Private Investments and Structures: formula sheet

Key formulas

Illiquidity premium (concept)

Required return on private asset ≈ return on comparable public asset + illiquidity premium

Use it to judge whether an expected return compensates for lack of liquidity. It is a framework, not an exact formula.

Unsmoothing appraisal-based returns (first-order adjustment)

Unsmoothed return(t) = [Observed return(t) − φ × Observed return(t−1)] ÷ (1 − φ)

This is the common first-order autoregressive (Geltner-type) unsmoothing approach. φ is the smoothing parameter between 0 and 1. Unsmoothing typically raises measured volatility and correlation with public markets. Use it only when the question gives φ or asks you to explain the idea.

Standard deviation after unsmoothing (direction)

Unsmoothed volatility > reported volatility, when returns are smoothed

Smoothing typically lowers reported volatility and correlation with public markets, so risk and diversification are overstated.

Unfunded commitment

Unfunded commitment = Total commitment − Capital called to date

Liquidity planning must cover the amount that may still be called.

Management fee

Fee = fee rate × fee basis (committed capital, invested capital or NAV, as per terms)

Check which basis applies in the period in question.

Standard waterfall order

1) Return of contributed capital → 2) Preferred return (hurdle) → 3) GP catch-up → 4) Carry split (e.g., 80% LP / 20% GP)

Each tier must be filled before the next one receives cash.

Full catch-up target

GP catch-up amount = carry% ÷ (1 − carry%) × preferred return paid (at 100% catch-up)

At 20% carry, the catch-up equals 25% of the preferred return paid. Check the catch-up rate in the question.

Carry on total profit

Carry = carry% × (total distributions − contributed capital), if the hurdle is cleared and the catch-up is full

A quick check on the waterfall result.

Net-to-LP return

LP net profit = total distributions to LP − contributions (including fees)

Fees are paid out of LP contributions, so they lower LP returns.

Equity value at exit

Exit equity = Exit enterprise value − Net debt at exit

Net debt = debt − cash. Use it to see how debt paydown adds to equity value.

Enterprise value from a multiple

EV = EBITDA × EV/EBITDA multiple

Apply the entry multiple at purchase and the exit multiple at sale.

Value creation drivers in an LBO

Equity gain = EBITDA growth effect + multiple expansion effect + net debt reduction

Each effect is the change in that item, valued at the relevant multiple. Leverage then magnifies the percent return on the smaller equity base.

Multiple of invested capital (MOIC)

MOIC = Total value returned ÷ Capital invested

Ignores timing. IRR adds timing.

IRR for a single in and out cash flow

IRR = (Exit equity ÷ Entry equity)^(1 ÷ years) − 1

Valid only when there are no interim cash flows.

Capital structure ranking

Senior secured > unitranche/senior > mezzanine > equity (priority of claim, highest first)

Higher priority means lower risk and lower expected return. Unitranche blends senior and junior risk into one loan at a blended rate.

Leveraged return on equity

Levered return = Asset return + (D ÷ E) × (Asset return − Cost of debt)

Applies when the return and cost of debt are in the same period, ignoring taxes and fees. Leverage cuts both ways when the asset return falls below the cost of debt.

Direct capitalization value

Value = NOI ÷ Cap rate

Use stabilized NOI. A higher cap rate means a lower value for the same NOI.

Debt service coverage ratio

DSCR = NOI ÷ Debt service

Higher means more cushion. Lenders use it to test whether property income supports the loan.

Loan-to-value ratio

LTV = Loan amount ÷ Property value

Lower LTV means more equity cushion for the lender.

Distressed debt return source

Return ≈ (Recovery value − Purchase price) ÷ Purchase price

A simplified view that ignores interim cash flows, costs and time value. Timing of recovery matters for annualized return.

DPI

DPI = cumulative distributions ÷ paid-in capital

Realised only. Cannot be changed by valuation opinion.

RVPI

RVPI = residual NAV ÷ paid-in capital

Unrealised and valuation-dependent.

TVPI

TVPI = DPI + RVPI = (distributions + residual NAV) ÷ paid-in capital

Total value multiple. Ignores timing.

Since-inception IRR

0 = Σ [CFt ÷ (1 + IRR)^t], with contributions negative, distributions positive, and ending NAV as a final inflow

Money-weighted. Depends on timing and valuation of ending NAV.

Kaplan-Schoar PME

PME = [Σ (distributions × FV factor to measurement date) + ending NAV] ÷ [Σ (contributions × FV factor to measurement date)]

Future value factors use index returns to the measurement date. The ending NAV is already at that date, so its factor is 1. PME > 1 means outperformance of the index.

Unsmoothed return (simple one-lag)

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

φ is the smoothing weight on the prior period, 0 ≤ φ < 1. Applied across the whole series, it widens the spread of returns, which raises standard deviation.

Unfunded commitment

Unfunded commitment = Total commitment − Capital called to date

Still callable by the GP. Count it as a future liquidity claim.

Total exposure (NAV + unfunded)

Total exposure = NAV + Unfunded commitment

This is the economic exposure, so it is higher than NAV alone. The target allocation is usually measured on NAV, so commitments are set above the target to reach it.

Capital called in a year

Call = Commitment × Contribution rate for that fund age

Apply the rate to the original commitment, not to NAV.

NAV roll-forward

NAV(end) = NAV(start) × (1 + growth rate) + Contributions − Distributions

Use the stated order: growth on opening NAV, then add calls and subtract distributions, unless the question says otherwise.

Distribution in a year

Distribution = Rate of distribution × NAV(start)

The rate of distribution applies to opening NAV, whereas contributions apply to commitment.

De-smoothing of appraisal returns

Unsmoothed return(t) = [Reported return(t) − φ × Reported return(t−1)] ÷ (1 − φ)

φ is the smoothing parameter, 0 ≤ φ < 1. Unsmoothed volatility is higher than reported volatility.

Quick revision

• Private assets are illiquid, opaque and priced by appraisal or models, so reported volatility and correlations look too low.
• Smoothed valuations understate risk; unsmoothing raises measured volatility.
• Manager dispersion is wide in private markets, so selection matters more than in public markets.
• Committed capital is called over time; investors must hold liquid assets to meet drawdowns.
• Management fees are usually charged on committed or invested capital; carried interest is a share of profits above a hurdle.
• A waterfall sets the order of distributions; know the difference between deal-by-deal and whole-fund approaches.
• Clawbacks protect limited partners if early carry paid turns out to be too high.
• Venture capital returns are driven by a few big winners; buyouts rely on operational improvement, leverage and exit.
• Private debt gives contractual income with credit and illiquidity risk; real assets offer cash flow and inflation links.
• IRR depends on timing of cash flows; multiples ignore time.
• TVPI = DPI + RVPI, where DPI is distributions to paid-in capital and RVPI is residual value to paid-in capital.
• Match every recommendation to the client's liquidity needs, horizon, risk tolerance and governance capacity.

Common mistakes

• Saying private assets have lower risk because reported volatility is low Fix: State that low reported volatility reflects smoothing. True risk is higher, and unsmoothing raises volatility and correlation.
• Treating the illiquidity premium as guaranteed Fix: Describe it as compensation investors require. Realised returns vary widely across managers.
• Charging the management fee on the wrong basis. Fix: Read the fee basis for the period. It often shifts from committed to invested capital after the investment period.
• Paying carry before the hurdle is cleared. Fix: Always fill return of capital and the hurdle first. Carry starts only after those tiers.
• Assuming VC uses heavy debt like an LBO. Fix: Remember that VC firms lack stable cash flow to service debt, so VC is funded mainly with equity.
• Counting only EBITDA growth as LBO value creation. Fix: Always check all three drivers: earnings growth, multiple expansion and net debt reduction.
• Treating mezzanine and distressed debt as the same thing. Fix: Mezzanine is subordinated debt to a going concern with coupon and upside. Distressed debt is bought at a discount from stressed issuers for recovery and control.
• Assuming leverage always raises returns. Fix: Leverage raises returns only when the asset return exceeds the cost of debt. Otherwise it magnifies losses.
• Using committed capital as the denominator for DPI, RVPI or TVPI. Fix: Use paid-in capital, meaning the capital actually called, unless the question says otherwise.
• Treating TVPI as realised performance. Fix: Split it. Only DPI is realised; RVPI is an estimate that can be revised.

Exam tips

• Read the client facts first. Most private market answers are judged on fit with liquidity needs, horizon and risk tolerance.
• When asked to explain a difference from public markets, name the direction (higher or lower) and give the reason in a short phrase.
• Show unsmoothing steps even for simple numbers. A correct number on its own earns credit, but steps protect you if you slip.
• In multiple-choice items, be wary of options claiming private assets are risk-free or guaranteed to outperform. These overstate the case.
• Answer only what the command word asks for and give the number of points requested, in the order given.
• Read the command word. 'Calculate' needs a number with workings, 'justify' needs a reason tied to the client's or LP's interest.
• Always state the waterfall type and fee basis from the vignette before computing.
• Check that LP plus GP amounts equal total distributions. It catches most arithmetic slips.