Modigliani and Miller with Tax, Market Imperfections and Gearing in Appraisal

The simple answer that gearing makes no difference to value is not true in the real world: most companies borrow, and borrow a lot. Something in the real world must make debt worth having. The first and most obvious thing is tax. Interest is a deductible expense, so the company pays less tax if it borrows, and that saving belongs to the owners of the company.

Modigliani and Miller, who proved that gearing was irrelevant without tax, then added corporate tax to their model and reached a very different result: the more debt, the higher the value.

This topic explains the model with tax (the value of the tax shield, the formulae for value, the cost of equity and the weighted average cost of capital (WACC)), the three-step method for finding the cost of capital at a new level of gearing and using it to appraise a project, and the market imperfections, especially the costs of financial distress, that stop companies from borrowing without limit, which leads to the idea of an optimum gearing from a balance of costs and benefits, with worked examples.

The next topic covers the theories about how a company should choose between ways of financing an investment.

Why companies adopt different levels of gearing was tested in November Diet, 2025, Question 5 (15 marks). Read the traditional view and Modigliani and Miller without tax first.

Modigliani and Miller with tax

With corporate tax, interest on debt reduces the company’s tax bill: a company that pays interest of I saves I × t in tax. If debt is permanent, the present value of these savings, the tax shield, is D × t, where D is the market value of the debt and t the tax rate. Modigliani and Miller then say:

• Proposition 1: the value of a geared company equals the value of an identical ungeared company plus the value of the tax shield: V_G = V_U + D × t.
  The WACC of the geared company is lower than that of the ungeared company: WACC_G = WACC_U × [1 − Dt ÷ (D + E)].
  The WACC falls continually as gearing rises, and the lowest WACC would be at 100% debt, were it not for financial distress.
• Proposition 2: the cost of equity still rises with gearing, now with a tax term: K_eG = K_eU + (1 − t)(K_eU − K_d) × D ÷ E, where K_d is the pre-tax cost of debt.

Example: an all-equity company is worth ₦200m with a cost of equity of 11%. It borrows ₦60m at 6% and uses it to buy back and cancel shares. Tax is 30%. The tax shield is 60 × 0.30 = ₦18m, so V_G = 200 + 18 = ₦218m: debt ₦60m and equity ₦158m. The WACC = 11% × [1 − 18 ÷ 218] = 11% × 0.9174 = 10.09%. The cost of equity = 11 + 0.70 × (11 − 6) × 60 ÷ 158 = 11 + 1.33 = 12.33%. As a check, (158 × 12.33% + 60 × 6% × 0.70) ÷ 218 = (19.48 + 2.52) ÷ 218 = 10.09%. Borrowing lifted the value by ₦18m, all for the shareholders.

Using the formulae to find the cost of capital at a new gearing

When a project is financed so that gearing changes but the business risk does not, the WACC to use is the one at the new gearing. The method, with the formulae:

1. Start with the company as it is: the values of its equity and debt, the cost of equity, the pre-tax cost of debt and the tax rate.
2. Find the equivalent ungeared company: V_U = V_G − Dt; K_eU from Proposition 2 solved backwards; WACC_U from Proposition 1 backwards.
3. Move to the new gearing: new debt and equity values, V_G = V_U + D_new × t, then the new K_eG and WACC_G.

Example: a company is worth ₦500m: equity ₦400m at 14% and debt ₦100m at a pre-tax cost of 8%; tax is 30%. It will borrow another ₦100m and use it to buy back equity.

• Ungeared value:
  V_U = 500 − 100 × 0.30 = ₦470m.
• Ungeared cost of equity:
  14 = K_eU + 0.70 × (K_eU − 8) × 100 ÷ 400, so 14 = 1.175K_eU − 1.4 and K_eU = 13.11%.
  (This is also WACC_U, since the ungeared company has only equity: the present WACC of (400 × 14 + 100 × 8 × 0.70) ÷ 500 = 12.32% divided by [1 − 30 ÷ 500] gives the same.)
• New values:
  debt ₦200m; V_G = 470 + 200 × 0.30 = ₦530m; equity = 530 − 200 = ₦330m.
• New cost of equity:
  13.11 + 0.70 × (13.11 − 8) × 200 ÷ 330 = 13.11 + 2.17 = 15.27%.
• New WACC:
  13.11% × [1 − 60 ÷ 530] = 13.11% × 0.8868 = 11.62%.

The WACC falls from 12.32% to 11.62%. A project costing ₦100m that gives after-tax cash flows of ₦28m a year for five years has, at the new WACC of 11.62% (annuity factor 3.639), a net present value (NPV) of 28 × 3.639 − 100 = +₦1.9m; at the old WACC of 12.32% (factor 3.576) it would have been only +₦0.1m, close to rejection. Using the right WACC for the new financing changes the answer.

This applies only if the business risk is unchanged; if it changes, the asset beta method of the earlier topic is used.

Market imperfections and the optimum

If tax relief were the only real-world factor, companies would borrow almost everything. They do not, because of other imperfections:

• Financial distress costs. The more debt, the greater the probability that the company cannot pay interest. Distress has direct costs (legal and administrative costs of insolvency, forced sale of assets) and indirect costs (customers and suppliers withdraw, good staff leave, investment is cut, credit becomes costly). Shareholders and lenders price the expected cost into the cost of equity and debt.
• Agency costs. Lenders protect themselves with covenants that restrict management; managers of a highly geared company may take excessive risks for the shareholders’ benefit at the lenders’ expense. The costs of monitoring and restrictions reduce value.
• Limits on tax relief: the shield is available only if the company has taxable profit, and the law may limit the deduction of interest.
• Information differences and personal taxes: managers know more than investors, so the way a company finances itself sends signals; investors’ own taxes may offset part of the corporate tax gain.
• Debt capacity: lenders will only lend against assets and cash flows they can see.

The static trade-off theory combines these: the value of a geared company is V_U + the value of the tax shield − the present value of the costs of financial distress (and agency costs). The tax shield rises steadily with debt; the expected distress costs rise slowly at first and then steeply.

The value is highest where the extra tax benefit of one more naira of debt equals the extra expected distress cost, the optimum. With V_U = ₦470m and t = 30%, illustrative figures: debt ₦0 gives value 470; ₦100m gives 470 + 30 − 2 = 498; ₦200m gives 470 + 60 − 10 = 520; ₦300m gives 470 + 90 − 30 = 530; ₦400m gives 470 + 120 − 70 = 520; ₦500m gives 470 + 150 − 130 = 490. The optimum is about ₦300m of debt.

Companies differ in their optimum: those with stable cash flows, tangible assets and high taxable profits can carry more debt; those with volatile cash flows, intangible assets or growth options carry less. This is why firms in different industries, and of different sizes and ages, have different gearing.

The ideas in more depth

Why the tax shield belongs to shareholders

The company, not the lender, pays less tax because of the interest, and the lender is paid its fixed interest. So the entire saving accrues to the equity, and the value of the equity rises by the present value of the shield.

In the example the equity of the geared company is ₦158m, against ₦200m for the ungeared one, after ₦60m was returned to shareholders by the buy-back, so shareholders hold ₦158m plus ₦60m in cash, ₦218m, which is ₦18m more than before.

Why the formula assumes permanent debt

The tax shield of D × t is the present value of a perpetual saving of interest × t discounted at the cost of debt (interest = D × K_d, saving = D × K_d × t, present value = D × t). For debt that will be repaid, the shield is smaller. The formulae are a good approximation for companies that keep a roughly constant level of debt.

Sources and references

• The tax relief on interest follows the Nigeria Tax Act 2025, which taxes companies other than small companies at 30% and limits the deduction of interest on loans from connected parties to 30% of earnings before interest, tax, depreciation and amortisation (the rule was extended to local connected-party loans from 1 January 2026; check the treatment of any excess); the Modigliani and Miller formulae with tax are among those given on the formula sheet of the paper.

These points are stated as at 5 November 2026.

Worked examples

1. Shield.
   60 × 0.30 = ₦18m
   V_G = 200 + 18 = ₦218m.

1. WACC.
   11% × (1 − 18 ÷ 218) = 11% × 0.9174 = 10.09%.

1. Equation.
   14 = K_eU + 0.70(K_eU − 8) × 0.25 = 1.175K_eU − 1.4.
2. Cost.
   K_eU = 15.4 ÷ 1.175 = 13.1%.

How to answer in the exam

• Follow the three steps: ungeared, then the new gearing, then the new cost and WACC.
• Use market values and the pre-tax cost of debt in the cost of equity formula.
• Discount the project at the new WACC and state the effect compared with the old one.
• Discuss the imperfections and why companies stop short of maximum debt.

Common mistakes in the exam

• Using the after-tax cost of debt in the Proposition 2 formula (the formula has (1 − t) already).
• Forgetting that the tax shield uses the market value of debt.
• Applying the no-tax formulae where tax is given.
• Using the old WACC to appraise a project that changes gearing.
• Saying that Modigliani and Miller recommend 100% debt without noting distress costs.
• Taking the cost of debt as constant at very high gearing.

Quick revision

• V_G = V_U + Dt; WACC_G = WACC_U[1 − Dt ÷ (D + E)].
• K_eG = K_eU + (1 − t)(K_eU − K_d)D ÷ E.
• Three steps: ungeared, new values, new cost of equity and WACC.
• Optimum: tax shield against distress and agency costs.

Practice questions

These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.

Recall check: multiple choice

Section B style: written questions

A further written question

Model answers

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