Level 3 — ProductionValuation Methods

Valuation Methods

⏱ 45 minutes60 marksprintable — key stays hidden on paper

Chapter: 2.6 Valuation Methods Level: 3 — Production (from-scratch derivations, explain-out-loud) Time limit: 45 minutes Total marks: 60

Show all working. Use $...$ notation for formulas. Round money to 2 decimals unless told otherwise.

Question 1 — Derive WACC from scratch (10 marks)

A firm has the following capital structure:

Market value of equity: $600$ million
Market value of debt: $400$ million
Cost of equity (from CAPM): risk-free $= 4\%$ , equity beta $= 1.25$ , market risk premium $= 6\%$
Pre-tax cost of debt: $7\%$
Corporate tax rate: $25\%$

(a) Compute the cost of equity using CAPM, stating the formula. (3) (b) Derive and compute the WACC, showing the weighting logic and the tax shield. (5) (c) Explain out loud, in 2–3 sentences, why the after-tax cost of debt is used and not the pre-tax figure. (2)

Question 2 — Two-stage DCF (from memory) (14 marks)

A company has free cash flow to firm (FCFF) of $100$ million in Year 1, growing at $10\%$ per year for Years 1→3 (i.e. Year 1 = 100, then growth applies). From Year 4 onward, FCFF grows at a stable $3\%$ forever. Use a discount rate (WACC) of $9\%$ .

(a) Write the terminal value formula and state at which year it is calculated. (2) (b) Project FCFF for Years 1, 2, 3 and the Year-4 FCFF used for terminal value. (3) (c) Compute the terminal value at end of Year 3. (3) (d) Discount all cash flows and the terminal value to get enterprise value. (4) (e) If net debt is $150$ million and shares outstanding are $50$ million, compute intrinsic value per share. (2)

Question 3 — Dividend Discount Model (10 marks)

A stock currently pays a dividend of $D_0 = \$ 2.00 $. Dividends are expected to grow at$ 6% $forever. The required return is$ 11%$.

(a) Derive the Gordon Growth formula starting from the infinite discounted dividend sum, showing the convergence condition. (4) (b) Compute the intrinsic value. (3) (c) If the market price is $\$ 50$, state whether the stock is over- or under-valued and by how much (%). (3)

Question 4 — Reverse DCF & implied growth (10 marks)

A stock trades at an intrinsic-equivalent enterprise value of $\$ 1{,}000 $million. Next year's FCFF is$ $60 $million and WACC is$ 8%$. Assume a single-stage (perpetual growth) model.

(a) Explain out loud what a reverse DCF answers that a forward DCF does not. (2) (b) Starting from the Gordon-style EV formula, solve algebraically for the growth rate $g$ implied by the market. (4) (c) Compute the implied $g$ and comment whether it is realistic given long-run GDP growth of ~ $3\%$ . (4)

Question 5 — Relative valuation & margin of safety (8 marks)

Company X has EPS of $\$ 4.00 $. Three comparable firms trade at P/E multiples of$ 15 $,$ 18 $, and$ 21$.

(a) Estimate a fair value per share using the median comparable multiple. (3) (b) Company X trades at $\$ 54$. Using your fair value, compute the margin of safety as a percentage. (3) (c) Explain in one sentence why an investor demands a margin of safety. (2)

Question 6 — Sensitivity / scenario reasoning (8 marks)

Using the single-stage EV formula $EV = \dfrac{FCFF_1}{r-g}$ with base case $FCFF_1 = 50$ , $r = 9\%$ , $g = 3\%$ :

(a) Compute base-case EV. (2) (b) Recompute EV if $g$ rises to $4\%$ (r unchanged). Express the change as a percentage. (3) (c) Explain out loud why DCF outputs are highly sensitive to the spread $(r-g)$ , and what this implies for terminal-value reliability. (3)

Answer keyMark scheme & solutions

Question 1 (10)

(a) Cost of equity — CAPM (3) Formula: $r_e = r_f + \beta(r_m - r_f)$ (1) $r_e = 0.04 + 1.25 \times 0.06 = 0.04 + 0.075 = 0.115 = 11.5\%$ (2)

(b) WACC (5) Weights: $E/V = 600/1000 = 0.6$ ; $D/V = 400/1000 = 0.4$ (1) Formula: $WACC = \frac{E}{V}r_e + \frac{D}{V}r_d(1-T)$ (1) After-tax cost of debt $= 0.07(1-0.25) = 0.0525$ (1) $WACC = 0.6(0.115) + 0.4(0.0525) = 0.069 + 0.021 = 0.090 = 9.0\%$ (2)

(c) Why after-tax (2) Interest is tax-deductible, so debt reduces the firm's tax bill (interest tax shield). The real economic cost of debt to shareholders is the interest net of the tax saving, so we use $r_d(1-T)$ .

Question 2 (14)

(a) Terminal value formula (2) $TV_n = \dfrac{FCFF_{n+1}}{WACC - g} = \dfrac{FCFF_n(1+g)}{WACC - g}$ , calculated at the end of the last explicit year (Year 3). (2)

(b) FCFF projection (3) Year 1 = $100.00$ ; Year 2 = $100 \times 1.10 = 110.00$ ; Year 3 = $110 \times 1.10 = 121.00$ . (2) Year 4 (terminal) = $121 \times 1.03 = 124.63$ . (1)

(c) Terminal value at end Year 3 (3) $TV_3 = \dfrac{124.63}{0.09 - 0.03} = \dfrac{124.63}{0.06} = 2077.17$ (3)

(d) Discount to EV (4) Discount factors at 9%: Yr1 $=1.09$ , Yr2 $=1.1881$ , Yr3 $=1.295029$ .

PV(Yr1) $= 100/1.09 = 91.74$
PV(Yr2) $= 110/1.1881 = 92.585$
PV(Yr3) $= 121/1.295029 = 93.436$
PV(TV) $= 2077.17/1.295029 = 1604.03$ (3)

$EV = 91.74 + 92.59 + 93.44 + 1604.03 = 1881.80$ million (1)

(e) Per share (2) Equity value $= 1881.80 - 150 = 1731.80$ ; per share $= 1731.80/50 = 34.64$ . (2)

Question 3 (10)

(a) Derivation (4) $P_0 = \sum_{t=1}^{\infty} \frac{D_0(1+g)^t}{(1+r)^t}$ . (1) This is a geometric series with ratio $\frac{1+g}{1+r}$ ; it converges when $g < r$ . (1) Sum $= D_0 \cdot \frac{(1+g)/(1+r)}{1-(1+g)/(1+r)} = \frac{D_0(1+g)}{r-g} = \frac{D_1}{r-g}$ . (2)

(b) Value (3) $D_1 = 2.00 \times 1.06 = 2.12$ ; $P_0 = \frac{2.12}{0.11-0.06} = \frac{2.12}{0.05} = 42.40$ . (3)

(c) Verdict (3) Intrinsic $42.40 <$ market $50.00$ → overvalued. (1) Overvaluation $= (50 - 42.40)/42.40 = 17.92\%$ above intrinsic value. (2)

Question 4 (10)

(a) Explain (2) A reverse DCF takes the current market price as given and solves for the growth (or margin) assumption the market is implicitly pricing in, letting you judge whether that expectation is plausible — rather than outputting a value from your own assumptions.

(b) Solve for g (4) $EV = \frac{FCFF_1}{r-g} \Rightarrow r-g = \frac{FCFF_1}{EV} \Rightarrow g = r - \frac{FCFF_1}{EV}$ . (3) for algebra (1) correct rearrangement.

(c) Compute (4) $g = 0.08 - \frac{60}{1000} = 0.08 - 0.06 = 0.02 = 2\%$ . (2) Implied $g = 2\%$ is below long-run GDP growth of ~3%, so the market is pricing conservative/realistic growth — the valuation is not demanding heroic assumptions. (2)

Question 5 (8)

(a) Fair value (3) Median multiple $= 18$ . Fair value $= 18 \times 4.00 = 72.00$ . (3)

(b) Margin of safety (3) $MoS = \frac{\text{fair} - \text{price}}{\text{fair}} = \frac{72 - 54}{72} = \frac{18}{72} = 25\%$ . (3)

(c) Reason (2) A margin of safety cushions against errors in assumptions and unforeseen adverse events, protecting capital if the true value is lower than estimated.

Question 6 (8)

(a) Base EV (2) $EV = \frac{50}{0.09-0.03} = \frac{50}{0.06} = 833.33$ . (2)

(b) g = 4% (3) $EV = \frac{50}{0.09-0.04} = \frac{50}{0.05} = 1000.00$ . (1) Change $= (1000 - 833.33)/833.33 = 20.0\%$ increase. (2)

(c) Explain (3) EV depends inversely on the spread $(r-g)$ ; a small 1pp change in $g$ shrinks the denominator sharply, causing a large swing in value. Because terminal value uses this same perpetuity, it dominates total value and is the most fragile input — so growth/discount assumptions must be stress-tested.

[
  {"claim":"WACC = 9.0%","code":"E=600;D=400;V=E+D;re=0.04+1.25*0.06;rd=0.07*(1-0.25);wacc=(E/V)*re+(D/V)*rd;result=abs(wacc-0.09)<1e-9"},
  {"claim":"DCF EV approx 1881.80","code":"w=0.09;f=[100,110,121];tv=121*1.03/(w-0.03);pv=sum(f[i]/(1+w)**(i+1) for i in range(3));pvtv=tv/(1+w)**3;ev=pv+pvtv;result=abs(ev-1881.80)<0.5"},
  {"claim":"DDM value = 42.40","code":"p=2.00*1.06/(0.11-0.06);result=abs(p-42.40)<1e-9"},
  {"claim":"Reverse DCF implied g = 2%","code":"g=0.08-60/1000;result=abs(g-0.02)<1e-12"},
  {"claim":"Sensitivity EV rises 20% when g goes 3% to 4%","code":"a=50/(0.09-0.03);b=50/(0.09-0.04);result=abs((b-a)/a-0.20)<1e-9"}
]