Optimize a policy without worshipping the optimizer

Learning goals

• Relate optimization, factor and liability approaches to actual constraints.
• Assess parameter uncertainty and rebalancing choices.

Draft · approximately 90 minutes, excluding further revision.

Decision case · fictional facts

A fictional pension investor compares A and B, each with 20% volatility and zero correlation. Expected returns are 7% and 5%. A proposed 50/50 allocation competes with a 100% A allocation. Near-term liabilities require 10% in liquid reserves, while the simple comparison initially ignores that constraint.

Optimization is conditional on estimates and constraints

Mean–variance optimization selects efficient return-risk combinations given estimated means, covariance and feasible weights. Small mean changes can create large weight changes; impose economically motivated constraints, shrink unstable inputs and compare robust alternatives. The market portfolio can provide a reference; factor allocation can reveal concentration hidden by asset labels. Risk budgets allocate contribution to total or active risk, not simply equal capital.

Monte Carlo and scenario tests can estimate the range of outcomes and shortfall probabilities, but inherit distribution, dependence and spending assumptions. Heuristics can be transparent and stable yet ignore liability specifics. Judge any method against the economic balance sheet, liquidity, horizon and client preferences, rather than choosing the most sophisticated mathematical label.

Integrate liabilities, goals and trading costs

Liability matching considers currency, timing, inflation linkage and rate exposure; goals-based allocation separates essential from aspirational spending and assigns appropriate shortfall tolerance. These policies can sacrifice unconstrained expected return to protect a commitment. Liquidity reserves and illiquid commitments require a cash-flow plan, not a static allocation pie chart.

Rebalance using cash flows where possible and compare calendar or tolerance bands against cost, tax and opportunity risk. A wider band reduces turnover but allows more policy drift. Changes in objectives or liabilities may warrant a new strategic policy rather than calling every move tactical rebalancing.

Risk budgets, simulation and factor-aware robust allocation

A global-market starting mix may require lower equity risk if human capital or the sponsor’s business is equity-like. For a100 portfolio with employer-linked capital, adding more domestic cyclical equity can amplify total economic risk even if account volatility looks acceptable. Optimize the economic balance sheet, not simply the traded account. Robust approaches shrink unstable means and compare feasible alternatives rather than hiding estimation sensitivity.

Simulation should jointly generate asset paths, inflation, calls, withdrawals and any liability changes, including adverse dependence and sequence risk. A95% modeled success probability is conditional on assumptions and is not a guarantee. Illiquidity constrains trades during stress. Absolute risk budgets allocate total risk; relative budgets allocate active risk against a benchmark. Equal money allocations are not equal risk contributions.

Factor analysis tests whether nominal diversification still concentrates growth, duration or liquidity risk. Incorporate the client’s loss tolerance, priorities and decision rights into constraints. A low estimated covariance based on stale marks should not justify excessive private commitments; test liquidity under calls arriving while public assets fall and private distributions stop.

Liability and goals allocations with explicit funding

Evaluate liability amount, timing, currency, inflation linkage, optionality and sensitivity. With a fixed100 payment in one year and a risk-free matching instrument yielding4%, reserving100/1.04=96.1538 today matches that nominal payment under the assumptions. A long-duration risky bond with the same current market value does not make an equivalent promise. Surplus optimization and hedging-portfolio/return-seeking-portfolio approaches expose different trade-offs.

Goals-based allocation can first fund essential near-term spending, then allocate remaining wealth to education or aspirational legacy with distinct acceptable shortfalls. This is not permission to ignore aggregate risk across buckets. Heuristics such as60/40 or age-based weights are transparent but may miss actual liabilities. Review rebalancing as returns, liability values, taxes, market impact and available cash change.

Make the cash constraint explicit

Extension case, independent of the earlier two-asset comparison: keep 10% in a one-year reserve with stipulated 2% return and zero volatility. Split the remaining 90% equally between A and B, whose expected returns are 7% and 5%, volatilities 20% each and correlation zero. All weights use beginning-of-period total assets; ignore taxes and trading costs. Calculate the feasible policy’s expected return and volatility, then repeat volatility if A/B correlation rises to 0.6.

Before choosing the policy, explain why an optimizer with no cash constraint answers a different question. The reserve is earmarked for a dated obligation, not a claim that market cash is risk-free in real purchasing-power terms. A higher expected return is not automatically a better policy if it sacrifices the ability to make the payment.

Model reasoning, sensitivity and self-check

Weights are 0.10, 0.45 and 0.45. Expected return = 0.10×2% + 0.45×7% + 0.45×5% = 5.60%. With zero A/B correlation and deterministic cash, variance = 2×0.45²×0.20² = 0.0162 and volatility = 12.7279%. Do not report the 6% return of the fully invested 50/50 risky portfolio as the return of this total policy.

At correlation 0.6, add 2×0.45×0.45×0.20×0.20×0.6 to variance. Total variance becomes 0.02592 and volatility 16.0997%. Expected return is unchanged under the stipulated means; uncertainty about covariance has changed risk. A policy review should also stress reserve timing, inflation and asset-sale liquidity.

Self-check: a complete response includes total-asset weights, both covariance calculations and a recommendation tied to the obligation. Merely naming diversification earns no evidence that the reserve has been financed. Neither the two forecasts nor the covariance stress establishes a probability of meeting the liability.

Formula and assumptions

E[Rp]=wᵀμ; σp²=wᵀΣw; feasible weights must satisfy liquidity and policy constraints

Errors to catch

• Calling an unconstrained efficient portfolio suitable for every investor.
• Reading simulated shortfall as guaranteed frequency.

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Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org