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How do you apply the treasury stock method to a DCF value per share?

From a 600,000,000 DCF equity value to a diluted value per share, with three option tranches and restricted stock units.

Julian R. Sterling · from Business Valuation

The treasury stock method adds the shares that in-the-money options would create and assumes the exercise cash buys some of them back at the value per share. Applied to a DCF it is circular, because the value per share depends on the share count, but it has an exact closed form: equity value plus the exercise proceeds of in-the-money options, divided by basic shares plus those options and the RSUs. On an illustrative 600,000,000 of equity value that is 635,000,000 / 26,900,000 = 23.61 a share, not the 25.00 that basic shares give.

Worked in full in Business Valuation by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

The case

A DCF and the bridge from enterprise value give an equity value of 600,000,000. The company has 24,000,000 basic shares outstanding, three tranches of employee options and 400,000 restricted stock units that will vest. All figures are illustrative.

Share capital and dilutive instruments

Instrument	Number	Strike	Exercise proceeds
Basic shares	24,000,000
Options, tranche 1	1,500,000	10.00	15,000,000
Options, tranche 2	1,000,000	20.00	20,000,000
Options, tranche 3	800,000	30.00	24,000,000
Restricted stock units	400,000	0.00	0

Step 1: the method, one tranche at a time

For each tranche whose strike is below the value per share, the options are assumed exercised; the company receives the strike times the number, and uses that cash to repurchase shares at the value per share. The net new shares are the options less the buyback. RSUs have no strike, so they add one share each. Tranches with a strike above the value per share add nothing.

Step 2: the circularity, and why one pass is not enough

The obvious starting point is the basic value per share, 600,000,000 / 24,000,000 = 25.00. At 25.00, tranche 1 nets 900,000 new shares and tranche 2 nets 200,000; tranche 3 is out of the money. With the RSUs that makes 25,500,000 diluted shares and 23.53 a share. But 23.53 is not the price the buyback was computed at. Run it again at 23.53, then at the new answer, and it settles after a few rounds:

Treasury stock method at the converged value of 23.61

Instrument	Number	Strike	Shares bought back	Net new shares
Tranche 1	1,500,000	10.00	635,433	864,567
Tranche 2	1,000,000	20.00	847,244	152,756
Tranche 3	800,000	30.00	0	0
RSUs	400,000		0	400,000
Diluted shares				25,417,323

600,000,000 / 25,417,323 = 23.6059, and at 23.6059 the buyback gives back exactly 25,417,323 shares. That is the fixed point.

Step 3: the closed form that removes the loop

The fixed point can be written without iteration. If P is the value per share, the treasury stock method says P × (basic + options + RSUs) − option proceeds = equity value. Rearranged:

The only check left is that the tranches flagged in the money really are: 10.00 and 20.00 are below 23.61, and 30.00 is above. If a strike sits right at the answer, test it both ways and keep the consistent one. Because the flags are inputs checked against the result, rather than formulas that read it, the method needs no circular reference switched on in Excel.

What if: the equity value moves

Diluted shares and value per share by DCF equity value

Equity value	Basic per share	Diluted shares	Diluted per share	Dilution
400,000,000	16.67	24,963,855	16.02	3.9%
500,000,000	20.83	25,145,631	19.88	4.6%
600,000,000	25.00	25,417,323	23.61	5.6%
700,000,000	29.17	25,619,048	27.32	6.3%
800,000,000	33.33	25,797,439	31.01	7.0%

Dilution rises with value, because more options are in the money and each buys back fewer shares. A sensitivity grid that divides every cell of equity value by one fixed share count is therefore wrong at both ends: too high in the upside cases, and slightly too low in the downside ones where tranche 2 falls out of the money.

The common mistakes

• Basic shares only. 25.00 a share, 5.9 per cent above the diluted 23.61. Employee equity is a claim on the company like any other.
• Every option, no buyback. 27,700,000 shares and 21.66 a share, 8.2 per cent too low. It counts the 30.00 options that would never be exercised and ignores the cash the others bring in.
• The market price instead of the DCF value. At a market price of 21.00 the method counts 25,233,333 shares and gives 23.78. The point of a DCF is that it disagrees with the market; its share count has to follow its own answer.

The share count is the last line of the bridge from enterprise value to equity per share; the items above it are in what goes in the bridge from enterprise value to equity value. The free workbook for this case runs the bridge for each of the book's three values, so that the gap between methods can be read in value per share as well as in enterprise value.

Takeaway

• Use the treasury stock method at the DCF's own value per share, not at the market price.
• Skip the iteration: (equity value + in-the-money option proceeds) / (basic + in-the-money options + RSUs).
• Here 635,000,000 / 26,900,000 = 23.61 a share on 25,417,323 diluted shares.
• Recompute the share count in every sensitivity cell; dilution grows with value.

Questions readers ask

Which share price should the treasury stock method use in a DCF?

The value per share the DCF itself produces, not the market price. Options are in or out of the money relative to what the valuation says the shares are worth. In this case the DCF gives 23.61; running the method at a market price of 21.00 counts fewer dilutive shares and overstates value at 23.78 a share.

Do out-of-the-money options count in diluted shares?

Not under the treasury stock method: an option with a strike above the value per share would not be exercised, so it adds no shares. Here the 800,000 options struck at 30.00 are excluded at a value of 23.61. Counting every option without the buyback gives 27,700,000 shares and 21.66 a share, 8.2 per cent too low.

Why does the treasury stock method need iteration in a valuation?

Because value per share depends on diluted shares, and diluted shares depend on value per share. A single pass at the basic value of 25.00 gives 25,500,000 shares and 23.53. Iterating converges on 23.61, which equals equity value plus option proceeds, 635,000,000, divided by 26,900,000 shares, with no iteration at all.

Also on this site

• What goes in the bridge from enterprise value to equity value?Debt at repayment amount, free cash only, leases, pensions, deferred items, minorities. On 120.0 of EV, equity is 53.32, not the 80.00 loans less cash gives.
• How do you calculate unlevered free cash flow for a DCF?EBIT taxed as if debt-free, plus D&A, less capex and the rise in working capital: 4,645,000 on 9,000,000 of EBITDA, a 51.6 per cent conversion.
• How do you calculate a control premium, and what must it earn?Offer over the unaffected price, minus one: 39.00 on 30.00 is 30.0 per cent, 180,000,000, which needs 15,600,000 a year of pre-tax synergies to repay.
• How does the capitalization of earnings method value a business?Next year's normalised cash flow over the discount rate less growth: 1,236,000 over 15.0 per cent (18.0 less 3.0) gives a value of 8,240,000. Worked example.
• How much does the mid-year convention add to a DCF?The square root of (1 + WACC), minus one: 4.40 per cent at 9 per cent. Shifting only the forecast years adds 1.20 per cent; the perpetuity adds the rest.
• How do you calculate WACC? A step-by-step worked exampleRelever the beta to 1.14, price equity at 10.50 per cent, debt at 4.88 per cent after tax, weight at 74/26: WACC is 9.04 per cent. Worked in full.

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