Valuation for acquisitions and mergers: formula sheet

Key formulas

Value of combined entity

V(A+B) = V(A) + V(B) + Synergy

Synergy is the extra value created by combining. It can be negative if integration goes badly.

Value of synergy

Synergy = PV of incremental cash flows from combining − PV of one-off integration costs

Use a discount rate that matches the risk of the synergy cash flows. Include tax effects.

Maximum price a bidder should pay

Maximum price = Standalone value of target + Value of synergies

Paying this leaves the bidder's shareholders no gain. Any lower price shares synergy with them.

Premium and synergy test

Premium paid = Offer price − Target's pre-bid market value. Bidder gains only if Synergy > Premium + Bid costs

Bid costs include advisers' fees. Compare against synergy after integration costs.

Gain to each party

Gain to target shareholders = Premium. Gain to acquirer shareholders = Synergy − Premium − Bid costs

Total gain equals synergy less costs. The price decides how it is split.

Capitalised value of a perpetual synergy

PV = Annual after-tax synergy ÷ discount rate (no growth); with growth g: PV = Synergy next year ÷ (r − g)

Only valid when r > g. Use a finite horizon if the synergy will fade.

Net asset value

NAV = Total assets − Total liabilities

Liabilities include debt and any preference shares if you want the value for ordinary shareholders. Book basis unless adjusted.

Adjusted NAV

Adjusted NAV = Σ revalued assets − Σ revalued liabilities

Use replacement cost or realisable value for each asset as the scenario requires. Include unrecorded assets and liabilities, such as contingent liabilities or a pension deficit.

Value per share

Value per share = Net asset value attributable to ordinary shares ÷ Number of ordinary shares

Deduct preference shares first. Use shares in issue.

Realisable value (net)

Net realisable value = Expected sale proceeds − Costs of disposal

Also deduct redundancy and closure costs in a break-up valuation.

Implied goodwill

Goodwill = Price paid or earnings-based value − Adjusted NAV

Shows how much of the price relates to earning power and not to identifiable assets.

P/E ratio

P/E = market price per share ÷ EPS = market capitalisation ÷ total earnings

Use earnings after tax and after preference dividends.

Earnings yield

Earnings yield = EPS ÷ share price = 1 ÷ P/E

A P/E of 8 gives an earnings yield of 12.5%.

Equity value using P/E

Equity value = maintainable earnings × P/E ratio

Divide by the number of shares for a value per share.

Equity value using earnings yield

Equity value = maintainable earnings ÷ earnings yield

Gives the same answer as the P/E method.

Enterprise value multiple

Enterprise value = equity value + debt − cash; EV/EBITDA = EV ÷ EBITDA

Useful when capital structures differ. Convert EV back to equity by deducting net debt.

Link between P/E and growth

P/E ≈ payout ratio × (1 + g) ÷ (ke − g)

Applies when dividends grow at constant rate g and earnings are the base. It shows why higher growth and lower risk raise P/E.

Dividend valuation model (constant growth)

P0 = D0(1 + g) ÷ (Ke − g) = D1 ÷ (Ke − g)

P0 is ex-div. Valid only when Ke > g. If the dividend just due has not been paid, add it to get the cum-div value.

Cost of equity from the DVM

Ke = D1 ÷ P0 + g

Use this to find Ke when you know the share price.

Historical dividend growth

g = (D latest ÷ D earliest)^(1 ÷ n) − 1

n is the number of growth periods, i.e. number of dividends minus 1.

Gordon growth model

g = b × r

b = retention ratio = 1 − payout ratio. r = return on retained funds, usually ROE or ROCE as the question states.

Retention ratio

b = 1 − (dividends ÷ earnings)

Payout ratio is dividends ÷ earnings.

Cum-div value

Cum-div value = ex-div value + dividend due

Be clear which basis the question asks for.

Free cash flow to the firm

FCFF = EBIT × (1 − t) + depreciation − capital investment − increase in working capital

Capital investment must cover both replacement and expansion. Use the tax rate on operating profit, not on profit after interest.

Free cash flow to equity

FCFE = FCFF − interest × (1 − t) + net new borrowing

Equivalent to profit after tax + depreciation − capital investment − increase in working capital + net new debt. Discount at the cost of equity.

Terminal value (growing perpetuity)

TV at year n = FCF(n+1) ÷ (r − g) = FCF(n) × (1 + g) ÷ (r − g)

Valid only if r > g. TV is a value at year n, so discount it using the year n factor. For a no-growth perpetuity set g = 0.

Enterprise value and equity value

EV = Σ FCFF(t) ÷ (1 + WACC)^t + TV ÷ (1 + WACC)^n ; Equity value = EV − market value of debt + surplus cash

Use market value of debt where given, not book value.

WACC

WACC = [E ÷ (E + D)] × ke + [D ÷ (E + D)] × kd × (1 − t)

Use market values for E and D, and the gearing the target will have after the deal if it changes.

Sustainable growth

g = retention (reinvestment) rate × return on new investment

Use it to check that your terminal growth rate is supported by the reinvestment in the forecast.

Ungeared cost of equity (Modigliani and Miller with tax)

ke(g) = ke(u) + (ke(u) − kd) × (1 − t) × D ÷ E

Rearrange to find ke(u). Alternatively, asset beta = equity beta × E ÷ [E + D(1 − t)] when debt beta is taken as zero.

Adjusted present value

APV = base case NPV at ke(u) + PV of tax shield on debt − issue costs

Tax shield per year = debt × interest rate × tax rate. Discount it at the pre-tax cost of debt unless told otherwise.

Relief from royalty

Value = Σ [Sales × royalty rate × (1 − tax rate)] ÷ (1 + r)^t

Use a royalty rate from comparable licences, and a discount rate that reflects the risk of the brand's cash flows. Add a terminal value if the life is indefinite.

Goodwill

Goodwill = Price paid − fair value of identifiable net assets (including separately valued intangibles)

The more intangibles you identify and value separately, the smaller the residual goodwill.

Value including options

Total value = Base DCF value + Value of real options

The option value is extra. It is never negative, because you only exercise when it pays.

Black-Scholes call value

c = S × N(d1) − X × e^(−rT) × N(d2)

S = PV of project inflows, X = investment cost, r = risk-free rate, T = years to decision. Used for expand or delay options.

Black-Scholes d1 and d2

d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × T] ÷ (σ × √T); d2 = d1 − σ × √T

σ is the annual volatility as a decimal. Use the normal distribution tables supplied in the exam.

Put-call parity (European options, no dividends)

c + X × e^(−rT) = p + S

Use it to get the put (abandonment) value from the call value.

Post-deal EPS

Combined EPS = (Bidder earnings + Target earnings + post-tax synergies − extra post-tax finance cost) ÷ (Bidder shares + new shares issued)

Use the same earnings basis for both companies. Synergies and finance costs must be after tax.

New shares issued in a share offer

New shares = Target shares × exchange ratio

Exchange ratio = bidder shares offered per target share, for example 2 for 3 is 0.667.

Share price from P/E

Share price = EPS × P/E ratio

State which P/E you assume after the deal. Often the bidder's existing P/E is used.

Maximum price the bidder can pay

Maximum price = Target standalone value + PV of synergies

Divide by target shares for a maximum price per share.

Premium

Premium = Offer price − Target's current market price

The premium is the target's gain in a cash offer.

Gain to each party

Total gain = Combined value after deal − (Bidder value + Target value before). Bidder gain = Total gain − Target gain

In a cash offer, target gain = premium × number of shares. In a share offer, target gain = value of its share of the combined company − its old value.

Gearing

Debt ÷ Equity, or Debt ÷ (Debt + Equity)

Say which measure you use. Use market values if given, otherwise book values.

Quick revision

• Value of target to the acquirer = stand-alone value + value of synergies.
• The maximum price you should pay is stand-alone value plus synergies; the minimum the seller will accept is its own stand-alone value.
• Asset-based methods ignore future earnings and often undervalue intangibles.
• Value from P/E: earnings × P/E ratio; the ratio must come from a comparable company and be adjusted for risk and size.
• Earnings yield is the inverse of the P/E ratio.
• Dividend valuation model with constant growth: P₀ = D₀(1 + g) ÷ (Ke − g), valid only if Ke > g.
• Free cash flow valuation discounts cash flows at a rate that reflects the risk of the cash flows and their financing.
• Terminal value usually forms a large share of a DCF value, so test its growth assumption.
• Real options such as expansion, abandonment or delay add value not captured by a plain DCF.
• Check EPS effect, share price effect and gearing effect separately; a rise in EPS does not prove value creation.
• Always end with a recommendation and state the key assumptions and risks behind it.

Common mistakes

• Treating the target's standalone value as the price the bidder should pay. Fix: Always state the value to the bidder as standalone value plus synergy, and then say the price must sit below that.
• Ignoring one-off integration and restructuring costs. Fix: Underline every cost in the scenario. Deduct the present value of these costs from the synergy before comparing it with the premium.
• Using book values when the question asks for replacement cost or realisable value. Fix: Underline the basis in the requirement first. Replace every asset the question gives a new figure for.
• Forgetting disposal costs, redundancy costs or tax in a realisable value valuation. Fix: Net every sale price of costs of disposal and deduct closure costs as liabilities. Check the data for tax on gains.
• Using the acquirer's P/E to value the target without comment. Fix: Say that the acquirer's P/E reflects its own risk and growth. Use it only if the target is similar, and adjust or explain otherwise.
• Using profit before tax or profit before interest as earnings. Fix: Use profit after tax and after preference dividends for an equity P/E. Use EBITDA only with an EV multiple.
• Using D0 instead of D1 in the formula. Fix: Always write D1 = D0(1 + g) as a separate line before dividing.
• Using the number of dividends as n when finding historical growth. Fix: Four dividends mean three years of growth. n = number of dividends − 1.
• Discounting FCFF at the cost of equity, or FCFE at WACC. Fix: Match the rate to the cash flow. Cash to all providers of finance uses WACC. Cash to shareholders only uses the cost of equity.
• Deducting interest in FCFF or ignoring tax on EBIT. Fix: Start from EBIT and tax it at the full rate. The tax benefit of debt is already in the WACC, so deducting interest would count it twice.

Exam tips

• Always separate standalone value, synergy and premium in your answer. Markers look for all three.
• Link every synergy to a fact in the scenario and comment on how reliable it is. These comments carry professional skills marks.
• State the discount rate you use for synergies and why. If the question gives a rate, use it.
• Finish with a price range and a recommendation. A calculation with no conclusion loses marks.
• Show costs and tax explicitly, even when the amounts are small, so you pick up method marks.
• Read the requirement for the basis and the purpose. Marks go for choosing the basis that fits the scenario, not only for arithmetic.
• Show every adjustment on its own line so the marker can award method marks even if one figure is wrong.
• Always add commentary on limitations tied to the target's industry. Asset values rarely capture brands, people or customer relationships.