FRM Part I · FRM Exam Part I

Fund Management: formula sheet

Full chapter guide

Key formulas

NAV per share: NAV = (Market value of assets − Liabilities) ÷ Shares outstanding; Use closing market values. Liabilities include accrued management fees and borrowings.
Premium or discount (closed-end fund): Premium/(Discount) = (Market price − NAV) ÷ NAV; Positive means premium, negative means discount. Applies to closed-end funds and ETFs, not to open-end mutual funds.
Fund return over a period: Return = (NAV_end − NAV_begin + Distributions) ÷ NAV_begin; Distributions are income and capital gains paid per share in the period.
Shares issued or redeemed: Shares = Amount invested ÷ NAV (net of any load); For a front-end load: shares = amount × (1 − load) ÷ NAV.
Front-end load offer price: Offer price = NAV ÷ (1 − Load %); The load is a percentage of the offer price, not of NAV.
Management fee: Management fee = fee rate × assets under management (AUM); Usually charged regardless of performance. Check whether the base is beginning, ending or average AUM.
Incentive fee with hurdle (hard hurdle): Incentive fee = incentive rate × max(0, profit − hurdle amount); With a hard hurdle the manager earns only on the excess over the hurdle. With a soft hurdle, once the hurdle is cleared the fee applies to the whole profit.
Incentive fee with high-water mark: Incentive fee = incentive rate × max(0, NAV − max(prior high-water mark, starting NAV)); Fee is charged only on new highs. Make sure you use NAV after management fee if the question says so.
Net return to investor: Net return = gross return − management fee − incentive fee; Work in currency amounts first, then divide by starting capital.
Convertible arbitrage hedge: Shares to short = number of bonds × conversion ratio × delta; Delta is the stock-price sensitivity of the conversion option; the short offsets equity moves.
Management fee: Management fee = fee rate × assets under management (AUM); Use beginning, average or ending AUM only as the question states.
Incentive fee with no hurdle: Incentive fee = incentive rate × max(0, profit); Profit may be measured before or after the management fee, as specified.
Incentive fee with hard hurdle: Incentive fee = incentive rate × max(0, profit − hurdle amount); Fee applies only to the excess over the hurdle.
Incentive fee with soft hurdle: If return > hurdle: fee = incentive rate × total profit; otherwise fee = 0; Once the hurdle is cleared, the fee applies to the full profit.
Incentive fee with high-water mark: Fee = incentive rate × max(0, ending NAV − max(beginning NAV, HWM)); Fee is due only on gains above the HWM (per unit NAV, with no hurdle).
Net return to investor: Net return = (ending NAV after fees ÷ beginning NAV) − 1; Equivalent to gross return minus management and incentive fees as a share of beginning value.
Survivorship bias (approximate): Bias ≈ Return of surviving funds − Return of all funds (including dead); Usually positive. Measured as the annual return gap between a survivor-only sample and the full sample.
Backfill bias (approximate): Bias ≈ Average return including backfilled months − Average return of post-listing months only; Usually positive. Fix: delete returns before the date the fund was added.
Alpha from a factor regression: Rₜ − R_f = α + β₁F₁ₜ + … + βₖFₖₜ + εₜ; Alpha is the intercept: return left after paying for factor exposures. Bias in the data biases alpha.
Direction summary: Survivorship: upward | Backfill: upward | Selection: ambiguous; Memorize this one-line summary for MCQs.
Management fee: Fee = fee rate × fee base (committed or invested capital); Check the base stated in the question. Committed and invested capital give different fees.
Carried interest (no hurdle): Carry = carry rate × fund profit; Profit is usually measured after returning contributed capital and, in some deals, fees.
Carry with hurdle and no catch-up: Carry = carry rate × (profit − hurdle amount), if profit exceeds the hurdle; Without catch-up, the GP shares only in profit above the hurdle.
Carry with full catch-up: Carry = carry rate × total profit, if profit is large enough to complete the catch-up; With full catch-up the GP receives 100% of profit above the hurdle until it holds the carry rate of total profit.
Multiple of invested capital: MOIC = total value returned ÷ capital invested; Ignores timing. IRR does consider timing.
LBO equity share: Equity = purchase price − debt; Debt-to-price ratio shows leverage. Equity return = (exit value − debt remaining − equity) ÷ equity.

Quick revision

Open-end funds issue and redeem shares at NAV; closed-end fund shares trade on an exchange and can sit at a premium or discount to NAV.
NAV per share = (assets − liabilities) ÷ shares outstanding.
Hedge funds use leverage, short selling and derivatives, and are less regulated than mutual funds.
Typical hedge fund fees combine a management fee on assets and an incentive fee on profits.
A hurdle rate is the return that must be beaten before an incentive fee is earned.
A high-water mark stops incentive fees being paid again until prior losses are recovered.
Survivorship bias arises because failed funds drop out of databases, which overstates average returns.
Backfill (instant history) bias arises when funds add past returns after joining a database, usually good ones.
Hedge fund returns are often smoothed by illiquid assets, which understates volatility and correlation.
Private equity commitments are drawn down over time and money is locked up for years.
Venture capital funds early-stage firms, while buyouts use heavy leverage on mature firms.
The J-curve shows negative early private equity returns from fees and write-downs before gains appear.

Common mistakes

Forgetting to subtract liabilities such as accrued fees from assets. Fix: Always write assets − liabilities first and check the question for any payables or borrowings.
Saying an open-end fund can trade at a premium or discount to NAV. Fix: Remember that open-end shares are bought and sold with the fund at NAV, so no premium or discount arises.
Charging the incentive fee on the full gain when a hard hurdle applies. Fix: Read the hurdle type. Hard: fee only on the excess. Soft: fee on the whole profit once the hurdle is beaten.
Ignoring the high-water mark after a loss. Fix: Track NAV against the prior peak. Fees apply only to gains above the peak.
Charging the incentive fee on the whole gain when a hard hurdle applies. Fix: Hard hurdle: fee on the excess only. Soft hurdle: fee on the full profit once the hurdle is cleared.
Charging an incentive fee after a loss recovery that has not reached the HWM. Fix: Compare ending NAV with the HWM, the highest NAV on which a fee was paid.
Treating survivorship bias and backfill bias as the same thing. Fix: Survivorship is about funds that disappear. Backfill is about history added when a fund enters. Ask: is the issue missing funds or added months?
Saying selection bias always overstates returns. Fix: Selection bias can be upward or downward, since some good funds also decline to report.
Charging carry on all profit when the fund has a hurdle and no catch-up. Fix: Read for the words hurdle, preferred return and catch-up before computing.
Applying the management fee to the wrong base. Fix: Use the base stated in the question. Default to committed capital only if told.

Exam tips

Questions often test structure: know who sets the price (NAV for open-end, the market for closed-end, arbitrage for ETFs).
In NAV questions, scan for hidden liabilities such as accrued fees or borrowings before dividing.
Expect conceptual items on liquidity risk: open-end funds face redemption pressure, closed-end funds do not.
Check whether a load is front-end or back-end and whether it is applied to NAV or offer price.
Watch the sign: a discount is negative and is measured relative to NAV, not market price.
Fee questions are arithmetic. Write amounts in currency and follow the order the question gives.
Always check hard versus soft hurdle and whether fees are before or after management fee.
Link each strategy to its signature risk: convertible arbitrage to liquidity and leverage, merger arbitrage to deal failure, managed futures to whipsaw markets.