A company is about to fund a big project with new debt. Its gearing will jump. Its cost of equity will rise because shareholders now carry more financial risk. Its WACC will move too. Judge the project at the old WACC and you get the wrong NPV. The same trap appears when a project is in a new industry.
Its business risk differs from the company’s, so the company’s own beta is the wrong yardstick. This topic gives the tools for both problems.
This topic covers how to restate the cost of equity and WACC when gearing changes (using the Modigliani and Miller formulae with tax), asset, equity and debt betas and the formula that links them, ungearing and regearing betas, using proxy companies to find a project-specific discount rate, and the steps from a proxy beta to a WACC. The previous topic covered financing theories and Modigliani and Miller.
The next covers adjusted present value. The November 2014 Pathfinder paper used this method in its compulsory question.
In 30 seconds
When a project changes gearing but not business risk:
- (1) find VU = VG − Dt, then the new VG = VU + Dnewt
- (2) find the ungeared WACC and ungeared cost of equity, then regear them to the new gearing with the MM formulae
- (3) discount at the new WACC
Asset beta (ungeared) reflects business risk only. Equity beta adds financial risk.
With risk-free debt: βA = βE × E ÷ [E + D(1 − t)], so βE = βA × [E + D(1 − t)] ÷ E. For a project in a different business: take proxy company equity betas, ungear to asset betas, average them, regear to the project’s gearing, use CAPM for the cost of equity, then build a WACC. The MM formula for value cannot be used when extra debt is raised, only when debt replaces equity.
Key terms in plain English
| Term | What it means |
|---|---|
| Asset (ungeared) beta | The beta of a business with no financing risk: business risk only |
| Equity (geared) beta | The beta of shares, including financial risk from borrowing |
| Debt beta | The beta of debt capital, usually assumed to be zero |
| Ungearing | Converting an equity beta into an asset beta |
| Regearing | Converting an asset beta into an equity beta at a given gearing |
| Proxy company | A listed company in the same business as the project |
| Project-specific rate | A discount rate that reflects the project’s own risk |
Syllabus check
This topic is in syllabus area C (financing decisions). It covers:
- the effect of a change in gearing on the cost of equity and WACC;
- asset and equity betas and their link;
- project-specific discount rates using the CAPM.
CAMA 2020 and risk-adjusted rates
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | The rate used for a project should match its real risk, for the company’s benefit |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the risk and gearing assumptions in an appraisal |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Changing from one level of gearing to another
If a project that keeps business risk the same changes gearing, the cost of equity changes (MM proposition 2) and WACC falls (proposition 1) while the cost of debt is assumed to stay unchanged at moderate gearing. So work out a new WACC before the NPV. The method:
- Step 1. Start with the current value, cost of equity and cost of debt.
- Step 2.
Find a comparable ungeared company: VU = VG − Dt; ungeared WACC from WACCG = WACCU × [1 − Dt ÷ (D + E)]; and ungeared cost of equity from keG = keU + (1 − t)(D ÷ E)(keU − kd). - Step 3. Regear to the new structure: VG = VU + Dnewt, then the new WACC and cost of equity.
ICAN’s example. A company is worth ₦100m (equity ₦80m, debt ₦20m). Cost of equity 10%, pre-tax cost of debt 6%, tax 25%. It borrows another ₦20m to buy back ₦20m of shares.
- VU = 100 − 20 × 0.25 = ₦95m.
New VG = 95 + 40 × 0.25 = ₦105m (debt ₦40m, equity ₦65m). - Current WACC.
(80 × 10% + 20 × 4.5%) ÷ 100 = 8.9%.
WACCU = 8.9% ÷ 0.95 = 9.368%.
New WACC = 9.368% × [1 − (0.25 × 40) ÷ 105] = 8.5%. - keU = 9.368% (from 10% = keU + 0.75 × (keU − 6) × 20 ÷ 80).
New ke = 9.368% + 0.75 × 3.368% × 40 ÷ 65 = 10.9%.
A limit
The formula VG = VU + Dt applies only when the size of the company is unchanged and debt replaces equity (or the reverse). If a project is paid for with extra debt, the formula cannot give the new value. ICAN then assumes, unsatisfactorily, that equity value is unchanged.
It also computes the expected new cost of debt from credit rating spreads over the risk-free yield, weighted by probability and by maturity, and keeps the cost of debt unchanged in the cost-of-equity step.
Asset, equity and debt betas
As gearing rises with business risk unchanged, equity beta must rise, because the cost of equity rises while the risk-free rate and market return do not change. Companies that differ only in gearing have different equity betas.
- Asset beta (ungeared or unlevered beta): the beta of the business if it had no debt. It measures business risk only and stays constant while the business stays the same. It applies to whole companies or single projects.
- Equity beta (geared beta): includes financial risk, so it is higher than the asset beta.
- Debt beta: much lower, and often taken as zero.
The link: βA = βE × E ÷ [E + D(1 − t)] + βD × D(1 − t) ÷ [E + D(1 − t)]. With a risk-free debt (βD = 0): βA = βE × E ÷ [E + D(1 − t)]. ICAN’s Plassid: equity beta 1.25, debt beta 0.05, E = ₦600m, D = ₦200m, tax 30%. Base = 600 + 200 × 0.7 = 740. βA = 1.25 × 600 ÷ 740 + 0.05 × 140 ÷ 740 = 1.0135 + 0.0095 = 1.023, or 1.0135 if debt is risk-free. The simplification makes little difference, so it is usually assumed.
Project-specific discount rates
Use a project-specific rate when (a) the project’s business risk differs from the company’s, or (b) the financing changes the capital structure a lot.
- Identify proxy companies: listed firms in the same industry.
- Get their equity betas and capital structures.
- Ungear each equity beta to an asset beta, with the proxy’s own gearing.
- Average the asset betas (simple, or weighted by market value) to get the project’s asset beta.
- If the project is all-equity, use the asset beta in CAPM. Otherwise regear: βE = βA × [E + D(1 − t)] ÷ E, using the gearing that will finance the project.
- Use CAPM for the project cost of equity and, where asked, weight it with the cost of debt to give a project WACC.
ICAN’s example. An all-equity company (beta 0.90) considers an industry whose average equity beta is 1.25 and debt to equity ratio is 40 : 60. The risk-free rate is 4%, the market return 7% and tax 30%. It will stay all-equity.
- βA = 1.25 × 60 ÷ (60 + 40 × 0.7) = 1.25 × 60 ÷ 88 = 0.85.
- Cost of equity.
4% + 0.85 × 3% = 6.55%. - Use 6.55% to appraise the project, not the company’s own rate.
The cost of capital for the November 2014 cement project followed these steps: the cement equity beta (2.5) was ungeared to about 2.0, regeared at a 30% debt to equity ratio to 2.36, and CAPM gave a cost of equity of 21.52% and a WACC of about 18%.
The rules in more depth
Why ungear before averaging
Proxies have different gearing, so their equity betas are not comparable. The asset betas strip out that difference.
Why regear
The project will be financed by the investing company’s mix, so the equity beta must carry that financial risk.
Why the (1 − t) is in the formula
Debt’s effect on equity risk is reduced by the tax relief on interest.
Why debt beta is usually zero
Corporate debt is far less risky than equity, and ignoring its small beta barely changes the asset beta.
Why the project rate may differ from the WACC
The WACC reflects the company’s average business and financial risk. A project that differs in either needs its own rate.
Effect on analysis
State the assumption that debt is risk-free, show each formula step, and compare the project rate with the company WACC.
Worked examples
Example 1: ungearing. A listed firm has an equity beta of 1.4. Debt is ₦50m and equity ₦100m. Tax is 30%. Assume debt is risk-free.
- E + D(1 − t) = 100 + 50 × 0.7 = 135.
- βA = 1.4 × 100 ÷ 135.
- βA = 1.037.
Example 2: regearing. The asset beta is 1.0. The project will be financed with debt of ₦50m and equity of ₦100m. Tax is 30%.
- βE = 1.0 × [100 + 50 × 0.7] ÷ 100.
- βE = 1.0 × 135 ÷ 100.
- βE = 1.35.
Example 3: a change in gearing. A company is worth ₦200m (equity ₦150m, debt ₦50m), with a cost of equity of 11%, a pre-tax cost of debt of 6% and tax of 25%. It borrows ₦30m to buy back shares.
- VU = 200 − 50 × 0.25 = 187.5.
New VG = 187.5 + 80 × 0.25 = ₦207.5m; equity ₦127.5m. - keU: 11 = keU + 0.75 × (keU − 6) × (50 ÷ 150), so 1.25 keU = 12.5 and keU = 10%.
- New ke = 10% + 0.75 × 4% × (80 ÷ 127.5) = 11.88%.
New WACC = (127.5 × 11.88% + 80 × 4.5%) ÷ 207.5 = 9.04%.
Example 4: an equity beta after a change. ICAN’s company: value ₦80m (equity ₦60m, debt ₦20m), equity beta 0.925, tax 30%, debt risk-free. It issues ₦10m of equity to repay ₦10m of debt.
- βA = 0.925 × 60 ÷ (60 + 20 × 0.7) = 0.925 × 60 ÷ 74 = 0.75.
- VU = 80 − 20 × 0.3 = 74.
New VG = 74 + 10 × 0.3 = 77
equity = 67. - New βE = 0.75 × (67 + 10 × 0.7) ÷ 67 = 0.75 × 74 ÷ 67 = 0.83.
How to score full marks
- Write each formula and then substitute.
- State the risk-free debt assumption.
- Ungear each proxy with its own gearing before averaging.
- Regear with the project’s own financing mix.
- Use the post-tax cost of debt in the WACC.
- Compare the result with the company’s WACC and explain the gap.
Common mistakes in the exam
- Averaging equity betas without ungearing.
- Using debt to equity when the formula needs the equity weight, or the reverse.
- Forgetting the (1 − t).
- Using the company’s gearing to ungear a proxy.
- Using VG = VU + Dt when extra debt is raised.
- Keeping the company’s WACC for a project in a different industry.
Quick revision
- Gearing change: VU, then new VG; ungeared and regeared WACC and ke.
- βA = βE × E ÷ [E + D(1 − t)] with risk-free debt.
- Proxy: ungear, average, regear, CAPM, WACC.
- Use a project rate if business or financial risk differs.
- MM value formula: not for extra debt.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. An asset beta measures:
- Business risk only
- Financial risk only
- Total risk
- Tax risk
Show the answer
A. With no debt.
Question 2. The equity beta of a geared company is:
- Higher than its asset beta
- Lower than its asset beta
- Equal to it
- Zero
Show the answer
A. It includes financial risk.
Question 3. With risk-free debt, asset beta equals:
- βE × E ÷ [E + D(1 − t)]
- βE × D ÷ E
- βE + D
- βE × (1 − t)
Show the answer
A. The ungearing formula.
Question 4. A proxy company is:
- A listed company in the same business as the project
- A subsidiary
- A bank
- A tax adviser
Show the answer
A. Its beta shows business risk.
Question 5. To use proxy betas you should first:
- Ungear each to an asset beta
- Average the equity betas
- Add the debt
- Ignore gearing
Show the answer
A. Gearing differs between proxies.
Question 6. A project rate is needed when:
- Business risk or gearing differs
- The project is small
- The project is similar
- Tax is zero
Show the answer
A. The WACC would be wrong.
Question 7. The debt beta is usually taken as:
- Zero
- One
- The equity beta
- Two
Show the answer
A. Debt is assumed risk-free.
Question 8. V_G = V_U + Dt may NOT be used when:
- Extra debt is raised
- Debt replaces equity
- Equity replaces debt
- Tax is 30%
Show the answer
A. The company’s size changes.
Question 9. Regearing converts:
- An asset beta into an equity beta
- An equity beta into an asset beta
- Debt into equity
- Tax into profit
Show the answer
A. At the target gearing.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Distinguish asset, equity and debt betas. (4 marks) (b) State when a project-specific discount rate should be used. (3 marks) (c) Explain why proxy equity betas are ungeared before they are averaged. (3 marks)
Question 12 (20 marks). Biso Plc is considering a project in a new industry. Two listed proxies: P1 has an equity beta of 1.20 and a debt to equity ratio of 20 : 80; P2 has an equity beta of 1.40 and 40 : 60. Tax is 30%. Debt is risk-free. The project will be financed 30% debt and 70% equity. The risk-free rate is 4%, the market return 10% and pre-tax borrowing costs 6%. (a) Calculate each proxy’s asset beta.
(6 marks) (b) Average them and regear to the project’s financing. (4 marks) (c) Calculate the cost of equity. (3 marks) (d) Calculate the project WACC. (4 marks) (e) Explain why the company’s own WACC would be unsuitable. (3 marks)
Question 13 (10 marks). A company is worth ₦200m (equity ₦150m, debt ₦50m) with a cost of equity of 11%, a pre-tax cost of debt of 6% and tax of 25%. It borrows ₦30m to buy back shares. (a) Calculate the new value of the company. (3 marks) (b) Calculate the ungeared cost of equity. (3 marks) (c) Calculate the new cost of equity. (4 marks)
Model answers
Show the model answer to Question 11
(a) Betas (4 marks). The asset beta measures business risk only (the beta of an ungeared company). (1.5) The equity beta includes financial risk, so it is higher in a geared company. (1.5) The debt beta is much lower and often taken as zero. (1)
(b) Use of a project rate (3 marks). When the project’s business risk differs from the company’s. (1.5) When the financing changes the capital structure significantly. (1.5)
(c) Ungearing (3 marks). Proxies have different gearing, so their equity betas include different financial risk and cannot be compared. (2) The asset betas show business risk alone, so they can be averaged. (1)
Show the model answer to Question 12
(a) Asset betas (6 marks).
- P1: 1.20 × 80 ÷ (80 + 20 × 0.7) = 1.20 × 80 ÷ 94 = 1.021. (3)
- P2: 1.40 × 60 ÷ (60 + 40 × 0.7) = 1.40 × 60 ÷ 88 = 0.955. (3)
(b) Average and regear (4 marks).
- Average = (1.021 + 0.955) ÷ 2 = 0.988. (1.5)
- Project D.
E = 30 : 70. βE = 0.988 × (70 + 30 × 0.7) ÷ 70 = 0.988 × 91 ÷ 70 = 1.284. (2.5)
(c) Cost of equity (3 marks). 4% + 1.284 × (10% − 4%) = 4% + 7.70% = 11.7%. (3)
(d) WACC (4 marks).
- Post-tax cost of debt.
6% × 0.7 = 4.2%. (1) - WACC = 0.7 × 11.7% + 0.3 × 4.2% = 8.19% + 1.26% = 9.45%. (3)
(e) Company WACC (3 marks). It reflects the company’s own business and financial risk. (1.5) The new industry has different business risk and the project has its own financing mix, so a different rate is needed. (1.5)
Show the model answer to Question 13
(a) Value (3 marks).
- VU = 200 − 50 × 0.25 = 187.5. (1.5)
- New VG = 187.5 + 80 × 0.25 = ₦207.5m, so equity = ₦127.5m. (1.5)
(b) Ungeared ke (3 marks). 11 = keU + 0.75 × (keU − 6) × 50 ÷ 150, so 1.25 keU = 12.5. (2) keU = 10%. (1)
(c) New ke (4 marks). keG = 10% + 0.75 × (10% − 6%) × 80 ÷ 127.5 = 10% + 1.88% = 11.88%. (4)
Pathfinder on Regearing WACC, ungearing betas and project-specific rates
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.
The 12-mark part asked for the cost of capital for a cement project. The model answer takes the proxy company’s equity beta of 1.82, strips out its 40% drinks division (equity beta 0.8) to find a cement equity beta of 2.5, ungears it using the proxy’s debt to equity ratio of 40% and tax of 25% to about 2.0, regears at the target ratio of 30% to 2.36, uses CAPM for a cost of equity of 21.52%, takes the after-tax cost of debt as 6% and weights them 100 : 30 for a WACC of about 18%. Candidates confused debt to equity with debt to total capital. Read the sections on project-specific rates and betas.
Revise first: Project-specific discount rates
Questions ask you to convert between equity and asset betas. See the section on asset, equity and debt betas.
Revise first: Asset, equity and debt betas
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: November 2014 question 1(a).
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