CFA Level III · Level III Core

Overview of Asset Allocation: formula sheet

Full chapter guide

Key formulas

Economic net worth: Economic net worth = Total economic assets − Total economic liabilities; Total assets = financial capital + human capital + other non-traded assets such as the present value of pensions.
Human capital: Human capital = PV of expected future labour income (after-tax); Discount at a risk-adjusted rate that reflects the riskiness of the income. Bond-like, stable income uses a lower rate; equity-like, volatile income a higher rate.
Total wealth composition: Total wealth = Financial capital + Human capital; Use this to judge the overall mix of bond-like and equity-like exposure, not only the investment portfolio.
Core allocation rule: Bond-like human capital → can hold more equity; equity-like human capital → hold less equity; A guide to direction, not a fixed number. It depends on correlation with markets and on the client's other constraints.
Surplus: Surplus = Asset value − PV of liabilities; Liability-relative risk is the risk that this falls. Discount liabilities at a rate suited to their nature.
Funded ratio: Funded ratio = Assets ÷ PV of liabilities; Above 1 means overfunded; below 1 means underfunded. It does not tell you the risk by itself.
Surplus return: Surplus return ≈ (Change in surplus) ÷ Assets at start; Measure surplus change against beginning assets when comparing surplus performance.
Surplus variance: σ²(S) = σ²(R_A) + (L/A)² × σ²(R_L) − 2 × (L/A) × Cov(R_A, R_L); R_A and R_L are the returns on assets and on liabilities, and L/A is the ratio of liabilities to assets at the start. Surplus return is measured over beginning assets, so both changes share the same base. For given volatilities, higher correlation between assets and liabilities lowers surplus risk. Use this to explain why hedging assets cut risk.
Mean-variance utility (asset-only): U = E(Rp) − 0.5 × λ × σp²; λ is risk aversion. Liability-relative versions replace portfolio return and variance with surplus return and variance.
Portfolio expected return: E(Rp) = Σ wi × E(Ri); Weights sum to 1 in a fully invested, long-only portfolio.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Lower correlation ρ lowers portfolio risk. Standard deviation is the square root.
Covariance and correlation: Cov(1,2) = ρ12 × σ1 × σ2; Use this to turn a correlation into the covariance the formula needs.
Reverse optimization (implied returns): Implied excess return vector = λ × Σ × w(market); λ is the risk aversion coefficient, Σ the covariance matrix, w the market weights. Know the logic: higher risk and covariance with the market implies higher required return. Calculation is rarely required.
Black-Litterman logic: Posterior returns = equilibrium returns tilted toward views, weighted by confidence; No calculation expected. Know the inputs and the result: stable, diversified weights that reflect views.
Marginal contribution to risk (MCTR): MCTR_i = (Cov(R_i, R_p)) ÷ σ_p = β_i,p × σ_p, where β_i,p = Cov(R_i, R_p) ÷ σ_p²; Change in portfolio risk from a small increase in asset i. β_i,p is asset i's beta to the portfolio, defined as Cov(R_i, R_p) ÷ σ_p².
Absolute contribution to risk: CTR_i = w_i × MCTR_i = w_i × Cov(R_i, R_p) ÷ σ_p; The CTR values of all assets add up to portfolio standard deviation σ_p.
Percentage contribution to risk: %CTR_i = CTR_i ÷ σ_p = w_i × Cov(R_i, R_p) ÷ σ_p²; These sum to 100%. A risk budget is stated in this form.
Risk-budget optimality (reward per unit of risk): (E(R_i) − R_F) ÷ MCTR_i is equal for all assets i, which is equivalent to (E(R_i) − R_F) ÷ β_i,p being equal across assets; At the optimum, no shift of risk between assets improves return per unit of risk. The two forms are equivalent because MCTR_i = β_i,p × σ_p and σ_p is the same for every asset. Unconstrained case.
Risk parity condition: w_i × MCTR_i = w_j × MCTR_j for all i, j; Equal risk contribution. If assets are uncorrelated, weights are proportional to 1/σ_i.
1/N weight: w_i = 1 ÷ N; Equal capital weight for each of N assets.
Sharpe ratio: SR = (E(R_p) − R_F) ÷ σ_p; Used to compare risk-budgeted portfolios.
Drift in weight: Drift = Actual weight − Target weight; Compare the drift with the permitted range or corridor to decide whether to trade.
Current weight: Weight of asset = Market value of asset ÷ Total portfolio value; Recompute total portfolio value after any market move before comparing weights.
Calendar rebalancing: Rebalance at fixed dates (e.g. quarterly), whatever the drift; Simple and easy to monitor. It may trade when it is not needed, or miss big moves between dates.
Percentage-range (corridor) rebalancing: Rebalance when weight < Target − band or weight > Target + band; Needs continuous monitoring. It trades only when drift is material.
Calendar-and-percentage-range: Check at set dates; trade only if a weight is outside its range; Combines low monitoring effort with trading only when needed.
Rebalance back to: Target weight, or the nearest edge of the range; Rebalancing to the target costs more trading but restores risk fully. Rebalancing to the edge trades less.
Corridor width rules of thumb: Wider corridor: higher transaction costs, higher risk tolerance, low correlation among assets, low volatility of assets. Narrower corridor: low costs, low risk tolerance, high correlation among assets, high volatility.; These are directional guides, not fixed numbers. In the CFA curriculum, high correlation among assets supports a narrower corridor and low correlation supports a wider one. Volatility works the same way: higher volatility supports a narrower corridor. Always link each factor to the client facts, and say so when factors conflict.

Quick revision

The economic balance sheet includes human capital and other non-traded assets, not just the investment portfolio.
Asset-only approaches focus on the asset side; liability-relative approaches focus on funding liabilities.
Liability-relative approaches suit clients with defined obligations to meet.
Mean-variance optimization needs expected returns, volatilities and correlations as inputs.
Mean-variance outputs are very sensitive to small changes in the inputs.
Unconstrained mean-variance optimization can produce concentrated, unintuitive portfolios.
Risk budgeting allocates risk, not just capital, across parts of the portfolio.
Every allocation approach must be matched to the client's objectives and constraints.
Rebalancing returns the portfolio to target weights and has costs as well as benefits.
Wider rebalancing ranges mean fewer trades but more drift from the target.
Follow the command word: state, justify or calculate each ask for something different.
There is no penalty for wrong answers, so always attempt every item.

Common mistakes

Treating human capital as part of the investable portfolio. Fix: Human capital cannot be traded. It shapes the choice of the financial portfolio but you do not allocate to it.
Assuming a young client should always hold the most equity. Fix: Check whether income is stable or correlated with markets. Equity-like human capital reduces the case for equity.
Treating asset volatility as the risk measure for a pension plan. Fix: For liability-relative investors, define risk as surplus risk or shortfall versus the liabilities.
Assuming a high funded ratio means low risk. Fix: Funded ratio is a level, not a risk. A plan with a high ratio but unhedged liabilities can still lose surplus quickly.
Saying MVO is unreliable because it assumes returns are certain. Fix: State that MVO treats the input estimates as if they were known exactly, so estimation error flows straight into the weights.
Claiming resampling creates better inputs. Fix: Resampling does not improve the estimates. It averages the optimal weights across simulated inputs, which gives more diversified and stable portfolios.
Treating risk budgeting as a capital allocation Fix: Compute risk contribution with weight × marginal contribution. A volatile asset's share of risk is usually larger than its share of capital.
Saying risk parity equals equal weights Fix: 1/N equalizes capital. Risk parity equalizes risk contribution, so low-volatility assets get larger weights.
Calling any deviation from policy tactical asset allocation. Fix: TAA is a deliberate active view. Drift is passive and caused by market moves. Say which one the question describes.
Saying rebalancing is done mainly to raise returns. Fix: State that the main aim is to keep risk in line with the policy. Any return effect depends on market behaviour.

Exam tips

When the command word is justify, name the specific balance-sheet feature (bond-like human capital, fixed liabilities) in your reason. That is where the points are.
Write the calculation steps for a present value of human capital. A correct number alone may earn full credit in constructed response, but steps protect you if a rate is slightly off.
Answer only the number of responses asked. If the question asks for two reasons, give exactly two.
In item sets, look for clues about job stability, industry and correlation with markets. They signal whether human capital is bond-like or equity-like.
Always tie the conclusion to ability to take risk, not only willingness.
Read the first lines of the vignette for the investor type. It usually decides the approach.
When a question says justify, give approach plus one client fact, such as fixed liabilities or flexible spending.
In calculations, show surplus and funded ratio separately. A correct number on its own earns credit, but show steps in case of a slip.