Geometry of Fund Multiples vs Rates

Multiples tell you how much. Rates tell you how fast. They live on different axes, and large samples make that difference visible. Same TVPI can sit next to very different IRRs depending on timing. That timing carves a shape in $(\text{TVPI}, \text{IRR})$ space, not a single curve ^[1].

Definitions and immediate consequences

Let $C_t \ge 0$ be contributions, $D_t \ge 0$ distributions, with terminal NAV $\text{NAV}_T \ge 0$.

• TVPI

  $$\text{TVPI} = \frac{\sum_{t=0}^{T} D_t + \text{NAV}_T}{\sum_{t=0}^{T} C_t}.$$

  TVPI is a scale statistic.

• IRR

  $$r \ \text{solves} \quad \sum_{t=0}^{T} \frac{-C_t + D_t}{(1+r)^t} + \frac{\text{NAV}_T}{(1+r)^T} \;=\; 0.$$

  IRR is a rate that makes the discounted cash flow polynomial equal zero.

When they coincide. If all contributions occur at $t=0$, all value comes back once at $t=T$, and $\text{NAV}_T=0$, then

so $r$ equals the simple CAGR of TVPI ^[2]. Depart from that lump-sum timing and the two metrics separate.

Continuous-time view. With a signed measure $d\text{CF}(t) = -C(t)\,dt + dD(t) + \text{NAV}_T \delta_T(dt)$,

IRR is a root of a Laplace transform of the cash-flow path ^[3]. Timing changes the transform, so it changes the root.

A toy slider: fix TVPI, move time

Fix TVPI and shift distributions earlier or later. IRR rises with earlier cash and falls with later cash, even though the wealth multiple is unchanged.

Key idea. At a given $r$, earlier positive cash flows have larger present value because $1/(1+r)^t$ is decreasing and convex in $t$ ^[4]. To keep NPV at zero, $r$ must adjust.

Two solved examples (same TVPI, different IRR)

Assume $C_0 = 100$, $\text{TVPI} = 1.5$, $\text{NAV}_T=0$.

• All-at-the-end (lower IRR bound for this horizon): $D_T = 150$.

  $$-100 + \frac{150}{(1+r)^T} = 0 \ \Rightarrow\ r = 1.5^{1/T} - 1.$$

  For $T=5$, $r \approx 8.43\%$.

• Split distributions earlier and later: $D_{2.5} = 75$, $D_5 = 75$. Let $x = (1+r)^{-2.5}$ so $(1+r)^{-5} = x^2$:

  $$-100 + 75x + 75x^2 = 0 \ \Rightarrow\ 3x^2 + 3x - 4 = 0 \ \Rightarrow\ x = \frac{-3 + \sqrt{57}}{6} \approx 0.7583.$$

  Then $(1+r)^{2.5} = 1/x \approx 1.3186$ and $1+r \approx 1.1169$, so $r \approx 11.7\%$.

Same TVPI. Different timing. Different IRR. This is the geometry in miniature.

Useful bounds and sanity checks

Assume $C_0>0$, $\text{NAV}_T=0$, and distributions occur in $[t_{\min}, t_{\max}]$ with $0 < t_{\min} \le t_{\max} \le T$.

• Lower bound: Lumping all $D_t$ at the latest date minimizes IRR:

  $$r \ge \text{TVPI}^{1/t_{\max}} - 1.$$

• Upper bound: Lumping all $D_t$ at the earliest date maximizes IRR:

  $$r \le \text{TVPI}^{1/t_{\min}} - 1.$$

  If $t_{\min}=0$ and $\text{TVPI}>1$, the upper bound is unbounded, reflecting that arbitrarily early positive cash can drive IRR arbitrarily high absent constraints. Realistic capital cycles impose $t_{\min}>0$.

• Evenly spaced $m$ installments over $[0,T]$ (contrib at $t=0$): Set $q = (1+r)^{-T/m}$. With equal installments $D_{jT/m} = C_0\,\text{TVPI}/m$ for $j=1,\dots,m$,

  $$\frac{\text{TVPI}}{m}\sum_{j=1}^{m} q^j = 1 \quad\Leftrightarrow\quad \text{TVPI}\cdot \frac{q(1-q^m)}{m(1-q)} = 1,$$

  which gives $q$ and hence $r$ by root-finding. As $m$ increases (holding TVPI and horizon fixed), IRR increases since more weight arrives earlier.

Sensitivity to timing (implicit function view)

Let

Here $\theta$ shifts distributions later when it increases. At the IRR, $F(r,\theta)=0$. By the implicit function theorem,

Since $\partial F/\partial \theta = -\sum_t D_{t+\theta}\,\ln(1+r)\,(1+r)^{-t} < 0$ and $\partial F/\partial r < 0$, we get $\frac{dr}{d\theta} < 0$. Making distributions later reduces IRR. Making contributions later raises IRR by the same logic. This is the algebra beneath the slider ^[4].

A convenient one-parameter stretch of the distribution clock is $D_t \mapsto D_{st}$ with $s>1$ meaning “later.” Then $\partial F/\partial s < 0$ and again $dr/ds<0$.

Statistical view: $p(\text{IRR} \mid \text{TVPI})$

Simulate smooth capital schedules (e.g., Beta-shaped calls and dists) and condition on TVPI. You get a distribution of IRRs at each multiple, not a single value.

• Near TVPI ~ 1.0, timing noise dominates and $r$ spreads widely.
• As TVPI rises, the spread narrows but remains a band, not a point.
• The map $(\text{TVPI} \to \text{IRR})$ is many-to-one. The shape matters ^[1].

The joint map in $(\text{TVPI}, \text{IRR})$

Estimate the joint geometry and color bins by average lateness of distributions.

• At fixed TVPI (vertical slices), you see IRR bands. Darker color means later distributions and lower IRR.
• Ridges tilt upward: larger multiples push IRR up, but the slope depends on timing, not just scale.

Subscription lines in one equation

For an LP facing delayed capital calls by $\Delta$ with line interest $i$, a simple one-lump toy rewrites the equation as

ignoring line cost on the GP balance sheet. The reduced time-in-seat $\Delta$ raises the LP’s IRR relative to no line. If you include financing cost as a reduction in effective TVPI (or an added outflow), the lift is partially offset, but the timing effect remains ^[6].

Concrete pair: fixed TVPI, diverging IRRs

Two stylized funds hit TVPI = 1.5 with different timing profiles.

Read them as two points on the same vertical slice of the map. The gap is timing, not scale.

Caveats

• NAV noise. Interim NAVs introduce estimation error into both metrics.
• IRR pathologies. Multiple sign changes can yield multiple or no IRRs.
• Model vs reality. Simulations use smooth shapes; real exits are lumpy with fees.

Wrap

Multiples summarize the scale of outcome. Rates summarize the pace of the path. Their relationship is not a function; it is a geometry shaped by time. Use both.

Notes