Shares, Ownership & Indices

Question 1

(a) Sequential corporate actions (9 marks)

Start: 8,000 shares, price \250$, FV$$10$. Investor value$= 8000\times250 = $2{,}000{,}000$.

(i) 5-for-1 split — multiply shares by 5, divide price by 5, divide FV by 5:

• Shares: $8000\times5 = 40{,}000$
• Price: 250/5 = \50$
• FV: 10/5 = \2$
• Value: 40000\times50 = \2{,}000{,}000$ ✓ (3)

(ii) 1:4 bonus — 1 free per 4 held → factor $\times\tfrac{5}{4}$; price divided by $\tfrac54$; FV unchanged (bonus from reserves, FV stays \2$):

• Shares: $40000\times\tfrac54 = 50{,}000$
• Price: 50\div\tfrac54 = 50\times\tfrac45 = \40$
• FV: \2$ (unchanged)
• Value: 50000\times40 = \2{,}000{,}000$ ✓ (3)

**(iii) Rights 1:10 at S=\30$** — cum-rights price$P=$40$:

• New rights shares: $50000/10 = 5{,}000$; total shares $= 55{,}000$
• TERP (from part b): \dfrac{nP+S}{n+1}=\dfrac{10(40)+30}{11}=\dfrac{430}{11}=\39.09$
• FV: \2$ (unchanged)
• Cash paid for rights: 5000\times30=\150{,}000$ (3)

(b) TERP derivation & wealth check (7 marks)

For every $n$ old shares (value $nP$) the holder buys 1 new share paying $S$. Post-action the holder has $n+1$ shares worth total $nP+S$ (frictionless, value = old value + cash injected): $\text{TERP}=\frac{nP+S}{n+1}\quad\text{(2)}$

Apply $n=10,P=40,S=30$: \text{TERP}=\dfrac{430}{11}=39.0909\ldots \approx\39.09$. (2)

Wealth invariance check (investor fully subscribes):

• After bonus: 50,000 shares \times \40 = $2{,}000{,}000$.
• Rights: pays \150{,}000$ cash, receives 5,000 shares.
• After rights: 55{,}000 \times 39.0909 = \2{,}150{,}000$.
• Net wealth = share value − cash paid = 2{,}150{,}000 - 150{,}000 = \2{,}000{,}000$. ✓

Wealth in equities rose only by the cash the investor themselves injected; underlying wealth unchanged. (3)

(c) Proof & balance-sheet distinction (6 marks)

Proof (split & bonus): Let value before $=N\cdot P$. Split/bonus multiplies share count by factor $k$ and (by value preservation) divides price by $k$: new value $=(kN)(P/k)=NP$. No cash enters or leaves; investor wealth invariant. (2)

Balance-sheet difference: In a split, only the FV and number of shares change; no accounting entry moves — equity composition unchanged (a mere subdivision). In a bonus issue, the company capitalises reserves: an amount equal to the additional shares' face value is transferred from free reserves (retained earnings) to share capital. FV stays the same but total paid-up share capital rises; reserves fall by the same amount. Net equity unchanged. (2)

Rights difference: A rights issue brings new external cash into the company and increases the number of outstanding shares in exchange for that cash. Before counting the cash paid, the per-share price falls to TERP because value is diluted across more shares while cash is being raised at a discount to market. It is not a pure repackaging: total company equity genuinely increases by cash raised. Investor wealth is preserved only because the discount they capture on new shares offsets the dilution of old ones. (2)

Question 2

(a) Building both indices (8 marks)

Price-weighted: sum of prices $=100+400+50=550$. Choose divisor $D_{pw}$ so index $=1000$: $\text{Index}=\frac{\sum P}{D_{pw}}=1000\Rightarrow D_{pw}=\frac{550}{1000}=0.55.$ (4)

Free-float market cap (Price × Shares × FF):

• A: $100\times500\times0.50=25{,}000$
• B: $400\times100\times1.00=40{,}000$
• C: $50\times800\times0.25=10{,}000$
• Total FF mcap $=75{,}000$ (mn units).

Base mcap $=75{,}000$, base index $=1000$: $\text{Index}=\frac{\text{current FF mcap}}{\text{base mcap}}\times 1000 = \frac{75000}{75000}\times1000=1000.$ (4)

(b) Split effect (7 marks)

Price-weighted: After A's 2-for-1 split, $P_A=50$. New price sum $=50+400+50=500$. Without adjustment: index $=500/0.55=909.09$ — an artificial drop of ~9% caused purely by the split, not economics. To keep continuity, solve for new divisor so index stays at pre-split value $1000$: $1000=\frac{500}{D_{pw}'}\Rightarrow D_{pw}'=\frac{500}{1000}=0.50.$ The divisor is lowered from $0.55$ to $0.50$. (4)

Free-float index: A's split doubles shares (500→1000 mn) and halves price (100→50). FF mcap of A $=50\times1000\times0.5=25{,}000$ — unchanged. Total mcap still $75{,}000$; index stays $1000$. A split does not change market cap, so a cap-weighted index needs no adjustment. (3)

(c) Code + methodology argument (7 marks)

def rebalance(prices, shares, floats, divisor_pw, base_mcap, base_index=1000):
    # Price-weighted (Dow-style)
    pw = sum(prices) / divisor_pw
    # Free-float market-cap weighted (Sensex/Nifty style)
    ff_mcap = sum(p * s * f for p, s, f in zip(prices, shares, floats))
    ff = (ff_mcap / base_mcap) * base_index
    return pw, ff
 
def split_adjust_divisor(new_prices, old_index):
    # keep index continuous after a corporate action
    return sum(new_prices) / old_index

(4 marks: correct sums, correct FF formula, correct divisor recompute.)

Argument (3): A free-float cap-weighted index (S&P 500, Sensex, Nifty) weights each stock by its investable market value, so it directly reflects the economic size of companies and the actual money at stake. A price-weighted index (Dow) weights by share price alone — a stock trading at \400$moves the index 8× more than a$$50$ stock even if the latter is a far larger company. This is arbitrary because price depends on how many shares a firm chose to issue (splits change price without changing size). Hence price-weighting is distorted by high-priced constituents and needs divisor patches for every split; cap-weighting is self-consistent.

Question 3

(a) Dividends & yields (6 marks)

Preferred dividend =2{,}000{,}000\times100\times0.08=\16{,}000{,}000$. **(2)** Residual for common$=40{,}000{,}000-16{,}000{,}000=$24{,}000{,}000$. **(1)** Dividend per common share$=24{,}000{,}000/10{,}000{,}000=$2.40$. Common dividend yield$=2.40/120=2.0%$. **(2)** Preferred yield on price$$96$: coupon$=8%\times100=$8$; yield$=8/96=8.33%$. (1)

(b) Buyback (6 marks)

Shares repurchased $=36{,}000{,}000/120=300{,}000$. (2) New common count $=10{,}000{,}000-300{,}000=9{,}700{,}000$. (1) Equity value before =10{,}000{,}000\times120=\1{,}200{,}000{,}000$; after cash out$=1{,}200{,}000{,}000-36{,}000{,}000=$1{,}164{,}000{,}000$. New price$=1{,}164{,}000{,}000/9{,}700{,}000=$120.00$. (2) (Price is unchanged here because buyback occurred at market price — value and shares fall proportionally.) EPS rises because the same earnings are divided over fewer shares, raising per-share metrics even without price change. (1)

(c) Skipped cumulative preferred dividend (4 marks)

Because the preferred is cumulative, an unpaid (skipped) dividend does not vanish — it accrues as dividends in arrears. In the following year the company must pay both the arrears and the current preferred dividend ($16m + 16m = 32m$) before any dividend can be paid to common shareholders. Priority ordering of distributions: preferred arrears → current preferred → common. Common shareholders have no such protection — a skipped common dividend is simply lost and never accrues. (4)

[
{"claim":"Q1 TERP for n=10,P=40,S=30 equals 430/11","code":"P=40; n=10; S=30; terp=(n*P+S)/(n+1); result = simplify(terp - Rational(430,11))==0"},
{"claim":"Q1 investor wealth invariant at 2,000,000 after all actions","code":"shares=8000*5*Rational(5,4); shares=shares+shares/10; terp=Rational(430,11); wealth=shares*terp - 5000*30; result = simplify(wealth-2000000)==0"},
{"claim":"Q2 price-weighted new divisor after split is 0.50","code":"new_prices=50+400+50; old_index=1000; d=Rational(new_prices,old_index); result = d==Rational(1,2)"},
{"claim":"Q2 free-float mcap unchanged after A split (still 75000)","code":"A=50*1000*Rational(1,2); B=400*100*1; C=50*800*Rational(1,4); total=A+B+C; result = total==75000"},
{"claim":"Q3 common dividend yield is 2% and preferred yield 8.33%","code":"pref=2000000*100*Rational(8,100); resid=40000000-pref; dps=resid/10000000; cy=dps/120; py=Rational(8,96); result = (cy==Rational(1,50)) and (simplify(py-Rational(1,12))==0)"},
{"claim":"Q3 buyback: 300000 shares, new price 120","code":"rep=Rational(36000000,120); newn=10000000-rep; newprice=(1200000000-36000000)/newn; result = (rep==300000) and (newprice==120)"}
]