Portfolio Management

inter-temporal rate of substitution

represents an investor's trade-off between real consumption now and real consumption in the future

Increase in rGDP and real interest rates and relationships with volatility

No difference in equity valuations because income CF and interest rates increase

Real risk-free rate rises with expected real growth and with growth volatility, because both change savers’ willingness to trade current for future consumption.

real

Shiller's CAPE (real cyclically adjusted P/E)

reduces the volatility of unadjusted P/E ratios by using real (i.e., inflation-adjusted) prices in the numerator and a 10-year moving average of real earnings in the denominator.

Commercial real estate characteristics`

• Bond-like characteristics. The steady rental income stream is similar to cash flows from a portfolio of bonds. Furthermore, just as the credit quality of issuers affects the value of a bond portfolio, the credit quality of tenants affects the value of commercial real estate.

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• Equity-like characteristics. The value of commercial real estate is influenced by many factors, including the state of the economy, the demand for rental properties, and property location. Uncertainty about the value of the property at the end of the lease term gives commercial properties an equity-like character.

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• Illiquidity. Real estate as an asset class is characterized by illiquidity; it could take years to exit a real estate investment at its fair value.

closet index fund

A closet index fund is a fund that is purported to be actively managed but in reality closely tracks the underlying benchmark index. These funds will have a Sharpe ratio similar to that of the benchmark index, a very low information ratio, and little active risk. After fees, the information ratio of a closet index fund is often negative.

What is the information ratio of a market-neutral long-short equity fund?

It equals the fund's Sharpe ratio — because with zero systematic risk and the risk-free rate as benchmark, active return equals portfolio return minus the risk-free rate, and active risk equals total risk, making the IR and Sharpe ratio calculations identical.

How does tripling the active weights of an unconstrained portfolio affect its information ratio?

It doesn't change the IR — both active return and active risk triple proportionately, so the ratio stays the same. The IR of an unconstrained portfolio is unaffected by the aggressiveness of the active weights.

What happens to the information ratio when you combine an actively managed portfolio with an allocation to the benchmark portfolio?

The blended portfolio has the same information ratio as the original active portfolio — increasing the benchmark weight decreases active return and active risk proportionately, leaving the IR unchanged.

What is the information coefficient (IC), and how does it differ from the ex-post information coefficient (ICR)?

IC is the ex-ante (expected), risk-weighted correlation between active returns and forecasted active returns — a measure of manager skill. ICR is the ex-post version, measuring the actual correlation between active returns and expected active returns.

What is the transfer coefficient (TC), and when does TC = 1?

TC is the correlation between actual active weights and optimal active weights (optimal weight is positively related to expected active return and negatively related to expected active risk). For an unconstrained portfolio, actual weights equal optimal weights, so TC = 1. For a constrained portfolio (e.g., short-sale or active-risk constraints), TC may be less than 1. Formula: TC = CORR(μi/σi, Δwiσi).

What is breadth (BR)?

The number of independent active bets taken per year. Example: a manager taking active positions in 10 securities each month has BR = 10 × 12 = 120.

authorized participants (APs)

APs are large broker-dealers that make the market in that ETF as primary market participants. APs are permitted to create additional shares, or redeem existing shares, for a service fee payable to the ETF manager. This creation/redemption process is in-kind: APs deliver a basket of securities (which may include cash) to the issuer in exchange for a number of ETF shares.

The in-kind creation/redemption process serves three purposes

1. Lower cost: The creation/redemption process does not force the ETF manager to sell/purchase portfolio investments; the manager does not incur any resulting transaction cost. The ETF manager usually collects a service charge from the AP to cover any incidentals.

2. Tax efficiency: A major benefit of the in-kind creation/redemption process is that it is not a taxable event. For a mutual fund, liquidity needs for redemption are often met by the fund manager by selling some of the fund's holdings, which triggers transaction costs as well as potential capital gains taxes. These costs are borne by all the shareholders of the fund (including those that did not redeem shares). Additionally, ETF managers can choose to publish customized redemption baskets, allowing them to target low-basis stocks that will be part of the redemption basket. This increases the tax efficiency (i.e., increases the basis) of the remaining holdings of the ETF.

3. Keeping market prices in line with NAV: APs will engage in arbitrage transactions if the ETFs trade at a price significantly different from their NAV. If the ETF trades at a premium, APs can sell the ETF, purchase the creation basket, and recreate those shares. Similarly, if the ETF trades at a significant discount to NAV, the APs can purchase the ETF and redeem the shares.

Sources of tracking error

1. Fees and expense. Fees reduce a fund's return.

2. Sampling and optimization. ETFs may use statistical techniques to replicate the performance of a benchmark without investing in all the securities that the index covers. Optimization techniques often favor higher-liquidity securities (and larger market cap companies) to minimize transaction cost. As a result, the ETF may impart a size bias relative to the benchmark.

3. Depository receipts (DRs). Foreign index ETFs often invest in DRs (rather than less-liquid securities traded on local exchanges). Any difference between the price of DRs and corresponding security (e.g., due to time zone differences in capturing price data) may contribute to tracking error of the ETF. Furthermore, sometimes an ETF may invest in other (sector) ETFs, and thus inherit the tracking errors of those ETFs.

4. Index changes. Index providers will occasionally rebalance or reorganize their indexes. ETF managers often use the creation/redemption process to rebalance the ETF portfolio to reflect this change in the index. The resulting delays from the use of the creation/redemption process contributes to tracking error. Because changes to an index are relatively infrequent, this component is often the smallest contributor to total tracking error.

5. Regulatory and tax requirements. In some countries, tax rates for foreign investors and domestic investors differ, leading to a difference in after-tax returns between an ETF and the index that it tracks.

6. Fund accounting practices. The time of the day when ETF NAV is calculated versus when the index provider performs this computation can lead to differences in calculated returns. Additionally, ETFs with foreign-currency-denominated holdings may use exchange rate values captured at a different time than the rate used by the index provider.

7. Asset manager operations. ETF managers may try to lower their cost by lending their shares to short sellers, and by foreign dividend capture (i.e., by working with foreign governments to minimize the taxes on distributions received). These methods tend to improve ETF performance relative to their benchmark.

The authorized participants (APs) in an ETF are most likely to be required to settle their ETF trades in

6 days

5 ETF Risks

• Counterparty risk: ETNs are unsecured bank debt, so investors lose money if the issuer defaults (e.g., Lehman in 2008), and a one-year CDS spread above 5% signals very high risk.

• Settlement risk: ETFs using OTC derivatives face the risk that a contract fails to settle, which they reduce through frequent settlement and collateral.

• Security lending: ETFs lend securities to short sellers for fees that offset operating expenses, and investors bear the small risk of the borrower defaulting.

• Fund closures: Closures force asset sales and cash distributions with possible tax costs, and ETN creation halts can push prices to a significant premium to NAV.

• Expectation-related risk: Leveraged and inverse ETFs reset daily, so returns compound (a 2x fund that gains 5% then loses 5% ends at $99, not $100), making them unsuitable for holding beyond about one month.

Efficient Portfolio Managemetn

• Portfolio liquidity management: Managers can quickly equitize excess cash in ETFs to reduce cash drag, and because ETFs are liquid and cheap to trade, ETF allocations can cover future cash outflows.

• Portfolio rebalancing: ETFs let managers cost-effectively move to target asset class weights, and shorting them quickly cuts exposure to a sector or asset class.

• Portfolio completion: ETFs fill temporary allocation gaps, such as when a manager leaves small-caps and the investor wants to keep that exposure, or when a fund's allocation differs from the investor's desired exposure.

• Transition management: A new manager can hold ETFs temporarily to maintain market exposure while winding down the old manager's positions.

• Limits for very large asset owners: SMAs may be cheaper than ETF fees, can be customized instead of having rigid allocations, and avoid the public disclosure of large holdings that regulators often require for ETFs.

Asset Class Exposure Management

• Asset class exposure management: The wide range of ETFs (asset class, subclass, sector) lets managers implement varied strategies, often at lower cost than holding the underlying securities (e.g., fixed-income ETFs are easier to trade than illiquid bonds).

• Core exposure: Passive ETFs cost-effectively provide broad diversification across equity sectors, commodities, bonds, etc., and can target a specific subsector for suitability (e.g., precious metals).

• Tactical strategies: Managers temporarily rotate into or out of sectors, or into thematic ETFs (e.g., ecommerce vs. broad tech), and choose ETFs by low trading cost and liquidity rather than low management fees, measuring liquidity as average dollar volume divided by average assets (higher is better).

Active Investment

Active investing: Newer ETFs with an active component, especially in fixed income, follow predefined rules rather than manager discretion (e.g., smart beta ETFs using quantitative screens or fundamental weights like dividend yield).

• Factor (smart beta) ETFs: They track rules-based indices built on return drivers such as size and momentum, and performance depends on whether the rules captured the factor and whether the market rewarded it. Competing funds differ by factor chosen or weighting (equal vs. cap-weighted), and multifactor ETFs blend several factors and can shift weights dynamically.

• Risk management: Some ETFs give higher or lower risk than a passive index (e.g., low volatility) or hedge specific risks (e.g., currency-hedged global equity, or duration-hedged high-yield bonds that keep only credit risk).

• Alternatively weighted ETFs: They use weights that differ from market cap, such as equal weighting or fundamentals-based weighting.

• Discretionary active ETFs: Actively managed and similar to closed-end mutual funds, with the largest being fixed-income ETFs (senior bank loans, mortgage securities, floating rate notes).

• Dynamic asset allocation and multi-asset strategies: Top-down ETFs shift between stocks and bonds based on risk/return forecasts, and are popular with global asset managers and hedge funds for discretionary allocation.

APT 3 Assumptions

1. Unsystematic risk can be diversified away in a portfolio. Investors have the choice of a large number of assets such that unsystematic risk can be diversified by forming portfolios of assets. This is a reasonable assumption and is supported by empirical evidence.

2. Returns are generated using a factor model. Unfortunately, the APT provides little practical guidance for the identification of the risk factors. The lack of clarity for the risk factors is a major weakness of the APT.

3. No arbitrage opportunities exist. An arbitrage opportunity is defined as an investment opportunity that bears no risk and no cost, but provides a profit. This assumption implies that investors will undertake infinitely large positions (long and short) to exploit any perceived mispricing, causing asset prices to adjust immediately to their equilibrium values.

Describe the three types of factor models (macroeconomic, fundamental, and statistical).

• Macroeconomic: Returns are explained by surprises (realized value minus consensus forecast) in macro factors like GDP, interest rates, and inflation. The expected value is already priced in.

• Fundamental: Returns are explained by firm-specific factors like P/E ratio, market cap, leverage, and earnings growth. For fixed income, the factors are duration, credit quality, currency, and geography.

• Statistical: Returns are explained using statistical methods. Factor analysis uses portfolios that explain covariance, and principal components uses portfolios that explain variance.

  • Weakness of statistical models: The factors are hard to interpret economically, so they are called "mystery factors."

“Priced Risk”

systematic factors represent priced risk (i.e., risk for which investors can expect compensation).

What is the difference between active factor risk and active specific risk?

Active factor risk is the risk from a portfolio's factor tilts (e.g., sector, size, style) differing from the benchmark's. Active specific risk is the risk from individual security selection differing from the benchmark, after controlling for those factor tilts.

What is tracking portfolio and active factor/specific risk

A tracking portfolio is a portfolio with a specific set of factor sensitivities. Tracking portfolios are often designed to replicate the factor exposures of a benchmark index like the S&P 500—in fact, a factor portfolio is just a special case of a tracking portfolio. One use of tracking portfolios is to attempt to outperform the S&P 500 by using the same factor exposures as the S&P 500 but with a different set of securities than the S&P 500

• low active factor risk

• high active specific risk.

What is factor portfolio and active factor/specific risk

A factor portfolio is a portfolio with a factor sensitivity of 1 to a particular factor and zero to all other factors. It represents a pure bet on that factor. For example, a portfolio manager who believes GDP growth will be greater than expected, but has no view of future interest rates and wants to hedge away the interest rate risk in her portfolio, could create a factor portfolio that is only exposed to the GDP factor and not exposed to the interest rate factor

• high active factor risk

• low active security risk

What are the advantages of VaR?

• Simple concept, easy to explain (even if methodology is complex)

• Allows comparison of risk across different portfolios, asset classes, or trading operations

• Useful for performance evaluation (return vs. risk taken, e.g., trading income/VaR ratio)

• Supports capital allocation/risk budgeting across trading units based on max VaR limits

• Accepted by global banking regulators as a risk measure (methodology not prescribed)

• Reliability can be verified through backtesting

What are the limitations of VaR?

• Requires many subjective choices (loss %, lookback period, distribution assumptions, parameter estimates) — results can be manipulated to show a low estimate

• Normality assumption underestimates tail risk, since actual return distributions often have fatter tails than normal

• Doesn't account for liquidity drying up when prices fall, understating actual liquidation losses

• Doesn't account for correlations spiking during stress, overestimating diversification benefits and underestimating potential losses

• As a single summary number, it doesn't capture all aspects of risk

• Focuses only on downside risk, ignoring right-hand tail (upside) outcomes that matter for the full risk-return picture

What are the different types/variations of VaR?

• Conditional VaR (CVaR) – also called expected tail loss or expected shortfall; the expected loss given that the loss equals or exceeds the VaR (average of losses in the left tail beyond VaR). Easy to calculate with historical simulation or Monte Carlo (just average tail losses); mathematically complex with the parametric method.

• Incremental VaR (IVaR) – the change in portfolio VaR resulting from a specific change in a security's weight in the portfolio (e.g., increasing a position by 2%).

• Marginal VaR (MVaR) – the slope of the curve plotting VaR as a function of a security's weight, at the point of its current weight; approximates the change in VaR for a 1% change in that security's weight.

• Ex ante tracking error / Relative VaR – the VaR of the difference between a portfolio's return and its benchmark's return; calculated as the VaR of a long position in the portfolio combined with a short position in the benchmark. Indicates likely magnitude of underperformance vs. benchmark at a given confidence level.

What are the different risk assessment models beyond VaR?

• Sensitivity analysis – measures the effect on portfolio value from a small change in one risk factor; helps understand risk factor by factor but does not predict probability of losses.

• Scenario analysis – estimates the effect on portfolio value from a set of significant changes across multiple risk factors simultaneously; usually designed to show a significant decline (but can also model gains).

• Historical scenario approach – a type of scenario analysis using risk factor changes that actually occurred in the past (e.g., 2008 subprime crisis, 1987 equity crash).

• Hypothetical scenario approach – a type of scenario analysis using any set of risk factor changes, including ones more extreme than historical events, as long as they have non-zero probability of occurring.

• Stress tests – examine the effect on value or solvency from a scenario of extreme risk-factor changes.

What is reverse stress testing?

A scenario analysis approach that starts with an unacceptable outcome (usually one that threatens the organization's survival) and works backward to identify the risk factor changes that would cause it, then assesses how likely those scenarios are.

What is surplus-at-risk?

A VaR-style risk measure used by defined benefit pension funds, applied to the plan's surplus (assets minus liabilities). A negative surplus must be made up by the firm unless higher-than-expected asset returns reduce it over time

What is a glidepath in a defined benefit pension fund?

A multi-year plan for adjusting pension fund contributions to reverse a significant overfunded or underfunded status

What are the four types of risk constraints used in managing market risk?

• Risk budgeting: Set the acceptable total risk (e.g., a maximum 5% VaR), then allocate it across business units, strategies, or asset classes.

• Position limits: Cap exposures (by currency amount, % of portfolio, or a liquidity measure) to ensure minimum diversification.

• Scenario limits: Limit the expected loss under a given scenario.

• Stop-loss limits: Require reducing or hedging an exposure once losses exceed a specified amount over a set period.

economic capital

Economic capital is the capital needed for a firm to survive if severe losses are experienced based on the risk the business is exposed to.

data-mining trap

those that perform well in a backtest are incorporated into the strategy—even if they don't have any logical or economic reason to be included

benchmark portfolio (BM)

which weights factors equally

risk parity portfolio (RP)

combines factors so that each factor contributes equally to risk

What is the rolling window (walk-forward) backtesting approach?

A method that moves through historical data in sequential steps—calibrating the model, trading/rebalancing, then rolling the window forward to recalibrate again—rather than splitting data into just one training and one testing sample.

What key decisions are made during Strategy Design?

• Define investment universe, return measurement, and benchmark

• Set rebalancing frequency and consider transaction costs

• Choose start/end dates, accounting for market regimes

• Select factors based on statistics AND economic logic (avoid data-mining)

• Decide how to combine factors: BM (equal-weighted) or RP (risk parity)

What happens during Historical Investment Simulation?

• Construct and rebalance the portfolio per the chosen strategy, subject to constraints (geography, liquidity, shorting limits)

• Apply rolling window (walk-forward) backtesting: calibrate → trade → roll forward → repeat

• For multifactor strategies, run rolling window twice: once for each long-short factor portfolio, once for RP covariances

• Evaluate results out-of-sample

What happens during Analysis of Backtesting Output?

• Use the long/short hedged portfolio approach (Fama-French): rank stocks by factor, split into quintiles, long top/short bottom

• Rebalance monthly and generate out-of-sample performance data

• Assess using Sharpe ratio, Sortino ratio, maximum drawdown, etc.

• Compare multiple methodologies, since results can differ—ideally they converge

What does the tail dependence coefficient measure?

The correlation (co-movement) between the tails of two random variables—i.e., whether extreme outcomes in one variable tend to occur alongside extreme outcomes in the other.

We are likely to underestimate downside risk if we fail to account for positive tail dependence.

What two features does the multivariate skewed t-distribution have that the multivariate normal distribution lacks?

Fat tails (excess kurtosis) and skewness.

Aside from the distributional assumption, how does the sensitivity analysis process compare to the Monte Carlo simulation process?

It follows the same overall process; the only difference is fitting the data to a multivariate skewed t-distribution rather than a multivariate normal distribution.

Standard rolling-window backtesting is most likely to fail to account for downside asset returns due to

negative skewness, excess kurtosis, and tail dependence.