Investments in Private Capital: Equity and Debt: formula sheet

Key formulas

Stages of venture capital financing (order)

Angel → Seed → Early stage → Later stage → Mezzanine

Angel, seed and early stage are the formative stages. Mezzanine comes just before IPO or sale. VC is mostly equity financed, but mezzanine financing often uses convertible or subordinated debt.

LBO financing profile

High debt + stable, mature cash flows + asset-rich target

Debt is repaid from the target's cash flows. This is the key contrast with VC.

VC financing profile

Mostly equity + young, high-growth, often unprofitable firm

Few winners drive fund returns; many investments fail.

GP vs LP liability

GP: unlimited liability, manages fund; LP: limited to capital committed, passive

LPs lose their investment at most; passive involvement protects limited liability.

GP compensation

Management fee (on committed or invested capital) + carried interest (share of profits)

Fee terms are covered in the topic on fees, terms and governance.

Management fee

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

Check the base in the question. Committed capital gives a fee even on undrawn money.

Carried interest, no hurdle

Carry = carry rate × profit

Profit is usually after returning capital. Typical rate is 20%.

Hard hurdle carry

Carry = carry rate × (profit − hurdle amount)

Only the excess over the preferred return is shared.

Soft hurdle carry

Carry = carry rate × total profit, if the hurdle is met, the catch-up is 100% and profit is large enough to complete the catch-up

Without a full catch-up, or if profit is too small to complete it, carry is lower. With no catch-up, only the excess over the hurdle is shared.

Hurdle amount

Hurdle = capital × [(1 + hurdle rate)^years − 1]

Compounding is typical. Use simple interest only if the question says so.

Clawback amount

Clawback = carry received − carry rate × cumulative total fund profit

Applies when carry paid exceeds the agreed share of whole-fund profit.

Waterfall order (European)

1) Return of capital and fees; 2) preferred return; 3) catch-up; 4) split, such as 80/20

Whole-fund basis. American waterfall applies the order deal by deal.

Post-money valuation

Post-money = Pre-money + Investment

Investment means the new money put in during the round.

Pre-money valuation

Pre-money = Post-money − Investment

Use this when the question gives the post-money figure.

Investor ownership share

Ownership % = Investment ÷ Post-money

Equals investor's shares ÷ total shares after the round.

Share price in a round

Price per share = Pre-money ÷ Existing shares outstanding

New shares issued = Investment ÷ Price per share.

Required value at exit (VC method)

Required exit value for the stake = Investment × (1 + target return)^n

Divide by expected ownership at exit to get the company value needed. Allow for later dilution.

Present value of exit (VC method)

Post-money today = Whole-company exit value × Retention ratio ÷ (1 + target return)^n

Use the value of the whole company at exit, not the value of the investor's stake. The retention ratio is the share of ownership left after later dilution; use 1 if there is no dilution. Subtract the investment to get pre-money, then ownership today = Investment ÷ Post-money.

Equity value in an LBO

Equity value = Enterprise value − Net debt

Debt paydown raises equity value even if enterprise value is flat.

Value creation drivers

Equity gain = EBITDA growth effect + multiple change effect + debt reduction

The three standard sources of LBO value creation.

Paid-in capital (PIC)

PIC = cumulative capital called from investors to date

Not the same as committed capital. Uncalled commitment = committed capital − paid-in capital.

DPI

DPI = cumulative distributions ÷ paid-in capital

Realized return only. Uses cash already paid out to investors.

RVPI

RVPI = residual value (NAV of remaining holdings) ÷ paid-in capital

Unrealized and dependent on GP valuation.

TVPI

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

Total value created per unit of paid-in capital. Often called MOIC for a fund. Usually stated net of fees.

IRR

0 = Σ CFt ÷ (1 + IRR)^t, with calls as negative flows and distributions and residual value as positive flows

Solved with a financial calculator. Use the cash flow worksheet.

Fully realized fund

RVPI = 0, so TVPI = DPI

Once all holdings are sold, only distributions remain.

Capital structure priority

Senior secured > Mezzanine/subordinated > Equity, with unitranche spanning senior and subordinated risk

Lower priority generally means higher risk and higher expected yield, though not always. Unitranche is one loan that blends senior and junior features, so it is not a separate priority rung. Its yield sits between pure senior and mezzanine.

Floating-rate loan coupon

Coupon rate = Reference rate + Credit spread

Typical for direct lending. Income rises when the reference rate rises.

PIK interest

New principal = Old principal × (1 + PIK rate)

Interest is added to principal instead of paid in cash, so cash flow is deferred and risk at maturity increases.

Distressed debt return on purchase

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

Ignoring interim cash flows. Recovery value is uncertain and is the key risk.

Required return decomposition

Private debt yield ≈ risk-free rate + credit spread + illiquidity premium + complexity premium

A conceptual build-up. It is not an exact pricing formula, so use it to reason about why yields exceed public bonds.

Expected loss

Expected loss = probability of default × loss given default

Loss given default = 1 − recovery rate. Senior secured loans have lower loss given default than subordinated debt.

Priority of claims

Senior secured > senior unsecured > subordinated/mezzanine > equity

Higher in the ranking means a lower yield and a higher expected recovery.

Smoothed-return effect

Reported standard deviation (smoothed) < true standard deviation

Appraisal-based valuations understate risk and correlation with public markets.

Quick revision

• Private equity funds are usually limited partnerships run by a general partner, with investors as limited partners.
• Committed capital is called down as drawdowns, and cash comes back later as distributions.
• Venture capital funds early-stage companies; buyouts acquire mature companies, often with heavy leverage.
• Management fees are based on committed or invested capital; carried interest is a share of profits.
• A hurdle rate is the return the fund must beat before the general partner earns carried interest.
• A clawback lets limited partners recover excess carry paid to the general partner earlier.
• Typical exit routes are IPO, sale to a strategic buyer, secondary sale and recapitalisation.
• The J-curve shows negative early returns because fees and costs come before gains.
• IRR is money-weighted and depends on the timing of cash flows.
• Multiples of invested capital ignore timing, so a high multiple can still mean a low IRR.
• Private debt includes direct lending, mezzanine, venture debt and distressed debt.
• Private debt is usually less liquid than public bonds, and part of its extra return compensates for that illiquidity.

Common mistakes

• Saying venture capital uses heavy debt. Fix: Leverage defines LBOs. VC firms lack steady cash flow to service debt, so they use mostly equity. Mezzanine-stage financing is the exception and often uses convertible or subordinated debt.
• Mixing up seed stage and early stage. Fix: Seed funds a company with an idea or concept that may still be developing its product. Early stage funds a company that is starting or about to start operations but is not yet in commercial production and sales.
• Treating the management fee as performance-based. Fix: The fee is charged regardless of results. Only carry depends on performance and hurdles.
• Charging carry on total profit under a hard hurdle. Fix: Hard hurdle: carry only on profit above the hurdle. Soft hurdle: carry on all profit only with a full catch-up and enough profit to complete it.
• Calculating ownership as investment ÷ pre-money. Fix: Always divide by post-money, which includes the new investment.
• Confusing a secondary sale with a secondary offering in public markets. Fix: In private equity, a secondary sale means selling the stake to another PE firm or financial investor, not listing shares.
• Dividing by committed capital instead of paid-in capital. Fix: Use paid-in capital as the denominator for DPI, RVPI and TVPI. Committed capital only matters for uncalled amounts.
• Treating DPI as the total return. Fix: DPI counts only cash already returned. Add RVPI to get TVPI.
• Treating unitranche as the same as mezzanine Fix: Unitranche is one loan blending senior and junior risk with a single rate. Mezzanine is a separate subordinated layer behind senior debt.
• Assuming all private debt is subordinated and high risk Fix: Direct lending is typically senior secured. Risk depends on the rung and the borrower.

Exam tips

• Questions are three-option MCQs with no penalty for guessing, so always eliminate role-reversal options first.
• Memorize the venture stage order and a one-line definition of each; items often describe a company and ask for its stage.
• The standard contrast is LBO (mature, debt-heavy) versus VC (young, equity-heavy). Stems often hide it in details such as 'no revenue yet' or 'stable cash flows'.
• For structure items, remember GP equals management and unlimited liability; LP equals capital and limited liability.
• Read for words like 'bridge' or 'prior to IPO' (mezzanine) and 'concept' or 'prototype' (seed).
• Always check three words first: committed or invested, hard or soft, European or American. They decide the answer.
• Expect conceptual questions on which party benefits: European favours LPs, American favours the GP's early cash flow.
• For clawbacks, the test is whole-fund profit, so work out entitled carry before comparing.