Valuation Methods

Question 1

(a) WACC (5 marks) Formula: $WACC = \frac{E}{E+D} r_e + \frac{D}{E+D} r_d (1-T)$ The $(1-T)$ term captures the tax shield: interest is tax-deductible, so the effective cost of debt to the firm is reduced. (1 formula, 1 explanation)

Weights: $E+D = 1200$. $w_e = 900/1200 = 0.75$; $w_d = 300/1200 = 0.25$. (1) $WACC = 0.75(0.12) + 0.25(0.06)(0.75) = 0.09 + 0.01125 = 0.10125 = 10.125\%$ (2 for correct computation)

(b) Terminal Value (5 marks) $TV_5 = \frac{FCF_5(1+g)}{WACC - g} = \frac{150(1.025)}{0.10125 - 0.025} = \frac{153.75}{0.07625} = 2016.39\text{m}$ (3 marks) Necessity of $g < WACC$: the geometric series $\sum (1+g)^t/(1+WACC)^t$ converges only if $\frac{1+g}{1+WACC}<1$, i.e. $g<WACC$; otherwise the perpetuity value is infinite/undefined (negative denominator). (2 marks)

(c) EV, equity value, per-share (8 marks) Discount factors $DF_t = 1.10125^{-t}$.

(4 marks for discounting method + values) Sum PV of explicit FCF $= 90.81+92.35+93.59+93.83+92.61 = 463.19$m. PV of TV $= 1245.02$m. $EV = 463.19 + 1245.02 = 1708.21\text{m}$ (2) Equity value $= EV - D = 1708.21 - 300 = 1408.21$m. Intrinsic price = 1408.21/40 = \35.21$ per share. (2)

(d) Margin of safety (3 marks) $MoS = \frac{\text{Intrinsic} - \text{Price}}{\text{Intrinsic}} = \frac{35.21 - 52.00}{35.21} = -0.4770 = -47.7\%$ Negative margin: market price ($52) **exceeds** intrinsic value ($35.21). Under a 25% MoS buy rule (buy only if price ≤ 75% of intrinsic = $26.41), this is not a buy — it is overvalued. (2 computation, 1 conclusion)

(e) Scaling proof (3 marks) Intrinsic price $= \frac{1}{N}\left(\sum_t \frac{k\,FCF_t}{(1+w)^t} + \frac{k\,TV}{(1+w)^5} - D\right)$ — note $D$ is fixed, but strictly if all cash flows scale the enterprise value scales: $EV(k) = \sum_t \frac{k\,FCF_t}{(1+w)^t} + \frac{k\,TV}{(1+w)^5} = k\cdot EV(1).$ Since discounting is linear, scaling every cash flow by $k$ scales EV (and hence intrinsic equity, absent fixed debt) by $k$. (Award full marks for demonstrating linearity of the PV operator.)

Question 2

(a) Reverse DCF (5 marks) $P = \frac{FCF_0(1+g)}{r-g}\Rightarrow P(r-g) = FCF_0(1+g)$ $Pr - Pg = FCF_0 + FCF_0 g \Rightarrow Pr - FCF_0 = g(P + FCF_0)$ $g = \frac{Pr - FCF_0}{P + FCF_0} = \frac{500(0.10) - 20}{500 + 20} = \frac{50-20}{520} = \frac{30}{520} = 0.05769 = 5.77\%$ (3 algebra, 2 numeric)

(b) Derivative (6 marks) Quotient rule on $P = FCF_0(1+g)(r-g)^{-1}$: $\frac{\partial P}{\partial g} = \frac{FCF_0(r-g) + FCF_0(1+g)}{(r-g)^2} = \frac{FCF_0(1+r)}{(r-g)^2}$ (3 derivation) At $g=0.05769$, $r-g = 0.04231$: \frac{\partial P}{\partial g} = \frac{20(1.10)}{(0.04231)^2} = \frac{22}{0.0017902} = 12289\text{ ($m per unit }g) Sign positive ⇒ higher growth raises price; magnitude large because $(r-g)^2$ in denominator is small — valuation is very sensitive near $r\approx g$. (3 evaluation + interpretation)

(c) Python (8 marks)

def implied_growth(fcf0, r, price):
    # Solve P = fcf0*(1+g)/(r-g) for g (closed form)
    g = (price*r - fcf0) / (price + fcf0)
    if not (r > g):
        raise ValueError("Invalid: requires r > g for convergence")
    return g
 
def sensitivity(fcf0, price, r_list):
    out = {}
    for r in r_list:
        try:
            out[r] = implied_growth(fcf0, r, price)
        except ValueError as e:
            out[r] = str(e)
    return out
 
# Example:
# implied_growth(20, 0.10, 500) -> 0.05769...

(4 correct closed form, 2 guard clause, 2 sensitivity loop)

(d) Qualitative (3 marks) From $g = \frac{Pr - FCF_0}{P + FCF_0}$, $\frac{\partial g}{\partial r} = \frac{P}{P+FCF_0} > 0$. So as the assumed discount rate $r$ rises, the implied growth rate rises linearly — a higher required return demands the market's price be justified by faster expected growth. (1 derivative, 1 sign, 1 interpretation)

Question 3

(a) Gordon DDM (3 marks) $P_0 = \frac{D_0(1+g)}{r-g} = \frac{3.00(1.04)}{0.09-0.04} = \frac{3.12}{0.05} = \$62.40$

(b) Relative valuation (4 marks) $P_{rel} = P/E \times EPS = 15 \times 4.20 = \$63.00$ (2) Difference: $63.00 - 62.40 = 0.60$ (~1%) — very close. Divergence reasons (any one, 2 marks): the peer P/E embeds different growth/payout/risk expectations; DDM depends only on dividends whereas P/E reflects total earnings (retained + paid), so buyback/reinvestment policy differences drive the gap.

(c) Sum-of-the-parts (7 marks)

• Grid segment value: $180$m (given). (1)
• Renewables EV $= 8 \times 30 = 240$m. (2)
• Total EV $= 180 + 240 = 420$m. (1)
• Equity value $= EV - \text{net debt} = 420 - 50 = 370$m. (1)
• Per share = 370/20 = \18.50$. (1)

Comparison & judgement (1): SOTP gives $18.50 vs standalone DDM$62.40 — wildly different because they value different things (DDM here priced a single-share dividend stream; SOTP values the whole enterprise). A value investor should trust SOTP when the firm has heterogeneous divisions with distinct risk/growth profiles, since a single blended DDM growth rate misprices a conglomerate; but must ensure inputs (segment EVs, net debt) are consistent before comparing.

[
  {"claim":"WACC = 10.125%","code":"E,D,re,rd,T=900,300,0.12,0.06,0.25; w=(E/(E+D))*re+(D/(E+D))*rd*(1-T); result=abs(float(w)-0.10125)<1e-6"},
  {"claim":"Terminal value = 2016.39m","code":"TV=150*1.025/(0.10125-0.025); result=abs(float(TV)-2016.393)<0.05"},
  {"claim":"Intrinsic price approx 35.21","code":"w=0.10125; fcf=[100,112,125,138,150]; TV=150*1.025/(w-0.025); ev=sum(f/(1+w)**(i+1) for i,f in enumerate(fcf))+TV/(1+w)**5; price=(ev-300)/40; result=abs(float(price)-35.21)<0.1"},
  {"claim":"Reverse DCF implied g = 30/520","code":"g=(500*0.10-20)/(500+20); result=abs(float(g)-Rational(30,520))<1e-9"},
  {"claim":"dP/dg at g=30/520 approx 12289","code":"g=Rational(30,520); d=20*1.10/(0.10-float(g))**2; result=abs(float(d)-12289)<50"},
  {"claim":"DDM price = 62.40","code":"P=3.00*1.04/(0.09-0.04); result=abs(float(P)-62.40)<1e-9"},
  {"claim":"SOTP per share = 18.50","code":"ev=180+8*30; eq=ev-50; ps=eq/20; result=abs(float(ps)-18.50)<1e-9"}
]