Shares, Ownership & Indices

Q1 (10 marks)

(a) Formula (words): Dividend per share (DPS) = dividend rate × face value; Dividend yield = DPS ÷ market price.

• DPS $= 40\% \times ₹10 = ₹4$. (2)
• Yield $= 4/250 = 0.016 = 1.6\%$. (2)

(b) Bonus $1:1$ doubles shares; market cap unchanged ⇒ theoretical price halves:

• New price $= 250/2 = ₹125$. (2)
• Post-bonus dividend still $40\%$ of face value $₹10 = ₹4$ per share, but now each old holder has 2 shares, so DPS-per-original-share = ₹4 stays ₹4 per (new) share.
• New yield $= 4/125 = 0.032 = 3.2\%$. (2)
• Explain: If the company continues paying ₹4 on every (now doubled) share, total payout doubles → yield doubles. The bonus alone (price halving, dividend halving) would leave yield unchanged; the yield rises here only because the ₹ dividend per share was maintained rather than halved — effectively a payout increase. (2)

Q2 (12 marks)

(a) Price-weighted $= (100+200+50)/3 = 350/3 = 116.67$. (2)

(b) Current free-float mcap:

• A: $100×1000×1.0 = 100{,}000$
• B: $200×500×0.5 = 50{,}000$
• C: $50×4000×0.8 = 160{,}000$
• Total $= 310{,}000$. (2)
• Index $= (310{,}000/300{,}000)×100 = 103.33$. (2)

(c) After B split (price $100$, shares $1000$):

• Price-weighted: sum of prices $= 100+100+50 = 250$. To keep index at pre-split $116.67$: new divisor $d$ satisfies $250/d = 116.67 ⇒ d = 250/116.667 = 2.143$. (3)
• Free-float: B's free-float mcap $= 100×1000×0.5 = 50{,}000$ (unchanged), total still $310{,}000$ ⇒ index stays $103.33$. (1)
• Explain: A split changes price without changing value; a price-weighted index sums prices, so it must adjust the divisor to remove the artificial drop. A free-float mcap index sums market values (price×shares), which are invariant to a split, so no adjustment is needed. (2)

Q3 (10 marks)

(a) TERP formula: $\text{TERP} = \dfrac{(N_{old}\times P_{cum}) + (N_{new}\times P_{sub})}{N_{old}+N_{new}}$.

• With $1:4$: $N_{old}=4, N_{new}=1$.
• TERP $= (4×120 + 1×90)/5 = (480+90)/5 = 570/5 = ₹114$. (4)

(b) Value of a right $= P_{cum} - \text{TERP} = 120 - 114 = ₹6$ per existing share. (Equivalently, right entitlement value = TERP − subscription = $114-90=24$ per new share, and $24/4 = 6$ per old share.) (3)

(c) With 400 shares: pre-value $= 400×120 = ₹48{,}000$. Post ex-rights value if not subscribing $= 400×114 = ₹45{,}600$. Dilution loss $= ₹2{,}400$ (= $400×6$, the value of the un-exercised rights). Explain: the investor's holding is diluted because new shares are issued below market; not subscribing (and not selling the rights) forfeits ₹2,400 of value. (3)

Q4 (10 marks)

(a) EPS before $= 50{,}000{,}000/10{,}000{,}000 = ₹5.00$.

• Shares after $= 9{,}500{,}000$. Net profit unchanged (₹50m).
• EPS after $= 50{,}000{,}000/9{,}500{,}000 = ₹5.263$. (4)

(b) EPS rose from ₹5.00 to ₹5.26 ⇒ accretive. Mechanism: same earnings spread over fewer shares. (Note: buyback price ₹220 > ₹200 uses cash, but with earnings fixed the share-count reduction dominates EPS.) (3)

(c)

One line each: buyback returns cash and shrinks share base; bonus capitalises reserves, expanding share base with no cash outflow. (3)

Q5 (10 marks)

(a) Authorized = max shares the charter permits issuing. Issued = shares actually created/allotted (≤ authorized). Outstanding = issued shares currently held by investors (= issued − treasury). Inequality: $\text{Outstanding} \le \text{Issued} \le \text{Authorized}$. Treasury shares (repurchased, not retired) are issued but NOT outstanding. (4)

(b)

(4)

(c) Dow Jones is price-weighted: a higher-priced stock has larger weight regardless of company size, so a big % move in a high-priced constituent moves the index more. Nifty 50 is free-float market-cap-weighted: weight depends on free-float market value, not price, so nominal price level alone doesn't grant outsized influence. (2)

Q6 (8 marks)

def free_float_index(prices, shares, freefloat, base_mcap, base_index):
    current_mcap = 0
    for p, s, f in zip(prices, shares, freefloat):
        current_mcap += p * s * f          # free-float market cap per stock
    return (current_mcap / base_mcap) * base_index

Marks: loop/sum of $p·s·f$ (3), ratio to base_mcap (2), scale by base_index & return (1).

Change for price-weighted (1 line): replace current_mcap += p * s * f with total += p and divide the summed price by a divisor instead of base_mcap (i.e. return sum(prices)/divisor). (2)

[
  {"claim":"Q1a dividend yield = 1.6%","code":"dps=0.40*10; yld=dps/250; result=abs(yld-0.016)<1e-9"},
  {"claim":"Q1b post-bonus yield = 3.2%","code":"yld=4/125; result=abs(yld-0.032)<1e-9"},
  {"claim":"Q2b free-float index = 103.333","code":"mcap=100*1000*1.0+200*500*0.5+50*4000*0.8; idx=(mcap/300000)*100; result=abs(idx-Rational(310,3))<Rational(1,1000)"},
  {"claim":"Q2c new divisor = 2.142857","code":"pre=Rational(350,3); d=250/pre; result=abs(d-Rational(15,7))<Rational(1,10000)"},
  {"claim":"Q3a TERP = 114","code":"terp=(4*120+1*90)/5; result=terp==114"},
  {"claim":"Q3b value of right = 6","code":"terp=Rational(4*120+90,5); vr=120-terp; result=vr==6"},
  {"claim":"Q4a EPS after buyback approx 5.263","code":"eps=50000000/Rational(9500000); result=abs(eps-Rational(100,19))<Rational(1,1000)"}
]