Private Markets Investing: formula sheet

Key formulas

Management fee

Fee = fee rate × fee base (committed or invested capital)

Check which base applies in the period. Commitments are common in the investment period, invested capital after.

Carried interest without hurdle

Carry = carry rate × total profit

Profit is distributions minus contributed capital, usually after fees as stated in the fund terms.

Hurdle amount (compounded)

Hurdle = contributed capital × [(1 + h)^t − 1]

This is the preferred return only. LPs first receive capital back plus this amount.

Full catch-up

Catch-up amount = carry rate ÷ (1 − carry rate) × preferred return

With a 100% catch-up to the GP, the GP gets this much so that it holds the carry rate of total profit. For 20% carry it is 25% of the preferred return.

Clawback

Clawback = carry received − carry rate × cumulative fund profit

Applies when carry paid early exceeds the entitlement at the end. Often capped, for example at after-tax carry.

Waterfall order

1) Return of capital → 2) Preferred return → 3) GP catch-up → 4) Split (e.g. 80/20)

European applies this to the whole fund. American applies it deal by deal.

Enterprise value

EV = Equity value + Net debt

Net debt = debt − cash. LBO purchase prices are usually quoted as EV/EBITDA.

Exit equity value in an LBO

Exit equity = Exit EBITDA × Exit multiple − Net debt at exit

Compare with equity invested at entry to get the gain.

Value creation split

Equity gain = EBITDA growth effect + Multiple expansion effect + Debt paydown effect

A common attribution. Interaction effects can be assigned to one component, so check how the question defines each.

Money multiple

MOIC = Exit proceeds ÷ Equity invested

Ignores time. IRR adds timing.

Post-money valuation (VC round)

Post-money = Pre-money + New investment; Investor stake = Investment ÷ Post-money

Later rounds dilute earlier holders unless they invest again.

IRR

Σ CFt ÷ (1 + IRR)^t = 0

Include contributions as negatives, distributions as positives and ending NAV as a final inflow. Money-weighted and sensitive to timing.

DPI

DPI = Cumulative distributions ÷ Paid-in capital

Realised cash returned. Not affected by NAV appraisals.

RVPI

RVPI = NAV ÷ Paid-in capital

Unrealised value. Depends on the manager's valuation.

TVPI

TVPI = DPI + RVPI = (Distributions + NAV) ÷ Paid-in capital

Ignores timing. Equals DPI once the fund is fully liquidated.

Kaplan-Schoar PME

KS-PME = FV(distributions + NAV at index) ÷ FV(contributions at index)

Both sides compounded with the index to the same date. Above 1 means the fund outperformed the index. Present values at the index give the same ratio.

Long-Nickels PME

Buy index with each contribution, sell with each distribution; IRR of the flows with the ending index position

Gives an index-based IRR to compare with the fund IRR.

Unfunded commitment

Unfunded = Total commitment − Cumulative paid-in capital (capital called)

Some practitioners also add recallable distributions. State the definition used.

Takahashi-Alexander capital call

Call(t) = RC(t) × (Commitment − Σ prior calls)

RC(t) is the rate of contribution in year t, applied to the remaining unfunded commitment.

Takahashi-Alexander distribution

Dist(t) = RD(t) × NAV(t−1) × (1 + G)

RD(t) is the rate of distribution; G is the assumed growth rate of the fund's assets. RD rises with age, capped at 100%, and is often set as the max of the age-based rate and a yield-based minimum.

NAV roll-forward

NAV(t) = NAV(t−1) × (1 + G) + Call(t) − Dist(t)

Use the same G as in the distribution formula.

Net cash flow to LP

Net cash flow(t) = Dist(t) − Call(t)

Negative in early years, positive later: this gives the J-shape in cash terms.

Secondary pricing

Discount = (NAV − Price) ÷ NAV

A positive value means the interest sold below reported NAV.

Floating-rate loan yield

Loan yield ≈ reference rate + credit spread

Direct loans reprice with the reference rate, so interest rate risk is low but borrower payment burden rises when rates rise.

Excess return over public credit

Illiquidity premium ≈ private credit spread − comparable public credit spread (same rating and seniority)

Only an approximation. Part of the gap may be compensation for credit risk, not illiquidity.

Expected loss

EL = PD × LGD × EAD

Use it to compare senior loans (low LGD) with mezzanine (high LGD).

Net credit return (approx.)

Net return ≈ yield − expected loss − fees

Fund fees and defaults reduce the headline spread.

Distressed debt return on a position

Return = (recovery value + interim cash − purchase price) ÷ purchase price

Driven by recovery and timing, so annualise if the holding period differs.

Leverage effect on equity return

Return on equity = asset return + (debt ÷ equity) × (asset return − cost of debt)

Fund-level leverage magnifies gains and losses.

Smoothed return (one-lag model)

R(obs,t) = (1 − α) × R(true,t) + α × R(obs,t−1)

α is the smoothing weight between 0 and 1. A higher α means more smoothing. Observed returns are a weighted average of true return and last observed return.

Unsmoothed return

R(true,t) = [R(obs,t) − α × R(obs,t−1)] ÷ (1 − α)

Rearranged from the model above. With a one-lag model, α is estimated by the first-order autocorrelation of observed returns.

Volatility effect of unsmoothing

σ(true) ≈ σ(obs) ÷ (1 − α) (approximation)

Holds approximately when true returns are uncorrelated through time, and gives a rough scaling only. True volatility is higher than observed volatility when α > 0.

Sharpe ratio

Sharpe = (R(p) − R(f)) ÷ σ(p)

Using smoothed volatility inflates the Sharpe ratio. Recompute with unsmoothed volatility.

Annualised volatility scaling

σ(annual) = σ(quarterly) × √4

Valid only if returns are independent. With autocorrelation, naive scaling understates annual risk.

Net asset value (NAV) of a fund

NAV = Fair value of assets − Liabilities (incl. accrued fees and carry)

Fair value inputs are the main source of GP discretion.

Management fee

Fee = fee rate × (committed or invested capital)

Fees on committed capital are paid even on undrawn money; fees on NAV reward higher marks.

Carried interest with hurdle (simple, no catch-up)

Carry = carry rate × (profit − hurdle amount)

With a full catch-up, carry is the carry rate × total profit once the hurdle is cleared. Read the LPA terms in the question.

Unsmoothing of returns (one-lag)

R(true, t) = [R(reported, t) − a × R(reported, t−1)] ÷ (1 − a)

a is the smoothing coefficient between 0 and 1. Unsmoothing raises measured volatility.

Autocorrelation red flag

High positive autocorrelation in reported returns suggests smoothing or stale pricing

It is a flag, not proof of misconduct.

Quick revision

• A general partner manages the fund; limited partners supply most of the capital and have limited liability.
• Management fees are usually charged on committed capital during the investment period; carried interest is a share of profits.
• A hurdle (preferred return) must be met before carry is paid; a catch-up lets the GP receive carry on earlier profits.
• A clawback returns excess carry to LPs if early payouts exceed what the final fund results justify.
• Venture capital has high failure rates and skewed returns; buyouts use leverage to acquire mature companies.
• IRR depends on timing and is sensitive to when capital is called and returned.
• TVPI = (distributions + residual value) ÷ paid-in capital; DPI counts only cash returned; RVPI counts only remaining value.
• TVPI = DPI + RVPI.
• PME compares private fund cash flows with what a public index would have delivered on the same timing.
• The J-curve: early returns are negative because of fees and costs, then rise as investments mature.
• Unfunded commitments are a liquidity risk: capital calls can arrive when the investor's other assets have fallen.
• Appraisal-based valuations are smoothed, which understates volatility and correlation with public markets.

Common mistakes

• Applying carry to total distributions instead of profit. Fix: Always subtract contributed capital before computing carry.
• Mixing up the fee base. Fix: Read the stated base for each period and apply it separately.
• Treating growth equity as the same as venture capital. Fix: Growth equity targets companies with proven revenue and often profits. VC targets firms with unproven models and high failure rates.
• Saying LBO value comes only from leverage. Fix: Leverage amplifies returns, but value comes from earnings growth, deleveraging and multiple expansion. Leverage does not create value by itself.
• Using committed capital as the denominator of DPI, RVPI or TVPI. Fix: The denominator is paid-in (called) capital. Committed but uncalled capital is not in it.
• Treating TVPI as a realised return. Fix: TVPI includes RVPI, which is unrealised NAV. Only DPI is realised.
• Applying the call rate to the total commitment instead of the unfunded commitment. Fix: Subtract all prior calls first. The call rate acts only on what is still uncalled.
• Forgetting to grow NAV by (1 + G) before applying the distribution rate. Fix: In the Takahashi-Alexander model, distribution = RD × NAV(t−1) × (1 + G).
• Treating mezzanine and distressed debt as the same thing. Fix: Mezzanine is a performing, subordinated financing with high coupon. Distressed debt is bought at a discount from a troubled issuer, with return from recovery.
• Saying direct loans have high interest rate risk. Fix: Floating-rate loans have low duration. The rate risk shows up as higher borrower debt service and default risk.

Exam tips

• Identify the waterfall type first. A European structure gives LPs more protection and makes a clawback less likely.
• Check whether the question uses commitments or invested capital as the fee base.
• Show the order of the waterfall tiers. Many wrong answers come from skipping the hurdle or catch-up.
• Link terms to incentives. Management fees are performance-independent, while carry, the hurdle and the clawback align interests.
• In case-style questions, name the risk: LPs bear interim carry risk in deal-by-deal structures.
• Read the company description first. Words like pre-revenue, cash burn or milestones signal VC. Stable cash flows and a debt package signal an LBO.
• In value creation questions, list all three LBO drivers and note which one the numbers show.
• Always subtract net debt when converting enterprise value to equity value.