The Finnerty average-strike put formula worked by hand, its grid by volatility and term, and how it sits beside the Chaffe and Longstaff models.
The Finnerty model prices the discount for lack of marketability as an average-strike put: DLOM = e−qT [N(v√T/2) − N(−v√T/2)], where v√T is an adjusted volatility built from σ²T. At an illustrative 45 per cent volatility and 2.5 years to liquidity, the Finnerty DLOM is 15.6 per cent, against 22.0 per cent for the Chaffe protective put on the same inputs. However high the inputs go, Finnerty never passes about 32 per cent.
Worked in full in The Private Markets Valuation Specialist by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Option models for marketability discounts answer one question: what would it cost to buy back the ability to sell? They disagree because they define that ability differently. Chaffe's protective put insures the end-of-period value against falling below today's. Longstaff's lookback assumes a perfect seller who would have sold at the peak. Finnerty assumes the holder could have sold at some point during the restriction and would have received, on average, the average price. That middle assumption is why many valuers prefer Finnerty for fair value work, and why it produces the lowest number of the three.
| Input | Value |
|---|---|
| Marketable value of the holding | 100.0 |
| Volatility, σ, from listed peers | 45% |
| Expected time to liquidity, T | 2.5 years |
| Dividend yield, q | 0% |
| Risk-free rate (used by Chaffe only) | 4% |
Step 1: the total variance over the period.
σ²T = 0.45² × 2.5 = 0.5063; eσ²T = 1.6591
Step 2: the adjusted variance of the average price. The average of a lognormal path is less volatile than its end point, and Finnerty's correction captures that.
v²T = σ²T + ln[2(eσ²T − σ²T − 1)] − 2 ln(eσ²T − 1)
= 0.5063 + ln(0.3056) − 2 × ln(0.6591) = 0.5063 − 1.1854 + 2 × 0.4169 = 0.1547
v√T = √0.1547 = 0.3933
Step 3: the put value as a share of marketable value.
DLOM = e−qT × [N(0.1967) − N(−0.1967)] = 1 × (0.5780 − 0.4220) = 15.6%
Excel, with σ²T in B1: =NORM.S.DIST(SQRT(B1+LN(2*(EXP(B1)-B1-1))-2*LN(EXP(B1)-1))/2,TRUE)*2-1
The marketable value of 100.0 becomes a non-marketable value of 84.4. Note what is absent: no risk-free rate, no strike, no share price. The 2012 formula depends only on σ, T and q, which makes it easy to audit and leaves volatility and the term as the only judgments to defend.
| Volatility | Finnerty | Chaffe protective put | Longstaff lookback |
|---|---|---|---|
| 30% | 10.7% | 13.5% | 43.8% |
| 45% | 15.6% | 22.0% | 70.6% |
| 60% | 20.0% | 30.2% | 101.0% |
| 90% | 26.7% | 45.2% | 173.5% |
Longstaff is an upper bound, not an estimate: at 60 per cent volatility it exceeds 100 per cent, a discount larger than the asset. Chaffe sits between the two and grows roughly with volatility. Finnerty grows more slowly and flattens: at σ²T of 4 it is 30.9 per cent, at 8 it is 32.2, and at 40 it is still 32.3. That ceiling is the model's best-known property and the most frequent reason a reviewer challenges a discount above it.
| Volatility | 1 year | 2 years | 2.5 years | 3 years | 5 years |
|---|---|---|---|---|---|
| 30% | 6.8% | 9.6% | 10.7% | 11.7% | 14.8% |
| 45% | 10.2% | 14.1% | 15.6% | 16.9% | 21.0% |
| 60% | 13.3% | 18.2% | 20.0% | 21.5% | 25.8% |
| 90% | 19.1% | 24.9% | 26.7% | 28.1% | 31.0% |
The two inputs work only through their product σ²T, so doubling the term is worth about the same as raising volatility by 41 per cent. A dividend shrinks the result in proportion: at a 3 per cent yield, e−qT is 0.9277 and the 15.6 per cent becomes 14.5.
The term should be the expected time to a realistic liquidity event for this holding, not the remaining fund life. Moving it from 2.5 to 5 years adds 5.4 points at 45 per cent volatility, almost as much as the 6.4-point gap between Finnerty and Chaffe that the choice of model is usually argued over.
Volatility is the input that does the work, and it has to be evidenced like any other. Take the equity volatility of the same listed peers used in the market approach, measured over a look-back window about as long as the expected term, and say why the subject should sit above or below their median: smaller, more concentrated or more leveraged companies are usually more volatile than their listed peers, not less. Record the figure and its source in the memorandum, because a reviewer who re-runs the formula will start there.
Treating the three models as interchangeable estimates and picking the one that delivers the number already in the draft. They are not three measurements of one quantity; they are three different contracts. A memorandum should say why the holder's position resembles an average-strike put (it could sell at some point during the period, at an uncertain time) rather than a fixed-date put, and should then use one model consistently from period to period. Switching from Finnerty to Chaffe between quarters moves this holding's discount from 15.6 to 22.0 per cent with no change in the facts, which an auditor will read as a change in method rather than a change in value.
The book treats the marketability discount as one of the hardest judgment calls in fair value; the free workbook for this case carries a protective-put sheet that argues with the conventional discount. What the protective-put construction can and cannot evidence is worked in what a marketability discount can actually reach.
Because it prices a different right. Chaffe's put protects the holder against the value at the end of the restriction falling below today's; Finnerty's average-strike put assumes the holder could have sold at some point during the period, at the average price, which is worth less. At 45 per cent volatility and 2.5 years, Finnerty gives 15.6 per cent and Chaffe 22.0.
It levels off at about 32 per cent however high the volatility or long the term. At σ²T of 8 the illustrative calculation gives 32.2 per cent, and at 40 still only 32.3. A memorandum that needs a discount above that level cannot rely on Finnerty and has to evidence it some other way.
No. The 2012 formula depends only on volatility, the term to liquidity and the dividend yield. A 3 per cent yield scales the discount by e^(-qT), 0.9277 over 2.5 years, taking the illustrative 15.6 per cent to 14.5. The Chaffe protective put, by contrast, does need a risk-free rate.
This article is one calculation from The Private Markets Valuation Specialist. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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