Funds, ETFs & Pooled Vehicles

Question 1

(a) NAV = (Assets − Liabilities)/Units = $500{,}000{,}000 / 25{,}000{,}000 = ₹20.00$. (2 marks: 1 formula, 1 value)

(b) Daily fee fraction $= e/365$. Over 365 days, net multiplier from fees alone $= (1 - e/365)^{365}$. Gross growth over the year (continuous) $= e^{r_g}$. Net NAV growth $= e^{r_g}\,(1 - e/365)^{365}$. Now $(1 - e/365)^{365} \to e^{-e}$ as the daily deduction is compounded; and for small $e$, $e^{-e} \approx 1 - e + \tfrac{e^2}{2}$, so to first order $(1-e/365)^{365}\approx 1-e$. Hence net growth $\approx e^{r_g}(1-e)$. Approximation stated: dropping the $O(e^2)$ term (fee-on-fee compounding), valid because $e=0.002\ll1$. (5 marks: 2 set-up multipliers, 2 limit/expansion, 1 explicit approximation statement)

(c) Total shortfall $= R_b - R_f = 12.00\% - 11.60\% = 0.40\%$.

• TER portion $= 0.20\%$.
• Structural drag (residual) $= 0.40\% - 0.20\% = 0.20\%$ (cash drag, rebalancing/transaction costs, sampling error, dividend timing).

Tracking error definition: the standard deviation of the fund's active return (fund return minus benchmark return) measured over many periods — a dispersion/volatility measure, not a level. The single-year shortfall of $0.40\%$ is a realised mean active return (tracking difference), one draw; tracking error requires the variability across a series of active returns. (6 marks: 2 TER/residual split, 2 numeric, 2 definition + distinction)

(d) Over 20 years the dominant, persistent, compounding drag is TER. Expected cumulative cost difference in TER $= (0.35\% - 0.05\%) = 0.30\%$/yr $\approx 0.30\% \times 20 = 6\%$ compounded ≈ a value drag of roughly $1-(1-0.003)^{20}\approx 5.83\%$ of terminal wealth in favour of Fund A. Tracking error is a dispersion term that does not accumulate as a directional cost and largely averages out over long horizons; the $0.15\%$ vs $0.04\%$ difference is second-order for a buy-and-hold trustee who does not trade on short-term deviations. Choose Fund A. Most important factor for a 20-year buy-and-hold: the expense ratio (TER), because it is a guaranteed recurring compounding cost. (7 marks: 3 quantitative TER argument, 2 tracking-error-averages-out argument, 2 decision + factor)

Question 2

(a) SIP buys $A/p_i$ units in month $i$.

• Total units $U = \sum_{i=0}^{n-1} A/p_i = A\sum 1/p_i$.
• Total invested $= nA$.
• Average cost $\bar c = \dfrac{nA}{A\sum 1/p_i} = \dfrac{n}{\sum_{i=0}^{n-1} 1/p_i} = H$, the harmonic mean of the prices.

By the AM–HM inequality, $H \le \text{AM} = \frac{1}{n}\sum p_i$, with equality iff all $p_i$ are equal. Proof of inequality (AM–HM): By Cauchy–Schwarz, $\left(\sum p_i\right)\left(\sum 1/p_i\right)\ge n^2$, so $\frac{\sum p_i}{n}\ge \frac{n}{\sum 1/p_i}$, i.e. AM ≥ HM. Equality when $p_i$ constant. Name: Arithmetic Mean–Harmonic Mean inequality. (8 marks: 2 units expression, 2 harmonic-mean identity, 3 AM–HM proof, 1 equality condition)

(b)

def sip_vs_lumpsum(prices, monthly_amount):
    n = len(prices)
    units_sip = sum(monthly_amount / p for p in prices)
    final_p = prices[-1]
    sip_value = units_sip * final_p
    budget = monthly_amount * n
    units_lump = budget / prices[0]
    lump_value = units_lump * final_p
    return {
        "sip_value": sip_value,
        "lumpsum_value": lump_value,
        "sip_outperformance": sip_value - lump_value,
    }

(6 marks: 2 SIP units/value, 2 lumpsum, 2 dict/return correctness)

(c) prices = [100,90,80,100,120], A = 12000, n=5, budget = 60000, final price 120. SIP units: $120 + 133.\overline{3} + 150 + 120 + 100 = 623.333\ldots$

• month0: 12000/100 = 120
• month1: 12000/90 = 133.3333
• month2: 12000/80 = 150
• month3: 12000/100 = 120
• month4: 12000/120 = 100 Total = 623.3333 units. SIP value = $623.3333 \times 120 = ₹74{,}800$. Lumpsum: 60000/100 = 600 units → value $600 \times 120 = ₹72{,}000$. SIP wins by ₹2,800. (Because average buy price via harmonic mean was below the lumpsum's single price 100 given the dip.) (5 marks: 3 SIP value, 1 lumpsum, 1 verdict + margin)

(d) Lumpsum invests all budget $nA$ at $p_0$, acquiring $nA/p_0$ units; SIP acquires $A\sum 1/p_i = nA/H$ units where $H$ is the harmonic mean. Both valued at $p_{n-1}$, so lumpsum beats SIP iff it holds more units: $\frac{nA}{p_0} > \frac{nA}{H} \iff p_0 < H.$ Condition: $p_0 < H$ (entry price below the harmonic mean of the price path). Intuition: if you enter cheaply relative to the average subsequent purchase cost — i.e. prices generally rose after entry — deploying capital immediately captures more of the rise, so lumpsum wins. (3 marks: 2 inequality derivation, 1 intuition)

Question 3

(a) (6 marks: 1 return source + 1 unique risk each)

• Gold ETF: return solely from gold price movement (no yield/coupon/dividend); unique risk = commodity price risk / no income, plus small physical-gold custody & tracking cost.
• Equity index ETF: return from constituent price appreciation + dividends passed through; unique risk = full equity/market systematic risk, tracking error from rebalancing.
• REIT: return from mandatory distributions (rental income, ≥90% payout) + capital appreciation of units; unique risk = property/interest-rate sensitivity and distribution cuts if occupancy/rents fall.

(b) Year-3 values, both at 12% compounded:

• ELSS: $150000 \times 1.12^3 = 150000 \times 1.404928 = ₹210{,}739.20$. Exit load N/A (past lock-in); lock-in already expired at year 3, so full value received.
• Non-ELSS: redeemed at year 3, exit load only applies within 1 year, so 0% load at year 3 → also $150000\times1.12^3 = ₹210{,}739.20$. Both = ₹210,739.20 at year 3 (ignoring taxes). Distinction: Lock-in = hard prohibition on redemption (you legally cannot exit for 3 years). Exit load = soft penalty (you may exit anytime but pay a fee within the window). ELSS constrains liquidity absolutely; exit load merely prices early liquidity. (6 marks: 2 each value, 2 lock-in vs load distinction)

(c) FoF investor bears both layers. Net-of-fee wealth factor over a year with gross growth factor $g$: $W = g\,(1 - e_{under})(1 - e_{fof}).$ Effective drag $E$ satisfies $(1-E) = (1-e_{under})(1-e_{fof}) = 1 - e_{under} - e_{fof} + e_{under}e_{fof}$. $E = e_{under}+e_{fof} - e_{under}e_{fof}.$

• First-order (additive): $E \approx e_{under}+e_{fof} = 0.60\% + 0.40\% = 1.00\%$.
• Second-order correction: $-e_{under}e_{fof} = -0.006\times0.004 = -0.000024 = -0.0024\%$, so exact $E = 0.9976\%$. Closed-end discount: a buyer purchasing units at, say, 10% below NAV gets exposure to full-NAV assets for 90% of the price; the extra NAV per rupee invested raises effective yield and can offset the expense drag (and if the discount narrows, provides additional return). (6 marks: 2 derivation of $(1-E)$ product, 2 additive first-order + second-order term, 2 closed-end discount reasoning)

[
  {"claim":"Day-0 NAV = 20", "code":"assets=500000000; units=25000000; result = (assets/units == 20)"},
  {"claim":"(1-e/365)**365 approx 1-e for e=0.002", "code":"e=0.002; approx=(1-e/365)**365; result = abs(approx-(1-e))<1e-5"},
  {"claim":"SIP value = 74800 for given path", "code":"prices=[100,90,80,100,120]; A=12000; u=sum(A/p for p in prices); result = abs(u*prices[-1]-74800)<1e-6"},
  {"claim":"Lumpsum value = 72000 and SIP outperforms by 2800", "code":"prices=[100,90,80,100,120]; A=12000; n=len(prices); u=sum(A/p for p in prices); sipv=u*prices[-1]; lumpv=(A*n/prices[0])*prices[-1]; result = (abs(lumpv-72000)<1e-6) and (abs(sipv-lumpv-2800)<1e-6)"},
  {"claim":"ELSS year-3 value = 210739.20", "code":"v=150000*(1.12**3); result = abs(v-210739.2)<1e-2"},
  {"claim":"FoF effective drag = 0.9976%", "code":"eu=0.006; ef=0.004; E=eu+ef-eu*ef; result = abs(E-0.009976)<1e-9"}
]