Private Markets III: Venture Valuation, Cap Tables and Private-Company Value

Branch: Private Wealth & Private Markets (Finance-Plus Ring) · Node PW1.04 · ~18 focused hours · Mastery gate ≥ 85%

A founder in Bengaluru signs a term sheet with three numbers on it. Pre-money ₹24 crore. Investment ₹6 crore. A new employee option pool of 15%, created before the round closes. She reads the first two carefully, because those are the numbers her friends will ask about, and she skims the third because a pool is for hiring and hiring is good.

That third clause moves ₹90 lakh from her side of the table to the investor's. The document never prints ₹90 lakh anywhere, and no line in it calls the transfer a fee or a discount. The whole of it is a consequence of where in the arithmetic the new pool shares get counted, and the two possible answers give the founders 65.00% and 68.00% of the same company at the same ₹30 crore post-money valuation.

Two hundred kilometres south, a precision-castings company in Coimbatore has never raised a round and never will. Its promoter pays himself ₹2.40 crore against a market salary of ₹0.90 crore, rents the factory from his own family trust at ₹1.80 crore against a market rent of ₹1.05 crore, and runs two cars and a club membership through the profit and loss account. Reported EBITDA is ₹7.20 crore. Rebuilt on the accounting policy its listed competitors use, EBITDA is ₹10.66 crore. Multiply both by the same 9.0x and the gap is ₹31.14 crore, which is 48% of the ₹64.80 crore the reported figure alone would have supported, and every rupee of it lives in a schedule the buyer constructs and the seller disputes.

And one level up again, a ₹800 crore Indian growth fund returns 1.6254x net of every fee and every rupee of carry, at a net internal rate of return of 11.04%. Those are respectable numbers, they are correctly computed, and an investor who had put the identical cash into a broad index on the identical dates would have finished with ₹229.79 crore more. The fund's public market equivalent is 0.88, which is the only one of the four figures that answers the question the investor actually had.

What links a term sheet, a promoter's salary and a fund benchmark is that none of them has a price. A listed share has a quote you can look up and argue with. Each of the three numbers above was manufactured instead. Someone chose the pool convention, someone chose the normalisation schedule, someone chose the index. The quality of the number is exactly the quality of the choices, and the choices are almost never printed next to the answer.

Eighteen hours from now you will build a cap table that survives a down round and price a liquidation preference across the exit values where the fight actually happens. You will normalise a promoter-run business into something a multiple can legally be applied to, put three defensible discount rates on the same cash flow, and compute a public market equivalent on a fund schedule you already know.

For every number you produce, you will also be able to say which clause or which convention would move it.

Learning objectives

By the end you can:

1. Place a private-markets commitment in its strategy family (buyout, growth, venture, secondaries), state what return each family underwrites, and run the venture-capital method end to end: required ownership at exit, the expected-dilution adjustment, ownership today, post-money and pre-money, with the implied discount rate reconciled against the target multiple.
2. Build a fully diluted cap table through a seed round, a Series A and a Series B down round, computing the price per share and the share count at every step, and read the founders' ownership off it at each stage.
3. Compute the option-pool shuffle both ways (pool created pre-money against pool created post-money), and state in rupees or dollars the value the convention transfers between founders and the incoming investor.
4. Price a liquidation preference: 1x non-participating, full participation, and participation subject to a cap, producing the payoff table across a range of exit values and identifying both the conversion-indifference point and the exit at which the cap first binds.
5. Compute anti-dilution protection under broad-based weighted average and under a full ratchet on the same down round, and state what each costs the founders in ownership points.
6. Normalise a private company's reported earnings, naming and sizing owner compensation, related-party charges, personal expenses and non-recurring items separately, and restate the result onto a listed peer's accounting policy before any multiple touches it.
7. Build a private discount rate three ways (the build-up method, the expanded capital asset pricing model, and total beta for an undiversified buyer), reconcile them, and say which buyer each one is the right rate for.
8. Value a private company by the income, market and asset approaches, reconcile the three rather than averaging them, and apply the control and marketability adjustments in the correct order, exactly once each.
9. Compute a Kaplan-Schoar public market equivalent on a real fund cash-flow schedule with the index substitution shown, handle residual net asset value correctly, and state what the ratio adds to an internal rate of return and a multiple.
10. Decompose a realised buyout return into the three financial levers and the operational levers underneath the EBITDA line, with the bridge tying to the equity gain, and know which part of all of the above an AI tool may draft and which part it must never originate (the R10 duality). Then pass the ≥85% gate.

The duality, stated once (R10). Objectives 1 to 9 are the understanding gate. Objective 10 carries the productivity payoff, and this branch's warning applies here with a specific edge. Cap-table arithmetic and preference waterfalls are mechanical, which makes them exactly the sort of thing a tool produces fluently and wrongly, with a plausible share count and a broken tie-out. Normalisation and discount selection are the opposite problem: no arithmetic is involved, so nothing looks wrong at all. A tool will hand you "a 30% DLOM is standard" with the same confidence it hands you a share count, and only one of those two errors is catchable by re-running the sum.

Prerequisites & connections

Builds on. The fund-economics node immediately before this one supplies the vocabulary of commitments, capital calls, distributions, the J-curve and the whole family of multiples, and it also supplies the actual Konkan Growth Partners Fund III cash-flow schedule that the public market equivalent here is computed on, so that the two nodes report the same fund. PW1.03 supplies the levels-of-value hierarchy, the control premium and its reciprocal, and the discount for lack of marketability with its three evidence families and the Mandelbaum factors, none of which is rebuilt here. M3.09 owns the leveraged-buyout engine and the value-creation bridge, including the point that the bridge has a free parameter in the cross term, and this node assumes both. M3.03 owns the cost of capital, the bottom-up beta and total beta. M3.06 owns comparable-company and precedent-transaction discipline. E11.02 owns deal structures, the announced-deal control premium measured off an unaffected price, and purchase accounting. M6.04's endowment effect and anchoring are the quiet background of every conversation with a founder about a down round.

Feeds forward. PW1.01's concentrated-position and estate-transfer work consumes the private-company bridge here as a direct input: an advisor cannot tell a business-owning family what their children will receive without first knowing which interest, at which level of value, at which discount, is being transferred. AL1.04 places private markets inside a portfolio allocation and uses this node's public market equivalent as the benchmarking instrument for an asset class with no index. AA1.03's performance-measurement principle that the measure follows the control of the cash flows is completed here, because the public market equivalent is what you reach for when neither a time-weighted nor a money-weighted return can be compared against anything. ES1.01 and ES1.02 supply the fiduciary and conflicts frame around the judgment calls in the second half.

Deliberate non-overlaps, stated so that nothing here is built twice. PW1.02 owns fund economics, both waterfall architectures, the clawback, and the fund-level multiples, and it hands this node the Konkan schedule rather than a duplicate of it; the public market equivalent is computed here and cross-referenced from there. PW1.03 owns the levels-of-value ladder, the control premium, the discount for lack of control and the discount for lack of marketability, together with the Chaffe and restricted-stock evidence; this node applies those tools in sequence to a full three-approach valuation and never re-derives them. M3.09 owns the LBO model itself, the sources-and-uses table, the debt waterfall and the bridge conventions; this node decomposes the EBITDA-growth lever into the operational moves that produced it, which M3.09 does not. M3.06 owns public comparables selection and the multiple-choice discipline; this node states only what changes when the subject is private. M3.03 owns total beta and its rules of application; this node uses it as a cross-check on a build-up rate. M3.04 owns discounted cash flow mechanics. AL1.04 owns the allocation and pacing decision for private markets as a whole.

Core teaching content

4.12 Reconciling three approaches, and the order of the adjustments

Three approaches will give three answers. Averaging them is the most common wrong move in the discipline, because an average buries the reason they disagree, and the reason is usually the finding.

Reconciliation means four things. Naming which approach best fits the subject and the purpose. Explaining the gap between the approaches mechanically, in terms of an assumption that differs. Deciding which approach carries the conclusion. And stating the one fact that would change the answer.

The asset approach deserves its own sentence, because its role is almost always misunderstood. For a profitable going concern, adjusted net asset value and orderly liquidation value are a floor, not a candidate answer. The floor matters when it binds: if a business's income-approach value falls below its liquidation value, the business is worth more dismantled than run, and the conclusion changes from a valuation to a recommendation.

Then the adjustments, and the order is not negotiable. Both levers move a value between the rungs of the levels-of-value ladder the private-markets node before this one builds, and each applies exactly once.

Establish first which level of value your base number sits on. A capitalized cash flow built on normalised earnings, with the owner's excess compensation and the related-party rent stripped out, is a control value, because only a controlling holder can make those changes. A guideline-public-company multiple is a marketable minority value.

To value a controlling interest, start from a control value and apply a discount for lack of marketability sized for a control block: smaller than a minority discount, because a controlling owner can force a sale process, but rarely zero, because the process takes twelve to eighteen months and costs money.

To value a minority interest, start from a marketable-minority value and apply a discount for lack of marketability sized for a minority block. If your base number was a control value, first step down with a discount for lack of control, and step down using the reciprocal:

DLOC = 1 − 1 ÷ (1 + control premium)

A 25% control premium implies a DLOC of exactly 20.00%, not 25%. Subtracting the premium as though the relationship were symmetric applies too large a discount and therefore understates the minority value, and at a 35% premium the error is 9.07 percentage points.

Read the answer back every time. The discount is always smaller than the premium it came from.

The prohibited move deserves naming, because it is easy to make with good intentions. Do not apply a discount for lack of marketability to a value already built from private-transaction comparables, because those transaction prices were paid for private companies and already embed the market's own illiquidity discount. The same discount then appears twice under two different names, and the second application is invisible in the write-up.

Where a valuation is being prepared for gift, estate, divorce or employee-ownership purposes, the appraiser's incentives point somewhere, and the ethical machinery this program teaches elsewhere is the frame. The technical protection is narrow and effective: write down every factor that moved a discount inside its empirical range, and do it before anyone tells you what number would be convenient.

Common mistakes & how experts think differently

1. Reading a term sheet's pre-money number without reading the pool clause. A ₹24 crore pre-money with a 15% pool created pre-money is a lower price than a ₹24 crore pre-money with the pool created post-money, by ₹90 lakh on a ₹30 crore round. Expert reflex: compute the founders' post-round percentage under both conventions before responding to any pre-money figure, and treat the pool convention as a price term rather than a hiring term.

1. Quoting ownership percentages as though they were additive. Percentages are quotients whose denominator changes at every round, at every anti-dilution trigger and at every option grant. Expert reflex: keep the model in share counts, derive percentages at the end, and never carry a percentage across a round.

1. Negotiating the preference multiple and ignoring participation. A 1x participating preference is worse for common shareholders across the realistic exit band than a 1.5x non-participating one. Expert reflex: ask three questions in order, participating or not, what cap, what seniority, then compute the two crossover points, because those are the only exit values at which the answer changes.

1. Treating anti-dilution as compensation. It issues no cash and creates no value; it moves ownership from founders, the pool and the new investor to the protected holder. Broad-based protection recovered ₹1.36 crore of a ₹14.00 crore paper loss for Anantara's Series A while costing the founders 0.67 points. Expert reflex: price the clause as a transfer, and check whether it is broad-based, narrow-based or a ratchet before pricing anything, since on Anantara's Series B those three issue 514,706, 1,182,432 and 4,375,000 extra shares on identical facts.

1. Building a normalisation schedule with no negative adjustments. A schedule containing only add-backs is an advocacy document. Vaishnavi's one-off export incentive of ₹0.62 crore had to come out, and it was the only item the seller did not volunteer. Expert reflex: require the schedule to run in both directions, and separately require a line naming what could not be adjusted, such as a promoter's personal guarantee.

1. Applying a peer multiple to a metric computed on a different accounting policy. Vaishnavi's tooling policy alone is worth ₹10.80 crore at the peer median multiple. Expert reflex: restate the subject's metric onto the peers' policy before the multiple is applied, and say in the write-up which policies were restated.

1. Double-counting size. A size premium inside the discount rate and a size-driven discount to the peer multiple are the same adjustment, applied in two places. Expert reflex: decompose the fundamental discount into named components and check each one against the discount-rate build, then charge for each risk exactly once.

1. Subtracting a control premium to get a discount for lack of control. The relationship is reciprocal. At a 35% control premium the arithmetic error is 9.07 percentage points, and it always runs in the direction that flatters a minority holder's value. Expert reflex: compute 1 − 1 ÷ (1 + premium) every time, by hand, and read the answer back against the premium to confirm it is smaller.

1. Averaging three approaches. An average buries the disagreement, and the disagreement is usually the most informative thing in the file. Expert reflex: name the approach that carries the conclusion, explain the gap mechanically in terms of one differing assumption, and state the fact that would change the answer.

1. Reporting a fund's internal rate of return against an index return. The two were computed over different capital at different times. Expert reflex: compute a public market equivalent on the fund's own dates, name the index and defend the substitution, and report the residual net asset value treatment alongside the ratio.

1. Treating a public market equivalent below 1.00 as an accusation. Konkan's 0.88 is a statement about a specific index over a specific window, and the same fund against a small-cap index or a different vintage window would report differently. Expert reflex: report the ratio with its index, its window and its net-or-gross basis attached, and report direct alpha next to it so the committee sees the annualised form.

1. Underwriting multiple expansion. Half a turn of exit multiple was worth $15.0 million to Summit Ridge, 9.5% of everything created and 1.87 points of internal rate of return, and the sponsor did nothing to earn it. Expert reflex: underwrite the flat-multiple case, show the contraction case as the honest downside, and treat any re-rating as a windfall to be explained rather than a plan.

Worked examples

Each entity in these examples is synthetic and labelled so. Every figure was produced by a Python computation run for this node and is reproducible from the inputs stated; share counts are carried unrounded and displayed to the precision shown. Where a figure is set by a statute, a regulator or a market and therefore moves, it carries a verify flag naming the primary source.

Worked example 1: Anantara Labs, seed to a Series B down round (India, ₹)

Anantara Labs Private Limited (synthetic) is a Bengaluru logistics-software company. Two founders hold 13,000,000 shares, 7,800,000 and 5,200,000, with nothing else outstanding. Peninsula Seed Fund I (synthetic) offers ₹6 crore at a ₹24 crore pre-money valuation, with a 15% post-money option pool created before the round. Build the cap table under both pool conventions, then run a Series A at ₹120 crore pre-money for ₹30 crore, then a Series B down round at ₹80 crore pre-money for ₹20 crore with broad-based weighted-average anti-dilution on the Series A.

Solution. Step 1, the seed with the pool created pre-money. Post-money is 24 + 6 = ₹30 crore, so the investor holds 6 ÷ 30 = 20.00% and the pool holds 15.00%, leaving the founders 65.00%. Their 13,000,000 shares are therefore 65.00% of the post-money count, so the total is 13,000,000 ÷ 0.65 = 20,000,000 shares, the pool is 3,000,000 and the investor's stake is 4,000,000. Price per share is ₹30 crore ÷ 20,000,000 = ₹15.00. Cross-check the pre-money: 16,000,000 pre-money fully diluted shares at ₹15.00 is ₹24,00,00,000, which is the term sheet's own number.

Step 2, the seed with the pool created post-money. Now the ₹24 crore pre-money is spread over the founders' 13,000,000 shares alone, so price per share is ₹18.4615385, and ₹6 crore buys 3,250,000 shares. The pool is then created out of the combined holding: 16,250,000 shares are 85% of the new total, so the total is 19,117,647 shares and the pool is 2,867,647. Founders hold 13,000,000 ÷ 19,117,647 = 68.00%; the investor holds 17.00%.

Step 3, price the convention. Post-money value is ₹30 crore under both conventions, so value per share in the second case is ₹30 crore ÷ 19,117,647 = ₹15.6923. The founders' stake is worth 13,000,000 × 15.00 = ₹19.50 crore under the first convention and 13,000,000 × 15.6923 = ₹20.40 crore under the second. The ₹90,00,000 difference is exactly 3.00 percentage points of a ₹30 crore company, and it lands in the investor's column: their ₹6 crore is worth ₹6.00 crore under the first convention and ₹5.10 crore under the second.

Step 4, the Series A. Pre-money fully diluted count is the 20,000,000 shares the seed left, so price per share is ₹120 crore ÷ 20,000,000 = ₹60.00, precisely 4.00x the seed price. The round issues ₹30 crore ÷ 60 = 5,000,000 shares and the post-round total is 25,000,000. Founders fall to 13,000,000 ÷ 25,000,000 = 52.00%, the seed investor to 16.00%, the pool to 12.00%, and the Series A holds 20.00%. Every existing holder was multiplied by the same retention ratio, 120 ÷ 150 = 0.80.

Step 5, the Series B down round. Price per share is ₹80 crore ÷ 25,000,000 = ₹32.00, a fall of 46.67% from ₹60.00, and the round issues ₹20 crore ÷ 32 = 6,250,000 shares.

Step 6, the anti-dilution adjustment. Broad-based weighted average: CSO = 25,000,000; CSP = ₹20 crore ÷ 60 = 3,333,333; CSN = 6,250,000. The ratio is 28,333,333 ÷ 31,250,000 = 0.9066667, so the new conversion price is 60 × 0.9066667 = ₹54.40. The Series A's ₹30 crore converts into 30 crore ÷ 54.40 = 5,514,706 shares, up 514,706. A full ratchet would have reset the conversion price to ₹32.00 and produced 9,375,000 shares, up 4,375,000, which is 8.50 times as many extra shares.

Step 7, the three post-Series-B cap tables.

Read it. The founders' track is 100% to 65.00% to 52.00% to 40.93%, and the largest single event in that sequence is not any of the three rounds. It is the pool convention in step 1, which cost three points before a single share was issued to an employee. Broad-based protection cost the founders 0.67 points and a full ratchet would have cost 5.11. What none of this says is anything about whether the founders are better off: their 40.93% of a ₹100 crore post-money company is ₹40.93 crore, against 65.00% of ₹30 crore, which is ₹19.50 crore. Dilution at a rising price is a good trade, and dilution at a falling price is the bill for a valuation that was set too high two rounds ago.

Worked example 2: Redwood Vector, the VC method and a preference table (US, $)

Redwood Vector, Inc. (synthetic) is a US enterprise-software company. The investor's committee assumes an exit value of $240 million in six years, requires a 10x target multiple on invested capital, will invest $8 million, and expects to retain 0.75 of today's ownership through later rounds. Compute the post-money and pre-money valuations and the implied discount rate. Then, on a cap table of 10,000,000 common and the Series A preferred bought at $1.00, tabulate the payoff under a 1x non-participating preference and under a participating preference capped at 2x, and identify the exit value at which the preferred holder is indifferent between converting and not converting.

Solution. Step 1, the venture-capital method. The stake must reach 8 × 10 = $80,000,000 at exit, so the required ownership at exit is 80 ÷ 240 = 33.3333%. Adjusting for expected dilution, the ownership required today is 33.3333 ÷ 0.75 = 44.4444%. Post-money is 8 ÷ 0.444444 = $18,000,000 and pre-money is $10,000,000.

Step 2, the discount-rate cross-check. A 10x over six years is 10^(1/6) − 1 = 46.7799% a year. Discounting the exit at that rate gives 240 ÷ 1.467799^6 = $24,000,000, which is the undiluted post-money, exactly exit ÷ target multiple. The $6,000,000 gap between $24.00 million and the $18.00 million actually offered is the expected-dilution adjustment priced in dollars.

Step 3, the cap table. Pre-money of $10,000,000 over 10,000,000 common shares gives a price of $1.00, so $8 million buys 8,000,000 preferred and the fully diluted total is 18,000,000. The preferred's as-converted ownership is 44.4444%, matching step 1.

Step 4, the two payoff functions. Non-participating: the holder takes max(preference, as-converted value), capped by the proceeds actually available. Participating with a 2x cap: the holder takes the $8 million preference plus 44.4444% of the remainder, subject to a total of $16 million, and may always convert instead.

Step 5, the three boundaries. Under the non-participating structure, conversion beats the preference when 0.444444E > 8, so at E = $18.000000 million, which is the post-money and is the sanity check that the table is right. Under the capped participating structure, the cap first binds where 8 + 0.444444(E − 8) = 16, so at E = $26.000000 million; and conversion beats the capped payout where 0.444444E = 16, so at E = $36.000000 million. That last figure is the indifference point the question asks for.

Step 6, size the structure. At the $18 million and $26 million exits the participating structure is worth $4,444,444 more to the preferred than the non-participating one, and every dollar of that comes out of the common shareholders. At $26 million the common receive $14,444,444 under one structure and $10,000,000 under the other, on the identical sale.

Read it. The preference argument is only ever about the middle of the distribution. Below $6 million the preferred takes everything under any structure; above $36 million the two structures are numerically identical. What the cap does is convert an unbounded claim into one that expires, and the exit at which it expires, $36 million, is exactly the cap divided by the as-converted ownership. What none of this prices is the behavioural effect: a management team whose common shares are worth nothing until $8 million of proceeds have gone elsewhere makes different decisions about a $10 million offer than a team without that overhang, and no payoff table captures that.

Worked example 3: Vaishnavi Engineering, a normalisation schedule (India, ₹ crore)

Vaishnavi Engineering Works Private Limited (synthetic) is a precision-castings maker near Coimbatore. Revenue ₹68.00, reported EBITDA ₹7.20, depreciation and amortisation ₹2.30, interest ₹0.80. The promoter draws ₹2.40 against a market cost of ₹0.90 for the equivalent functions, and the factory is rented from a family trust at ₹1.80 against a market rent of ₹1.05. A further ₹0.35 of personal expenses runs through the accounts, a one-off export incentive of ₹0.62 sits in other income, and ₹0.28 of legal costs on a settled land dispute were charged this year. The company expenses tooling as incurred, spending ₹1.20 this year, while its listed peers capitalise tooling and would carry ₹0.95 of amortisation on a comparable pool. Build the normalised earnings and show what the schedule is worth at a 9.0x peer median multiple.

Solution. Step 1, tie out the reported statement first. EBIT is 7.20 − 2.30 = ₹4.90, and profit before tax is 4.90 − 0.80 = ₹4.10, which matches the filed accounts. A normalisation schedule that does not begin by reproducing the reported figure has nowhere to stand.

Step 2, the five adjustments.

The uplift is 31.3889% and the margin moves from 10.5882% to 13.9118%.

Step 3, restate onto the peers' accounting policy. Capitalising tooling raises EBITDA by the ₹1.20 of spend and raises depreciation and amortisation by the ₹0.95 the peer policy would charge. Normalised EBITDA becomes ₹10.66, D&A becomes ₹3.25, and EBIT becomes ₹7.41 against ₹7.16 on the company's own policy, a difference of ₹0.25 at the EBIT line and ₹1.20 at the EBITDA line.

Step 4, price the schedule. At the 9.0x peer median: reported EBITDA gives an enterprise value of ₹64.80; own-policy normalised EBITDA gives ₹85.14; peer-policy normalised EBITDA gives ₹95.94. The full swing is ₹31.14, or 48.06% of the reported-EBITDA answer, and ₹10.80 of it is the tooling policy alone.

Step 5, carry the result down to after-tax earnings. India's concessional corporate rate of 22%, grossed up by a 10% surcharge and a 4% cess, gives an effective rate of 25.168%. That rate, its surcharge and its cess are set by the Finance Act and change, so verify Section 115BAA against the current Act on the valuation date. Normalised NOPAT is 7.41 × (1 − 0.25168) = ₹5.5451, and normalised net income to equity is (7.41 − 0.80) × (1 − 0.25168) = ₹4.9464.

Read it. Nearly half of this company's enterprise value on a comparable-multiple basis is decided before the multiple is chosen, in a schedule with five adjustment lines and one accounting-policy restatement. The most contested line will be owner compensation, because the promoter genuinely believes ₹2.40 crore is what he is worth and, as the person who built the business, may be right about his contribution while still being wrong about the adjustment, which prices the functions a successor would have to hire. What this schedule does not do is prove any of it. Every line needs a source in the file: a salary survey, two market rent quotes, the ledger detail behind the personal expenses, and the peers' own accounting-policy notes.

Worked example 4: Cardinal Fluid Systems, three discount rates on one cash flow (US, $m)

Cardinal Fluid Systems, Inc. (synthetic) is a US industrial distributor with $42 million of revenue, a top customer at 28% of revenue, and a founder who is also head of sales. Its normalised net cash flow to equity is $3.10 million, growing at 3.0%. Build its cost of equity by the build-up method, cross-check it with the expanded capital asset pricing model on a peer levered beta of 1.15, then compute the total-beta rate on a peer average regression R-squared of 0.16, and value the cash flow at the build-up and total-beta rates.

Solution. Step 1, the build-up. Risk-free 4.30% (20-year Treasury on the valuation date; re-pull and date-stamp), plus equity risk premium 5.00%, plus size premium 3.40%, plus industry risk premium 0.60%, plus a company-specific premium of 3.00% for customer concentration, key-person dependence and a short audited history. Total: 16.30%.

Step 2, the expanded capital asset pricing model. 4.30 + 1.15 × 5.00 + 3.40 + 3.00 = 16.45%. The two constructions differ by 0.15 percentage points, and the reason is mechanical rather than lucky. The expanded model folds industry risk into the beta instead of charging it as a separate layer, so the whole gap is (1.15 − 1.00) × 5.00 = 0.75 points of beta effect set against the 0.60-point industry premium the build-up charges.

Step 3, total beta. The peer regressions average an R-squared of 0.16, so the implied correlation is √0.16 = 0.40 and total beta is 1.15 ÷ 0.40 = 2.875. The cost of equity for a buyer who cannot diversify is 4.30 + 2.875 × 5.00 = 18.675%.

Step 4, read the three against each other. The total-beta rate exceeds the build-up rate by 2.375 percentage points. Restated in the build-up's own vocabulary, the total-beta construction is asserting that the correct premium above the risk-free rate, the market premium, the size premium and the industry premium is 5.375 points, not the 3.000 the appraiser charged.

Step 5, value the cash flow both ways. Next year's cash flow is 3.10 × 1.03 = $3.193 million. At 16.30% the value is 3.193 ÷ 0.1330 = $24.0075 million, an implied 7.744x on current cash flow. At 18.675% it is 3.193 ÷ 0.15675 = $20.3700 million, an implied 6.571x. The gap is $3.6375 million, or 15.15%, with no disagreement whatever about the business.

Read it. Both numbers are correct and they are answers to different questions. A private-equity buyer holding twenty positions, or a listed strategic acquirer, is correctly priced at the market-beta rate and can pay $24.01 million rationally. An individual buyer for whom this business will be most of their net worth is correctly priced at $20.37 million, and the $3.64 million difference is the market value of diversification. That is also the mechanical explanation for something founders experience as unfairness: the strategic buyer who outbids them for a competitor is not smarter, it is diversified. What total beta must never do is price a listed minority stake, because the marginal buyer of a listed share is diversified by construction.

Worked example 5: Kaplan-Schoar public market equivalent on the Konkan schedule (India, ₹ crore)

Konkan Growth Partners Fund III (synthetic) is the ₹800 crore Indian growth fund whose full schedule is built in the fund-economics node immediately before this one. Its five capital calls and its four distributions net to limited partners, after fees and after the European waterfall's carry, are given below, together with a broad Indian equity total-return index. Compute the Kaplan-Schoar public market equivalent, the direct alpha, and the index-equivalent internal rate of return on the same call schedule.

The index levels used here are an illustrative synthetic path, chosen to compound at a realistic 13.63% a year over the fund's life. Rerun the whole computation with the actual closing total-return levels of your chosen index on these dates before any of these figures leaves your desk, and name the index and its dates in the output.

Solution. Step 1, the flows and the index factors. Compound each flow forward to the terminal date, 2024-11-01, at which the index stands at 17,900.00, by multiplying by I_T ÷ I_t.

Step 2, the ratio. Calls compound to ₹1,937.174951 crore and distributions to ₹1,707.383449 crore. The fund is fully realised, so residual net asset value is zero and nothing is added to the numerator. KS-PME = 1,707.383449 ÷ 1,937.174951 = 0.881378.

Step 3, cross-check the fund's own numbers before interpreting the ratio. The same net cash flows solve to a net internal rate of return of 11.0434%, reproducing the fund-economics node's 11.043% to within four ten-thousandths of a percentage point. They also give a net total-value-to-paid-in of 1,300.32 ÷ 800 = 1.6254x, which reproduces its figure exactly. The public market equivalent is therefore being computed on a schedule that has been independently verified, which is the only condition under which a benchmarking ratio means anything.

Step 4, the index counterfactual in cash terms. Running the identical five calls into the index produces terminal wealth of ₹1,937.174951 crore against the fund's ₹1,707.383449 crore, a shortfall of ₹229.791502 crore, or 11.86% of the index-equivalent wealth. Solving for the rate that turns the call schedule into ₹1,937.17 crore gives an index-equivalent internal rate of return of 14.2857%, against the fund's 11.0434%, a gap of 3.24 percentage points a year.

Step 5, direct alpha. Taking the index-adjusted flows in the last column of the table above at their original dates and solving for their internal rate of return gives −2.6876%, which is the annualised underperformance in the form a committee can act on. Its sign agrees with the ratio, as it must.

Step 6, decompose the shortfall. Compounding the pre-carry distributions instead (₹369.20, ₹512.20, ₹475.50 and ₹68.50) gives ₹1,857.638381 crore and a gross-of-carry ratio of 0.958942. So carry and the post-investment-period fees together cost 0.0776 of public market equivalent, and the remaining 0.0411 of the shortfall to 1.00 was there before the general partner was paid anything.

Step 7, the interim reading, for contrast. At 2022-11-01, seven years into a ten-year fund, calls compound to ₹1,455.586765 crore and realised distributions to ₹906.306349 crore. Adding the ₹345 crore residual mark on the two companies still held gives an interim ratio of 0.859658, against 0.622640 on realised cash alone. The final ratio of 0.881378 is 0.021720 above the interim figure, so in this case the mark was close to fair and the fund's relative position barely moved.

Read it. Konkan returned 1.6254x net and 11.04% net, and an investor who had put the identical cash into the index on the identical dates would have finished ₹229.79 crore richer on ₹800 crore of calls. Both statements are true and neither is a criticism of the general partner's stock picking, because the index compounded at 13.63% over the same window and the fund carried an illiquidity constraint and a fee load the index did not. What the ratio adds to an internal rate of return and a multiple is the only comparison that holds timing constant, and what it cannot do is choose the index for you. A large-cap total-return index and a small-cap total-return index would give different answers on the same fund, and the answer must therefore always be quoted with its benchmark, its window and its net-or-gross basis attached.

Worked example 6: Vaishnavi valued three ways, with the discounts in order (India, ₹ crore)

Take Worked Example 3's normalised figures for Vaishnavi Engineering: peer-policy EBITDA ₹10.66, EBIT ₹7.41, D&A ₹3.25, NOPAT ₹5.5451, normalised net income to equity ₹4.9464. Maintenance capital expenditure is ₹2.60 and the working-capital charge consistent with long-run growth is ₹0.54. Net debt is ₹8.00. Adjusted net tangible assets on an equity basis are ₹21.00. Value the company by the income, market and asset approaches, reconcile them, and conclude a value for a 100% controlling interest and for a 26% minority stake.

Solution. Step 1, the discount rate. Build-up cost of equity: risk-free 6.90% (the 10-year government security yield on the valuation date; this moves daily, check against the published benchmark before use), plus an India equity risk premium of 6.50%, plus a size premium of 3.50%, plus a company-specific premium of 2.00%, giving 18.90%. Pre-tax cost of debt 9.50%, so after tax at 25.168% it is 7.1090%. At target weights of 25% debt and 75% equity the weighted average cost of capital is 0.75 × 18.90 + 0.25 × 7.1090 = 15.9523%.

Step 2, the income approach. Free cash flow to the firm is 5.5451 + 3.25 − 2.60 − 0.54 = ₹5.6551. At a long-run nominal growth rate of 5.5%, the capitalized cash flow value of invested capital is 5.6551 × 1.055 ÷ (0.159523 − 0.055) = 5.966079 ÷ 0.1045226 = ₹57.0793. Less net debt of ₹8.00, the equity value is ₹49.0793, on a control, marketable basis, because the normalisation removed the owner's excess compensation and the related-party rent, and only a controlling holder can do that. The implied enterprise multiple is 57.0793 ÷ 10.66 = 5.355x.

Step 3, the excess earnings method, for completeness. Tangible assets of ₹21.00 at a required return of 12.0% demand ₹2.5200. Normalised net income to equity of ₹4.9464 leaves excess earnings of ₹2.4264, capitalised at 22.0% to give intangibles of ₹11.0291 and a total equity value of ₹32.0291. That is ₹17.0503 below the capitalized-cash-flow answer, and the gap is a modelling difference rather than a finding: capitalising at a flat 22% with no growth term values the intangibles as a no-growth perpetuity while the income approach assumes 5.5% growth. Repairing that single inconsistency, by capitalising next year's excess earnings of 2.4264 × 1.055 at (0.22 − 0.055), gives ₹36.5142, still short. The flat capitalisation rate that would reconcile the two methods is 8.64%, below even the 12.0% the tangible assets are required to earn, and the honest conclusion is that the excess-earnings method does not carry this valuation. Revenue Ruling 68-609 says the same thing in its own words.

Step 4, the market approach. Listed Indian precision-engineering peers trade at a median enterprise-value-to-EBITDA multiple of 9.0x on the peers' own accounting policy. The subject warrants a 45% fundamental discount, decomposed as roughly 18 points for size, 12 for customer concentration, 8 for the growth differential and 7 for liquidity, giving a subject multiple of 4.95x. Enterprise value is 4.95 × 10.66 = ₹52.7670 and equity value is ₹44.7670, on a marketable minority basis.

Step 5, the asset approach as a floor. Adjusted net asset value on a going-concern basis is plant at ₹16.50 plus net working capital at ₹18.00 plus other assets at ₹2.50 less debt of ₹8.00, or ₹29.00. On an orderly liquidation, plant realises 45% (₹7.425), receivables and inventory realise 70% (₹12.600), other assets realise ₹1.00, and after debt of ₹8.00 and wind-down costs of ₹2.00 the floor is ₹11.025, which is 22.46% of the income-approach value. The floor does not bind.

Step 6, reconcile rather than average. Grossing the marketable-minority market value to a control basis at a 25% control premium gives 44.7670 × 1.25 = ₹55.9588. The income approach's control value of ₹49.0793 sits 9.63% above the marketable-minority figure and 12.29% below the control-adjusted market figure, so the two approaches bracket each other rather than contradict. The income approach carries the conclusion, because the subject's capital intensity, customer concentration and growth are modelled explicitly in it and are only proxied by a fundamental discount in the market approach. The fact that would move the answer is the top customer: a signed multi-year contract would reduce the company-specific premium and narrow the fundamental discount at the same time, and the value would move on both approaches together.

Step 7, the two conclusions, with the adjustments in order. For a 100% controlling interest with no buyer currently in sight, start from the control value of ₹49.0793. Apply a discount for lack of marketability sized for a control block at 15%, justified by an expected twelve-to-eighteen-month sale process and its transaction costs, and the answer is ₹41.7174.

For a 26% minority stake being transferred inside the family, step down from control to marketable minority first, using the reciprocal of the 25% control premium, which is a discount for lack of control of exactly 20.00%: 49.0793 × 0.80 = ₹39.2635. The 26% pro-rata slice is ₹10.2085, and a minority discount for lack of marketability of 30% gives ₹7.1459. Cross-checking by the other route, 26% of the marketable-minority market value of ₹44.7670 is ₹11.6394, and after the same 30% discount, ₹8.1476. The two routes differ by ₹1.0016, or 14.02%, which is the same gap between the two approaches expressed on a quarter of the company.

Read it. A naive answer of 26% of the control value, ₹12.7606, overstates the defensible figure by 78.57%, and every rupee of that overstatement comes from skipping two adjustments that each have a name and a formula. The mirror-image error is worth as much: applying the marketability discount twice, once inside a private-transaction multiple and once explicitly, would take the ₹7.1459 down to ₹5.0022 and destroy ₹2.1438 of value that nothing justifies. What no part of this example proves is the 15% and the 30%. Those are judgments, and their entire defence is a written record naming which Mandelbaum factors moved them and in which direction, produced before anyone said what number would be welcome.

Worked example 7: Summit Ridge Fasteners, the bridge and the layer below it (US, $m)

Summit Ridge Fasteners (synthetic) is a US industrial fastener distributor. A sponsor buys it at 8.0x an entry EBITDA of $18.0 million, funded with $86.4 million of net debt. Five years later it exits at 8.5x an EBITDA of $30.0 million with net debt of $40.0 million. Revenue grew from $120.0 million to $169.475 million, of which $19.0 million was organic volume, $3.475 million came from a 2.5% price increase, and $27.0 million came from two bolt-ons acquired for $16.0 million of cash at a 17.0% EBITDA margin. Net working capital fell from 22.0% to 16.5% of revenue. Build the value-creation bridge and then decompose the EBITDA-growth lever.

Solution. Step 1, the equity outcome. Entry enterprise value is 8.0 × 18.0 = $144.0m, so the sponsor's cheque is 144.0 − 86.4 = $57.6m at entry leverage of 4.80x. Exit enterprise value is 8.5 × 30.0 = $255.0m, so exit equity is 255.0 − 40.0 = $215.0m at exit leverage of 1.33x. That is a MOIC of 215.0 ÷ 57.6 = 3.7326x and, over five years, an internal rate of return of 30.1377%.

Step 2, the three financial levers. Pricing EBITDA growth at the entry multiple and multiple expansion on exit EBITDA: growth = (30.0 − 18.0) × 8.0 = $96.0m; expansion = (8.5 − 8.0) × 30.0 = $15.0m; net-debt reduction = 86.4 − 40.0 = $46.4m. The three sum to $157.4m, which equals 215.0 − 57.6 exactly, so the bridge ties.

State the convention with the answer. The mirror convention, pricing growth at the exit multiple and expansion on entry EBITDA, gives $102.0m and $9.0m, and the two differ by the cross term of $6.0m, which belongs to neither lever. The "luck" line is therefore either $9.0m or $15.0m depending on a choice the presenter makes, a swing of 66.67% in the number that separates skill from a rising market.

Step 3, decompose the EBITDA-growth lever. Organic volume of $19.0m of revenue at the entry margin of 15.0% contributes $2.850m, an implied organic revenue growth of 2.98% a year. The 2.5% price increase on the post-volume base of $139.0m contributes $3.475m with full flow-through. The bolt-ons contribute 27.0 × 17.0% = $4.590m. The cost programme is the residual, $1.085m. The four sum to $12.000m, which is exactly the EBITDA delta.

Exit margin is 30.0 ÷ 169.475 = 17.70% against 15.00% at entry, and the entire margin expansion is attributable to pricing and to the bolt-ons' higher margin.

Step 4, the two levers that hide in the debt line. Cash generated over the hold was $46.4m of net-debt reduction plus the $16.0m spent on bolt-ons, so $62.4m in total. Net working capital at entry was 22.0% of $120.0m = $26.400m; at exit it was 16.5% of $169.475m = $27.963m. In absolute terms working capital rose by $1.563m, and yet against the counterfactual of an unchanged 22.0% ratio, which would have required $37.285m, the improvement released $9.321m, or 14.94% of all the cash generated. The bolt-ons were bought at 16.0 ÷ 4.590 = 3.49x EBITDA and carried out at 8.5x, worth $39.015m against $16.0m of cash, a multiple arbitrage of $23.015m before any integration cost.

Step 5, price the re-rating. Exiting at the entry multiple of 8.0x instead of 8.5x gives equity of $200.0m, a MOIC of 3.4722x and an internal rate of return of 28.2689%, so the half-turn was worth 1.87 percentage points of annual return. At 7.0x the numbers are $170.0m, 2.9514x and 24.1667%; at 9.0x they are $230.0m, 3.9931x and 31.9049%.

Read it. The headline is a 30.1% internal rate of return. Underneath it, 9.53% of the value created came from a multiple the sponsor did not earn and 29.48% from deleveraging, which is rented from lenders rather than created. The remaining 60.99% came from EBITDA growth, whose largest single component is bolt-on acquisitions bought cheaply and carried out inside a bigger platform. Pricing, at 28.96% of that growth, is the lever hardest to repeat, because a market absorbs one price increase more easily than two.

Put the question to the next fund's pitch in exactly that form. Which of the four levers is still available in the asset you are about to buy?

Practice set

Work each problem fully before reading its solution. A calculator is assumed and every figure below was computed in Python for this node. Share counts are exact; currency answers are correct to the precision shown, and anything within 0.5% of the stated figure counts as right unless the problem says otherwise. The guided problems come first and carry their scaffolding in the solution; from P5 onward you are on your own.

P1 (guided). The venture-capital method with dilution. A fund will invest $5 million in a company it believes can exit for $90 million in five years, and requires an 8x target multiple. It expects to retain 70% of today's ownership through later rounds. Compute the required ownership at exit, the ownership required today, the post-money and pre-money valuations, and the implied annual discount rate.

Solution. The stake must reach 5 × 8 = $40 million, so required ownership at exit is 40 ÷ 90 = 44.4444%. Ignoring dilution, post-money would be 5 ÷ 0.444444 = $11,250,000 and pre-money $6,250,000. Adjusting for expected dilution, ownership required today is 44.4444 ÷ 0.70 = 63.4921%, so post-money is 5 ÷ 0.634921 = $7,875,000 and pre-money is $2,875,000. The implied rate is 8^(1/5) − 1 = 51.5717% a year. The dilution adjustment cost the founders $3.375 million of pre-money valuation, which is why the retention assumption is worth arguing about.

P2 (guided). The pool shuffle, both ways. Founders hold 9,000,000 shares. An investor offers ₹4.50 crore at an ₹18 crore pre-money valuation with a 12% post-money option pool. Compute the founders' ownership with the pool created pre-money and with it created post-money, the price per share in each case, and the value transferred.

Solution. Post-money is 18 + 4.5 = ₹22.50 crore, so the investor holds 4.5 ÷ 22.5 = 20.00% either way. With the pool pre-money the founders hold 100 − 20 − 12 = 68.00%, and price per share is 22.5 crore ÷ (9,000,000 ÷ 0.68) = ₹17.00. With the pool post-money, price per share is 18 crore ÷ 9,000,000 = ₹20.00, the investor takes 2,250,000 shares, and the 12% pool is cut from the combined 11,250,000 shares, giving a total of 12,784,090.91 and founders 70.40%. The gap is 2.40 percentage points, worth 0.024 × ₹22.50 crore = ₹54,00,000.

P3 (guided). Price per share and the round mechanics. A company with 15,000,000 fully diluted shares raises $12 million at a $45 million pre-money valuation, with no new option pool. Compute the price per share, the new shares issued, the post-round share count, the investor's percentage and the post-money valuation.

Solution. Price per share is 45,000,000 ÷ 15,000,000 = $3.00. New shares are 12,000,000 ÷ 3.00 = 4,000,000. Post-round shares are 19,000,000, so the investor holds 4,000,000 ÷ 19,000,000 = 21.0526%, and the post-money valuation is 19,000,000 × 3.00 = $57,000,000, which is also 45 + 12. The two routes to post-money must agree, and if they do not, the pre-money share count is wrong.

P4. Three rounds of dilution. A founder starts with 60.00% after the seed round's pool. The company then raises at ₹20 crore pre-money for ₹5 crore, then ₹60 crore pre-money for ₹15 crore, then ₹150 crore pre-money for ₹50 crore. Where does the founder finish, and what is the cumulative retention factor?

Solution. Retention ratios are 20 ÷ 25 = 0.80, 60 ÷ 75 = 0.80, and 150 ÷ 200 = 0.75. The founder goes 60.00% to 48.00% to 38.40% to 28.80%. Cumulative retention is 0.80 × 0.80 × 0.75 = 0.48, and 60.00 × 0.48 = 28.80 confirms it. Ownership compounds multiplicatively, so the order of the rounds does not matter to the endpoint.

P5. A participating preference at one exit. An investor put in $10 million for 30% on an as-converted basis, with a 1x preference. The company sells for $25 million. Compute the split under a 1x non-participating preference and under an uncapped participating preference.

Solution. Non-participating: the holder chooses max($10m, 0.30 × 25 = $7.5m) = $10,000,000, leaving common $15,000,000. Participating: the holder takes $10m plus 30% of the remaining $15m, or 10 + 4.5 = $14,500,000, leaving common $10,500,000. Participation is worth $4,500,000 at this exit, all of it taken from the common shareholders.

P6. Both crossover points. Same $10 million preference and 30% as-converted stake. Find the exit at which a 1x non-participating holder is indifferent between converting and taking the preference; then, for a participating preference capped at 3x, find the exit at which the cap first binds and the exit at which converting becomes better.

Solution. Non-participating indifference: 0.30E = 10, so E = $33,333,333. Capped participating, cap = 3 × 10 = $30m: the cap binds where 10 + 0.30(E − 10) = 30, so E − 10 = 66.667 and E = $76,666,667. Conversion beats the capped payout where 0.30E = 30, so E = $100,000,000. Between $76.67m and $100m the holder is stuck at $30m while the company's value rises, which is the whole economic content of a cap.

P7. Broad-based weighted average. A Series A bought ₹24 crore of preferred at ₹40 a share. The company now issues ₹12 crore of new stock at ₹25 a share, with 12,000,000 shares outstanding on a broad fully diluted basis immediately before. Compute the new conversion price and the extra shares the Series A receives.

Solution. CSP = 12 crore ÷ 40 = 3,000,000; CSN = 12 crore ÷ 25 = 4,800,000. NCP = 40 × (12,000,000 + 3,000,000) ÷ (12,000,000 + 4,800,000) = 40 × 15,000,000 ÷ 16,800,000 = ₹35.714286, a fall of 10.7143%. The Series A's shares go from 24 crore ÷ 40 = 6,000,000 to 24 crore ÷ 35.714286 = 6,720,000, an extra 720,000.

P8. The same facts under a full ratchet. Using P7's numbers, compute the extra shares under a full ratchet and express the result as a multiple of the broad-based answer.

Solution. A full ratchet resets the conversion price to the new round's ₹25.00, so the Series A's shares become 24 crore ÷ 25 = 9,600,000, an extra 3,600,000. That is 5.00 times the broad-based answer of 720,000 extra shares, on identical facts, from a clause that differs by two adjectives. The ratchet also ignores the size of the new issue entirely: a ₹1 lakh round at ₹25.00 would trigger exactly the same adjustment.

P9. A normalisation schedule. A company reports EBITDA of ₹5.40 crore. Owner compensation is ₹1.10 crore above the market cost of the functions; ₹0.40 crore of personal expenses run through the accounts; a one-off insurance recovery of ₹0.55 crore sits in other income; ₹0.20 crore of non-recurring legal costs were charged. Compute normalised EBITDA and the value of the schedule at a 7.5x peer multiple.

Solution. Normalised EBITDA is 5.40 + 1.10 + 0.40 − 0.55 + 0.20 = ₹6.55 crore, an uplift of 21.2963%. At 7.5x, enterprise value moves from 7.5 × 5.40 = ₹40.50 crore to 7.5 × 6.55 = ₹49.125 crore, so the schedule is worth ₹8.625 crore. The insurance recovery is the line most sellers omit, and it is the only one that runs against them.

P10. A build-up rate and what a company-specific premium costs. Risk-free 4.50%, equity risk premium 5.25%, size premium 3.10%, company-specific premium 2.50%. Normalised cash flow to equity is $2.40 million growing at 3.0%. Compute the cost of equity, the capitalized cash flow value, and the value the company-specific premium removes.

Solution. Cost of equity is 4.50 + 5.25 + 3.10 + 2.50 = 15.35%. Next year's cash flow is 2.40 × 1.03 = $2.472 million, so value is 2.472 ÷ (0.1535 − 0.03) = 2.472 ÷ 0.1235 = $20.0162 million. Without the 2.50% premium the rate is 12.85% and the value is 2.472 ÷ 0.0985 = $25.0964 million. The premium costs $5.0803 million, which is a fifth of the company's value resting on one unsourced judgment, and it is why that judgment must be written down.

P11. Total beta. A peer group has an average levered beta of 1.05 and an average regression R-squared of 0.20. Risk-free is 4.50% and the equity risk premium is 5.25%. Compute the total beta, the market-beta cost of equity, the total-beta cost of equity, and the gap.

Solution. The implied correlation is √0.20 = 0.447214, so total beta is 1.05 ÷ 0.447214 = 2.347871. Market-beta cost of equity is 4.50 + 1.05 × 5.25 = 10.0125%; total-beta cost of equity is 4.50 + 2.347871 × 5.25 = 16.8263%. The gap is 6.8138 percentage points, and it is entirely a statement about the buyer, not about the company. Use the first for a diversified acquirer and the second for an individual putting most of their wealth into this one asset.

P12. Discount for lack of control. Compute the implied discount for lack of control at control premiums of 20%, 30% and 35%. Then take a ₹60 crore control value with a 30% control premium and compute the marketable-minority equivalent both correctly and by the common error, and state which direction the error runs.

Solution. DLOC = 1 − 1 ÷ (1 + premium), so the three answers are 16.6667%, 23.0769% and 25.9259%. At 35% the error from subtracting the premium is 9.0741 percentage points. On ₹60 crore at a 30% premium the correct marketable-minority value is 60 ÷ 1.30 = ₹46.1538 crore; subtracting 30% gives 60 × 0.70 = ₹42.00 crore. Subtracting therefore applies too large a discount and understates the minority value, here by ₹4.1538 crore. Read the answer back every time: the correct discount is always smaller than the premium it comes from.

P13. Excess earnings. Net tangible assets on an equity basis are ₹14.0 crore, normalised net income to equity is ₹3.10 crore, the required return on tangible assets is 11.0% and the intangible capitalisation rate is 20.0%. Compute the equity value and the implied price-to-earnings ratio.

Solution. The tangible assets demand 14.0 × 0.11 = ₹1.54 crore, leaving excess earnings of 3.10 − 1.54 = ₹1.56 crore. Capitalised at 20.0% the intangibles are worth ₹7.80 crore, so equity value is 14.0 + 7.80 = ₹21.80 crore, an implied price-to-earnings ratio of 7.0323x. Raise the intangible rate to 25% and the answer falls to ₹20.24 crore; nothing observable pins that rate down, which is exactly why the method is a check rather than a conclusion.

P14. A public market equivalent from four numbers. A fund called $50 million on 1 January 2018 when the index stood at 1,000 and $50 million on 1 January 2019 at 1,100, and distributed $150 million on 1 January 2023 at 1,600, with nothing left in the portfolio. Compute the total-value-to-paid-in multiple, the Kaplan-Schoar public market equivalent, and the fund's internal rate of return against the index-equivalent rate.

Solution. TVPI is 150 ÷ 100 = 1.50x. Calls compound to 50 × 1.6 + 50 × (1600 ÷ 1100) = 80 + 72.727273 = $152.727273 million; the distribution is already at the terminal date, so it stays at $150 million. KS-PME = 150 ÷ 152.727273 = 0.982143. The fund's internal rate of return is 9.3982% against an index-equivalent 9.8347%, a shortfall of 0.4364 percentage points a year. A 1.50x multiple and a 9.4% rate both look respectable and the fund still lost narrowly to the index, which is the entire reason the ratio exists.

P15. A value-creation bridge that must tie. A sponsor buys at 7.5x an EBITDA of ₹22 crore with ₹99 crore of net debt, and exits five years later at 7.5x an EBITDA of ₹31 crore with net debt of ₹55 crore. Compute the multiple on invested capital, the internal rate of return, and the three bridge components, and show that they tie.

Solution. Entry enterprise value 7.5 × 22 = ₹165 crore, so entry equity is ₹66 crore. Exit enterprise value 7.5 × 31 = ₹232.5 crore, so exit equity is ₹177.5 crore. MOIC is 177.5 ÷ 66 = 2.6894x and the internal rate of return is 21.8796%. Bridge: EBITDA growth (31 − 22) × 7.5 = ₹67.5 crore; multiple expansion (7.5 − 7.5) × 31 = ₹0; deleveraging 99 − 55 = ₹44 crore. The three sum to ₹111.5 crore, which equals 177.5 − 66. Deleveraging supplied 39.46% of everything created, and it was rented from the lenders rather than produced by the business.

P16. Exit route and the multiple-versus-rate conflict. A fund invested ₹100 crore on 1 June 2019. Route A sells the whole position for ₹250 crore on 1 June 2024. Route B sells a slice for ₹75 crore on 1 June 2024 into a continuation vehicle and the balance for ₹210 crore on 1 June 2027. Compute the multiple and the internal rate of return for each and say who prefers which.

Solution. Route A: TVPI 2.50x, internal rate of return 20.0884%, and distributions to paid-in reach 2.50x on the 2024 date. Route B: TVPI 2.85x, internal rate of return 15.9468%, and distributions to paid-in stand at only 0.75x on the 2024 date. Route B returns 0.35x more capital and 4.14 percentage points a year less. An investor measured on multiples prefers B, an investor measured on rate of return prefers A, and a general partner approaching the end of a fund's term has reasons of their own, which is why continuation-vehicle pricing is set by a third party rather than by the general partner.

Applied mini-project

Value one private company three ways, then dilute it.

Two deliverables, five artefacts, and every artefact carries a numbered test it must pass before it counts toward a score. Use a spreadsheet or Python, show every tie-out rather than asserting it, label every invented entity synthetic, and trace every market input to a named primary source with the date you pulled it.

Pick a real private company you can get information about, or build a synthetic one with a full set of accounts. A family business you know, a company whose accounts are filed with the Registrar of Companies or Companies House, or a portfolio company disclosed in a listed fund's report all work.

Choose something real if you possibly can. Synthetic accounts teach the mechanics and never the argument.

The first artefact is a normalisation schedule running in both directions, with owner compensation, related-party charges, personal expenses, non-recurring items and at least one accounting-policy restatement onto the comparable peers' policy, each line carrying its own evidence. Test 1: the schedule opens at the reported figure and reproduces it from the filed accounts. Test 2: at least one adjustment is negative, and the schedule carries a line naming what could not be adjusted.

The second artefact is a discount rate built three ways, by the build-up method, by the expanded capital asset pricing model on a stated peer beta, and by total beta on a stated peer R-squared. Test 3: the build-up and the expanded model agree within one percentage point, or the write-up explains the gap in terms of a named input. Test 4: every layer is sourced and dated, and no risk is charged in two places.

The third artefact is a three-approach valuation with a written reconciliation: an income approach, a market approach with the fundamental discount decomposed into named components, and an asset or liquidation floor. Test 5: the income and market approaches are labelled with their level of value and the labels are correct. Test 6: the reconciliation names which approach carries the conclusion, explains the gap mechanically, and states the one fact that would move the answer. Averaging the three scores zero on this row.

The fourth artefact applies the discounts in order, once each, for two interests: a 100% controlling stake and a minority stake of your choosing. Test 7: the discount for lack of control is computed as the reciprocal of the control premium and the two routes to the minority value are both shown, with the gap between them explained.

The fifth artefact is a cap table after a hypothetical round, with a stated pre-money valuation, a stated raise and a stated option pool. Show the pool created pre-money and post-money, the founder dilution separated from the pool effect, and a liquidation-preference payoff table across at least six exit values under two structures. Test 8: proceeds sum to the exit value at every row. Test 9: the two crossover points are computed algebraically and confirmed on the table.

Write a one-page reconciliation memo carrying the concluded value for each interest, the five test results, a named list of every assumption, and a closing paragraph on the single assumption the conclusion is most sensitive to, with that sensitivity quantified.

Rubric (21 points; pass at 17 with no zero on any row).

Any artefact whose test fails scores zero on its row, whatever the memo looks like. A valuation is a chain of adjustments and the chain has the strength of its weakest tie-out, which is why presentation earns nothing here and a failed identity costs everything.

Reading & resources

Read each entry here as an anchor to verify against, never as a source to lift a number from without tracing it yourself.

Private-company valuation, the standing references:

• Shannon Pratt and Roger Grabowski, Cost of Capital: Applications and Examples. The exhaustive treatment of the build-up method, size premia and the private-company adjustments, and the reference behind most appraisal practice in this area. [Paid] [Advanced]
• Shannon Pratt, Valuing a Business. The standard desk reference for the three approaches, the levels-of-value ladder and the discount literature. [Paid] [Intermediate to Advanced]
• Aswath Damodaran's private-company valuation material and his published datasets on betas, total betas by sector, country risk premia and industry costs of capital. The datasets are updated each January and the accompanying posts explain how each column was built. [Free] [All levels]
• US Internal Revenue Service Revenue Ruling 59-60 and Revenue Ruling 68-609. The first requires a reasoned, factor-based approach to valuing closely-held stock; the second sets out the excess earnings formula and calls it a method of last resort. Both are short, both are free, and both are still cited in litigation. [Free] [Intermediate]
• ICAI Valuation Standards and the SEBI and Companies Act provisions governing registered valuers in India, which set out who may sign a valuation and what the report must contain. Verify the current version, since the registered-valuer regime and its thresholds have been amended repeatedly. [Free] [Intermediate]

Venture terms and cap tables:

• Brad Feld and Jason Mendelson, Venture Deals. The clearest available account of what each term-sheet clause does economically, including the pool shuffle and both anti-dilution families. [Paid] [Beginner to Intermediate]
• The National Venture Capital Association model legal documents, and the Indian Venture and Alternate Capital Association materials for the Indian equivalents. Reading a real certificate of incorporation's conversion-price section once is worth more than any summary of it. [Free] [Intermediate]
• SEBI (Alternative Investment Funds) Regulations, 2012 and the Companies Act, 2013 provisions on preference shares and share issue pricing, which shape what an Indian term sheet can and cannot say. Verify the current amended text. [Free] [Intermediate]

Fund benchmarking:

• Steven Kaplan and Antoinette Schoar, "Private Equity Performance: Returns, Persistence, and Capital Flows", the paper that introduced the public market equivalent, freely available through SSRN and NBER. Read it for the construction rather than for its now-dated return figures. [Free] [Advanced]
• Oleg Gredil, Barry Griffiths and Rüdiger Stucke, "Benchmarking Private Equity: The Direct Alpha Method", which converts the wealth ratio into an annualised excess return and explains the short-position problem the PME+ and modified-PME variants exist to fix. [Free] [Advanced]
• Robert Harris, Tim Jenkinson and Steven Kaplan's series on private-equity performance, which applies public market equivalents across large samples and is the honest place to calibrate what an ordinary result looks like. [Free] [Advanced]
• The ILPA Reporting Template and the CFA Institute's Global Investment Performance Standards provisions for private-markets composites, which govern how these figures must be presented to investors. [Free] [Intermediate]

One exercise beats another chapter here. Take any term sheet you can find, in a model-document set or a published example, and compute two numbers before you read another word of it: the founders' post-round percentage under both pool conventions, and the exit value at which the preferred holder becomes indifferent between converting and taking its preference. Twenty minutes with a calculator on a real document will teach you more about venture economics than any summary of one, because you will discover how many of the inputs you need are not stated anywhere on the page.

Teach it back & journal

Feynman prompt. Sit down with a founder who is about to sign a seed term sheet and explain the option-pool clause using only the two share counts. Do not use the word dilution. Draw the pre-money denominator twice, once with the pool inside it and once without, and let them compute 13,000,000 divided by each total themselves. When they arrive at 65.00% and 68.00%, ask them what the three-point difference is worth on a ₹30 crore post-money, and wait for them to say ₹90 lakh rather than saying it for them. Then ask the question that turns the arithmetic into a negotiating position: what hiring plan justifies this pool size, and what would a pool sized to that plan actually be?

Second half, and it needs a different listener. Explain to a family that has been offered a price for their 26% of a private engineering company why the answer is not 26% of what the whole business is worth. Make them say the two questions out loud, whether the holder can direct the cash flows and whether the holder can convert the stake to cash, and make them name which of the two each adjustment answers. If they can afterwards tell you why a 30% control premium implies a 23% discount rather than a 30% one, and can say which direction the error runs, they have the concept.

Journal prompt. For two weeks, every time you meet a private-markets number in a pitch, a report or a conversation, write down three things before you use it. What level of value is it (marketable minority, control, nonmarketable), what convention produced it (which pool treatment, which bridge convention, which index, which accounting policy), and what single fact would move it most. Then answer in writing: how many of them came with their convention attached, and how many public market equivalents named their index? Count separately the discounts for lack of marketability that named an evidence family rather than a round number.

Then write a paragraph on the uncomfortable part. This node's second half is where the program's process-over-outcome philosophy meets a genuinely hard case, because the process here does not produce a number that can be checked against reality until years later, and sometimes never. A cap table can be audited; a company-specific risk premium cannot. Which of the four disciplines here (a schedule that runs in both directions, a discount charged in exactly one place, a reconciliation that explains rather than averages, a benchmark quoted with its index) are you most likely to skip when the client has already told you what they hope the answer is? Write down what will go wrong when you skip it, and one thing you will change about how you write a valuation file so that skipping is harder than doing.