The Denominator Effect Playbook

Watching policy bands break when $w_t = \tfrac{NAV_t}{NAV_t + \sum_j L_{j,t}}$ spikes and forces secondaries.

9/7/2025

Large institutional investors (pensions, endowments, insurers, sovereign funds) run with explicit policy bands for private equity. A typical investment policy statement sets a target PE weight with upper and lower limits, for example:

target $w_{\text{PE}} = 12\%$
min $= 8\%$ , max $= 16\%$

These bands are enforced at the total-portfolio level. When public markets suffer a fast drawdown while private NAVs lag, the measured PE percentage of total assets can jump above the max band even if nothing has changed in the underlying funds. That breach can force secondary sales of otherwise good assets.

Public markets tank.
The PE percentage of total assets rises because private marks lag.
Policy bands get breached, so otherwise good PE assets must be sold.
A motivated secondaries team buys those assets at a discount.
To systematize it, keep a live book of potential forced sellers and quote liquidity when shocks hit.

Controls

Starting mix (must sum to 100)

Public (%)

Fixed income (%)

Private equity (%)

Public drawdown (%)

One-shot shock applied to public assets

PE policy cap (%)

Max allowed PE weight of total portfolio

Baseline discount at 0% drawdown (%)

Discount increase per 10% drawdown (pp)

If 10% public drawdown, add this many percentage points

Discount noise (sd, %)

Sample size (points)

Horizon T (years)

Random seed

Definitions

Let total assets split into public and private:

A_{\text{total}} = A_{\text{public}} + A_{\text{PE}},\qquad w_{\text{PE}} = \frac{A_{\text{PE}}}{A_{\text{public}} + A_{\text{PE}}}.

Apply a public shock $s \in [0,1)$ while PE NAV is still marked at yesterday’s levels:

A_{\text{public}}' = (1-s)\,A_{\text{public}},\qquad A_{\text{PE}}' = A_{\text{PE}},

so the new PE weight is

w_{\text{PE}}' = \frac{w_{\text{PE}}}{1 - s(1-w_{\text{PE}})}.

What this says in plain language. When the denominator (publics) shrinks faster than the numerator (PE NAV), the ratio goes up. Bands are often hard caps:

w_{\text{PE}} \le w_{\text{PE}}^{\max}.

If $w_{\text{PE}}'$ crosses $w_{\text{PE}}^{\max}$ , you must sell PE exposure. That forced supply meets secondaries buyers.

Back‑of‑envelope: for small $w_{\text{PE}}$ , use $w' \approx w/(1-s)$ . A 20% drawdown moves a 10% weight to about $0.10/0.80 = 12.5\%$ (exact formula ~12.2%). Handy for quick checks; use the exact expression for sizing.

Portfolio weights before and after a shock

Start with 60/30/10 across Public/Fixed/PE. Hit Public with −20%. PE becomes a bigger wedge of the pie even if NAV is flat. If you cross the policy line, you must sell. If the sale clears at discount $d$ , selling $x$ of PE NAV changes total AUM by $d x$ . The minimal sale that restores the band solves

\frac{A_{\text{PE}}-x}{A_{\text{total}}' - d x} = w_{\text{PE}}^{\max} \quad\Rightarrow\quad x = \frac{A_{\text{PE}} - w_{\text{PE}}^{\max} A_{\text{total}}'}{1 - d\,w_{\text{PE}}^{\max}}.

With $d=0$ , this reduces to $x = A_{\text{total}}'(w_{\text{PE}}' - w_{\text{PE}}^{\max})$ .

Discounts widen when drawdowns deepen

In secondaries, price is NAV times $(1-d)$ with $d \in [0,1)$ :

P_{\text{sec}} = \text{NAV}\,(1 - d).

A stylized link between discounts and public drawdown severity $\Delta$ :

d = \alpha + \beta\,\Delta + \varepsilon.

What this chart is really saying. Each dot is a synthetic “print”: given a public market drawdown, what discount might a secondary sale clear at? The quantile bands show that as drawdowns deepen, the whole distribution of discounts shifts upward. This is a deliberately simple approximation of placement price vs stress level.

Ways to make it stronger in practice

Taken together, this view approximates the function LPs and buyers care about:

\text{Drawdown size} \;\mapsto\; \text{expected discount to clear}

That’s the wedge systematic secondaries can harvest.

IRR uplift from buying at a discount

When you buy a secondary interest, the fund’s future cash flows do not change. You are paying less up front for the same eventual distributions. That mechanical wedge shows up as a higher internal rate of return (IRR).

Setup.

Primary buyer: pays $1.00 for $1.00 of NAV, then receives $1.00 back over time.
Secondary buyer at discount $d$ : pays $(1-d)$ for that same NAV, but still receives the full $1.00.

IRR equation. By definition, IRR is the annualized rate $r$ that makes net present value zero. If all proceeds come back as a lump at horizon $T$ ,

-(1-d) + \frac{1}{(1+r)^T} = 0 \quad\Rightarrow\quad r_{\text{disc}} = \Big(\tfrac{1}{1-d}\Big)^{1/T} - 1.

Example. With $d=20\%$ and $T=3$ , the discounted buyer’s IRR is about $7.7\%$ per year above the base fund return.

Rule of thumb. Expand $-\ln(1-d) \approx d + d^2/2 + \dots$ to get a first-order approximation:

r_{\text{disc}} \approx \frac{d}{T}.

That’s why people say “20% discount over three years ≈ 7% uplift.” The exact is always a bit higher because compounding works in your favor.

IRR uplift from buying at a discount

Discount d (%)

Illustrative T (years)

Highlights this horizon on the chart

T = 3y

Exact ≈ 7.72%

d/T ≈ 6.67%

Gap ≈ 1.06 pp

Exact = (1/(1−d))^(1/T) − 1. Approx = d/T. The exact curve sits above the linear rule because −ln(1−d) ≥ d (convexity). For staged cash flows, replace T with duration D to get a first-order rule: uplift ≈ d / D.

General cash-flow patterns. If payouts are staggered instead of lump-sum, replace $T$ with the Macaulay duration $D$ (the present-value weighted average time you get your money back). Then the uplift is roughly

\Delta r \;\approx\; \frac{d}{D}.

Why duration matters. Duration is your time-in-seat. The sooner distributions arrive (shorter $D$ ), the more powerful the discount wedge is when annualized. Later distributions (longer $D$ ) dilute the effect.

Mathematical view of duration

Duration $D$ is the present-value weighted average time of your cash flows. If flows stay the same but your purchase price falls by $d$ , the fixed-income sensitivity gives

\frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta r \quad\Rightarrow\quad \Delta r \approx \frac{d}{D_{\text{mod}}} \approx \frac{d}{D}

for moderate base rates. This is bond math repurposed: a one-time price wedge divided by duration gives the approximate rate bump.

A threshold rule that scales

if discount >= d* and supply >= S*: deploy else: hold

$d^*$ is your minimum acceptable discount, calibrated to your return target and horizon (duration math helps set it).
$S^*$ is a minimum deal flow or supply level — without enough volume, you can’t put meaningful dollars to work.
To act quickly, keep a watchlist of LPs likely to breach policy bands after big public drawdowns, so you can quote liquidity right when they need it.

Computed as uplift ≈ (expected discount / horizon) × 10,000 bps, with discount = α + β × drawdown (clipped to [0, 50%]).

How to read the heatmap. Each cell shows the approximate annualized IRR uplift (in basis points) from buying at a discount, given two factors:

Horizontal axis (public drawdown): as markets fall, policy breaches force more sales and discounts widen.
Vertical axis (horizon): the number of years over which NAV is paid back; the longer the wait, the more the discount’s effect gets diluted.
Colors (uplift): green/yellow/red indicates how attractive the setup is. Bottom-right (deep drawdown, short horizon) is where the liquidity premium is strongest.

Traffic-light framing.

Green (discounts < 10%, limited supply): stay focused on primaries.
Yellow (10–20% discounts, medium supply): tilt more capital into secondaries.
Red (≥ 20% discounts, heavy supply): prioritize secondaries and consider structured liquidity.

This is constraint math turned into a playbook: market stress dictates discounts, policy breaches dictate supply, and your rules dictate when to lean in.

Where the uplift math comes from

The uplift is computed with a simple rule of thumb:

Discount model. $d = \alpha + \beta \times \text{drawdown}$ , clipped between 0% and 50%. $\alpha$ is a baseline frictional discount; $\beta$ is sensitivity to market stress.
IRR uplift. If you pay $(1-d)$ for $1$ of NAV and get it back in $T$ years,
$r_{\text{disc}} = (1/(1-d))^{1/T} - 1.$
For small $d$ , this expands to $r_{\text{disc}} \approx d/T$ .
Heatmap calculation. Each cell shows
$\text{uplift (bps)} \approx \frac{d}{T} \times 10{,}000,$
which is the annualized rate bump from buying at discount $d$ over horizon $T$ .

This is bond math in disguise: a one-time price wedge divided by duration gives an approximate yield change.

Why the effect exists at all

Measurement lag. Publics mark fast; private NAVs mark slow.
Hard bands. The denominator shrinks first, the ratio rises, the cap is breached, selling starts.
Liquidity wedge. Buyers demand a discount to underwrite uncertainty and time. That wedge is the premium the secondaries sleeve earns.

Connections. This is the same geometry as deleveraging spirals when vol spikes and as policy-band rebalances in public portfolios.

Neptune's Arkestra

The Denominator Effect Playbook

Controls

Definitions

Portfolio weights before and after a shock

Discounts widen when drawdowns deepen

IRR uplift from buying at a discount

IRR uplift from buying at a discount

A threshold rule that scales

Why the effect exists at all

Denominators move the world.