", "Multiple expansion is the only component the sponsor does not control. A bridge in which it is the largest bar is a bridge that describes a market, not an operating result.", "Debt paydown and EBITDA growth are not independent: growth generates the cash that pays the debt. Presenting them as separate levers overstates how substitutable they are.", "Splitting EBITDA growth further into revenue growth and margin expansion is the standard next level: revenue effect = (Rev_t - Rev_0)*margin_0*x_entry, margin effect = Rev_t*(margin_t - margin_0)*x_entry. Those two sum to the growth bar only if the cross-term is assigned explicitly." ] }, { "id": "debt-paydown", "term": "Debt paydown and the cash sweep", "definition": "The amount of debt retired over the hold, which is the free cash flow after cash interest, cash taxes, capital expenditure, working capital, and mandatory amortisation, multiplied by the sweep percentage in the credit agreement.", "formula": "D_n = D_(n-1) - sweep% * (uFCF_n - cash interest_n - mandatory amortisation_n); cash interest_n = r_d * D_(n-1)", "fields": [ { "label": "Worked, year 1", "value": "uFCF = 0.60 * 105.00 = 63.00; interest = 0.06 * 500.00 = 30.00; paydown = 33.00; D_1 = 467.00", "mono": true }, { "label": "Worked, year 5", "value": "uFCF = 0.60 * 127.6282 = 76.5769; interest = 0.06 * 335.3230 = 20.1194; paydown = 56.4575; D_5 = 278.8655", "mono": true }, { "label": "Cumulative", "value": "D_0 - D_5 = 500.0 - 278.8655 = 221.1345 of equity value created", "mono": true } ], "notes": [ "Paydown accelerates over the hold even at constant EBITDA, because interest falls as the balance falls. The year-5 sweep in the base case is 71 percent larger than the year-1 sweep on EBITDA only 22 percent higher.", "This is exactly why the sweep is the last thing a lender concedes: every dollar of sweep the borrower keeps compounds into future interest the borrower keeps paying.", "Floating-rate debt makes the schedule a function of the forward curve, not a constant. A model with a flat rate assumption understates the dispersion of the equity outcome far more than it misstates the base case.", "Interest deductibility is capped in the United States at a percentage of adjusted taxable income under IRC section 163(j). Above the cap, incremental leverage costs the full pre-tax rate and the tax shield stops scaling, which flattens the IRR benefit of leverage at exactly the point a model naively shows it accelerating." ], "source": "IRC s.163(j)" }, { "id": "required-exit-multiple", "term": "Exit multiple required for a target IRR", "definition": "Inverting the return calculation to solve for the exit multiple that delivers a specified IRR, holding the operating case and the debt schedule fixed. This is the single most useful number in an LBO model because it converts a return target into a testable claim about the world.", "formula": "x_exit = [E_0 * (1 + IRR_target)^t + D_t] / EBITDA_t", "fields": [ { "label": "Worked, 20% target", "value": "[530.0 * 1.20^5 + 278.8655] / 127.6282 = [1,318.8096 + 278.8655] / 127.6282 = 1,597.6751 / 127.6282 = 12.518x", "mono": true }, { "label": "Worked, 15% target", "value": "[530.0 * 1.15^5 + 278.8655] / 127.6282 = 1,344.9241 / 127.6282 = 10.538x", "mono": true }, { "label": "Worked, 25% target", "value": "[530.0 * 1.25^5 + 278.8655] / 127.6282 = 1,896.3010 / 127.6282 = 14.858x", "mono": true } ], "notes": [ "Stated this way the conversation changes. A 20 percent target on this base case requires buying at 10.0x and selling at 12.5x - a 25 percent multiple expansion - which is a bet on the exit environment, not on the company.", "The mirror question, required EBITDA at a constant multiple, is EBITDA_t = [E_0*(1+IRR)^t + D_t] / x_entry. At the 20 percent target that is 159.77, or 9.8 percent annual growth against the 5.0 percent base case.", "Because D_t itself depends on the operating case, solving for exit multiple and solving for growth are not independent. Do one at a time and say which is held fixed." ] }, { "id": "leverage-and-irr", "term": "Why leverage raises IRR less than it looks like it should", "definition": "Additional debt reduces the equity check, which mechanically raises the return multiple on a given exit equity value. It also raises cash interest, which reduces the sweep and leaves more debt outstanding at exit, which reduces exit equity. The net effect is positive but much smaller than the first effect alone.", "formula": "dE_t/dD_0 = -1 + (cumulative interest saved or incurred), so IRR gain is second-order in D_0", "fields": [ { "label": "Worked, 3.0x to 6.0x", "value": "E_0 falls 730.0 to 430.0 (-41%), but E_t falls 1,265.1 to 863.6 (-32%). MOIC rises 1.733x to 2.008x and IRR rises 11.62% to 14.97%", "mono": true }, { "label": "Naive expectation", "value": "Holding E_t at 1,265.1 while cutting E_0 to 430.0 would imply MOIC 2.942x and IRR 24.1% - which is the error", "mono": true } ], "notes": [ "Tripling the debt multiple from 3.0x to 6.0x moves IRR by 335 basis points in this base case, not by the 1,250 basis points the equity-check effect alone suggests. Interest eats most of it.", "Leverage widens the distribution symmetrically while shifting the mean modestly. The reason sponsors use it is not that it raises expected IRR much; it is that it lets a given fund own more enterprise value.", "Below roughly 3.0x in this base case the sweep exhausts the debt before exit and the model starts accumulating cash instead. At that point additional deleveraging stops adding to equity value and the leverage lever is simply switched off.", "The leverage benefit also compresses when the exit is bad. At an 8.0x exit the 6.0x structure returns 7.19 percent against 6.70 percent for the 3.0x structure - a 49 basis point advantage, down from 335 basis points at a 10.0x exit - while the absolute equity value at exit falls from 1,009.8 to 608.3. Leverage pays least in exactly the outcome it makes more likely." ] }, { "id": "recap-irr-moic-divergence", "term": "Dividend recapitalisation - IRR up, MOIC down", "definition": "Re-levering the company mid-hold to distribute cash to the sponsor pulls proceeds forward in time. Because the distributed cash is replaced with debt that accrues interest, total dollars returned fall while the time-weighted return rises. The two headline metrics move in opposite directions.", "formula": "IRR solves -E_0 + Div/(1+IRR)^s + E_t'/(1+IRR)^t = 0, where E_t' = E_t - Div*(1+r_d)^(t-s)", "fields": [ { "label": "Base case", "value": "-530.0 at t=0, +997.4161 at t=5. MOIC 1.8819x, IRR 13.48%", "mono": true }, { "label": "With a 150.0 recap at t=2", "value": "Exit equity falls by 150.0 * 1.06^3 = 178.6524 to 818.7637. Cash flows: -530.0, +150.0 at t=2, +818.7637 at t=5", "mono": true }, { "label": "Result", "value": "MOIC = (150.0 + 818.7637)/530.0 = 1.8279x (down from 1.8819x); IRR = 14.52% (up from 13.48%)", "mono": true } ], "notes": [ "This is the clearest case where quoting one metric is a choice about what to hide. A recap improves IRR and destroys MOIC, and both are true.", "The magnitude of the divergence scales with the gap between the deal IRR and the cost of the new debt. Recapping at 6 percent to advance cash on a 13 percent deal helps the rate; recapping at 12 percent barely moves it.", "A recap also permanently raises the exit leverage, which raises the probability that the exit equity is zero. The IRR improvement is priced in expectation; the risk increase is not visible in either metric.", "Recap capacity is a covenant question, not a modelling question. Restricted payment baskets and the leverage test that gates them determine whether the recap in a model can actually be done." ] } ] }