Secondaries as Temporal Arbitrage

Thesis. A secondary trade is mostly a trade in time, not in assets. The buyer and the seller value the same future cashflows using different discount curves. That difference in when money matters creates a price wedge and, if you are disciplined, a repeatable edge.

How to read this page. The flow is deliberate. Each section introduces one concept, shows the math, and then lets you touch it with an interactive. The thread is time:

1. Price one stake as a calendar.
2. Stack stakes into a curve.
3. Construct a portfolio on the curve.
4. Separate value from frictions.
5. Translate discounts into IRR.
6. Encode a deploy rule.
7. Add microstructure.
8. Stress the whole thing.

Act I — One stake is a calendar, not a ticker

Objective. Make time visible and measurable in a single position.

Setup. Work in discrete time points $t_1,\dots,t_n$ after deal date $t_0=0$. Let the signed net cashflow at $t_i$ be

with the convention calls are negative and distributions are positive. Let $m(t)$ be a stochastic discount factor (SDF) that maps one unit of cash delivered at $t$ into present dollars at $t_0$.

Pricing. The present value of the position is

A practical parameterization is $m(t) = \text{DF}(t)\cdot \text{SR}(t)$, where $\text{DF}(t)$ is a pure time-discount curve (for example $(1+r)^{-t}$ or $e^{-rt}$) and $\text{SR}(t)$ scales cash paid in risky states. In many underwriting models $\text{SR}(t)$ is taken piecewise-constant by horizon.

Two clocks, one calendar. For a seller $S$ and a buyer $B$:

If $V_B>V_S$ there exists a clearing price $P$ with $V_S\le P\le V_B$. The wedge $V_B-V_S$ exists even when beliefs about $X_i$ agree. It is a time preference spread.

IRR as the market quote. Given a trade price $P$ at $t_0$, IRR is the rate $r^\star$ that solves

Under standard PE cashflow shapes (one sign change), $r^\star$ is unique. When there are multiple sign changes, IRR can be ill-defined; use NPV at a curve of rates and quote implied yield instead.

Sensitivity. For a flat buyer curve $(1+r)^{-t}$, the PV sensitivity to $r$ is approximately

$D_{\text{mod}}$ is modified duration on net flows; it is large when cash comes back late.

How to use the interactive. Start with buyer at 10% and seller at 18%, set price near 0.8x NAV, then:

• Push a distribution 1 year later and watch PV fall more for the seller than the buyer if their curve is steeper.
• Flatten the seller curve to 12% and watch the wedge close.
• Note how the implied IRR at price sits between the two discount rates.

Thread to Act II. A single calendar is helpful, but the world is a book of calendars. Plot them on one axis to see the curve.

Act II — The private-market term structure

Objective. Map stakes onto a curve with x = remaining duration and y = implied yield.

Remaining duration. For positive legs only (distributions), define a Macaulay-like duration at buyer rate $r$:

For net flows, replace $D_{t_i}$ by $X_i$ when the denominator stays positive; otherwise quote $D^+$ to avoid nonsense.

Implied yield. For position $i$, implied yield $y_i$ is the solution to

where $P_i$ is the paid price today. Think of $y_i$ as a single-number summary of all timing and risk.

What the interactive shows. A scatter of $(RD_i,y_i)$ pairs. Toggle a trend to see the slope. Expect young vintages to sit at long $RD$ and higher $y$, and tail-end stakes to sit at short $RD$ with lower $y$ when exits are near.

Practical caveats:

• Selection matters. Tail-end assets are not random; they can be tougher or simply slow.
• Marks are sticky. If NAVs are stale, $y_i$ can be biased. Haircut stale sectors before reading the curve.

Thread to Act III. Once you see a curve, you can trade it: target where you want to live on the time axis and harvest roll-down.

Act III — Portfolio construction on a time axis

Objective. Combine stakes to hit a target time profile while respecting liquidity.

Duration math for a sleeve. Let $w_i$ be dollars invested in stake $i$ and $P_i$ its PV on your curve. Define PV weights $\omega_i = \frac{w_i P_i}{\sum_j w_j P_j}$. A sleeve-level remaining duration is

For near-term liquidity risk, define a 4-quarter call measure

Target $RD_{\text{port}}$ for time exposure and cap $\text{NTL}$ for operational liquidity.

Roll-down. If the curve is downward sloping at your point (yields fall as time shortens), holding a stake for $\Delta t$ earns:

• distributions received (carry),
• plus a mark-up from moving left along the curve (roll),
• plus selection/market noise.

A stylized PnL attribution over $\Delta t$ is

What the interactive shows. Pick two stylized stakes, see individual and combined calendars, IRRs, remaining durations, and near-term calls. Try a steepener: long a long-duration 2021 growth and pair with a shorter 2018 buyout to keep sector exposure balanced.

Thread to Act IV. The curve helps you position, but what are you actually paying for when someone quotes 0.82x NAV?

Act IV — What price really buys: value vs frictions

Objective. Separate fundamentals from liquidity and convexity.

A useful decomposition is

SDF factorization. Write $m(t)=\text{DF}(t)\cdot \text{SR}(t)$. DF captures pure time preference (funding curve, horizon). SR captures state risk. A CAPM-like linearization makes this concrete:

so cashflows that co-move with bad times (high $R_{m,t}$ when you are hurting) are discounted harder.

Implication. Two stakes with the same $RD$ can have different $y$ because their state loadings differ. Do not trade the time curve blind to states.

Thread to Act V. Suppose you buy the same calendar at a discount to NAV. How much IRR does that wedge deliver?

Act V — From discount to IRR: explicit and approximate

Objective. Translate price wedges into annualized return lift.

Exact lump-sum case. If $1$ of NAV returns at horizon $T$ and you pay $1-d$ today,

General staggered cashflows. A first-order approximation uses modified duration $D_{\text{mod}}$ at your base rate $r$:

when rates are moderate and $D_{\text{mod}}\approx D/(1+r)$. This is fixed income logic applied to PE calendar math.

What the interactive shows. Slide $d$ and $T$ to see how uplift scales. Notice how the rule $d/T$ understates uplift slightly because of compounding.

Thread to Act VI. Turn that mapping into a deploy rule that scales with market stress and capacity.

Act VI — A deploy rule that respects time and supply

Objective. Encode a go/no-go policy in two variables you can monitor in real time: market drawdown and horizon.

Policy.

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

• $d^\*$ is the minimum discount that clears your hurdle for a given horizon $T$ or duration $D$.
• $S^\*$ is a capacity floor; without supply you cannot scale a strategy.

Operationalizing. Tie expected discount to public stress via $d = \alpha + \beta \cdot \text{drawdown}$, clipped to $[0,0.5]$. Calibrate $\alpha,\beta$ on prints. Put a watchlist on LPs at risk of breaching policy bands so you can meet forced supply when it appears.

What the interactive shows. A heatmap of uplift in bps vs drawdown (x) and horizon (y). Tune $\alpha,\beta$ and see where the policy turns from hold to deploy.

Thread to Act VII. Prices are one thing; payoff geometry is another. Fees and carry create kinks that change time sensitivity.

Act VII — Microstructure bends time: waterfalls and convexity

Objective. Quantify how fees and carry skew $P(\tau)$ when exits are delayed or accelerated.

Model sketch.

• Fees: charge $f(t)$ annually on NAV until a step-down year $t_s$.
• Carry: European carry with percent $c$ on surplus above a hurdle $h$ applied to LP cashflows.

Let $\tau$ shift positive cashflows in time. The timing curvature is

When waterfalls introduce kinks (step-downs, catch-up), $\Gamma_\tau$ is often positive: delays hurt more than equivalent accelerations help.

What the interactive shows. Toggle fee step-downs and carry, then sweep $\tau$ to see $P(\tau)$. Read off how convexity changes with parameters.

Thread to Act VIII. Now add states. Link public stress to discounts, exit delays, and exit sizes, then look at distributions of money time.

Act VIII — Stress that respects states

Objective. Tie public drawdown to three channels: the discount you pay, the exit delay you suffer, and the exit size you realize. Inspect timing distributions and IRR percentiles.

Linking equations.

• Discount: $d = \alpha + \beta \cdot \text{drawdown}$, clip to $[0,0.5]$.
• Delay: $\Delta t \approx \kappa \cdot \text{drawdown}$ (months per 10% drawdown scaled to years).
• Size trim: $\text{multiplier} \approx 1 - \lambda \cdot \text{drawdown}/0.5$.

Outputs to read.

1. Drawdown to discount mapping. Points colored by exit delay. You should see a clean line with warmer colors at deeper drawdowns.
2. Cumulative net cashflow bands. P10 to P90 envelope over time. The median line tells you how the calendar shifts; the band width tells you uncertainty.
3. IRR histogram with p10/p50/p90. This is the meeting slide: under this stress model, where do outcomes land.

What the interactive shows. All three at once, using your theme, full width, with calibrated controls.

Checklist — Before you wire dollars

1. Compute $V_B$, $V_S$, wedge, and implied IRR at the quoted price.
2. Measure remaining duration $D$ for distributions and net flows.
3. Place the stake on the $(RD,y)$ curve and assess roll-down.
4. Decompose price into fundamentals, liquidity, convexity, and selection; note any NAV lag.
5. Convert discount to expected IRR uplift using $d/D$.
6. Apply the deploy rule with your calibrated $\alpha,\beta$ and live supply.
7. Inspect waterfall convexity for timing asymmetry.
8. Run the state stress and record p10, p50, p90 IRR.

Broader connections

• Fixed income: duration, roll-down, curve positioning. The difference is your curve is made of exit probabilities.
• Consumption-based asset pricing: $m(t)$ is marginal utility through time; secondaries let you trade against other institutions’ liquidity utility.
• Housing finance: prepayment and lock-in are timing risks that rhyme with exit risk.
• Infrastructure and climate: the market prices time to benefit and time to risk across decades; PE secondaries are the same equation in a closer time window.

Glossary

• SDF (stochastic discount factor). A function $m(t)$ with $\mathbb{E}[m(t)\cdot \text{return}(t)] = 1$.
• Duration. PV-weighted average time of cashflows.
• Roll-down. Price change from aging along the curve, holding credit/selection constant.
• GP-led vs LP-led. GP-leds reorganize assets and often extend duration; LP-leds transfer an LP stake as-is.

Closing

The thread is time. Price the calendar, see the curve, build on it, separate value from frictions, translate discounts into rates, encode a rule, account for kinks, and then stress it like you mean it. That is a coherent playbook for secondaries as temporal arbitrage.