CFA Level III · Level III Core

Principles of Asset Allocation: formula sheet

Full chapter guide

Key formulas

Economic net worth: Economic net worth = PV(economic assets) − PV(economic liabilities); Economic assets include financial assets, human capital and the PV of expected benefits. Liabilities include debts and the PV of future spending or promised benefits.
Funding ratio: Funding ratio = PV of assets ÷ PV of liabilities; Above 1 is a surplus, below 1 is a deficit. Liability-relative investors manage this ratio or the surplus.
Surplus: Surplus = Assets − Liabilities; Liability-relative risk is often the volatility of the surplus, not of assets alone.
Surplus return (approximate): Surplus return ≈ (ΔA − ΔL) ÷ A = (Asset return × A − Liability return × L) ÷ A, where A = beginning assets, L = beginning liabilities, ΔA = change in assets and ΔL = change in liabilities; The base is beginning assets A: surplus return is the change in surplus divided by beginning assets. Asset return is the percentage change in asset value. Liability return is the percentage change in the PV of the liabilities. It is negative when rates rise, because the PV falls, and positive when rates fall, because the PV rises. So a fall in rates raises the PV of liabilities and can hurt the surplus even when assets rise.
Asset-only vs liability-relative: Asset-only: choose assets by return, risk, correlation. Liability-relative: choose assets to fund or hedge the liability.; Liability-relative is the better fit when liabilities are large, fixed and dated.
Surplus: Surplus = Value of assets − Present value of liabilities; The risk measure in a liability-relative approach is the variability of the surplus, not of assets alone.
Funded ratio: Funded ratio = Value of assets ÷ Present value of liabilities; Above 1 means overfunded. Below 1 means underfunded.
Asset-only portfolio variance (two assets): σp² = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2; Used inside mean-variance optimization. Asset-only risk is portfolio volatility.
Goals-based framework (not a calculation): Personal risk → Market risk → Aspirational risk; A framework, not a formula. Personal risk protects against downside to the client's personal situation and funds essential needs with the highest required probability. Market risk maintains lifestyle. Aspirational risk is wealth enhancement with the lowest required probability. Each module gets its own sub-portfolio, time horizon and required probability of success.
Portfolio expected return: E(Rp) = Σ wi × E(Ri); Weights sum to 1. Use the same time unit for all inputs.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Standard deviation is the square root. Lower correlation gives lower risk.
Covariance and correlation: Cov(1,2) = ρ12 × σ1 × σ2; MVO needs a covariance matrix. The number of inputs grows quickly with the number of assets.
Efficient frontier rule: Efficient if no portfolio has higher E(R) at equal or lower σ; Portfolios below the frontier are dominated.
Black-Litterman starting point: Equilibrium returns come from reverse optimization of market-cap weights; Views then tilt these returns, scaled by confidence in each view.
Portfolio variance (two assets): σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Portfolio risk is σp = √σp². Weights are capital weights.
Marginal contribution to risk (MCTR): MCTRi = Cov(Ri, Rp) ÷ σp; Change in portfolio risk from a small increase in asset i's weight.
Contribution to risk (CTR): CTRi = wi × MCTRi = wi × Cov(Ri, Rp) ÷ σp; CTRs sum to σp. Percentage contribution = CTRi ÷ σp.
Equal-risk-contribution condition: w1 × MCTR1 = w2 × MCTR2 = …; Defines risk parity. It uses no expected returns.
Optimal risk budget condition: (E(Ri) − Rf) ÷ MCTRi equal for all i; Excess return per unit of marginal risk is equal across components at the optimum.
Parametric VaR (normal): VaR = (z × σ − μ) × portfolio value; z = 1.65 for 95% one-tail, 2.33 for 99%. Match μ and σ to the same horizon. Normality may understate tail risk.
Tracking risk (active risk): TE = standard deviation of (Rp − Rb); Used to budget active risk across managers or strategies.
Reverse optimization (implied returns): Implied expected return vector = λ × Σ × w(market); λ is the market's risk aversion, Σ the covariance matrix, w(market) the market-cap weights. The returns are those that make the market weights optimal.
Asset class criteria: Homogeneous within; diversifying between; mutually exclusive; can hold a large share of investable wealth; capacity for investment; List these when asked to justify or critique a proposed asset class.
Drift of a weight: Current weight = (Target weight × (1 + class return)) ÷ (1 + portfolio return), where portfolio return = Σ(target weight × class return); Use the one-period return of each class to find the portfolio return first, then the post-move weight. Example: 60% equity returning 20% and 40% bonds returning 0% give a portfolio return of 0.6 × 20% = 12%, so the equity weight is 72 ÷ 112 = 64.29%.
Rebalancing bands: Rebalance if current weight < target − band or > target + band; Band width depends on transaction costs, risk tolerance, correlations and volatility of the class. Wider: high costs, low risk aversion, momentum markets, low correlation with the rest of the portfolio. Narrower: high volatility of the class, high risk aversion, mean-reverting markets. High correlation with the rest of the portfolio is not a reason for wider bands.
Hedged currency return (approx.): Hedged return ≈ foreign asset local return + (domestic risk-free rate − foreign risk-free rate); Under covered interest parity, the forward premium or discount reflects the interest rate differential. The hedge removes the currency spot move and leaves the forward premium or discount, so you earn or pay the rate differential on top of the local return.
Tactical (active) weight: Active weight = Actual weight − Strategic weight; Sum of active weights across assets is zero if the portfolio is fully invested with no leverage.
Value added by tactical allocation: Value added = Σ (Actual weight − Strategic weight) × (Asset class return − Strategic benchmark return); Using asset class returns minus the total benchmark return is one common form. Using the asset class return alone gives the same total because active weights sum to zero.
Portfolio return difference: Active return = Portfolio return − Strategic benchmark return; Compare to the tactical contribution. The remainder comes from security selection or implementation.
Tactical decision rule: Deviate only if expected gain > costs and extra risk, and the deviation is within IPS limits; A rule of thumb for answers, not a formula to compute.

Quick revision

The economic balance sheet includes human capital and other non-tradable items, not just financial assets.
Asset-only focuses on risk and return of assets; liability-relative focuses on funding liabilities; goals-based splits money into goal buckets.
Liability-relative allocation is the natural fit when there are clear, measurable liabilities.
Goals-based approaches assign different risk levels to different goals based on their importance.
Mean-variance optimization is sensitive to input estimates and can give concentrated portfolios.
Optimizer weights are only as good as the expected return, volatility and correlation inputs.
Risk budgeting allocates risk, not just capital, across asset classes or strategies.
Always check that a risk measure, such as standard deviation or value at risk, fits the client's real concern.
Strategic allocation is the long-term policy mix; tactical allocation is a short-term deviation from it.
Rebalancing keeps the portfolio near policy weights and has costs to weigh against drift.
Behavioral biases can lead to poor allocation choices, so the adviser should recognise and manage them.
In every answer, link the choice back to the client's objectives and constraints.

Common mistakes

Counting only the financial portfolio as the investor's assets. Fix: Always scan for human capital, expected pension or social benefits, and expected sponsor contributions. Include them at present value.
Mixing present values with future values or undiscounted amounts. Fix: Discount every future item at an appropriate rate before you subtract. Do not net a future figure against a present one.
Saying asset-only ignores liabilities and goals completely. Fix: Say they are not modelled explicitly. They still shape risk tolerance and constraints.
Measuring risk in a liability-relative approach as asset volatility. Fix: Use surplus risk: how asset and liability values move together.
Saying MVO is wrong because the math is flawed. Fix: Say the math works but outputs are very sensitive to estimated inputs, especially expected returns.
Picking a portfolio below the efficient frontier because it matches the client's risk exactly. Fix: For the same risk, choose the portfolio with higher return on the frontier.
Treating capital weights as risk weights. Fix: Always compute contribution = weight × marginal contribution. Compare it with the capital weight.
Adding standalone volatilities to get portfolio risk. Fix: Use the variance formula with correlation. Only the contributions to risk add up, not standalone volatilities.
Treating the optimizer output as the final policy portfolio Fix: Check against constraints, then note estimation error and extreme weights, and apply limits or other methods.
Using reported private asset volatility at face value Fix: Say that valuations are smoothed, so risk and correlation to public markets are understated. Unsmooth or adjust the inputs.

Exam tips

Read the vignette for hidden economic assets and liabilities such as pensions, future earnings and bequests, and include each at present value.
When asked to justify an approach, name the liability features (size, certainty, timing) in one sentence. That earns the points.
Link human capital to risk: say whether it is bond-like or equity-like, then state the effect on the portfolio's risk capacity.
Show your net worth or funding ratio calculation in the answer space. Type the number alone only if the question asks for a single value.
Answer only the number of responses asked for, in the order requested, as extra answers are not evaluated.
Tie the approach to the investor type first. Pensions and insurers point to liability-relative, individuals with separate goals point to goals-based.
In essay sets, give the approach and one reason from the case. Extra theory earns no extra points.
A correct number alone earns full credit for a calculation, such as the funded ratio or surplus. Only the number of responses asked for is evaluated, in the order given, so give exactly that many answers in that order.