Economic Moats & Macro

Question 1

(a) [5 marks] $V(n)=k\,n(n-1)$. Percentage increase $= \dfrac{V(2500)-V(2000)}{V(2000)}\times100$ — $k$ cancels. (2 for symbolic cancellation)

$V(2000)=k(2000)(1999)=3{,}998{,}000k$. $V(2500)=k(2500)(2499)=6{,}247{,}500k$. (1)

Increase $=\dfrac{6{,}247{,}500-3{,}998{,}000}{3{,}998{,}000}=\dfrac{2{,}249{,}500}{3{,}998{,}000}=0.56266$. (1)

≈ 56.27% increase (a 25% increase in members yields a ~56% value increase — super-linear). (1)

(b) [5 marks] $\Delta V = k[(n+1)n - n(n-1)] = k[n^2+n - n^2+n] = 2kn$. (3 for correct expansion) This is linear in $n$ (slope $2k$). (1) Because each additional member adds value proportional to the current size $n$, the moat widens as the firm grows: the incumbent's per-member value advantage over a small rival grows without bound, raising the entry barrier over time rather than staying fixed. (1)

(c) [4 marks]

• Threat of new entrants — weakened by the network effect: a new entrant starts with small $n$, hence tiny $V$, and cannot match incumbent value. (2)
• Bargaining power of buyers (customers) — weakened by the ₹4.2 lakh switching cost, which locks members in and reduces their leverage to demand price cuts or leave. (2) (Accept "rivalry among competitors" for the network point if well argued, but entrants/buyers is the tightest mapping.)

(d) [6 marks] Any three, 2 marks each:

1. Positive — value creation: ROIC 14% > WACC 9% (a 5-point spread) means capital is deployed above its cost; buybacks funded here are accretive because the moat sustains high returns. (2)
2. Positive — alignment: 85% equity, 5-year vesting aligns CEO with long-term shareholder value and discourages short-termism. (2)
3. Caution/negative — leverage risk: debt-funded buybacks increase financial leverage; sustainable only while ROIC>WACC and moat holds — if rates rise or network erodes, the leveraged structure amplifies downside. Flag as a risk to monitor. (2)

Question 2

(a) [10 marks] Pre-cut rate $r=6.50\%$. Coupon = ₹70/yr, face ₹1,000.

$P = \dfrac{70}{1.065}+\dfrac{70}{1.065^2}+\dfrac{1070}{1.065^3}$ $=65.727+61.716+885.05 = ₹1012.49$ (≈). (3)

Post-cut $r=6.00\%$: $P=\dfrac{70}{1.06}+\dfrac{70}{1.06^2}+\dfrac{1070}{1.06^3}=66.038+62.300+898.29=₹1026.63$. (2)

Exact change $=1026.63-1012.49 = +₹14.14$. (1)

Macaulay duration at 6.50%: $D=\dfrac{1}{P}\sum t\cdot CF_t/(1+r)^t$ $=\dfrac{1(65.727)+2(61.716)+3(885.05)}{1012.49}=\dfrac{65.727+123.43+2655.14}{1012.49}=\dfrac{2844.30}{1012.49}=2.809$ yrs. Modified duration $D_{mod}=D/1.065=2.638$ yrs. (2)

Approx price change $=-D_{mod}\cdot\Delta r\cdot P = -2.638\times(-0.005)\times1012.49 = +₹13.35$. (1)

Comparison: approximation (+₹13.35) underestimates exact (+₹14.14) by ~₹0.79 because duration is a linear (first-order) estimate ignoring positive convexity, which makes the true price gain slightly larger. (1)

(b) [6 marks] Pre-cut: $r=6.50\%+5.5\%=12.0\%$. $P=\dfrac{12}{0.12-0.05}=\dfrac{12}{0.07}=₹171.43$. (2) Post-cut: $r=6.00\%+5.5\%=11.5\%$. $P=\dfrac{12}{0.115-0.05}=\dfrac{12}{0.065}=₹184.62$. (2) % change $=\dfrac{184.62-171.43}{171.43}\times100 = 7.69\%$. (2)

(c) [6 marks] $P=\dfrac{D_1}{r-g}\Rightarrow \dfrac{\partial P}{\partial r}=-\dfrac{D_1}{(r-g)^2}=-\dfrac{P}{r-g}$. (3) The magnitude $\left|\dfrac{\partial P}{\partial r}\right|=\dfrac{P}{r-g}$ blows up as $(r-g)\to0$, i.e. when growth $g$ is close to $r$ (long-duration/high-growth priced-for-the-future stocks). So a small rate rise (higher $r$) causes a proportionally larger price fall for these than for high-yield value stocks where $r-g$ is large and the denominator is stable. (2) Because $\partial P/\partial r<0$, prices move inversely to rates — identical in sign to bonds, whose PV of fixed cash flows also falls when discount rates rise. (1)

Question 3

(a) [6 marks] Rule: $i_t = 2 + \pi_t + 0.5(\pi_t-4) + 0.5 g_t$. (1 mark each row)

(b) [6 marks] (1 mark each)

• t1: $g<0$, low $\pi$, low rate → contraction / trough.
• t2: $g$ still negative but rising, $\pi$ rising, rate rising → early recovery.
• t3: $g>0$, $\pi$ high & rising, rate rising sharply → expansion.
• t4: $g$ peak (2.5), $\pi$ peak (6.0), highest rate → peak / overheating.
• t5: $g$ falling, $\pi$ falling, rate falling → slowdown.
• t6: $g<0$ again, $\pi$ near target, rate down to 6.50% → early contraction / late slowdown.

(c) [6 marks] Function [3], application [3]:

def recommend_sector(g, pi, i_prev, i_now):
    rising_rates = i_now > i_prev
    if g < 0 and rising_rates:            # early recovery
        return "cyclicals/industrials"
    if g > 0 and rising_rates:            # expansion -> peak
        return "financials (rate-sensitive) / energy"
    if g > 0 and not rising_rates:        # slowdown begins
        return "consumer staples/utilities"
    if g < 0 and not rising_rates:        # contraction, easing
        return "tech/growth (long-duration benefits from cuts)"
    return "balanced"

Application t3→t4: $g=2.5>0$, $i$ rose 8.00→10.25 (rising) → returns "financials / energy". Reasoning: at the overheating peak with rising rates, banks earn wider net interest margins and commodity/energy names benefit from inflation, while long-duration growth is penalised by rising discount rates. (3)

[
  {"claim":"Q1a network value increase ~56.27%","code":"V=lambda n:n*(n-1); r=(V(2500)-V(2000))/V(2000)*100; result=abs(r-56.2656)<0.01"},
  {"claim":"Q1b marginal value equals 2kn","code":"from sympy import symbols,expand,simplify; n,k=symbols('n k'); dV=expand(k*(n+1)*n - k*n*(n-1)); result=simplify(dV-2*k*n)==0"},
  {"claim":"Q2a pre-cut bond price ~1012.49","code":"P=70/1.065+70/1.065**2+1070/1.065**3; result=abs(P-1012.49)<0.05"},
  {"claim":"Q2a exact change ~14.14","code":"P0=70/1.065+70/1.065**2+1070/1.065**3; P1=70/1.06+70/1.06**2+1070/1.06**3; result=abs((P1-P0)-14.14)<0.1"},
  {"claim":"Q2b equity % change ~7.69","code":"P0=12/0.07; P1=12/0.065; ch=(P1-P0)/P0*100; result=abs(ch-7.6923)<0.01"},
  {"claim":"Q3a Taylor rate t4 = 10.25","code":"i=2+6+0.5*(6-4)+0.5*2.5; result=abs(i-10.25)<1e-9"},
  {"claim":"Q3a Taylor rate t2 = 4.75","code":"i=2+3.5+0.5*(3.5-4)+0.5*(-1.0); result=abs(i-4.75)<1e-9"}
]