The model is a standard option formula; the mark is decided by the share value, the term the warrant is expected to live, and the dilution its own exercise causes.
Price each warrant as a call option on the ordinary share's fair value, over the time it is expected to stay alive rather than its contractual life, then adjust for the dilution its exercise causes. On illustrative terms, a warrant with a 5.00 strike over a share worth 4.00, at 55 per cent volatility and a three-year expected term, is worth 1.30 a warrant, or 0.65 million for 0.5 million warrants. Using the seven-year contractual term instead gives 2.13, an overstatement of 63.7 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 →
Warrants appear in private company capital structures as the sweetener on venture debt, as a fee to a placement agent, or as part of a rescue. They are out of the money more often than not, so their intrinsic value is usually zero, and a fund that holds them must still carry them at fair value. The method is a standard option model; the judgement is in three of its inputs.
| Input | Value | Source of the judgement |
|---|---|---|
| Fair value of one ordinary share | 4.00 | The equity allocation, not the last round price |
| Strike | 5.00 | Warrant instrument |
| Contractual term | 7 years | Warrant instrument |
| Expected term | 3 years | Lapses if unexercised on a sale, expected in about three years |
| Volatility | 55% | Listed peers' equity volatility over the term |
| Risk-free rate | 4.0% | Government yield matching the term |
| Warrants; shares outstanding, fully diluted | 0.5m; 20.0m | Capitalisation table |
C = S × N(d1) − K × e−rT × N(d2), with d1 = [ln(S/K) + (r + σ²/2)T] ÷ σ√T and d2 = d1 − σ√T
ln(4.00/5.00) = −0.2231; (0.04 + 0.55²/2) × 3 = 0.5738; σ√T = 0.9526. So d1 = 0.3680 and d2 = −0.5846, N(d1) = 0.6436 and N(d2) = 0.2794.
C = 4.00 × 0.6436 − 4.4346 × 0.2794 = 2.5743 − 1.2391 = 1.3352.
In Excel: =S*NORM.S.DIST(d1,TRUE)-K*EXP(-r*T)*NORM.S.DIST(d2,TRUE), with d1 and d2 in their own cells so a reviewer can follow them.
A listed call option is written by another investor; a warrant is written by the company, and exercising it issues new shares at less than their value. The holders of the existing shares bear that dilution, and so, through the share price, does the warrant itself.
Warrant value = C × N ÷ (N + M), with N shares outstanding and M warrants
20.0 ÷ (20.0 + 0.5) = 0.9756, so 1.3352 × 0.9756 = 1.30 a warrant, and 0.5 million warrants are worth 0.65 million.
The factor is small here because the warrants are 2.5 per cent of the share count. On a rescue financing where warrants run to a fifth or more of the shares, it is not.
| Term used | Option value | After dilution | Total, 0.5m warrants | Share of share value |
|---|---|---|---|---|
| Expected, 3 years | 1.34 | 1.30 | 0.65m | 32.6% |
| Contractual, 7 years | 2.19 | 2.13 | 1.07m | 53.3% |
The intrinsic value is zero: the share is worth 4.00 and the strike is 5.00. The fair value is not, because the share has three years in which to move above the strike. The term drives the answer more than anything else, and the warrant instrument usually settles it: many venture debt warrants must be exercised on a sale of the company or lapse, so their life ends at the expected exit, not at the date printed on them. Using the seven years overstates the mark by 0.83 a warrant, 0.41 million in total.
| Volatility | 2 years | 3 years | 5 years | 7 years |
|---|---|---|---|---|
| 45% | 0.77 | 1.05 | 1.48 | 1.81 |
| 55% | 0.99 | 1.30 | 1.78 | 2.13 |
| 65% | 1.21 | 1.55 | 2.06 | 2.42 |
Ten points of volatility move the three-year value by about 0.25, or 19 per cent; four more years of term move it by 64 per cent. Both belong in the valuation memo, and the term needs the stronger evidence: the sale process, the sponsor's holding period, the lapse clause.
Warrants over preferred shares. Many venture debt warrants are over the latest preferred series, not ordinary shares. Then the underlying carries a liquidation preference, and a plain option model on a single share price misses it. Value the warrant inside the equity allocation, as an additional claim in the waterfall, the same way the ordinary and preferred classes are valued.
Using the last round price as the share value. The round priced a preferred share with rights the ordinary share does not have. At an illustrative round price of 6.00, the same warrant would come out at 2.71, more than double the 1.30, and 1.36 million in total. The share value going into the option model must be the ordinary share's own fair value, which is the subject of why common stock is worth less than the preferred round price. The option formula itself is worked line by line in a Black-Scholes price by hand.
Value a private company warrant as a call on the ordinary share's fair value, over its expected life, diluted for its own exercise: 1.30 a warrant here, 0.65 million in total, against 2.13 if the contractual term is used. The free workbooks for the book walk the bridge from enterprise value through the preference stack to each security, which is where the share value in this calculation comes from.
Over the period it is expected to remain outstanding. Many venture debt warrants must be exercised on a sale or lapse, so their life ends at the expected exit. On illustrative terms, a three-year expected term gives 1.30 a warrant; the seven-year contractual term gives 2.13, an overstatement of 63.7 per cent.
Multiply the option value by N divided by N plus M, where N is shares outstanding and M the warrants, because exercise issues new shares below their value. With 20.0 million shares and 0.5 million warrants the factor is 0.9756, taking an illustrative option value of 1.3352 to 1.30.
No. An out-of-the-money warrant still has time value. A warrant with a 5.00 strike over an illustrative share worth 4.00 has zero intrinsic value, but at 55 per cent volatility over three years it is worth 1.30, about 32.6 per cent of the share's value.
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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