What is Adjusted Present Value (APV)?
– Adjusted Present Value (APV) is a valuation method that breaks a firm’s value into two parts: (1) the value of the operating business if it had no debt (the “unlevered” or all-equity value), and (2) the value added (or subtracted) by financing choices—most commonly the tax benefits from interest deductibility and the costs from financial distress or subsidies.
– Key idea: value the asset on its own merits, then add the cash effects of financing.
Definitions (short)
– Unlevered firm value: present value (PV) of expected free cash flows (FCFs) discounted at the unlevered cost of capital (the return required by equity if the firm had no debt).
– Tax shield: the annual tax saving from interest deductibility = interest expense × corporate tax rate.
– Net effect of debt (NE): PV of all financing-related cash flows (tax shields, subsidies, issuance costs, distress costs, hedging benefits, etc.).
– APV formula: APV = Unlevered firm value + Net effect of debt
When APV is useful
– Capital structures that change over time (e.g., leveraged buyouts).
– Complex or subsidized financing (below-market-rate loans, government subsidies).
– Cross-border deals where tax regimes and financing risks differ by jurisdiction.
– Cases where separating business risk from financing effects gives clearer decision-making.
Step-by-step checklist to compute APV
1. Forecast unlevered free cash flows (FCFs) for the project/company.
2. Choose an unlevered discount rate appropriate for business risk (often called unlevered cost of equity).
3. Discount the unlevered FCFs to get the unlevered firm value (present value). Subtract any upfront required investment to get base-case NPV if evaluating a project.
4. Estimate annual interest expense(s) and any other financing-related cash flows (tax shields, subsidies, issuance costs, expected costs of financial distress).
5. Choose discount rates for each financing effect (tax shields are often discounted at the cost of debt if their risk is similar to debt; other effects may require different rates).
6. Discount each financing effect to present
value using the chosen rates. 7. Sum the present value of all financing effects and add them to the unlevered firm value (or base-case NPV) to get the APV. 8. Perform sensitivity checks (vary discount rates, tax rate, debt schedule, and bankruptcy costs) to see how APV responds.
Key formulas (definitions)
– Unlevered firm value (VU): VU = Σt (FCFt / (1 + ru)^t) where FCFt = unlevered free cash flow in year t, and ru = unlevered discount rate (cost of capital for the business without debt).
– PV of a financing effect: PV(effect) = Σt (Effectt / (1 + reffect)^t) where Effectt is the cash flow attributable to financing (tax shield, issuance costs, subsidies, distress costs) and reffect is the discount rate appropriate to the risk of that effect.
– Adjusted Present Value (APV): APV = VU + Σ PV(financing effects)
– If evaluating a project with an upfront investment I0, you can express base-case NPV = VU − I0, and then APV = (VU − I0) + Σ PV(financing effects).
Worked numeric example (step-by-step)
Assumptions
– Project requires an upfront investment I0 = 1,000.
– Unlevered free cash flow is constant at 200 per year in perpetuity (starting one year from now).
– Unlevered discount rate ru = 12%.
– Company issues debt D = 1,000 at interest rate rd = 6%.
– Corporate tax rate Tc = 21%.
– Debt is permanent (for the simplified example) and tax shields carry debt-like risk, so discount tax shields at rd.
– Issuance costs (one-time) = 20.
Step A — Compute unlevered firm value (VU)
– VU = FCF / ru = 200 / 0.12 = 1,666.67
Step B — Compute base-case NPV (unlevered value minus investment)
– Base-case NPV = VU − I0 = 1,666.67 − 1,000 = 666.67
Step C — Compute annual tax shield
– Annual interest = D × rd = 1,000 × 0.06 = 60
– Annual tax shield = interest × Tc = 60 × 0.21 = 12.6
Step D — PV of tax shields (perpetuity discounted at rd)
– PV(tax shields) = annual tax shield / rd = 12.6 / 0.06 = 210
Step E — Include issuance costs (PV = 20, paid upfront)
Step F — Compute APV
– APV = base-case NPV + PV(tax shields) − issuance costs
– APV = 666.67 + 210 − 20 = 856.67
Interpretation: On these assumptions, financing adds value (net tax
shield benefit minus issuance costs) of 210 − 20 = 190. In other words, the tax advantage from debt (PV = 210) more than offsets the upfront financing cost (20), so the APV (856.67) is larger than the all-equity base-case value (VU − I0 = 666.67).
Key formulas recap
– Base-case (unlevered) value, VU: compute by discounting unlevered free cash flows at the unlevered cost of capital rU (example used a perpetuity: VU = FCF1 / rU).
– Annual interest = D × rd.
– Annual tax shield = Annual interest × Tc.
– PV of perpetual tax shields (constant debt) = Annual tax shield / rd = D × Tc.
– APV = (VU − I0) + PV(tax shields) − PV(issuance costs) ± PV(other financing effects).
When to discount tax shields at rd versus other rates
– If the tax shield cash flows are as risky as the debt (i.e., creditor-like risk), discount at rd. That is the default assumption in many APV applications.
– If tax shields are riskier (e.g., rely on operating performance or are unsecured), you may need a higher discount rate that reflects that risk.
– If debt is expected to change with firm value, the appropriate treatment can be more complex; see limitations below.
Worked alternative: finite-term debt example
Same base-case numbers as earlier (VU = 1,666.67, I0 = 1,000 → base-case NPV = 666.67). Now assume the firm issues D = 1,000 for 5 years at rd = 6%; Tc = 21%; issuance costs = 20.
Step 1 — Annual interest and tax shield
– Annual interest = 1,000 × 0.06 = 60
– Annual tax shield = 60 × 0.21 = 12.6
Step 2 — PV of 5-year tax shields (discount at rd = 6%)
– PV = 12.6 × [(1 − (1 + 0.06)^(−5)) / 0.06] ≈ 12.6 × 4.21237 ≈ 53.08
Step 3 — APV
– APV = 666.67 + 53.08 − 20 = 699.75
Interpretation: Financed value rises only slightly relative to all-equity (699.75 vs 666.67) because tax shields expire after five years; perpetual debt produced a much larger PV(tax shields) = 210.
Step-by-step checklist to perform APV
1. Project unlevered free cash flows (FCF) to the horizon you use.
2. Estimate
2. Estimate the discount rate for unlevered cash flows (r0).
– r0 is the required return on the firm’s assets if it had no debt (also called the unlevered cost of equity or cost of capital for an all-equity firm).
– Common approach: use the Capital Asset Pricing Model (CAPM): r0 = rf + βU × (rm − rf), where rf is the risk-free rate, rm is the expected market return, and βU is the unlevered beta (asset beta).
– To unlever a observed equity beta (βL): βU = βL / [1 + (1 − Tc) × (D/E)], where Tc is the corporate tax rate and D/E is the target debt-to-equity ratio. Note assumptions: tax shield impact and debt level are constant in this formula.
3. Discount unlevered free cash flows to get the base (all-equity) firm value.
– Project unlevered free cash flows (FCF) for each forecast year. Unlevered FCF means cash available to all providers (debt and equity) before interest but after taxes.
– PV(unlevered firm) = Σ (FCF_t / (1 + r0)^t) + PV_terminal_unlevered. Use a terminal value formula consistent with your growth and r0 assumptions (e.g., Gordon growth: TV = FCF_n+1 / (r0 − g)).
4. Value the tax shields and other financing effects separately. Main options for tax shields:
– If debt policy is explicit and debt schedule is known (e.g., a fixed outstanding debt amount for each year), compute annual tax shield = Interest_t × Tc and discount those at the cost of debt rd (if tax shields carry the same risk as debt). That’s what the worked example did.
– If debt is perpetual and maintained at a constant market value, and tax shields are assumed to be as safe as government debt, a common simplification is PV(tax shields) = Tc × D (market value of debt). This assumes perpetual, risk-free-like shields.
– If tax shields share business risk (i.e., their cash flows vary with firm earnings), you may discount them at r0 or an intermediate rate reflecting their risk. Be explicit about which assumption you use.
5. Add other financing side effects (each discounted appropriately). Examples:
– Flotation or issuance costs: one-time cash outflows when raising capital.
– Subsidies, grants, or tax credits: present value
of expected receipts or reductions in operating cash outflows (discount each at a rate that reflects the risk of those benefits).
Other financing side effects to consider:
– Bankruptcy or financial distress costs: expected present value of direct (legal/administrative) and indirect (lost sales, higher input costs) costs that rise with leverage. Discount at a rate appropriate to the risk of those costs.
– Agency benefits or costs: value changes from manager/shareholder conflicts (e.g., monitoring benefits of debt). These are usually scenario-dependent and should be valued explicitly when material.
– Guarantees and covenants: fees or constraints that change cash flows or risk; include PV of fees and any expected costs from covenant breaches.
– Re-leveraging effects on firm value: if leverage is changed over time (e.g., target capital structure), model how debt issuance/repayment and associated tax shields evolve.
– Flotation/issuance costs: one-time cash outflows when raising debt or equity (model as immediate cash outflow or amortize if required by accounting/analysis preferences).
Worked APV example (step-by-step)
Assumptions
– Project initial investment (outflow at t=0): I0 = $1,500
– Unlevered free cash flow (UFCF) perpetuity, starting t=1: UFCF = $200/year
– Unlevered discount rate (cost of unlevered equity / project risk), r0 = 10%
– Debt issued immediately at market value D = $500, cost of debt rD = 5%
– Corporate tax rate Tc
= 21% (Tc = 0.21).
Step-by-step APV calculation (numeric)
1) Value of the unlevered project (Vu)
– Formula: Vu = UFCF / r0
– Numbers: Vu = $200 / 0.10 = $2,000
2) Annual interest and tax shield
– Interest each year = rD × D = 0.05 × $500 = $25
– Annual tax shield = Tc × interest = 0.21 × $25 = $5.25
3) Present value of tax shields (assume perpetual, debt is permanent and tax shields have debt-like risk)
– If tax shields are as safe as debt, discount at cost of debt rD:
PV(tax shield) = annual tax shield / rD = $5.25 / 0.05 = $105
(Equivalent shortcut for perpetual risk-free debt: PV(tax shield) = Tc × D = 0.21 × $500 = $105.)
4) Base APV (no flotation or distress costs)
– APV = Vu + PV(tax shield) = $2,000 + $105 = $2,105
5) Add/subtract other effects (examples)
– Flotation (issuance) cost example: if debt issuance costs = 2% of debt = 0.02 × $500 = $10, then APV_adj = $2,105 − $10 = $2,095.
– Expected present value of bankruptcy costs or covenant costs should be subtracted if material; include PV of costs from covenant breaches, fees, or expected distress probability.
Alternative discounting approach (when to use each)
– Discount tax shields at rD (cost of debt) when tax shields are nearly risk-free / have the same default risk as debt.
– If tax shields are as risky as the firm’s unlevered cash flows, discount at r0 (unlevered cost of capital). That yields:
PV(tax shield) = $5.25 / 0.10 = $52.50 → APV = $2,052.50
– If debt carries default risk different from bond-like debt, determine an appropriate discount rate between rD and r0 or model expected default and tax shield survival explicitly.
Worked sensitivity examples
– Change in corporate tax rate: if Tc = 0.25, PV(tax shield) = 0.25 × $500 = $125 → APV = $2,125.
– Change in debt level: if D = $800, PV(tax shield) = 0.21 × $800 = $168 → APV = $2,168 (holding Vu constant).
Checklist for implementing APV in practice
– Define assumptions clearly: timing and permanence of debt, tax rate, debt schedule, issuance costs, distress probabilities.
– Compute Vu using unlevered cash flows and r0.
– Model tax shields cash flows year-by-year if debt is temporary or changing; discount each tax shield using an appropriate rate reflecting its risk.
– Add PV of other financing effects