Level 3 — ProductionFunds, ETFs & Pooled Vehicles

Funds, ETFs & Pooled Vehicles

⏱ 45 minutes60 marksprintable — key stays hidden on paper

Level 3 — From-Scratch Derivations & Explain-Out-Loud

Time Limit: 45 minutes
Total Marks: 60

Instructions: Derive all formulas from first principles where asked. Show every step. For "explain-out-loud" prompts, write as if teaching a peer. Use $...$ for math.

Q1. [NAV from scratch] (10 marks)

A mutual fund scheme holds the following at end of day:

Equity holdings market value: $₹480{,}000{,}000$
Cash & receivables: $₹25{,}000{,}000$
Accrued liabilities (fees, payables): $₹5{,}000{,}000$
Total units outstanding: $20{,}000{,}000$

(a) Derive the NAV-per-unit formula from the definition of net assets, then compute today's NAV. (4)

(b) An investor places a purchase order of $₹150{,}000$ . Given an entry mechanism where units are allotted at NAV (no entry load, per SEBI), compute units allotted. (3)

(c) Explain out loud why NAV is struck once per day for open-ended funds but ETFs trade at continuously varying market prices. (3)

Q2. [Expense ratio & tracking error] (12 marks)

(a) Derive an expression for the annual drag on returns caused by an expense ratio $e$ , and state why a fund's realized return $\approx r_{\text{gross}} - e$ . (3)

(b) An index returns $12.0\%$ in a year. Fund A (passive) has expense ratio $0.20\%$ and additional frictional slippage of $0.15\%$ . Compute Fund A's expected return and its tracking error contribution from these two costs. (4)

(c) Fund A reported these annual return differences (fund − index) over 5 years (%): $-0.35,\ -0.40,\ -0.30,\ -0.55,\ -0.25$ . Compute the tracking error (standard deviation of return differences, sample). (5)

Q3. [SIP vs Lumpsum derivation] (12 marks)

(a) Derive the future value formula for a SIP of amount $P$ invested at the end of each period for $n$ periods at periodic rate $i$ . (4)

(b) An investor does a SIP of $₹10{,}000$ /month for $12$ months at an assumed monthly return of $1\%$ . Compute the future value at the end of month 12. (4)

(c) Explain out loud one scenario where lumpsum outperforms SIP and one where SIP outperforms lumpsum, referencing rupee-cost averaging. (4)

Q4. [Direct vs Regular plans] (10 marks)

A fund's Regular plan has expense ratio $1.50\%$ ; its Direct plan has $0.80\%$ .

(a) Explain out loud what causes this gap. (2)

(b) Assuming a gross annual return of $11\%$ compounded annually, derive and compute the terminal value of $₹100{,}000$ after $10$ years under each plan (net of expense ratio). (5)

Q5. [ELSS + Exit loads] (8 marks)

(a) State the lock-in period of ELSS and explain out loud why exit loads are irrelevant during that period. (3)

(b) A non-ELSS equity fund charges a $1\%$ exit load if redeemed within 1 year. An investor redeems units worth $₹200{,}000$ (redemption value) after 8 months. Compute the exit load and the net amount received. (3)

Q6. [REITs, InvITs & pooled structures] (8 marks)

(a) Explain out loud how a REIT differs from a real-estate mutual fund in terms of underlying assets and income source. (3)

(b) A REIT owns property generating annual distributable cash flow of $₹90{,}000{,}000$ and has $30{,}000{,}000$ units. It distributes $90\%$ of cash flow (regulatory minimum). Compute distribution per unit and the yield if the unit trades at $₹300$ . (3)

Answer keyMark scheme & solutions

Q1

(a) Net Assets $=$ Assets $-$ Liabilities. NAV $=$ Net Assets / Units. (formula 1 mark)

$\text{Net Assets} = 480{,}000{,}000 + 25{,}000{,}000 - 5{,}000{,}000 = 500{,}000{,}000$ (1)

$\text{NAV} = \frac{500{,}000{,}000}{20{,}000{,}000} = ₹25.00 \text{ per unit}$ (2)

(b) Units $= 150{,}000 / 25 = 6{,}000$ units. (3 — formula + compute)

(c) Open-ended funds transact only against the fund's underlying portfolio, valued once when markets close → single NAV strike. ETFs are exchange-listed; buyers/sellers trade with each other in real time, so price fluctuates intraday with supply/demand, kept close to iNAV by authorized-participant arbitrage (creation/redemption). (3 — one for open-end mechanism, one for ETF trading, one for AP arbitrage)

Q2

(a) Fees are deducted from fund assets daily, accruing to $e$ over a year. Since it directly reduces the fund's asset base, net return $= r_{\text{gross}}(1-e) -$ small terms $\approx r_{\text{gross}} - e$ for small $e$ . (3)

(b) Total cost $= 0.20 + 0.15 = 0.35\%$ . (2) Expected return $= 12.0 - 0.35 = 11.65\%$ . (1) Tracking-error contribution from cost = $0.35\%$ (systematic drag). (1)

(c) Differences: mean $\bar{d} = (-0.35-0.40-0.30-0.55-0.25)/5 = -1.85/5 = -0.37$ . (1) Deviations: $0.02, -0.03, 0.07, -0.18, 0.12$ ; squares: $0.0004, 0.0009, 0.0049, 0.0324, 0.0144$ ; sum $=0.0530$ . (2) Sample variance $= 0.0530/(5-1) = 0.01325$ . (1) Tracking error $=\sqrt{0.01325} = 0.1151\% \approx 0.12\%$ . (1)

Q3

(a) Each payment $P$ compounds for remaining periods. Payment at end of period $k$ grows for $(n-k)$ periods: $FV = P\sum_{k=1}^{n}(1+i)^{n-k} = P\bigl[(1+i)^{n-1}+\dots+1\bigr]$ Geometric series with ratio $(1+i)$ : $FV = P\cdot\frac{(1+i)^n - 1}{i}$ (4 — series setup 2, closed form 2)

(b) $P=10{,}000$ , $i=0.01$ , $n=12$ : $(1.01)^{12} = 1.126825$ (1) $FV = 10{,}000\cdot\frac{1.126825 - 1}{0.01} = 10{,}000 \cdot 12.6825 = ₹126{,}825$ (3)

(c) Lumpsum outperforms when markets rise steadily from day one (all capital deployed early captures full upside). SIP outperforms in volatile/falling-then-rising markets — rupee-cost averaging buys more units when prices are low, lowering average cost. (4 — 2 each)

Q4

(a) Regular plans embed distributor/broker commission (trail) in the expense ratio; Direct plans are bought straight from the AMC with no commission → lower ratio. (2)

(b) Net return $=$ gross $-$ expense ratio (approx):

Regular: $11 - 1.50 = 9.50\%$ → $TV = 100{,}000(1.095)^{10}$
Direct: $11 - 0.80 = 10.20\%$ → $TV = 100{,}000(1.102)^{10}$ (formula 1)

$(1.095)^{10} = 2.47823$ → Regular $TV = ₹247{,}823$ (2) $(1.102)^{10} = 2.64155$ → Direct $TV = ₹264{,}155$ (2)

(c) Difference $= 264{,}155 - 247{,}823 = ₹16{,}332$ . (2) As % of Direct: $16{,}332 / 264{,}155 = 6.18\%$ . (1)

Q5

(a) ELSS lock-in $= 3$ years (36 months). Units cannot be redeemed during lock-in, so an exit load (which penalizes early redemption) has nothing to apply to. (3)

(b) Redeemed within 1 year → load applies. Load $= 1\% \times 200{,}000 = ₹2{,}000$ . (2) Net received $= 200{,}000 - 2{,}000 = ₹198{,}000$ . (1)

(c) Exit loads discourage short-term churn/redemptions, protecting long-term investors from liquidation costs and stabilizing the fund's assets. (2)

Q6

(a) A REIT directly owns income-producing physical real estate; income comes from rent/lease, and it must distribute ~90% of cash flow. A real-estate MF invests in securities of real-estate companies (equity/debt), earning from those securities rather than owning buildings directly. (3)

(b) Distributable $= 90\% \times 90{,}000{,}000 = 81{,}000{,}000$ . (1) Per unit $= 81{,}000{,}000 / 30{,}000{,}000 = ₹2.70$ . (1) Yield $= 2.70 / 300 = 0.90\%$ . (1)

(c) FoF: invests in units of other mutual funds (a fund of funds). Closed-end fund: fixed number of units issued at NFO, listed and traded on exchange, no ongoing creation/redemption. (2)

[
  {"claim":"Q1 NAV = 25 and units allotted = 6000","code":"net=480000000+25000000-5000000; nav=net/20000000; units=150000/nav; result=(nav==25 and units==6000)"},
  {"claim":"Q2c tracking error approx 0.1151%","code":"d=[-0.35,-0.40,-0.30,-0.55,-0.25]; m=sum(d)/len(d); var=sum((x-m)**2 for x in d)/(len(d)-1); te=sqrt(var); result=abs(float(te)-0.11511)<0.001"},
  {"claim":"Q3b SIP FV = 126825","code":"P=10000; i=Rational(1,100); n=12; FV=P*(((1+i)**n)-1)/i; result=abs(float(FV)-126825.03)<1"},
  {"claim":"Q4 terminal values and 6.18% gap","code":"reg=100000*(1.095**10); dir=100000*(1.102**10); diff=dir-reg; pct=diff/dir*100; result=(abs(reg-247823)<5 and abs(dir-264155)<5 and abs(pct-6.18)<0.05)"},
  {"claim":"Q6b per unit 2.70 and yield 0.90%","code":"dist=0.90*90000000; per=dist/30000000; yld=per/300*100; result=(abs(per-2.70)<0.001 and abs(yld-0.90)<0.001)"}
]