FRM Part II · FRM Exam Part II

Hedge Fund Investment Strategies: formula sheet

Full chapter guide

Key formulas

Management fee: Management fee = fee rate × assets under management (AUM); Check whether it is charged on beginning, average or ending NAV, and whether it is deducted before the incentive fee.
Incentive fee with HWM: Incentive fee = incentive rate × max(0, NAV before incentive fee − HWM); Applies after the management fee, unless the question says otherwise.
Hard hurdle incentive fee: Incentive fee = incentive rate × max(0, return − hurdle) × starting NAV; Only the excess over the hurdle is charged.
Soft hurdle incentive fee: If return > hurdle: incentive fee = incentive rate × return × starting NAV; otherwise 0; Once the hurdle is cleared, the fee applies to the whole profit.
Incentive fee as an option: Payoff ≈ incentive rate × max(NAV − HWM, 0); A call option on NAV with strike equal to the HWM. Value rises with volatility.
Net return: Net return = gross return − management fee rate − incentive fee ÷ starting NAV; Compute in currency terms first, then convert to a percentage.
Net exposure: Net exposure = (Long value − Short value) ÷ Capital; Directional long/short funds run positive net; short bias funds run negative; market neutral targets about zero.
Gross exposure: Gross exposure = (Long value + Short value) ÷ Capital; Measures leverage. A gross exposure above 100% means the fund is levered.
Portfolio beta: β_p = (Σ w_long × β_long − Σ w_short × β_short) ÷ Capital, with weights in value terms; Beta neutral means β_p ≈ 0. Dollar neutral does not guarantee this.
Long/short return (approximate): R ≈ Long return × Long weight − Short stock return × Short weight + Rebate on short proceeds − Borrow cost; A gain on the short book comes when the shorted stocks fall or lag.
Alpha decomposition: R_p − R_f = α + β_p × (R_m − R_f) + ε; For market neutral, the return should be mostly α, with β_p near zero.
Deal spread (cash deal): Spread = Offer price − Target market price; Gross profit per share if the deal closes. Express as % of market price for comparison.
Annualised spread: Annualised return ≈ (Spread ÷ Target price) × (365 ÷ days to close); Simple annualisation. Needed to compare deals with different closing dates.
Implied deal-completion probability: Market price = p × Offer price + (1 − p) × Break price, so p = (Price − Break price) ÷ (Offer − Break price); Ignores time value and dividends. Use it to read what the market believes.
Stock deal hedge: Short exchange ratio × acquirer shares per target share bought; If the target gets 0.5 acquirer shares per share, short 0.5 acquirer shares per target share held.
Payoff profile: Merger arbitrage ≈ short out-of-the-money put on the target; Gain is capped at the spread. Loss is Price − Break price if the deal fails.
Distressed return: Return = (Recovery value − Purchase price) ÷ Purchase price; Recovery is measured in present value terms, after costs and delay.
Convertible bond hedge (delta hedge): Shares to short = Delta × Conversion ratio × Number of bonds; Delta is the option delta per share of the embedded call, between 0 and 1. The hedge makes the position roughly neutral to small stock moves.
Conversion value: Conversion value = Conversion ratio × Stock price; Compare with the convertible's price. Premium = Convertible price − Conversion value, often divided by conversion value to give a percentage.
Delta-hedged P&L (approximate): P&L ≈ ½ × Gamma × (ΔS)² + Theta × Δt + Vega × Δσ; Long gamma gains from large moves in either direction and pays theta. Vega gain comes from implied volatility rising.
Break-even move: Daily break-even move ≈ S × σ_implied ÷ √252; This is approximately the one-day move implied by the volatility. A single day's move larger than this gives a gain for that day on a gamma-theta basis. Over a period, a delta-hedged long-gamma position profits when realised volatility exceeds the implied volatility embedded in theta. Use 252 trading days.
Leverage: Leverage = Total position value ÷ Equity capital; A spread return r on positions gives return ≈ leverage × r on equity, before funding costs. Losses are magnified the same way.
Convertible arbitrage return sources: Return = Coupon income + Short rebate − Borrow and funding cost + Gamma trading profit ± Spread changes; Use this to identify what drives profit and which risks need separate hedges.
Trend-following signal (moving-average rule): Long if short-term average price > long-term average price; short if <; Typical systematic rule. Direction follows the recent trend, not a forecast of fundamentals.
Volatility-scaled position size: Position = target risk contribution ÷ asset volatility; Higher-volatility markets get smaller positions so each market adds similar risk.
Lookback call payoff: S(T) − min S(t) over the period; Always ≥ 0. Buys at the lowest price in hindsight.
Lookback put payoff: max S(t) − S(T); Always ≥ 0. Sells at the highest price in hindsight.
Lookback straddle payoff: max S(t) − min S(t); Equals the call plus the put. It is the price range over the period, so it is always ≥ 0 and rewards large moves in either direction.
Multi-factor return model: R(fund) = α + β₁F₁ + β₂F₂ + … + βₖFₖ + ε; α is the intercept (return not explained by factors). Betas are estimated by regression. Alpha is only as good as the factor set.
Survivorship bias: Bias = average return of surviving funds − average return of all funds (including dead funds); Positive in sign: databases that drop dead funds overstate performance.
Smoothed return (MA model): R(observed, t) = θ₀R(t) + θ₁R(t−1) + θ₂R(t−2), with θ₀ + θ₁ + θ₂ = 1; Weights sum to 1, so the mean is unchanged but variance is reduced. Smaller θ₀ means more smoothing.
Effect of smoothing on volatility: σ(observed) < σ(true) when θs are positive; Observed beta and correlation to markets are also biased down. Sharpe ratio is biased up.
Annualising with autocorrelation: σ(annual) ≠ σ(monthly) × √12 when returns are autocorrelated; With positive autocorrelation, the √12 rule understates true annual volatility.
Fung-Hsieh seven factors: Equity market, size spread, bond market, credit spread, bond trend-following, currency trend-following, commodity trend-following; The three trend-following factors are lookback straddles, which capture option-like payoffs.
Net return of a FoF to investor: Net return = Gross return of underlying funds − underlying fund fees − FoF management fee − FoF performance fee; Fees stack in two layers. Fee on each fund's gain is paid separately, so total fees can exceed those implied by the net portfolio return.
Fund management and incentive fee: Fees = management fee % × assets + incentive fee % × profit above hurdle (and above high-water mark, if applicable); A high-water mark means the manager earns incentive fees only on gains above the previous peak NAV.
Leverage ratio (simple): Leverage = Total assets (or gross exposure) ÷ Equity capital; Return on equity ≈ leverage × asset return − (leverage − 1) × borrowing cost.
Return on equity with leverage: ROE = L × R − (L − 1) × c; L is assets ÷ equity, R the asset return, c the borrowing rate. Losses are also magnified by L.
Gate: Maximum redemption per period = gate % × fund NAV; Requests above the cap are usually paid pro rata and the rest carried forward.

Quick revision

Hedge funds typically charge a management fee on assets and an incentive fee on profits.
A high-water mark means incentive fees are paid only on gains above the fund's previous peak value.
A hurdle rate means incentive fees apply only to returns above a set threshold.
Long/short equity earns from stock selection and keeps net market exposure managed; equity market neutral aims for near-zero beta.
Merger arbitrage typically earns a small spread with a risk of large loss if the deal fails.
Distressed debt investing relies on analysis of recovery value and the restructuring process, and carries credit and liquidity risk.
Relative value strategies use leverage to magnify small spread gains, so they are exposed to liquidity shocks and spread widening.
Convertible arbitrage is typically long the convertible bond and short the underlying stock, to capture mispricing and volatility.
Managed futures and CTAs often follow trends and can perform well in prolonged market moves, but suffer in choppy markets.
Survivorship bias and backfill bias tend to overstate reported hedge fund returns.
Illiquid or stale prices can smooth returns and understate volatility and correlation.
Due diligence covers both investment and operational factors, since operational failures and fraud are major causes of fund losses.

Common mistakes

Charging the incentive fee on all gains even when NAV is below the HWM. Fix: Compare NAV with the HWM first. Only the amount above the HWM is fee-eligible.
Confusing hard and soft hurdles. Fix: Hard: fee on the excess over the hurdle only. Soft: fee on the full profit once the hurdle is cleared.
Assuming dollar neutral means no market risk. Fix: Weight each side by beta. Market neutral requires beta near zero, not just equal values.
Saying market neutral funds have no risk. Fix: List the residual risks: factor tilts, crowding, leverage, liquidity and model risk. The August 2007 quant losses show this.
Saying merger arbitrage is a bet on the market direction. Fix: It is a bet on deal completion. Beta is low in normal times, but losses cluster in market stress.
Describing the payoff as a long put or a long call. Fix: The fund collects a small premium (spread) and bears the large loss: it is short a put.
Treating relative value as risk-free because it is hedged. Fix: Remember the residual risks: spread widening, liquidity, credit, financing and model risk. Leverage makes them large.
Confusing convertible arbitrage with volatility arbitrage. Fix: Convertible arbitrage holds a bond with a call and has credit and rate exposure. Volatility arbitrage uses pure options and trades implied vs realised volatility.
Treating global macro and CTAs as the same strategy. Fix: Macro is discretionary and view-driven. CTAs are systematic and rule-driven, mostly on trends.
Computing the lookback straddle payoff using the final price. Fix: Use max − min over the whole path. The final price only matters for each leg separately.

Exam tips

Read the order of fees carefully. Many questions state whether the incentive fee is before or after the management fee.
Do all arithmetic in currency terms, then convert to a return percentage at the end.
For conceptual questions, link the HWM to the strike of the call option and say volatility raises the option value.
Know the contrast between hedge funds and mutual funds: liquidity, leverage, disclosure and fee structure.
Link lockups, gates and notice periods to liquidity mismatch and run risk in stress.
Always compute beta-weighted exposure. Questions often give equal long and short values to tempt you into calling it market neutral.
For quant quake questions, the key words are crowding, common factors, leverage and forced deleveraging, not a market crash.
For short bias, expect risk answers on unlimited loss, short squeeze, borrow cost and negative long-run drift.