Portfolio Management

Question 1

Grinold-Kroner Model

Answer 1

ERP =
(+) Dividend Yield
(+) Expected Inflation
(+) Real GDP Growth
(+) % change in P/E
(-) % change in S/O
——–
(-) Real risk-free

Question 2

Optimal active risk

Answer 2

IR / SR_B * sd_B

Question 3

How must economic factors affect market values according to the PV model?

Answer 3

To impact value, factors must affect: 1) Default-free interest rates (discount rate component), 2) Timing and/or magnitude of expected cash flows (numerator), or 3) Risk premiums (discount rate component).

Question 4

What is the Intertemporal Rate of Substitution (mt​)?

Answer 4

Measures the marginal utility of consuming one unit in the future relative to consuming one unit today.

Question 5

Price of risk free bond in terms of mt

Answer 5

Price of a risk-free bond P0​=E(mt​)
mt is intertemporal rate of substitution

Question 6

Risk-free rate in terms of mt

Answer 6

Rf = (1/E(mt))-1
mt is intertemporal rate of substitution

Question 7

How does the business cycle affect short-term interest rates and the yield curve slope?

Answer 7

Expansion: Economy overheats, inflation rises. Central bank tightens policy (raises short-term rates). Yield curve often flattens or inverts as LT rates may rise less (or fall) due to lower future growth expectations/inflation control. Recession: Economy weakens, inflation falls. Central bank loosens policy (lowers short-term rates). Yield curve typically steepens (bull steepening).

Question 8

How does the business cycle affect credit spreads?

Answer 8

Credit spreads typically narrow during expansions (lower default risk, higher recovery rates) and widen during recessions (higher default risk, lower recovery rates). Lower-rated bonds are more sensitive.

Question 9

How does the business cycle affect earnings growth expectations and valuation multiples?

Answer 9

Earnings Growth: Expectations are pro-cyclical; rise during expansions, fall during recessions. Valuation Multiples (e.g., P/E): Tend to expand during early expansion (anticipating recovery) and contract during late expansion/recession (higher risk aversion, lower growth expectations). Shiller CAPE often counter-cyclical.

Question 10

What is Value Added by active management?

Answer 10

The difference between the portfolio’s return (RP​) and the benchmark’s return (RB​). RA​=RP​−RB​. Can be decomposed ex-post into asset allocation and security selection returns.

Question 11

Define the Information Ratio (IR).

Answer 11

Measures active return per unit of active risk. IR=E(RA​)/σA​=(RP​−RB​)/σ(RP​−RB​). Ex-ante IR measures expected skill; ex-post measures realized skill-adjusted return.

Question 12

What are the components of the Fundamental Law of Active Management?

Answer 12

Basic Law (Unconstrained): E(RA​)=IC×BR​×σA​, or IR=IC×BR​. Expanded Law (Constrained): E(RA​)=TC×IC×BR​×σA​, or IR=TC×IC×BR​. IC: Information Coefficient (skill, correlation of forecasts & outcomes). BR: Breadth (number of independent bets). TC: Transfer Coefficient (efficiency, constraint impact, 0 to 1). σA​: Active Risk (aggressiveness).

Question 13

How is the optimal level of active risk determined?

Answer 13

σA∗​=SRB​IR​×σB​, where SRB​ is Sharpe Ratio of benchmark. Optimal portfolio Sharpe Ratio SRP∗​=SRB2​+IR2​.

Question 14

Describe the ETF creation/redemption process and the role of Authorized Participants (APs).

Answer 14

Creation: AP delivers a basket of underlying securities (creation basket) to ETF issuer, receives ETF shares. Redemption: AP returns ETF shares to issuer, receives underlying securities (redemption basket). APs act as market makers, ensuring ETF price stays close to NAV via arbitrage. Process is typically “in-kind” (shares for shares), enhancing tax efficiency.

Question 15

What causes ETF tracking error?

Answer 15

Fees/expenses, sampling/optimization instead of full replication, cash drag, index changes, fund accounting practices, tax rules, depository receipts timing differences.

Question 16

What affects ETF bid-ask spreads?

Answer 16

Liquidity of the ETF itself, liquidity of the underlying securities, risk premium for AP hedging, AP’s arbitrage costs. Wider for illiquid underlying or during market stress.

Question 17

Define Arbitrage Pricing Theory (APT) assumptions.

Answer 17

1) Asset returns described by a factor model. 2) Asset-specific risk can be diversified away. 3) No arbitrage opportunities exist among well-diversified portfolios. Unlike CAPM, does not assume investors are rational mean-variance optimizers or specify the factors.

Question 18

Contrast Macroeconomic, Fundamental, and Statistical Factor Models.

Answer 18

Macro: Factors are observed macro variables (e.g., inflation, GDP growth); sensitivities estimated via time-series regression. Fundamental: Factors are company attributes (e.g., P/E, size, industry); sensitivities derived from attributes. Statistical: Factors derived statistically (PCA, factor analysis) from historical returns; factors may lack economic meaning.

Question 19

Define Active Risk (Tracking Error) and its sources.

Answer 19

Standard deviation of active returns (σA​=σ(RP​−RB​)). Sources: Active Factor Tilts (difference in portfolio vs benchmark sensitivities to factors) and Security Selection (difference in portfolio vs benchmark exposure to specific/residual risk).

Question 20

Define Value at Risk (VaR).

Answer 20

Minimum loss expected over a specific time horizon at a given probability level (e.g., 5% daily VaR of $1m means 5% chance of losing at least $1m in one day).

Question 21

Compare VaR estimation methods (Parametric, Historical, Monte Carlo).

Answer 21

Parametric: Assumes distribution (e.g., normal), uses mean/std dev to calculate VaR (VaR=μ+zσ). Simple but relies on assumptions. Historical: Uses past return distribution; ranks historical returns to find VaR cutoff. No distribution assumption but depends on lookback period. Monte Carlo: Simulates many possible future paths using models/distributions for risk factors; calculates VaR from simulated outcomes. Flexible but complex/model-dependent.

Question 22

Define extensions of VaR (CVaR, IVaR, MVaR).

Answer 22

Conditional VaR (CVaR): Expected loss given that the VaR threshold has been exceeded (average tail loss). Incremental VaR (IVaR): Change in VaR from adding/removing a position. Marginal VaR (MVaR): Change in VaR for a small change in a position’s size (sensitivity).

Question 23

Compare Sensitivity Risk Measures and Scenario Risk Measures to VaR.

Answer 23

Sensitivity (Beta, Duration, Greeks): Measures impact of small change in one risk factor. Simple but ignores correlations, non-linearities, and probability.

Scenario: Measures impact of large move in multiple factors (historical or hypothetical event). Captures correlations/non-linearities but doesn’t give probabilities.

VaR: Probabilistic measure of potential loss, considers portfolio effects but can be complex and model-dependent.

Question 24

Describe the process and potential problems in backtesting investment strategies.

Answer 24

Process: Design strategy (hypothesis, rules, rebalancing), Simulate over historical data (often using rolling windows: in-sample parameter estimation, out-of-sample testing), Evaluate results (returns, risk, ratios, plots). Problems: Survivorship bias (excluding failed firms), Look-ahead bias (using future info), Data snooping (overfitting to historical data). Use point-in-time data and cross-validation to mitigate

Question 25

Rolling Return Assessment

Answer 25

Also know as tracking differences gives an informative picture of the investment outcome for an investor in an ETF. Can see cumulative impact of portfolio management and expense ratio

Question 26

Sources of ETF tracking error

Answer 26

Fees and expenses Representative sampling/optimization Depositary receipts and other ETFs Index changes Fund accounting practices Regulatory tax requirements Asset manager operations

Question 27

ETF Capital gain distribution

Answer 27

Tax fairness, unlike mutual funds do not need to sell underlying security to pay the investor in cash, typically ETFs sold to another investor in the secondary market. Selling activities of individual investors do not create a taxable event, if AP redeems ETF shares redemption is in kind. Tax efficiency, redemption process allows managers to manage fund's tax liability, can choose to deliver shares with largest unrealised capital gains, ETF managers can use in-kind redemption to reduce potential capital gain

Question 28

ETF Bid-Ask Spread components

Answer 28

Creation/redemption fees and other trading costs Bid-ask of underlying securities Compensation for the risk of hedging or carrying positions Market maker's desired profit spread Discount related to likelihood of receiving an offsetting ETF order

Question 29

ETN

Answer 29

if a large bank that wants to issue unsecured debt at a fixed interest rate finds that the rate demanded by the market is sifnificantly higher than the swap rate for the same maturity, they can instead issue an ETN that pays the return on an equity swap. The bank simultaneously enter into an equity swap as the equity return receive and the swap fixed rate payer, the index return received is used to service the ETN and the bank’s effective borrowing cost becomes the swap fixed rate

Question 30

VaR calculation

Answer 30

Daily to monthly: Volatility scales with square root of time: s.d. x sqrt(n days) Calculate monthly return: daily return x n-days Determined z-score, for 5% confidence this is -1.645 Expected monthly return – (1.65 x monthly sd)

Question 31

Combined active risk

Answer 31

σC = [σx^2 -2σx * σy * rxy + σy^2]1/2

Question 32

Taylor's Rule

Answer 32

r = Rn + π + 0.5(π – π*) + 0.5(y – y*)

Question 33

Sharpe Ratio impact of adding cash/leveraging

Answer 33

No impact because, SR is ratio of excess return over Rf / sd relative to Rf. Adding cash would reduce the excess return and risk by a proportionate amount

Question 34

Information Ratio impact of adding cash/leverage

Answer 34

Generally reduces the information ratio, reduction in active risk is more significant than the reduction in active return

Question 35

Information Ratio impact of changing active weights

Answer 35

No impact, so adding or removing benchmark does not impact information ratio

Question 36

Total risk in managed portfolio

Answer 36

𝜎2𝑃 =𝜎2𝐵 + 𝜎2𝐴

Question 37

General mean-variance optimality condition in active management

Answer 37

Ratio of expected return in excess of benchmark to active variance (square active risk) is equal to ratio of expected benchmark return in excess of risk-free rate to benchmark return variance (square of benchmark risk). This is how the optimal active risk equation is derived

Question 38

Correlation Triangle - Value Added

Answer 38

Correlation between active weights and realised active return of a security

Question 39

Correlation Triangle - Signal Quality

Answer 39

Correlation between forecasted active returns and realised active returns Termed information coefficient

Question 40

Correlation Triangle - Portfolio Construction

Answer 40

Correlation between active weights and forecasted active returns Termed transfer coefficient Transfer coefficient can also be expressed as the risk-weighted correlation between the optimal active weights and the actual active weights

Question 41

Expected active return of portfolio Grinold's

Answer 41

IC x √BR x active risk

Question 42

Information ratio unconstrained portfolio

Answer 42

IC x √BR

Question 43

Expected return extended

Answer 43

TC x IC x √BR x active risk

Question 44

Covariance of covariance between a risk-averse investor’s inter-temporal rate of substitution and the expected future price of a risky asset is typically

Answer 44

When expected future price of a risky asset is rising, it can be interpreted that economy is in its good times and expanding, so Marginal utility of current consumption > MU of Future consumption, hence inter-temporal rate of substitution is low. Thus, -ve covariance.