#EXAMPLE

1. Determine the manager that is best suited to meet the needs. Justify your response.
   1. Manager A is using a value-weighted approach which requires no input from the fund manager’s beliefs. Candidates: Look for key words. The 2nd and 3 rd choice ==mention “strategically” which should quickly hint at active strategies compared to passive strategies.== Note, all passive strategies are discussed in readings 15 and 16 while active strategies are discussed in readings 17 and 18. Get great at identifying them!
2. Calculate the change in contract value expected. Show your calculations
   1. 𝐶𝐶ℎ𝑎𝑎𝑖𝑖𝑎𝑎𝑌𝑌 𝑌𝑌𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑟𝑟𝑎𝑎𝑖𝑖𝑟𝑟 𝑖𝑖𝑎𝑎𝑌𝑌𝑟𝑟𝑌𝑌 = 250 × (2,353 − 2,350) = 750 Or 𝐶𝐶𝑟𝑟𝑟𝑟𝑟𝑟𝑌𝑌𝑖𝑖𝑟𝑟 𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑟𝑟𝑎𝑎𝑖𝑖𝑟𝑟 𝑟𝑟𝑖𝑖𝑌𝑌𝑟𝑟 𝑖𝑖𝑎𝑎𝑌𝑌𝑟𝑟𝑌𝑌 = 2,350 × 250 = 587,500 𝐸𝐸𝐶𝐶𝑝𝑝𝑌𝑌𝑖𝑖𝑟𝑟𝑌𝑌𝑌𝑌 𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑟𝑟𝑎𝑎𝑖𝑖𝑟𝑟 𝑟𝑟𝑖𝑖𝑌𝑌𝑟𝑟 𝑖𝑖𝑎𝑎𝑌𝑌𝑟𝑟𝑌𝑌 𝑌𝑌𝑖𝑖 𝑟𝑟𝑅𝑅𝑖𝑖 𝑌𝑌𝑎𝑎𝑦𝑦𝑏𝑏 = 2,353 × 250 = 588,250 𝐶𝐶ℎ𝑎𝑎𝑖𝑖𝑎𝑎𝑌𝑌 𝑌𝑌𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑟𝑟𝑎𝑎𝑖𝑖𝑟𝑟 𝑖𝑖𝑎𝑎𝑌𝑌𝑟𝑟𝑌𝑌 = 588,250 − 587,500 = 750 The expected change in contract value is 750.
3. Calculate at what futures price a margin call will be triggered. Show your calculations.

Scenario Details

Let’s assume an investor goes long (buys) 1 S&P 500 E-mini Futures Contract.

1. How Much Can the Price Fall Before Triggering a Margin Call?

A margin call occurs when your cumulative losses exceed the buffer between your initial margin ($12,000) and your maintenance margin ($10,000).

$$\text{Maximum Allowed Loss} = \text{Initial Margin} - \text{Maintenance Margin} = $12,000 - $10,000 = \mathbf{$2,000}$$ Now, convert that dollar loss into index points:

$$\text{Max Allowed Drop (in Points)} = \frac{\text{Maximum Allowed Loss}}{\text{Contract Multiplier}} = \frac{$2,000}{$50/\text{point}} = \mathbf{40\text{ points}}$$

Break-Even & Margin Call Thresholds:

$$\text{Margin Call Trigger Price} = 5,000 - 40 = \mathbf{4,960\text{ points}}$$

$$\text{Percentage Drop Required} = \frac{40}{5,000} = \mathbf{0.80%}$$

Takeaway: The futures price only needs to fall by 0.80% (40 points) for the investor to get a margin call.

1. Explain the ==high tracking error of Manager D with two reasons.== Do not use the row mentioning tracking error as an explanation.
   1. ==Dividends are reinvested the next day rather than same day which creates cash drag.== Cash balances create tracking error.
   2. The portfolio contains a ==significantly smaller number of securities held than the index== ==leading to tracking error.==

A. FACTOR BASED STRATEGIES

Index strategies use ==rules-based replication== of an index (e.g., S&P 500) → ↑transparency, ↓costs, diversification, tax efficiency (low turnover).

CAPM limitation: Only market beta explains returns. Evidence supports additional systematic risk factors (and associated premiums): “SSM-QDV”

• Size: Small-cap premium (compensation for higher risk)
• Style (Value): Low P/E & P/B outperform growth (high P/E, high EPS growth)
• Momentum: Prior winners continue to outperform (short-term)
• Quality: Consistent earnings + low debt → outperform in downturns
• Dividend Yield: High-yield stocks outperform in declining rate environments
• Volatility (Low-Vol): Lower-vol stocks outperform in high-uncertainty periods

Passive Factor-Based (PFB) Strategy:

• Factor premiums treated as given (passive).
• Only active decision = ==desired factor betas/exposures.==
• Simple model: $r=βfactor×Premiumfactor r = \beta_{\text{factor}} \times \text{Premium}_{\text{factor}} r=βfactor​×Premiumfactor​$

Factor-based index (valid benchmark) requires transparent rules on: factor selection, betas/exposures, rebalancing.

vs Cap-Weighted Index:

• Factor index weights based on fundamentals (size, value, earnings, yield, etc.) → ==less prone to mispricing (overvalued stocks not overweighted)==. ==Cap-weighted indexes are prone to mispricing== because overvalued stocks are over-weighted based on price alone.
• Cap-weighted dominated by largest stocks; ==any outperformance of factor index often = extra pay for extra risk (not alpha).==
• Result: ==High tracking error (TE)== vs cap-weighted.
• Fees: higher than cap-weighted but lower than pure active. #EXAMPLE

HHI = Σ wᵢ² = 0.0286.
Effective # stocks = 1 / HHI = 1 / 0.0286 ≈ 34.97 (approximately 35).
(This is the number of equal-weighted stocks that would match the concentration.)

Price Weighted Index #EXAMPLE Price-weighted index: weight = price per share / sum of all prices → equivalent to holding one share of each constituent stock.

#EXAMPLE Factor-based strategies (including momentum) ==concentrate risk exposures vs. broad market-cap== weighting; investors vulnerable when the factor (momentum) is out of favor.

PFB Structures: #imp

• Return-Oriented: Tilt to high-dividend, momentum, etc. #EXAMPLE Momentum = amount of a stock’s excess price return over a period → classic return-oriented factor strategy.
• Risk-Oriented: Low-vol tilt or optimize whole-portfolio volatility (==requires correlation estimates==).
• Diversification-Oriented: Multi-factor exposure + factor rotation. ==Risk: Premiums decay as exploited (e.g., size premium squeezed as small-caps become expensive).==

B. APPROACHES TO PASSIVE EQUITY INVESTING

Three main approaches:

1. Pooled Funds

• Mutual Funds (MFs):
  • Easy unit purchase, record-keeping by sponsor, many options available.
  • Manager handles all ops: rebalancing, adds/deletes, reinvest cash flows, proxy voting.
  • CONS: Tax-inefficient (must sell stocks on redemptions → capital gains); cannot trade intra-day; higher fees.
• ETFs:
  • Trade like stocks (margin if bullish, short if bearish).
  • ==In-kind creation/redemption → highly tax-efficient (no cap gains triggered on ETF share sales).==
  • Wider variety of indexes (easier factor exposure, e.g., long small-cap ETF + short large-cap ETF).
  • ==Lower management fees than MFs==.
  • Tradeoff: ==ETF (lower fee) vs MF (lower trading costs — no commissions/spreads).==
  • CONS: Trading costs (bid-ask spreads, commissions); potential large discount to NAV if underlying basket illiquid (e.g., EM ETFs).

2. Derivative-Based Strategies

• Synthetic exposure: T-bill + long index futures = index return.
• Advantages: ==Very low cost, leverage (margin only), high liquidity for tactical shifts/rebalancing/equitizing cash, efficient overlays.==
• Disadvantages: #imp
  • OTC (forwards/swaps): counterparty risk + illiquidity.
  • ==Long options: lose full premium if OTM.==
  • Daily ==settlement== → cash flow management + margin calls.
  • Must ==roll contract==s for long-term exposure (best used as overlay, not core holding).
  • ==Basis risk== (imperfect correlation with index). #EXAMPLE
  • Regulations may restrict availability; mostly institutional only. #EXAMPLE

Bonus = USD5,750,000. Policy weight US equity = 15% → need USD862,500 exposure.
E-mini multiplier = 50; futures price = 2,464.29 → contract value = 2,464.29 × 50 = USD123,214.50.
Contracts needed = 862,500 / 123,214.50 ≈ 7.00. Buy 7 E-mini contracts (equitizing cash inflow while respecting policy weights).

3. Separately Managed Accounts (SMAs)

• Large institutions self-manage index portfolio.
• Requires: ==accounting/compliance systems, broker relationships, direct index data access.==

C. CONSTRUCTION OF PASSIVELY MANAGED EQUITY PORTFOLIOS

#EXAMPLE

The three requirements for an index to be the basis for an equity investment strategy are: (a) rule-based, (b) transparent, and (c) investable. Buffering (Statement 2) makes benchmarks more investable by gradual/ orderly index transitions.
Free-float adjustment increases (does not lower) investability/liquidity because it excludes non-tradable shares (founders/government/etc.), accurately reflecting true liquidity.

Four approaches:

1. Full Replication ==Hold every index constituent at exact index weights==. Obstacles: Thousands of stocks, ==large AUM== required, ==illiquid== names → ==↑TE==, constant data feed needed from index provider. #EXAMPLE
2. Stratified Sampling Partition index into mutually exclusive, exhaustive, proportional strata (by industry, size, style, domicile, etc.). Hold a subset of stocks in each stratum to match factor exposures. ✓ Works with ==smaller portfolios and illiquid/large indexes.== #EXAMPLE ==Stratified sampling is used for indexes with large # constituents or low AUM; it holds fewer securities than the index while still tracking closely and reducing brokerage costs==
3. Optimization Mathematical program: minimize TE subject to constraints (e.g., max 100 most liquid stocks). Relies on historical correlations. ==Use historical $\rho$ (correlations) to minimize $TE$ with fewer stocks. match index characteristics/performance== Risk: ==Correlations break down → frequent re-optimization → high turnover & trading costs.==
4. Blended Approach Full replication on liquid/large-cap portion (e.g., top 1,000); stratified sampling or optimization on remainder (e.g., next 2,000). ==Index split into sub-indexes, each tracked optimally.==

D. MANAGING TRACKING ERROR

• Numerical Step-Logic (Worked Example):
  • Step 1: $AR_t = R_{PF} - R_{Index}$.
  • Step 2: $\overline{AR} = \frac{\sum AR_t}{n}$.
  • Step 3: $TE = \sqrt{\frac{\sum(AR_t - \overline{AR})^2}{n-1}}$.
• Numerical Mapping (Fund ABC): Mean $AR = 2%$, $TE = 0.7%$.
  • Math: $2% \pm (1 \times 0.7%)$.
  • Result: 68% probability active returns fall between 1.3% and 2.7%.

Goal of index manager: ==Non-negative active return + low TE.==

Factors that increase TE:

• Larger number of index constituents (↑ trading costs).
• Higher management fees.
• Sampling or optimization (vs full replication).
• Cash drag (portfolio not fully invested while index is); mitigate by equitizing cash with index futures.
• Intra-day trading (index uses closing prices); mitigate by trading near market close.

TE control: minimize trading costs, net investor flows, equitization via derivatives to offset cash drag

#EXAMPLE

Tracking error rises with (1) more constituents and (2) higher expense ratios. EFA (MSCI EAFE ETF) has 933 constituents (vs. 60 and 506) + highest expense (0.33%) → highest expected tracking error under full replication.

#EXAMPLE

Lowest tracking error comes from lowest cash holdings + lowest fees. Portfolio 3 has the lowest of both (cash 2.13%, fees 0.07%) vs. the others and the index fund.

#EXAMPLE

Tracking error rises with ==(1) more constituents and (2) higher expense ratios.== EFA (MSCI EAFE ETF) has 933 constituents (vs. 60 and 506) + highest expense (0.33%) → highest expected tracking error under full replication.

E. SOURCES OF RETURN AND RISK ON A PASSIVE PORTFOLIO

Attribution Analysis — decomposes total return by decision:

• Asset allocation (mix across classes)
• Domestic vs international
• Style exposures (size, value/growth, momentum, quality, etc.)
• Security selection (company-specific)

#EXAMPLE contribution analysis Sector | Portfolio Contribution | Benchmark Contribution | Difference
Information technology | 2.01% | 2.05% | –0.04% (largest negative)

Ways to enhance passive returns:

• Securities Lending: Earn lender fee on shares lent out. Risks: Counterparty fails to return shares, collateral value drops (market risk), lose voting rights during loan (but borrower must compensate dividends). #EXAMPLE
• Shareholder Activism: Pressure portfolio companies (sell unprofitable divisions, cut costs). Index funds/pensions have voting power. Trade-off: Absolute returns may rise, but higher costs → portfolio likely lags index net of costs.