Comparing Portfolio Decisions: A Capstone

Question: how do different portfolio objectives and input assumptions change capital allocation, modeled risk, and the trading required to maintain a portfolio?

Extend the historical three-industry illustration in Chapter 2.3, using the risk attribution and cost accounting from Chapter 2.4. Work in a separate notebook or adapt the editable cells in Chapter 2.3. Include the required imports and data loading so your analysis can run from a fresh session.

1. Specify the Decision Before Evaluating It

Use the bundled value-weighted industry returns for Food, Steel, and Finance (Food, Steel, Fin). Estimate inputs from January 2000–December 2009, then evaluate over January 2010–December 2018. Keep all portfolios long-only, fully invested, and unlevered.

For the base case, use the chapter's 50% constant-correlation shrinkage covariance, annualized by multiplying the monthly estimate by 12. Use its end-2009 capitalization reference normalized within the three industries, risk aversion $\delta =2.5$, and $\tau =0.05$.

Retain the two hypothetical views: Food exceeds Steel by 2 percentage points, and Finance has a 6% expected excess return, both on the annualized arithmetic basis. Use the chapter's diagonal view-error covariance $\mathrm{\Omega }=\mathrm{diag}(P(\tau \mathrm{\Sigma })P^T)$. For allocation, use the original conditional return covariance $\mathrm{\Sigma }$ consistently; keep the predictive covariance returned by Black–Litterman separately labeled.

Record these choices before computing evaluation-period performance. This is a retrospective teaching comparison: the selected industries and views do not establish a forecasting process available in 2009.

2. Compare Objectives, Capital Weights, and Risk Shares

Construct five allocations using the same training data:

Allocation	Construction
Equal weight	One-third in each industry
Reference	End-2009 capitalization fractions within the selected universe
GMV	Minimize variance under the base covariance and constraints
ERC	Target one-third of modeled risk from each industry
Black–Litterman	Maximize mean–variance utility using posterior excess means and the base covariance

Reuse weight_minvar(), risk_parity_weigths(), black_litterman(), and bl_portfolio_weights() from the lessons. Common covariance scaling leaves GMV and ERC weights unchanged; Black–Litterman inputs must share the stated annualized basis.

Report capital weights, modeled annual volatility, effective holdings, and relative Euler risk shares under the same base covariance. Explain why equal capital need not mean equal risk, and why GMV and ERC solve different problems. Relate any concentrated allocation to the exposure-versus-holdings distinction in Chapter 2.1.

3. Separate Input Sensitivity from Subsequent Performance

Before examining the evaluation returns, perform two sensitivity checks:

• Replace the shrinkage covariance with the sample covariance. Recompute GMV, ERC, and Black–Litterman. Keep the reference, views, $\delta$, and $\tau$ fixed; rebuild the implied prior and default $\mathrm{\Omega }$ from the new covariance. Explain that this changes both the risk estimate and the Black–Litterman information model.
• Return to the base covariance and halve the base $\mathrm{\Omega }$, holding all other inputs fixed. Report how the posterior means and Black–Litterman weights respond to more precise views.

For each covariance choice, verify ERC's budgets under the matrix used to construct it. Also assess both ERC allocations under the base covariance: equal shares under one estimate need not remain equal under another. Keep these sensitivity results separate from the base-case performance comparison below.

4. Evaluate Frozen Targets with Rebalancing Costs

Freeze the five base-case target allocations and rebalance monthly throughout the 108 evaluation months. As in Chapter 2.3, fixed targets mean monthly rebalancing, not buy-and-hold quantities.

Use Chapter 2.4's drifted holdings to calculate one-way turnover. Assume each account starts invested in its first target, exclude initial funding costs equally, and charge 10 basis points per dollar bought or sold on later rebalances. Solve the self-financing equation for remaining wealth before applying each month's returns.

Report gross and net CAGR, annualized realized volatility, maximum drawdown as a positive loss magnitude, and mean monthly one-way turnover. Label gross and net statistics explicitly. Plot gross and net wealth from 100 at the end of December 2009, using the same dates and strategy names throughout.

Deliverables and Interpretation

Submit a reproducible calculation, an allocation/risk table, a performance/cost table, gross/net wealth charts, and a short explanation of the sensitivity results. In roughly 300 words, identify the allocation you would choose for a stated objective and explain what evidence could change that choice. Distinguish estimated risk at the decision date from risk realized afterward.

Before interpreting the results, check that weights and full-universe risk shares each sum to one, base-case ERC shares are one-third each, and zero costs recover gross returns to numerical precision. Higher nonnegative costs should not increase terminal wealth for fixed targets and returns. Chart values must agree with the corresponding calculations, including dates, asset order, and percent versus percentage-point units.

The strongest conclusion explains the trade-offs among objectives, input sensitivity, concentration, and implementation. A historical performance ranking alone does not identify a universally best portfolio.

Prepare for the Next Course

The Machine Learning for Asset Management overview introduces the next learning sequence. Begin with its information-timing exercise: explain how you would distinguish fitting an investment model from evaluating a genuinely new prediction.