CFA Level III · Level III Core

Portfolio Management for Institutional Investors: formula sheet

Full chapter guide

Key formulas

IPS structure: Objectives (return, risk) + Constraints (liquidity, time horizon, taxes, legal/regulatory, unique circumstances); Use this frame for every institutional investor. Write one line per element, each linked to the liabilities.
Spending-rule return requirement: Required nominal return = (1 + spending rate + cost rate) × (1 + inflation) − 1, approximately spending rate + cost rate + inflation; Use it for endowments and foundations that must preserve real capital. The additive approximation is slightly lower than the exact figure; state it as an approximation unless asked for the exact value.
Funded ratio: Funded ratio = Plan assets ÷ Present value of plan liabilities; Below 100% means a deficit. A weak funded ratio reduces the plan's ability to take risk.
Surplus: Surplus = Assets − Liabilities; Liability-relative investors manage the surplus, not the assets alone.
Bank net interest margin idea: Net interest income = Interest earned on assets − Interest paid on liabilities; Banks care about the spread and about asset-liability mismatch in duration and liquidity.
Funded status: Funded status = Plan assets − Present value of plan liability; A negative value is a deficit (underfunded). A positive value is a surplus.
Funded ratio: Funded ratio = Plan assets ÷ Present value of plan liability; Below 1 means underfunded. Use the same liability measure the question specifies.
Liability sensitivity to rates: %ΔLiability ≈ −Duration × ΔDiscount rate; Approximation for small changes. Falling rates raise the liability.
Risk tolerance rule: Risk tolerance = lower of ability and willingness; If one is below average and the other above, the overall result is the more conservative one.
Risk allocation in DB plans: DB: sponsor bears investment, longevity and funding risk; Benefit is fixed by formula. The sponsor must fund any shortfall.
Risk allocation in DC plans: DC: participant bears investment, longevity and decision risk; Sponsor bears fiduciary risk in menu and default design, not funding risk.
Cash balance account roll-forward: Ending balance = Beginning balance + Pay credit + Interest credit; Interest credit is a stated rate, for example a fixed rate or an index-linked rate, not the actual portfolio return. Pay credit is usually a percentage of pay.
Interest credit: Interest credit = Beginning balance × Credit rate; Use the beginning balance unless the question says contributions earn interest in the year.
Nominal required return: (1 + spending rate + cost rate) × (1 + inflation) − 1; Use this geometric form when the exam asks for precision. It is the CFA curriculum's convention: the spending and cost rates are treated as real rates, so their sum is the real need, which you then compound with inflation.
Approximate required return: Spending rate + cost rate + inflation; Fine as a quick check. Say which version you used.
Real required return: Spending rate + cost rate; The return above inflation needed to preserve real capital. This holds because the spending and cost rates are treated as real, as in the nominal formula above.
Simple spending rule: Spending = spending rate × market value at start of period; Volatile spending that tracks the market.
Rolling-average rule: Spending = spending rate × average market value of last n periods; Smooths spending but lags market moves.
Geometric (hybrid) rule: Spending(t) = w × [Spending(t−1) × (1 + inflation)] + (1 − w) × [spending rate × MV(t−1)]; w is the smoothing weight on the inflation-adjusted prior spending. Use the weight given in the question.
Net interest margin: NIM = (Interest income − Interest expense) ÷ Average interest-earning assets; Bank profitability measure; the core return driver for a bank.
Duration gap: Duration gap = Duration of assets − (Liabilities ÷ Assets) × Duration of liabilities; The gap is leverage-adjusted, because liability duration is scaled by Liabilities ÷ Assets. A positive gap means asset dollar duration exceeds liability dollar duration, so a rise in rates reduces surplus (equity) and a fall in rates increases it. A negative gap works the other way: a rise in rates increases surplus and a fall reduces it.
Change in surplus (approx.): ΔSurplus ≈ −D_A × A × Δy + D_L × L × Δy; Surplus = Assets − Liabilities. Δy is a parallel shift in yields, the same for assets and liabilities. D is modified duration. Use consistent duration measures for both sides.
Change in surplus from the duration gap: ΔSurplus ≈ −(Duration gap) × A × Δy; This is the same result as the line above, written with the leverage-adjusted gap. For example, a gap of −1.2 with A = 500 million and Δy = +0.005 gives −(−1.2) × 500 × 0.005 = +3.0 million.
Combined ratio (non-life): Combined ratio = Loss ratio + Expense ratio; Above 100% means an underwriting loss; investment income must cover it.
Liability matching rule: Match asset duration and cash flows to liabilities; add liquidity where liabilities have options; Short, uncertain liabilities need liquid, short, high-quality assets; long, predictable ones allow longer, less liquid assets.
SWF type to profile: Stabilization: high liquidity, low risk, short horizon | Savings: long horizon, higher risk, illiquids allowed; This is a guide to reasoning. Always check the facts given in the vignette.
IPS structure: Objectives (return, risk) + Constraints (liquidity, time horizon, tax, legal and regulatory, unique circumstances); Use this checklist for any institution.
Required return for a spending fund: Required nominal return ≈ spending rate + inflation + costs; A simple approximation. Use it only when the question gives these inputs.
Surplus: Surplus = Market value of assets − Present value of liabilities; Funded ratio = Assets ÷ PV of liabilities. A ratio below 1 means a deficit.
Surplus return: Surplus return = (Change in surplus) ÷ (Beginning assets); Check which base your question uses. Surplus return is often measured relative to beginning assets.
Immunization conditions: PV(assets) = PV(liabilities); Duration(assets) = Duration(liabilities); Convexity(assets) ≥ Convexity(liabilities); Protects against small parallel yield shifts only. Rebalance as time and yields change.
Duration matching with weights: w1 × D1 + w2 × D2 = D(liabilities), where w1 + w2 = 1; Solve for w1 using w1 = (DL − D2) ÷ (D1 − D2) when two assets are used.
Approximate price change: %ΔPrice ≈ −Modified duration × ΔYield + ½ × Convexity × (ΔYield)²; Use it to compare asset and liability value changes after a yield move.

Quick revision

Objectives: return and risk. Constraints: liquidity, horizon, taxes, legal and regulatory, unique circumstances.
Risk capacity is the ability to take risk. Risk willingness is the attitude to it. Overall risk tolerance is generally set by the lower of the two.
DB plans: funded status is assets compared with the present value of liabilities. Surplus risk is the risk that assets fall short of liabilities.
A stronger sponsor and a younger plan generally allow more risk. A weak sponsor and a mature plan call for less.
In a DB plan, the participant has a promise and the sponsor bears the investment risk. In a DC plan, the participant bears it.
DC plans: the design focus is the investment menu, defaults, and participant education, not liability matching.
Endowments aim to maintain purchasing power while funding spending, so return needs to cover spending plus inflation plus costs.
Spending rules smooth the budget and link spending to market value. Know the main types and their trade-offs.
Banks focus on interest rate risk, liquidity and credit. Insurers match liabilities by type, and life and non-life differ.
Sovereign wealth funds vary by mandate, such as stabilisation, savings or development. Match the policy to the stated purpose.
LDI: use assets to hedge the liabilities, then seek return with the remainder. Cash-flow matching is tighter but costlier than duration matching.

Common mistakes

Treating endowments and foundations as identical. Fix: Remember the differences: foundations often face a legal minimum payout and may lack new contributions, while endowments often receive gifts and set a spending rule linked to the institution's budget.
Basing the risk objective on willingness alone. Fix: Assess both ability and willingness, then state that the lower of the two governs the risk tolerance.
Treating the plan like an individual investor and setting the objective as maximum growth. Fix: Anchor the return objective to the liability and required contributions. Growth matters only to the extent it funds the promise.
Saying a high equity allocation is fine because the time horizon is long. Fix: Check funded status, sponsor strength and the share of retirees first. A long horizon does not offset a large deficit.
Treating a cash balance plan as a DC plan because it shows an account balance. Fix: Remember the account is hypothetical. The sponsor bears investment risk and the plan is a DB plan in form.
Saying the sponsor has no risk in a DC plan. Fix: The sponsor still faces fiduciary, legal and reputational risk from poor menu or default design.
Leaving inflation out of the return objective. Fix: Always add inflation, and add fund costs. Preserving real capital is the goal.
Treating a high spending rate as a liquidity need. Fix: Spending is usually small relative to assets. Liquidity needs come from actual cash demands such as capital calls, debt or large grants.
Treating a bank's investment portfolio as return-maximising. Fix: Anchor on liquidity, capital rules and interest rate risk; the portfolio supports lending and withdrawals.
Saying life and non-life insurers have the same horizon and liquidity needs. Fix: Life: long, predictable liabilities. Non-life: shorter, uncertain claims needing more liquidity and higher quality.

Exam tips

Link every IPS element to the liability or mission. Unlinked statements rarely earn full credit.
When asked for risk tolerance, show ability and willingness separately, then state which governs.
Match the investor to the answer set first. Many item-set options mix up endowment, foundation and insurer features.
In essay answers, give calculations in full and keep justifications to one sentence each.
Read the vignette for details on legal minimums, funded status and governance. Those are usually the hidden clues.
Read the command word. Calculate means show the number. Determine or justify means state the conclusion and the reason.
In risk tolerance answers, name ability and willingness separately, then the lower of the two. This earns the points quickly.
Use case facts as evidence: funded status, retiree share, sponsor strength and correlation. Generic statements earn little.