LM2 Analysis of Active Portfolio Management Question Notes

Variant: Decide Whether a Benchmark Is Fit for the Job

Abstract: A benchmark is the cheap passive alternative the manager is trying to beat. It must represent the mandate, be investable at low cost, and have observable rules and returns.

A global small-cap manager is judged against a private index of 40 large-cap domestic stocks. Its weights are disclosed only after year-end, and no passive fund can replicate it. Is this a valid benchmark? Explain each failure and whether the zero-sum argument for active management applies cleanly.

1. Run the three-part benchmark check

• Representative? No. Large-cap domestic stocks are not the manager's global small-cap opportunity set.
• Replicable at low cost? No. The client cannot actually hold this “passive alternative.”
• Observable? No. Hidden weights prevent ex ante verification and prompt performance measurement.

2. Handle the zero-sum claim carefully

Active management is zero-sum before costs when the benchmark is a float-adjusted capitalization-weighted portfolio covering the complete relevant market. This narrow, mismatched index does not meet that setup because managers can own assets outside it.

Variant: Calculate Portfolio, Benchmark, and Active Return

Abstract: Build both portfolios from weight times return, then subtract benchmark return from managed return. A positive portfolio return alone proves nothing about active skill.

Stocks return 14% and bonds return 2%. The benchmark is 60% stocks and 40% bonds; the manager holds 70% and 30%. Calculate both returns and value added.

1. Price the passive alternative

2. Price what the manager actually held

3. Measure the manager, not the market

The manager added 1.20 percentage points relative to the passive portfolio.

Variant: Calculate Active Weights and Security Contributions

Abstract: Active weight is the manager's deviation from the benchmark. Multiply that bet by the asset return; overweights in winners and underweights in losers create value.

A country portfolio has benchmark/managed weights of UK 17%/16%, Japan 25%/14%, France 11%/8%, Germany 9%/24%, and Other 38%/38%. Returns are −7.6%, −9.0%, −3.5%, −15.8%, and −0.1%. Find the largest bets and total value added.

1. Subtract benchmark weights from managed weights

Germany is the largest overweight at \(+15\%\); Japan is the largest underweight at \(-11\%\).

2. Add each bet's contribution

The Japan underweight helped, but the much larger German overweight got crushed.

Variant: Separate Active Return from Beta-Adjusted Alpha

Abstract: Active return merely subtracts the benchmark; alpha also adjusts for benchmark sensitivity. They match only when portfolio beta equals one.

A portfolio returns 8.6%, its benchmark returns 7.4%, and portfolio beta is 1.10. Calculate active return and the module's simplified beta-adjusted alpha.

1. Calculate the plain benchmark gap

2. Charge the portfolio for its extra benchmark exposure

The portfolio beat the benchmark by 1.20%, but 0.74 percentage point of that gap is associated with beta above one.

Variant: Decompose Value Added into Allocation and Selection

Abstract: Allocation asks whether the manager put extra weight in the right asset classes. Selection asks whether the chosen funds beat each class benchmark.

Actual weights are 68% equities and 32% bonds versus policy weights of 60% and 40%. Equity fund/benchmark returns are −5.6%/−4.5%; bond fund/benchmark returns are −0.3%/0.0%. Decompose value added using the curriculum convention.

1. Find active class weights and within-class active returns

2. Allocation contribution uses benchmark returns

3. Selection contribution uses actual weights

Variant: Solve Backward for a Missing Attribution Component

Abstract: Attribution pieces must reconcile to total active return. If two pieces are known, the missing piece is simply the plug that closes the accounting loop.

A portfolio added 1.40% in total. Asset allocation contributed 0.60%. Find security-selection contribution and verify the total.

1. Use the attribution identity

2. Close the loop

This is not a new performance source. It is just Equation 4 run backward.

Variant: Calculate Active Share and Diagnose Closet Indexing

Abstract: Active share measures how different the holdings are, not whether the differences are intelligent. Tiny active share plus tiny tracking risk is the closet-indexing smell test.

Benchmark weights are 40%, 30%, 20%, and 10%; portfolio weights are 42%, 28%, 21%, and 9%. Calculate active share and interpret it.

1. Find absolute active weights

2. Divide the double-counted difference by two

Every overweight must be funded by an underweight, so the raw absolute sum counts the reshuffling twice.

Variant: Calculate and Annualize a Sharpe Ratio

Abstract: Sharpe pays you for excess return over cash per unit of total volatility. Keep return and volatility on the same time scale before dividing.

A portfolio has average monthly return 0.60%, monthly risk-free return 0.10%, and monthly return standard deviation 2.00%. Calculate its annualized ex post Sharpe ratio using the curriculum convention.

1. Annualize average excess return arithmetically

2. Annualize volatility with square-root time

3. Divide reward by total risk

Variant: Calculate an Ex Post Information Ratio

Abstract: Information ratio is benchmark-relative reward per unit of benchmark-relative risk. Use active return in the numerator and tracking risk in the denominator.

A manager averages 0.15% active return per month, and the monthly standard deviation of active returns is 0.90%. Calculate the annualized information ratio.

1. Annualize both active ingredients

2. Calculate consistency of value added

This says the manager generated about 0.58 unit of average active return per unit of active risk.

Variant: Add Cash to Match a Target Volatility

Abstract: Cash shrinks a risky portfolio's return above cash and its volatility by the same proportion, so its Sharpe ratio survives the resize.

Small caps have expected return 10.3%, volatility 19.2%, and Sharpe ratio 0.42 when the risk-free rate is 2.3%. Large caps have volatility 14.6% and expected return 8.2%. Mix small caps with cash to match 14.6% risk and calculate the combined return and Sharpe ratio.

1. Find how much risky portfolio fits inside the risk budget

Cash weight is \(23.96\%\).

2. Blend the returns

3. Confirm Sharpe survives

At equal risk, small caps plus cash still beat the 8.2% large-cap forecast.

Variant: Blend an Active Fund with Its Benchmark

Abstract: Adding the benchmark turns down both active return and active risk by the same knob, so the information ratio stays unchanged.

An active fund has expected active return 2.0% and active risk 5.0%. Invest 60% in the fund and 40% in its benchmark. Find the blend's active return, active risk, and information ratio.

1. Calculate the original information ratio

2. Scale only the active piece

Variant: Distinguish Cash Scaling from Benchmark Scaling

Abstract: Cash is the anchor for Sharpe; the benchmark is the anchor for information ratio. Scaling with the wrong anchor changes the ratio you care about.

A fund returns 10%, its benchmark returns 8%, cash earns 2%, total volatility is 16%, and active risk is 5%. What happens to Sharpe and information ratio if half the fund is replaced by cash rather than by the benchmark?

1. Original ratios

2. Replace half with cash

But the benchmark is still 8%, so the blend now underperforms it by 2%. Its IR is not preserved.

Variant: Diagnose Closet Indexing and a Market-Neutral Fund

Abstract: A closet index can mimic benchmark Sharpe while producing almost no active efficiency. A market-neutral fund using cash as benchmark makes IR and Sharpe the same calculation.

Fund C earns −0.10% active return with 0.50% active risk and otherwise hugs its benchmark. Fund M is market neutral, earns 4% above cash, and has 8% total risk. Calculate the useful ratios and interpret both funds.

1. Closet-index information ratio

Low active risk does not rescue negative active return. A benchmark-like Sharpe ratio would merely confirm that the fund behaves like the index.

2. Market-neutral ratio

Cash is the benchmark, so excess return equals active return and total risk equals active risk:

Variant: Select the Better Active Manager Using Information Ratio

Abstract: The manager with the highest expected IR can create the best combined portfolio. Stand-alone return, volatility, or Sharpe can point at the wrong winner.

Fund I has expected active return −1.4% and active risk 5.1%. Fund II has expected active return 1.2% and active risk 6.2%. Both use the same benchmark. Which fund should be combined with that benchmark?

1. Calculate both information ratios

2. Choose efficiency, not raw risk

Fund II produces positive expected value added per unit of active risk. Fund I does not.

Variant: Calculate the Maximum Combined Sharpe Ratio

Abstract: An efficient active strategy adds a perpendicular source of reward. Square benchmark Sharpe and IR, add them, then take the square root.

A benchmark has expected Sharpe ratio 0.53. The chosen active manager has expected information ratio 0.20. Find the highest Sharpe ratio available from an optimal combination.

1. Use the optimal-combination identity

The improvement is real but modest: \(0.57-0.53\approx0.04\).

Variant: Find Optimal Active Risk and Benchmark Weight

Abstract: The benchmark sets the scale; IR tells us how hard to lean into active management. Then compare desired active risk with the fund's built-in active risk.

A benchmark has \(SR_B=0.53\) and volatility 14.4%. An active fund has \(IR=0.20\) and active risk 6.2%. Find optimal active risk and the weights in the active fund and benchmark.

1. Find the Sharpe-maximizing tracking risk

2. Resize the active fund

The fund brings too much tracking risk on its own, so the benchmark dilutes it.

Variant: Recognize a Leveraged Active-Fund Allocation

Abstract: If optimal active risk exceeds what the fund supplies, the active-fund weight goes above 100% and the benchmark weight turns negative.

Indigo has \(IR=0.15\) and active risk 8.0%. Its benchmark has \(SR_B=0.333\) and volatility 18.0%. Find optimal active risk and both portfolio weights.

1. Calculate target active risk

2. Scale the fund

So invest 101.4% in Indigo and finance the extra 1.4% by shorting the benchmark.

Variant: Calculate Simple and Beta-Adjusted Active Security Return

Abstract: Simple active return subtracts the benchmark one-for-one. Residual return first scales the benchmark by the security's beta.

A security returns 12%, the benchmark returns 8%, and the security beta is 1.25. Calculate its simple active return and its single-factor beta-adjusted residual return.

1. Simple benchmark-relative return

2. Beta-adjusted version

The simple version calls all 4% “active.” The risk-model version recognizes that a high-beta security was expected to move more with the benchmark.

Variant: Scale Scores into Expected Active Returns

Abstract: Grinold scaling turns a direction score into a return forecast: skill times active volatility times score. Riskier names receive larger raw forecasts for the same score.

Four securities have scores \(+1,+1,-1,-1\), active volatilities 25%, 50%, 25%, 50%, and expected \(IC=0.20\). Calculate each expected active return.

1. Apply the scaling rule

Variant: Decide When Breadth Equals the Number of Bets

Abstract: Breadth counts independent decisions, not rows in a spreadsheet. Repeated or correlated bets are duplicates wearing different name tags.

A manager follows four securities and refreshes forecasts quarterly. What is breadth if all 16 security-period decisions are independent? What if quarterly forecasts merely repeat the annual view?

1. Fully independent case

2. Repeated-signal case

If each quarter repeats the same annual decision, time adds no independent information:

Variant: Calculate Unconstrained Optimal Active Weights

Abstract: The optimizer rewards forecast per unit of variance, then scales all bets to the active-risk budget. A bigger forecast helps; volatility gets squared and hits harder.

Four uncorrelated securities have expected active returns of 5%, 10%, −5%, and −10%; active volatilities of 25%, 50%, 25%, and 50%; \(IC=0.20\), \(BR=4\), and target active risk 9%. Calculate optimal active weights.

1. Use the curriculum weight formula

The common scaling piece is:

2. Size each position

By symmetry:

Variant: Convert Active Weights and Returns into Total Portfolio Inputs

Abstract: Active quantities are deviations. Add them back to the benchmark quantities to recover the actual portfolio the client owns.

An equally weighted four-security benchmark has 25% in each security and expected return 10%. Active weights are 18%, 9%, −18%, and −9%; expected active returns are 5%, 10%, −5%, and −10%. Find total weights and total return forecasts.

1. Rebuild actual weights

2. Rebuild total security returns

Variant: Calculate Portfolio Active Return and Active Risk Directly

Abstract: Expected active return adds weight times forecast. With uncorrelated active returns, active variance adds squared weight times squared volatility.

Use total weights 43%, 34%, 7%, and 16%; total return forecasts 15%, 20%, 5%, and 0%; active weights 18%, 9%, −18%, and −9%; and active volatilities 25%, 50%, 25%, and 50%. Find portfolio return, active return, and active risk.

1. Find the managed return

Against a 10% benchmark:

2. Add independent variance contributions

Variant: Verify the Basic Fundamental Law

Abstract: Skill, independent opportunities, and aggressiveness multiply into expected value added. Breadth helps through a square root, not one-for-one.

An unconstrained strategy has \(IC=0.20\), \(BR=4\), and active risk 9%. Calculate its information ratio and expected active return; compare with a direct active return of 3.6%.

1. Calculate active efficiency

2. Attach the chosen risk budget

The law matches the direct portfolio calculation, which is the required checksum.

Variant: Solve the Fundamental Law Backward for Breadth

Abstract: When IR and IC are given, strip out implementation loss, divide by skill, and square. The answer is effective independent bets, not automatically the number of holdings.

A manager has \(IR=0.75\), \(IC=0.1819\), and \(TC=1.0\). Assuming independent security decisions, solve for breadth.

1. Rearrange the full law

2. Substitute

Variant: Calculate the Risk-Weighted Information Coefficient

Abstract: IC checks whether forecasts lined up with outcomes after both are divided by active risk. It measures forecasting skill, not portfolio construction.

Three managers forecast four securities. Their risk-weighted forecast vectors are M1 \((0.176,0.400,0.417,0.240)\), M2 \((0.235,0.100,0.000,0.080)\), and M3 \((0.147,0.150,0.042,0.060)\). Realized risk-weighted active returns are \((0.353,0.700,0.333,0.080)\). Which manager has the highest IC?

1. Correlate each forecast vector with realized outcomes

The cross-sectional correlations are:

2. Select the strongest forecaster

Variant: Calculate the Risk-Weighted Transfer Coefficient

Abstract: TC checks whether forecasts made it into positions after risk adjustment. It measures implementation efficiency, not whether the forecasts came true.

For the same managers, risk-weighted forecast vectors are M1 \((0.1765,0.4000,0.4167,0.2400)\), M2 \((0.2353,0.1000,0,0.0800)\), and M3 \((0.1471,0.1500,0.0417,0.0600)\). Risk-adjusted weight vectors are M1 \((-0.0213,0.0025,0.0090,0.0063)\), M2 \((0.0340,0,-0.0120,-0.0250)\), and M3 \((-0.0085,0.0050,0.0060,-0.0125)\). Which manager has the highest TC?

1. Match forecasts with implemented weights

The correlations are:

2. Select the cleanest implementation

Variant: Verify the Expanded Fundamental Law under Constraints

Abstract: Constraints bend actual weights away from ideal weights. TC measures that damage and scales down the active return the forecasts could otherwise produce.

Expected active returns are 5%, 10%, −5%, and −10%; actual active weights are 6%, 4%, 7%, and −17%; active volatilities are 25%, 50%, 25%, and 50%. Given \(TC=0.58\), \(IC=0.20\), and \(BR=4\), verify active return and risk directly and through the expanded law.

1. Direct active return

2. Direct active risk

3. Fundamental-law check

Variant: Find Constrained Optimal Risk and Maximum Sharpe

Abstract: A constraint penalty lowers both the best active-risk target and the attainable Sharpe improvement. Apply TC before deciding how aggressively to invest.

A constrained strategy has \(TC=0.50\) and unconstrained \(IR^*=0.30\). Its benchmark has \(SR_B=0.40\) and volatility 16%. The fund itself carries 8% active risk. Find constrained optimal active risk, maximum Sharpe, and fund/benchmark weights.

1. Shrink optimal active risk by TC

2. Calculate the constrained maximum Sharpe

3. Dilute the fund to the target

Variant: Decompose Ex Post Performance into Skill and Noise

Abstract: Replace expected IC with realized IC to measure what forecasting delivered that period; whatever remains in actual active return is noise.

A strategy has \(BR=100\), expected \(IC=0.05\), \(TC=0.80\), and active risk 4%. Realized \(IC_R=-0.10\) and actual active return is −2.6%. Find ex ante return, conditional ex post return, noise, and variance shares.

1. Ex ante expectation

2. Condition on what forecasting actually did

3. Back out noise

4. Split performance variance

Noise softened a bad forecasting period; it did not turn bad forecasting into skill.

Variant: Compare Security Selection with Sector Selection

Abstract: More skill can lose to more breadth. Compare strategies through \(IC\sqrt{BR}\), then attach the same risk budget.

A stock selector has \(IC=0.05\) across 100 independent securities. A sector selector has \(IC=0.15\) across nine independent sectors. Both are unconstrained and target 3% active risk. Compare IR and expected active return.

1. Stock selection

2. Sector selection

The sector manager is three times as accurate, but the stock manager has enough extra independent bets to win narrowly.

Variant: Build a Quarterly Credit-Timing Strategy

Abstract: First measure the risk of IG minus HY, then annualize it, convert hit rate into IC, and scale the active tilt to the permitted tracking risk.

Quarterly IG and HY volatilities are 2.84% and 4.64%, with correlation 0.575. A manager makes four independent annual credit calls, is correct 55% of the time, targets 2% annual active risk, and starts from 70% IG/30% HY. Calculate differential risk, IC, active tilt, allocations, IR, and expected active return.

1. Risk of the return difference

2. Turn hit rate into skill

3. Size the tilt

Pro-credit allocation: IG \(70-26.3=43.7\%\), HY \(30+26.3=56.3\%\). Defensive allocation: IG 96.3%, HY 3.7%.

4. Apply the law

Variant: Test Whether More Rebalancing Really Raises Breadth

Abstract: A faster calendar raises breadth only if it produces new information. Repeating the same signal twelve times is still one decision wearing twelve timestamps.

A four-asset strategy has annual breadth 3.2 because some active returns are correlated. What is the maximum breadth under monthly independent rebalancing? What happens if monthly signals merely repeat the annual forecast?

1. Truly independent monthly decisions

2. Persistent signals

If all twelve months repeat the same view:

Higher turnover can also lower TC, so even genuinely higher breadth does not guarantee a proportional IR improvement.

Variant: Measure How Constraints Damage a Global Equity Strategy

Abstract: Holding IC and breadth fixed isolates the effect of constraints: lower TC means less forecast reaches the portfolio and IR falls.

A global strategy has \(IC=0.099\) and \(BR=24.5\). Compare: (A) \(TC=0.995\), active risk 2%; (B) long-only and weight limits with \(TC=0.694\), risk 2%; and (C) tighter binding at \(TC=0.567\), risk 2.74%. Find IR and expected active return.

1. Nearly unconstrained

2. Constrained at the same risk

3. More aggressive, but more constrained

More active risk raised return slightly from B to C, but efficiency fell because constraints bit harder.

Variant: Calculate Effective Breadth with Correlated Bets

Abstract: Common positive correlation turns many names into fewer independent bets. Strong negative correlation can make effective breadth exceed the number of securities.

Use \(BR=N/[1+(N-1)\rho]\). First take \(N=80\) and common correlation 0.04. Then take \(N=2\) and correlation −0.80.

1. Duplicated positive-correlation bets

Eighty holdings deliver only about 19 independent bets.

2. Near-arbitrage negative correlation

The two positions offset risk so strongly that effective breadth exceeds two.

Variant: Adjust the Fundamental Law for Uncertain Skill

Abstract: If realized IC jumps around, forecasting uncertainty creates strategy risk. The useful skill measure becomes IC divided by the volatility of IC.

A strategy covers 100 independent securities. The standard deviation of realized IC is 0.04, risk-model tracking risk is 1.5%, and expected IC is 0.05. Assume \(TC=1\). Calculate realized active risk and expected active return using Equations 17 and 18.

1. Calculate strategy risk

2. Reward skill relative to its uncertainty

Variant: Debug an Absurdly High Information Ratio

Abstract: Huge breadth can manufacture fantasy performance when correlated holdings and repeated signals are counted as independent. Debug the assumptions before admiring the answer.

A monthly S&P 500 stock selector claims \(IC=0.05\) and sets \(BR=12\times500=6{,}000\). At 3% active risk, calculate the claimed IR and return, then give the main reasons they are likely overstated.

1. Reproduce the tempting calculation

2. Find what broke

• Stocks sharing sector and factor exposures are not 500 independent bets.
• A stock forecast often persists month to month, so 12 observations are not 12 new decisions.
• The ex ante IC may be biased upward or unstable through time.
• Long-only, turnover, position, and risk limits reduce TC.

Variant: Compare Managers When Skill, Breadth, and Constraints All Differ

Abstract: A manager can win with fewer bets if those bets are more accurate and reach the portfolio more cleanly. Put every input into the same formula before choosing.

Manager 1 has \(IC=0.15\), \(TC=1.0\), and 50 independent security decisions. Manager 2 has \(IC=0.10\), \(TC=0.80\), and 100 independent decisions. Both target 5% active risk. Which manager has the higher expected active return?

1. Run Manager 1 through the full law

2. Run Manager 2 through the same law

Manager 2 has twice as many bets, but breadth only helps through a square root. Manager 1's stronger skill and perfect transfer more than make up for the smaller opportunity set.

Variant: Evaluate Offsetting Changes in IC and TC

Abstract: Strategy changes can push different parts of the law in opposite directions. Recalculate the whole product instead of declaring one change dominant by inspection.

A strategy begins with \(IC=0.10\), \(TC=0.50\), \(BR=100\), and active risk 3%. Research changes lower IC to 0.08, while relaxed constraints raise TC to 0.75; breadth and risk stay fixed. Compare expected active return before and after.

1. Original strategy

2. Revised strategy

The loss of forecasting skill is more than offset by better implementation.