CFA Level I · CFA Level I Exam

Basics of Portfolio Planning and Construction: formula sheet

Full chapter guide

Key formulas

Portfolio expected return: E(Rp) = Σ wi × E(Ri); Weighted average of asset expected returns. Weights sum to 1. Diversification does not change this calculation.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Portfolio standard deviation is √σp². The correlation term drives the diversification benefit.
Covariance and correlation: Cov(1,2) = ρ12 × σ1 × σ2; Correlation ranges from -1 to +1.
Perfect positive correlation case: σp = w1σ1 + w2σ2 when ρ12 = +1; No diversification benefit. For ρ < +1, σp is less than the weighted average of the standard deviations.
Risk decomposition: Total risk = Systematic risk + Nonsystematic risk; Diversification reduces nonsystematic risk only. Systematic risk remains.
Components of an IPS: IPS = Objectives (return, risk) + Constraints (liquidity, time horizon, tax, legal/regulatory, unique circumstances) + Guidelines; Objectives say what the client wants. Constraints say what limits the portfolio.
Risk tolerance rule: Overall risk tolerance = lower of (ability to take risk, willingness to take risk); Use this as the standard approach when the two conflict. It is a conservative convention, not a mathematical law.
Return objective types: Absolute return (a stated % or amount) vs Relative return (a stated margin over a benchmark); Required return is what is needed to meet goals. Desired return is what the client would like in addition.
Constraint memory aid: LTTLU = Liquidity, Time horizon, Tax, Legal/regulatory, Unique circumstances; Check each fact in a question against these five labels.
Overall risk tolerance: Overall risk tolerance = the lower of (ability, willingness); If ability is low and willingness is high, the client's tolerance is low. Use the more conservative of the two.
Required nominal return (approximate): Required return ≈ real spending rate + inflation (+ fees and taxes if stated); Exact form: (1 + real) × (1 + inflation) − 1. Use the approximation only if the question allows it.
Tracking error (relative risk): Tracking error = standard deviation of (portfolio return − benchmark return); This is the usual measure of a relative risk objective.
Active return (relative return): Active return = portfolio return − benchmark return; A relative return objective is stated as a margin above the benchmark.
Constraint checklist (TTLLU, the last five letters of RRTTLLU): Time horizon, Tax concerns, Liquidity, Legal and regulatory, Unique circumstances; Constraints limit choices. Return and risk are objectives, not constraints.
Horizon and risk ability: Longer horizon → higher ability to take risk; shorter horizon → lower ability; This is a general tendency, not a fixed rule. Willingness to take risk is a separate matter.
Liquidity and asset choice: Larger or nearer cash needs → hold more cash and liquid assets; fewer illiquid assets; Applies to both known and unexpected cash needs.
After-tax return: After-tax return = pre-tax return × (1 − tax rate); Applies when the whole return is taxed at one rate. Different tax rates on income and gains need separate treatment.
Active weight: Active weight = Actual weight − Strategic (target) weight; Positive means overweight versus the strategic target. Tactical allocation creates active weights. Active weights across asset classes sum to zero when both the actual and strategic portfolios are fully invested (weights sum to 100%).
Portfolio expected return: E(Rp) = Σ wᵢ × E(Rᵢ); Use the target weights and long-term expected returns to test whether an allocation meets the return objective.
Two-asset portfolio variance: σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ₁,₂ σ₁ σ₂; Use it to check that the allocation fits the risk objective. Lower correlation lowers portfolio risk.
Asset class criteria: Homogeneous within, different between, diversifying, mutually exclusive, investable; Remember this checklist for selecting asset classes.
Active return: Active return = Portfolio return − Benchmark return; Positive means the portfolio beat the benchmark. Tracking error is the standard deviation of active returns.
Core-satellite portfolio return: Rp = wcore × Rcore + wsat × Rsat; Weights sum to 1. Use it as the usual weighted average.
Top-down vs bottom-up order: Top-down: macro → asset class/sector → security. Bottom-up: security → portfolio; The order of the decisions is the distinguishing feature.
Active vs passive trade-off: Active: higher fees, higher active risk, potential alpha. Passive: low fees, low tracking error, no alpha; Active return after fees must beat the benchmark for active to add value.

Quick revision

A portfolio is judged by its total risk and return, not asset by asset.
Diversification works because asset returns are not perfectly correlated.
The IPS is a written plan that records objectives and constraints and guides decisions.
Return and risk objectives are the goals; constraints are the limits.
Ability to take risk is about financial capacity; willingness is about attitude.
When ability and willingness differ, the more conservative one usually governs.
Constraints are liquidity, time horizon, tax, legal and regulatory, and unique circumstances.
A longer time horizon generally allows more illiquid and higher-risk assets.
Strategic asset allocation sets long-run target weights based on the IPS.
Tactical changes are short-term deviations from the strategic weights.
Top-down starts with macro views; bottom-up starts with security selection.
ESG can be applied by exclusion, integration, thematic investing or engagement.

Common mistakes

Averaging the standard deviations to get portfolio risk Fix: Use the variance formula with the correlation term. The weighted average of standard deviations holds only when ρ = +1.
Saying diversification eliminates all risk Fix: Diversification removes nonsystematic risk only. Systematic risk stays, and the correlation is rarely -1.
Treating time horizon and liquidity as the same constraint. Fix: Liquidity is the need for cash, often at short notice or in amounts. Time horizon is the length of the investment period. A long horizon can still carry a large near-term cash need.
Using willingness to take risk when it is higher than ability. Fix: Take the lower of the two. A bold client with a small cushion and near-term needs should have a lower risk objective.
Averaging ability and willingness Fix: Use the more conservative of the two. The IPS risk tolerance is the lower one.
Treating a young age or long horizon as willingness Fix: Horizon, income and wealth are ability factors. Willingness is about attitude and behaviour.
Treating liquidity and time horizon as the same thing. Fix: Horizon is when the main money is needed. Liquidity is the cash needed along the way, and its size and timing.
Listing risk tolerance or required return as a constraint. Fix: Return and risk are objectives. Only time horizon, tax, liquidity, legal and regulatory, and unique circumstances are constraints.
Treating tactical asset allocation as the same as strategic allocation. Fix: SAA is the long-term neutral mix from the IPS and long-run expectations. TAA is a short-term deviation from it based on market views.
Choosing weights from expected return alone and ignoring risk tolerance. Fix: The allocation must meet the return objective within the risk tolerance. When ability and willingness to take risk differ, use the lower.

Exam tips

Expect conceptual items on what diversification can and cannot do. Wrong options usually say it removes all risk or guarantees returns.
For numerical items, compute the weighted-average bound first. It often eliminates one or two options quickly.
Remember the three correlation cases: +1 gives no benefit, 0 gives some benefit, -1 allows zero risk with the right weights.
Read whether the question asks for variance or standard deviation, and whether returns are given in percent or decimals.
Link this topic to systematic and nonsystematic risk questions, which often test the same idea from another angle.
Expect scenario questions that ask you to classify a fact. Practise labelling facts quickly with the five constraints.
Remember the ability versus willingness rule. It is a favourite test point, and the lower of the two is the standard answer.
Watch for time horizon wording such as multi-stage horizons, where one client has near-term and long-term needs.