FNCE30011 · Essentials Of Corporate Valuation
The Vanilla and Recursive WACC
This is the model-selection heart of the subject. Where the Standard WACC handles a constant target ratio, the Vanilla WACC kv = (1−L)ke + Lkd handles a target debt schedule — fixed dollar debt, so the ratio L drifts. Here the interest tax shield lives in the cash flow: FCF (which already contains the shield) pairs with kv, mirroring the standard world where FCFU pairs with ks. The bridge between them is ks = kv − kdTL and FCF = FCFU + Y·T. A magic special case appears when the tax-shield risk equals ku: then kv = ku, a rate that is independent of leverage — which is exactly what lets you value a project whose leverage changes over its life. When leverage really is a changing target ratio (or ku varies year to year), the right tool is the recursive standard WACC: backward induction Vt = (FCFUt+1 + Vt+1)/(1 + ks,t+1), rolling V2→V1→V0. This recurrence is the single most exam-tested DCF type in the subject.
What this chapter covers
- 014.1 The Vanilla WACC kv = (1−L)ke + Lkd for a target debt schedule
- 024.2 ITS in the cash flow (FCF↔kv) vs ITS in the rate (FCFU↔ks)
- 034.3 The bridge: ks = kv − kdTL and FCF = FCFU + Y·T
- 044.4 The magic case kv = ku when the tax-shield risk equals ku
- 054.5 Same V0 two ways — the equivalence demonstration
- 064.6 The recursive standard WACC: backward induction V2→V1→V0
Worked example: the magic kv = ku on a perpetuity
- +1Identify. Tax-shield risk = ku, so the Vanilla WACC collapses to kv = ku — a rate independent of leverage. Discount FCF (which holds the shield) at kv.
- +1Rate. kv = ku = 10%.
- +1Value. V0 = FCF / kv = 100 / 0.10 = $1,000m.
- +1Interpret. Because kv = ku does not move with L, a changing debt schedule leaves the value unchanged — which is exactly why this rate is used for projects whose leverage drifts.
Key terms
- Vanilla WACC (kv)
- kv = (1−L)ke + Lkd, the rate for FCF (which already contains the interest tax shield) when debt follows a fixed dollar schedule. No (1−T) factor, because the shield is in the cash flow, not the rate.
- Target debt schedule
- A fixed dollar debt path (e.g. repay $50m a year), so the leverage ratio L drifts as value changes. This is the cue for the Vanilla WACC rather than the Standard WACC.
- The WACC bridge
- ks = kv − kdTL and FCF = FCFU + Y·T — the algebra linking the standard and vanilla worlds, so a problem set up one way can be answered the other.
- The magic case kv = ku
- When the interest tax shields are as risky as the operations (kits = ku), the vanilla WACC equals ku and is independent of leverage — the rate that values projects whose leverage changes.
- Recursive standard WACC
- Backward induction Vt = (FCFUt+1 + Vt+1)/(1 + ks,t+1), rolling value back period by period. Used when L is a changing target ratio or ku varies by year — the most exam-tested DCF type.
The Vanilla and Recursive WACC FAQ
Vanilla or Standard WACC — how do I tell which the question wants?
Read the leverage story. A constant target ratio (L held at a fixed percentage of value) is the Standard WACC, and you discount unlevered FCF at ks = (1−L)ke + Lkd(1−T). A fixed dollar debt schedule (so L drifts) is the Vanilla WACC, and you discount FCF — shield inside — at kv = (1−L)ke + Lkd. The phrase that fixes it is whether debt is set as a ratio or as dollars.
Why does pairing FCF with kv (not FCFU) avoid double-counting the tax shield?
FCF already contains the interest tax shield, and the vanilla rate kv deliberately omits the (1−T) factor, so the shield is counted once — in the cash flow. If you discounted FCF at the standard ks (which also embeds the shield via (1−T)), you would count it twice. The mirror rule is FCFU ↔ ks: shield in the rate, not the cash flow.
When must I use the recursive standard WACC instead of a single WACC?
When leverage is a changing target ratio or ku varies year to year, a single constant WACC is wrong because the rate differs each period. Then you roll value back by backward induction: Vt = (FCFUt+1 + Vt+1)/(1 + ks,t+1), starting from a terminal value and working V2→V1→V0. This is the flagship exam type, so practise the rollback until it is automatic.
What is the intuition behind kv = ku?
If the interest tax shields are exactly as risky as the firm's operations, then adding debt does not change the risk profile of the total cash flows — so the rate that values them is the unlevered cost of capital, independent of how much debt there is. That leverage-proof property is precisely what lets you value a project whose leverage changes over its life with a single rate.
Exam move
This chapter is where model selection is won or lost, so drill the cue-to-model map: constant ratio → Standard WACC (FCFU at ks); fixed dollar schedule → Vanilla WACC (FCF at kv); changing leverage or ku → recursive standard WACC by backward induction. Keep the two matched pairs and the bridge (ks = kv − kdTL, FCF = FCFU + Y·T) on your A4. Above all, rehearse the recursive rollback — it is the single most-tested DCF type — until you can set up Vt = (FCFUt+1 + Vt+1)/(1 + ks,t+1) and roll it back from a terminal value without hesitation.
