Illiquid Assets: formula sheet

Key formulas

Liquidity premium

Liquidity premium ≈ Expected return (illiquid asset) − Expected return (comparable liquid asset)

Compensation for illiquidity. Compare assets with similar cash flow risk.

Net return after transaction costs

Net return ≈ Gross return − (Round-trip cost ÷ Holding period in years)

Costs matter less the longer you hold. Illiquid assets suit long holding periods.

Round-trip cost from a spread

Round-trip cost = Bid-ask spread ÷ Mid price (one full buy and sell)

Half-spread is the cost of one side only. Check whether the question wants one-way or round-trip.

Smoothed (appraisal) return

R(reported, t) = α × R(true, t) + (1 − α) × R(reported, t−1), with 0 < α ≤ 1

One common smoothing model. Smaller α means more smoothing and more understated volatility.

Volatility effect of smoothing

σ(reported) < σ(true) when 0 < α < 1

Reported returns are autocorrelated. Reported risk is biased down.

Smoothing model

r(t) = (1 − α) × r*(t) + α × r(t−1)

r(t) is the reported return, r*(t) the true return, α the smoothing parameter between 0 and 1. A larger α means more smoothing.

Geltner unsmoothing

r*(t) = [r(t) − α × r(t−1)] ÷ (1 − α)

You lose the first observation because it has no prior return. Use α estimated from the data.

Estimating α

α ≈ ρ1, the first-order autocorrelation of reported returns

Works when true returns are roughly uncorrelated over time and smoothing is first order.

Volatility adjustment

σ* ≈ σ(reported) × √[(1 + α) ÷ (1 − α)]

Holds if true returns are independent over time. Since the factor exceeds 1, true volatility is higher than reported.

Reported variance under smoothing

σ²(reported) = σ*² × (1 − α) ÷ (1 + α)

The same relationship rearranged. It shows how much variance smoothing removes.

Liquidity premium

Liquidity premium = E(R illiquid) − E(R comparable liquid asset)

Compare assets with similar cash-flow and credit risk. Otherwise you also capture other risk premiums.

Required return on illiquid asset

Required return = Risk-free rate + Market risk premium + Liquidity premium

Use this as the discount rate when valuing illiquid cash flows. A higher premium lowers present value.

Unsmoothing returns (Geltner, first-order)

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

φ is the smoothing (autocorrelation) parameter, with 0 ≤ φ < 1. It is often estimated from first-order autocorrelation.

Volatility after unsmoothing

σ(true) ≈ σ(obs) × √[(1 + φ) ÷ (1 − φ)] ... approximately, for a first-order smoothing process

Unsmoothed volatility is higher than observed. Higher φ means a bigger correction.

Illiquidity-adjusted Sharpe ratio

Sharpe = (R − Rf) ÷ σ

Use unsmoothed σ. Using reported σ overstates the ratio.

Liquidity-adjusted VaR

LVaR = VaR + ½ × Position value × Bid-ask spread

Exogenous spread cost for liquidating a position. Useful for tradable but less liquid assets.

Smoothed (appraisal) return

r(reported,t) = (1 − α) × r(true,t) + α × r(reported,t−1)

α is the smoothing weight between 0 and 1. Higher α means more stale pricing.

Unsmoothing

r(true,t) = [r(reported,t) − α × r(reported,t−1)] ÷ (1 − α)

Rearranged from the line above. Needs α below 1.

Effect of smoothing on volatility

σ(reported) < σ(true), and the Sharpe ratio is overstated

Autocorrelation in reported returns is a warning sign. This is the general direction, not an exact ratio.

Secondary market discount

Price = NAV × (1 − discount)

Discount = (NAV − price) ÷ NAV. It widens in stress.

Unfunded commitment

Unfunded = Total commitment − Capital called to date

Treat it as a contingent liquidity outflow.

Gate payout

Paid per investor = Requested × (Gate amount ÷ Total requested)

Applies when requests exceed the gate and the fund prorates. Check the fund's terms.

Total illiquid exposure

Illiquid exposure = NAV of illiquid assets + unfunded commitments

Use this for limits and stress tests, not NAV alone.

Stressed liquidity coverage

Coverage = stressed liquid assets ÷ stressed cash needs

Below 1 means a shortfall and a risk of forced sales.

Stressed net cash need

Net need = spending + capital calls − distributions

In a crisis, assume calls higher and distributions lower.

Illiquid share of portfolio

Illiquid share = illiquid NAV ÷ total portfolio value

Rises when liquid assets fall; this is the denominator effect.

Liquid assets after stress

Liquid after = liquid assets × (1 − market shock) − net cash need

Check this stays positive over the horizon.

Quick revision

• Illiquid assets are costly or slow to sell without a price concession.
• Smoothed returns show positive autocorrelation.
• Smoothing understates volatility, and often understates correlation with liquid markets.
• Understated risk makes the Sharpe ratio look better than it really is.
• Unsmoothing should raise estimated volatility compared with reported volatility.
• Appraisal-based and stale prices are common causes of smoothing.
• A liquidity premium is extra expected return for bearing illiquidity.
• Lockups, gates and notice periods limit an investor's ability to exit.
• Private equity investors face capital calls, so they need liquid reserves.
• Illiquidity risk is greatest in stress, when selling and funding needs coincide.
• Match illiquid allocations to liabilities and spending horizons.
• Always state the measure, the method and the interpretation in your answer.

Common mistakes

• Treating reported volatility of private assets as true risk. Fix: Remember appraisal smoothing biases volatility and correlation down. Unsmooth before using the data in risk models.
• Saying the liquidity premium is a guaranteed extra return. Fix: It is expected compensation for bearing illiquidity risk. Realised returns can be lower, especially in a crisis.
• Saying smoothing lowers the average return. Fix: Smoothing mainly changes the timing and variability of returns. The long-run average is not the main effect. The distortion is in volatility, correlation, beta and Sharpe ratio.
• Dividing reported volatility by √[(1 + α) ÷ (1 − α)] instead of multiplying. Fix: True volatility is larger. Multiply by √[(1 + α) ÷ (1 − α)], which is greater than 1 for positive α.
• Using reported volatility of private assets as true risk Fix: Unsmooth first. Expect higher volatility, higher correlation with equities and a lower Sharpe ratio.
• Treating the whole excess return as a liquidity premium Fix: Compare with a liquid asset of similar risk, or remove the market risk component first.
• Saying the J-curve means private equity loses money overall. Fix: The J-curve is a timing pattern. Fees and write-downs come first, gains arrive later.
• Ignoring unfunded commitments when assessing liquidity. Fix: Add unfunded commitments as contingent cash outflows in any liquidity stress.
• Measuring illiquid exposure by NAV only. Fix: Add unfunded commitments to NAV when judging total exposure and liquidity needs.
• Assuming distributions continue in a crisis. Fix: Assume lower distributions and sometimes higher calls in stress scenarios.

Exam tips

• Expect scenario questions: given an asset description, name the liquidity feature and its risk effect.
• Know the direction of bias: smoothing lowers volatility and correlation and raises the apparent Sharpe ratio.
• In cost questions, check one-way versus round-trip and annualise by holding period.
• Pick answers that call the liquidity premium compensation, not a free or certain return.
• Link this topic to liquidity-adjusted VaR: illiquid positions need a longer liquidation horizon.
• Expect questions that give autocorrelation and ask for corrected volatility, Sharpe ratio or VaR. Memorise the square-root factor.
• Know the direction of every effect: volatility, correlation and beta go up after unsmoothing; Sharpe ratio goes down.
• In conceptual questions, link the cause (appraisal-based, stale pricing) to the evidence (positive autocorrelation) and then to the risk consequence.