Impact of financing on investment decisions and adjusted present values: formula sheet

Key formulas

APV

APV = Base-case NPV + PV of financing side effects

Side effects include the tax shield, subsidised loan benefit and, as a negative, issue costs. Accept if APV > 0.

Asset (ungeared) beta

βa = βe × E ÷ [E + D(1 − T)] (debt beta = 0)

Use market values of E and D. If a debt beta is given, use the full weighted version: βa = βe × E ÷ [E + D(1 − T)] + βd × D(1 − T) ÷ [E + D(1 − T)].

Ungeared cost of equity (CAPM)

Keu = Rf + βa × (Rm − Rf)

This is the base-case discount rate.

Ungeared cost of equity (Modigliani and Miller with tax)

Keg = Keu + (Keu − Kd)(1 − T) × D ÷ E

Rearrange to find Keu from an observed geared cost of equity. Check which form your formula sheet gives.

Annual tax shield

Tax shield = Interest × T

Only count interest on debt the project supports, and only while the tax relief is available.

Tax shield on permanent debt

PV of tax shield = D × T

Valid when debt is perpetual and discounted at the pre-tax cost of debt.

Discount rate for tax shield

Use the pre-tax cost of debt (Kd)

The tax shield has the risk of the debt. State this assumption in your answer.

CAPM

Ke = Rf + β × (Rm − Rf)

Rm − Rf is the equity risk premium. Check whether the question gives Rm or the premium.

Asset beta (general form)

βa = βe × E ÷ [E + D(1 − T)] + βd × D(1 − T) ÷ [E + D(1 − T)]

E and D are market values. T is the tax rate. βd is the debt beta.

Asset beta (debt beta zero)

βa = βe × E ÷ [E + D(1 − T)]

This is the usual exam form. Use it unless a debt beta is given.

Regearing the beta

βe = βa × [1 + (1 − T) × D/E]

Assumes a zero debt beta. Use only if the question asks for a geared beta or WACC.

Ungeared cost of equity from CAPM

Ke(u) = Rf + βa × (Rm − Rf)

This is the APV base-case discount rate.

MM with tax: geared cost of equity

Ke = Ke(u) + (Ke(u) − Kd) × (1 − T) × D/E

Kd is the pre-tax cost of debt. Use market value D/E.

MM with tax: ungeared cost of equity

Ke(u) = [Ke × E + Kd × (1 − T) × D] ÷ [E + D(1 − T)]

Use when you are given the geared cost of equity, not a beta.

APV

APV = Base-case NPV + PV of financing side effects

Base-case NPV uses the ungeared cost of equity and the project's operating cash flows only.

Annual tax shield

Tax shield = Interest × Tax rate = Debt × Interest rate × Tax rate

Use the interest on debt the project supports. Check when tax is paid and whether a tax cap or limit applies.

PV of tax shield

PV = Σ Tax shield ÷ (1 + Kd)^t

Usual discount rate is the pre-tax cost of debt. Use the rate the question tells you if it states one, such as the risk-free rate.

Gross finance raised

Gross amount = Net amount needed ÷ (1 − issue cost %)

Issue costs are a percentage of the gross amount, not of the net amount.

Issue costs in APV

Issue costs = Gross amount × issue cost %; after-tax cost = Issue costs × (1 − tax rate) if deductible

Paid at time 0, so no discounting. Include as a negative.

Subsidised loan benefit

Benefit = Loan − PV of after-tax interest and repayment at the market pre-tax cost of debt

Equivalent to the PV of the interest saving plus the tax effect. Use the market rate as the discount rate.

APV

APV = base-case NPV + PV of financing side effects

Base-case NPV uses the ungeared cost of equity. Side effects include the tax shield, issue costs and subsidised loan benefits.

Debt capacity

Debt = debt capacity % × stated base (initial cost or project value)

Use the base the question names. If it gives the debt amount directly, use that.

Annual interest

Interest = debt outstanding × pre-tax interest rate

Use the debt outstanding at the start of the period unless told otherwise.

Annual tax shield

Tax shield = interest × tax rate

Count it only in years when the tax relief is actually received. Follow any stated tax lag.

PV of tax shield

PV = Σ tax shield ÷ (1 + Kd)^t

Kd is the pre-tax cost of debt, or the risk-free rate if the question tells you to use it.

Permanent debt shortcut

PV of tax shield = debt × tax rate

Applies only if debt is permanent and discounted at the same rate as the interest rate.

After-tax issue costs

Issue cost × (1 − tax rate) if tax deductible

Deduct from APV. If the question says issue costs are not deductible, deduct them in full.

Adjusted present value

APV = base-case NPV (at Keu) + PV of financing side effects

Financing side effects: tax shield on debt, issue costs, subsidised loan benefit. Issue costs are negative.

Ungeared cost of equity (with tax)

Keu = [Ve × Keg + Vd × Kd(1 − T)] ÷ [Ve + Vd(1 − T)]

Assumes debt is risk-free-like and permanent (Modigliani and Miller with tax). Keg is the geared cost of equity.

Geared cost of equity (with tax)

Keg = Keu + (Keu − Kd)(1 − T) × Vd ÷ Ve

Use to re-gear when a project has a different debt level.

Asset beta

βa = βe × Ve ÷ [Ve + Vd(1 − T)] + βd × Vd(1 − T) ÷ [Ve + Vd(1 − T)]

If debt beta is taken as zero, the second term drops out. Then Keu = Rf + βa × (Rm − Rf).

Tax shield on debt

Annual tax shield = debt × interest rate × T

Discount at the pre-tax cost of debt Kd, as ACCA examiners normally do, unless the question says otherwise.

WACC

WACC = Ke × Ve ÷ (Ve + Vd) + Kd(1 − T) × Vd ÷ (Ve + Vd)

Valid only for projects with the firm's business risk and capital structure.

Subsidised loan benefit

Annual benefit = (market rate − subsidised rate) × loan × (1 − T)

Discount at the market (pre-tax) cost of debt. Do this if the question gives a below-market loan.

Quick revision

• APV = base-case NPV + PV of financing side effects.
• The base-case NPV discounts operating cash flows at the ungeared cost of equity.
• Ungear the proxy beta before using it, then use the ungeared cost of equity in the base case.
• Business risk of the project, not the parent company, decides the base-case rate.
• The interest tax shield = interest × tax rate, counted only when the interest is tax deductible and there are profits to use it.
• The tax shield is usually discounted at the pre-tax cost of debt, since its risk is linked to the debt. State your assumption.
• Issue costs are a cash outflow at the start. Deduct them and note whether they are tax deductible.
• Debt capacity sets the amount of debt that supports the project and so the size of the tax shield.
• A positive APV means the project adds value. Accept it.
• APV is suited to projects with a changing financing mix or special finance. NPV with a single rate suits stable gearing.
• Always list your assumptions in the answer and say what happens if they fail.
• End with a clear recommendation and the main risks.

Common mistakes

• Discounting base-case cash flows at WACC instead of the ungeared cost of equity. Fix: Remember the first stage has no financing in it. Use Keu only. WACC does not appear in APV.
• Forgetting to ungear the proxy firm's beta, or ungearing it without the (1 − T) term. Fix: Always ask whether the beta reflects the proxy firm's gearing. If so, convert it to βa with E ÷ [E + D(1 − T)] before applying CAPM.
• Using the company's own equity beta or cost of equity to discount the base case. Fix: Check whether it reflects the project's business risk and no financial risk. Usually you need a proxy asset beta or the MM ungeared rate.
• Leaving out (1 − T) when degearing. Fix: Write the divisor as E + D(1 − T) every time. The tax shield on debt reduces its effect on equity risk.
• Discounting the tax shield at the ungeared cost of equity or WACC. Fix: Discount the tax shield at the pre-tax cost of debt unless the question tells you otherwise. Only the base-case cash flows use the ungeared cost of equity.
• Calculating issue costs on the net amount needed instead of the gross amount raised. Fix: Gross up first: net needed ÷ (1 − issue cost %). Then apply the percentage to the gross figure.
• Discounting the tax shield at the WACC Fix: Use the pre-tax cost of debt for the shield, unless the question gives another rate. The WACC already contains the tax benefit.
• Multiplying the whole debt by the tax rate each year Fix: The annual shield is debt × interest rate × tax rate. Only the permanent-debt shortcut gives debt × tax rate, and it needs a perpetuity.
• Discounting the base-case cash flows at the WACC instead of Keu. Fix: In APV the base case is all-equity. Always use the ungeared cost of equity for operating cash flows.
• Discounting the tax shield at Keu. Fix: The tax shield is as risky as the debt that creates it. Discount at the pre-tax cost of debt unless told otherwise.

Exam tips

• Show the base-case NPV and the financing effects as separate sub-totals. Marks are given for each, so a wrong tax shield will not cost you the base case.
• State your assumptions in a line each: debt amount, discount rate for the tax shield, and tax deductibility of issue costs. Examiners reward clear, sensible assumptions.
• In the discussion part, compare APV with NPV using WACC. Say APV copes better with changing gearing and special financing, but needs reliable estimates of debt and side effects.
• Use the professional skills marks. Give a clear recommendation, say which numbers it depends on, and comment on the risk that the tax shield may not be fully usable if profits are too low.
• Check the time frame of debt. If debt is repaid at the end of year 4, the shield stops at year 4. If it is permanent, use D × T.
• Read the data for hints on the route. A beta and a proxy company point to degearing and CAPM. A geared cost of equity with Kd points to the MM formula.
• Show the degearing line clearly, with E + D(1 − T) written out. Method marks are awarded even if a number slips.
• State your assumptions, such as a zero debt beta and market values. These gain professional skills credit in Section A.