If a company borrows more, it replaces some expensive equity with cheaper debt, so its average cost of capital ought to fall and its value ought to rise. But the shareholders who are left face more risk, so they demand a higher return. Does the cheap debt win, or does the rising cost of equity cancel it? Two famous answers have been given.
The traditional view says that the cheap debt wins at first and the cost of equity catches up later, so there is a best level of gearing at which the cost of capital is lowest and the company is worth most. Modigliani and Miller showed that in a simple world, without tax, the two effects exactly cancel, and gearing makes no difference to value at all.
This topic explains both views: how the weighted average cost of capital (WACC) behaves as gearing rises under each, the formulae of Modigliani and Miller without tax and how to use them in both directions, the assumptions behind the result and why it is a benchmark, not a description of the real world, with worked examples. The next topic adds corporate tax and the imperfections of real markets.
The effect of a new bank loan on the cost of equity, the cost of debt and the weighted average cost of capital, under the traditional model, was tested in May Diet, 2026, Question 2(b) (9 marks). Read capital gearing, business risk and financial risk first.
In 30 seconds
Traditional view: as gearing rises, the cost of equity rises, but debt is cheaper, so at first the WACC falls; beyond a point the rising cost of equity (and of debt) outweighs it and the WACC rises. So there is an optimum gearing that minimises the WACC and maximises company value.
Modigliani and Miller without tax: Proposition 1: the WACC and the total value are the same at all gearing levels (VG = VU); Proposition 2: the cost of equity rises with gearing: KeG = KeU + (D ÷ E)(KeU − Kd), so that the rise exactly offsets the cheaper debt. Assumptions: perfect markets, no tax, no bankruptcy costs, same borrowing rates for investors and companies. The result is a benchmark.
The traditional view
The traditional view of gearing is that a company has an optimum level of gearing, at which the WACC is lowest and the value of the company highest. Its reasoning has three stages.
- Low gearing: the company is mostly equity. Adding some debt, which is cheaper than equity, has a large effect on the average. The cost of equity rises only a little because the extra risk is small, and the lenders’ risk is small. The WACC falls.
- The optimum: as gearing rises the cost of equity rises faster, and lenders begin to ask for more. At some point the effect of the dearer equity and debt exactly balances the effect of the larger share of debt. The WACC is at its minimum.
- High gearing: the risk of default is serious; shareholders demand much more, and lenders raise the rate. The WACC rises.
An illustration with assumed figures:
| Debt ÷ total value | Cost of equity | Cost of debt | WACC |
|---|---|---|---|
| 0% | 12.0% | – | 12.0% |
| 20% | 12.5% | 6.0% | 11.2% |
| 40% | 13.5% | 6.5% | 10.7% |
| 60% | 16.5% | 8.5% | 11.7% |
| 80% | 22.0% | 11.0% | 13.2% |
The WACC is lowest, 10.7%, at gearing of about 40%. For a company whose annual earnings before interest are ₦100m, a perpetual value is earnings ÷ WACC: ₦833m at 12.0% (no debt) against ₦935m at 10.7%, a rise of ₦102m from using the right amount of debt.
A new bank loan that moves gearing from 20% to 40% would therefore, on this view, raise the cost of equity (12.5% to 13.5%), raise slightly the cost of debt (6.0% to 6.5%) and lower the WACC (11.2% to 10.7%); a loan that took it from 40% to 60% would raise the WACC.
Modigliani and Miller without tax
Modigliani and Miller (MM) built a model on simplifying assumptions to see what would be expected in a simple world, and then compared it with reality.
The assumptions:
- capital markets are perfect (all investors value securities in the same way, there are no dealing costs, and information is perfect and instant)
- there is no taxation
- there is no bankruptcy risk or cost of financial distress
- investors can borrow on the same terms as companies, so they are indifferent between borrowing themselves and investing in a geared company
In such a world:
- Proposition 1: the WACC is the same at all levels of gearing, and the total market value of a geared company equals that of an identical ungeared company with the same earnings and business risk: VG = VU and WACCG = WACCU. Gearing has no effect on the total value.
- Proposition 2: the cost of equity rises with gearing by just enough to offset the cheaper debt: KeG = KeU + (D ÷ E) × (KeU − Kd), where KeU is the cost of equity of the ungeared company, D and E are the market values of debt and equity of the geared company, and Kd is the cost of debt. The extra return is a financial risk premium.
Worked example: using the formulae
An all-equity company is worth ₦400m with a cost of equity of 12% (earnings ₦48m a year). It borrows ₦100m at 7% and uses it to buy back equity. By Proposition 1 the total value is still ₦400m, made up of debt ₦100m and equity ₦300m. By Proposition 2: KeG = 12 + (100 ÷ 300) × (12 − 7) = 12 + 1.67 = 13.67%. The WACC = (300 × 13.67% + 100 × 7%) ÷ 400 = (41.0 + 7.0) ÷ 400 = 12.0%, unchanged. With debt of ₦200m: equity ₦200m, KeG = 12 + 1.0 × 5 = 17%, and the WACC = (200 × 17% + 200 × 7%) ÷ 400 = 12.0% again.
| Debt ₦m | Equity ₦m | D ÷ E | Cost of equity | WACC | Value ₦m |
|---|---|---|---|---|---|
| 0 | 400 | 0 | 12.0% | 12.0% | 400 |
| 100 | 300 | 0.33 | 13.7% | 12.0% | 400 |
| 200 | 200 | 1.0 | 17.0% | 12.0% | 400 |
The formulae also work in reverse. A geared company has equity of ₦600m and debt of ₦150m with KeG = 15% and Kd = 9%. Then 15 = KeU + (150 ÷ 600)(KeU − 9), so 1.25KeU = 15 + 2.25 = 17.25 and KeU = 13.8%, which is also the WACC of the geared company: (600 × 15 + 150 × 9) ÷ 750 = 13.8%.
Why the result holds: homemade gearing
Modigliani and Miller’s argument is that if a geared company were worth more than an identical ungeared one, investors could do better by buying the ungeared company’s shares and borrowing personally to gear their own holding, at the same rate as the company would borrow.
This “homemade gearing” makes the cheaper route the preferred one, and arbitrage (buying the cheaper and selling the dearer) would push the values together. So the company cannot create value for its shareholders by changing its mix, since they can do the same themselves.
Comparing the two views
- Traditional: there is an optimum; the market does not price risk exactly; the WACC curve is U-shaped; the financing decision matters.
- Modigliani and Miller (no tax): no optimum; the WACC is flat; the financing decision is irrelevant to value; only investment decisions matter.
- Reality: taxes, costs of financial distress, agency costs and information differences all exist, so the truth lies between the two. These imperfections are the subject of the next topic.
The ideas in more depth
Why Proposition 2 follows from Proposition 1
If the WACC is constant while the proportion of cheaper debt rises, the cost of equity must rise to compensate. The formula gives exactly the rise needed: each extra naira of debt adds the difference between the equity and debt rates, spread across a smaller equity base. It is the financial risk premium: equity holders bear the risk of the fixed interest.
Why the cost of debt is assumed constant
In the simple formula the cost of debt stays the same however much is borrowed, because default is assumed impossible. At very high gearing debt would cost more, and the cost of equity would not rise in a straight line. The model is accurate only where debt is nearly risk-free, which is why the result is a benchmark for thinking, not a rule for setting gearing.
Sources and references
- The Modigliani and Miller propositions (1958) and the traditional view are theories, not rules. Check the formula sheet of the paper for the form of the formulae provided.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the cost of equity. KeU 12%; Kd 7%; D ₦100m; E ₦300m. Required: KeG.
- Premium.
(100 ÷ 300) × (12 − 7) = 1.67%. - Cost.
12 + 1.67 = 13.67%.
Example 2: the WACC. E ₦300m at 13.67%; D ₦100m at 7%. Required: the WACC.
- Weighted.
(300 × 13.67 + 100 × 7) ÷ 400 = 48.0 ÷ 4. - WACC. 12.0%, as for the ungeared company.
Example 3: the ungeared cost. KeG 15%; Kd 9%; D ÷ E = 0.25. Required: KeU.
- Equation.
15 = KeU + 0.25(KeU − 9)
1.25KeU = 17.25. - Cost. KeU = 13.8%.
How to answer in the exam
- State which model you are applying and its assumptions.
- Use market values for D and E in the formula, and substitute step by step.
- Show the effect on the cost of equity, the cost of debt and the WACC separately.
- Comment on realism: taxes, distress costs and the other imperfections.
Common mistakes in the exam
- Saying that Modigliani and Miller claim that gearing is harmful, or good, instead of irrelevant (without tax).
- Using book values for D and E.
- Forgetting that the total value stays the same when equity is bought back.
- Mixing up KeU and KeG when solving in reverse.
- Describing the traditional view without the rise in the WACC at high gearing.
- Applying the no-tax formula where tax is given.
Quick revision
- Traditional: U-shaped WACC; an optimum gearing.
- Modigliani and Miller (no tax): WACC and value constant; VG = VU.
- KeG = KeU + (D ÷ E)(KeU − Kd).
- Assumptions: no tax, no distress costs, perfect markets.
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
Question 1. The traditional view holds that:
- Gearing is irrelevant
- More debt always lowers the WACC
- There is an optimum gearing that minimises the WACC
- Debt is costlier than equity
Show the answer
C. The WACC is U-shaped.
Question 2. Under Modigliani and Miller without tax, the WACC:
- Falls with gearing
- Is the same at all gearing levels
- Rises with gearing
- Is zero
Show the answer
B. Proposition 1.
Question 3. Under Proposition 2 the cost of equity:
- Rises as gearing rises
- Falls as gearing rises
- Is constant
- Equals the cost of debt
Show the answer
A. By a financial risk premium.
Question 4. According to Modigliani and Miller without tax, the value of a geared company equals:
- Twice the ungeared value
- Half the ungeared value
- Its book value
- The value of the identical ungeared company
Show the answer
D. VG = VU.
Question 5. Debt is usually cheaper than equity because:
- It is riskier
- Lenders face less risk and are paid first
- It has no interest
- It is never repaid
Show the answer
B. And has tax relief in practice.
Question 6. The cost of equity of a geared company is:
- KeU + (D ÷ E)(KeU − Kd)
- KeU − Kd
- Kd + KeU
- KeU × D
Show the answer
A. The ungeared cost plus the risk premium.
Question 7. One assumption of Modigliani and Miller without tax is:
- Heavy corporate tax
- High bankruptcy costs
- No taxes and no bankruptcy costs
- Imperfect information
Show the answer
C. Together with perfect markets.
Question 8. In the traditional view, beyond the optimum the WACC:
- Falls
- Stays flat
- Is zero
- Rises
Show the answer
D. Risk of default outweighs the cheaper debt.
Question 9. Modigliani and Miller concluded (no tax) that the financing decision is:
- Critical
- The same as the dividend
- Irrelevant to value
- Negative
Show the answer
C. Only the investment decision matters.
Question 10. “Homemade gearing” means that:
- Investors can borrow themselves, so companies cannot create value by gearing
- Companies cannot borrow
- Banks decide gearing
- Shares are tax free
Show the answer
A. The basis of the arbitrage argument.
Section B style: written questions
Question 11 (10 marks). (a) Describe the traditional view of gearing and the WACC. (4 marks) (b) State the two propositions of Modigliani and Miller without tax. (4 marks) (c) State TWO assumptions. (2 marks)
Question 12 (15 marks). An all-equity company is worth ₦300m with a cost of equity of 10%. It borrows ₦90m at 6% to buy back shares. Assume Modigliani and Miller without tax. (a) State the new value of equity. (3 marks) (b) Calculate the cost of equity. (5 marks) (c) Calculate the WACC. (5 marks) (d) State the effect on total value. (2 marks)
A further written question
Question 13 (20 marks). A geared company has equity of ₦600m and debt of ₦200m, a cost of equity of 16% and a cost of debt of 8%. Assume Modigliani and Miller without tax.
(a) Calculate the WACC. (4 marks) (b) Calculate the cost of equity of the same company if ungeared. (6 marks) (c) The company doubles its debt to ₦400m, with equity ₦400m; calculate the new cost of equity and WACC. (7 marks) (d) State why the real world differs. (3 marks)
Model answers
Show the model answer to Question 11
(a) There is an optimum gearing (1). At first the cheaper debt lowers the WACC (1); beyond a point the rising costs of equity and debt raise it (1); so value is maximised at the optimum (1).
(b)
- Proposition 1: WACC and total value are the same at all gearing levels (2).
- Proposition 2: the cost of equity rises with gearing by the financial risk premium KeG = KeU + (D ÷ E)(KeU − Kd) (2).
(c) Any two: perfect markets; no taxes; no bankruptcy costs; same borrowing terms for investors and companies (1 each).
Show the model answer to Question 12
(a) 300 − 90 = ₦210m (3).
(b) KeG = 10 + (90 ÷ 210) × (10 − 6) = 10 + 1.71 = 11.7% (5).
(c) (210 × 11.71% + 90 × 6%) ÷ 300 = (24.6 + 5.4) ÷ 300 = 10.0% (5).
(d) No effect: the total value is still ₦300m (2).
Show the model answer to Question 13
(a) (600 × 16 + 200 × 8) ÷ 800 = 11,200 ÷ 800 = 14.0% (4).
(b)
- 16 = KeU + (200 ÷ 600)(KeU − 8)
- 1.333KeU = 16 + 2.667 = 18.667
- KeU = 14.0% (6), the same as the WACC.
(c)
- KeG = 14 + (400 ÷ 400)(14 − 8) = 20.0% (4)
- WACC = (400 × 20 + 400 × 8) ÷ 800 = 14.0% (3).
(d) Taxes, bankruptcy and distress costs, agency costs and unequal information exist in practice, so gearing does affect value (3).
Pathfinder on The traditional view and Modigliani and Miller without tax
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Part (b) (9 marks) asks for the effect of a new bank loan on the cost of equity, the cost of debt and the weighted average cost of capital under the traditional model of gearing.
Revise first: The traditional view
Source: ICAN Pathfinder, Financial Management. Diets covered: Every diet in the Pathfinder, November Diet, 2025 and May Diet, 2026, was reviewed here. The diets listed above test this topic directly..
Spotted a mistake? Tell us and we will check it.
