Estimating Capacity Loss in Nonprofit Networks under Correlated Funding Shocks

1. The problem

Regional nonprofit networks are usually described as though each organisation faces its own funding risk. In practice a small number of funders appear in a great many portfolios at once, so the failure of one funder is not one failure but many simultaneous ones.¹ The question this paper asks is narrow: given a network and a shock, how much delivered capacity survives?

2. Model

Let $F$ be the set of funders and $P$ the set of providers. Write $w_{fp} \geq 0$ for the annual funding flowing from funder $f$ to provider $p$, and let $c_p = \sum_{f \in F} w_{fp}$ be provider $p$‘s total funded capacity. A shock is a subset $S \subseteq F$ of funders that withdraw together.

Delivered capacity after the shock is not simply the residual funding, because a provider below its operating threshold $\tau_p$ delivers nothing at all. Writing $[\,\cdot\,]_+$ for the positive part, surviving capacity is

$$C(S) \;=\; \sum_{p \in P} \Big[\, c_p - \sum_{f \in S} w_{fp} \,\Big]_+ \cdot \mathbb{1}\!\left( c_p - \sum_{f \in S} w_{fp} \;\geq\; \tau_p \right).$$

The indicator is what makes the problem interesting. Without it, capacity loss is linear in the withdrawal and concentration does not matter. With it, a provider that loses a third of its funding may deliver nothing, while one that loses half may continue at half strength.

Define the shared-funder concentration of the network as the normalised second moment of provider overlap,

$$\kappa \;=\; \frac{1}{|P|^2} \sum_{p, q \in P} \frac{|F_p \cap F_q|}{|F_p \cup F_q|},$$

where $F_p$ is the set of funders supporting provider $p$. Networks in which providers draw on disjoint funders have $\kappa$ near zero; networks in which every provider draws on the same handful have $\kappa$ near one.

3. Result

Our main claim is that for a shock of fixed total value, expected capacity loss grows monotonically in $\kappa$, and that the growth is superlinear once $\kappa$ exceeds roughly $0.4$. The intuition is that concentration aligns the shocks: a withdrawal that would have been survivable spread across a network instead lands on the same providers at once, pushing many of them below $\tau_p$ together.

The table below reports simulated capacity retention for a synthetic network of 40 providers and 12 funders, under a withdrawal equal to 20% of total network funding.

Concentration $\kappa$	Providers below threshold	Capacity retained	Retention vs. linear prediction
0.12	2 of 40	81.4%	−1.6 pts
0.28	5 of 40	76.9%	−6.1 pts
0.41	11 of 40	68.2%	−14.8 pts
0.57	19 of 40	54.7%	−28.3 pts
0.73	27 of 40	41.1%	−41.9 pts

A linear model predicts 83% retention in every row. The gap in the final column is the entire contribution of this paper: it is the capacity that concentration destroys and that a per-organisation risk assessment cannot see.

4. Network structure

The figure shows the two extremes schematically. On the left, four providers draw on four largely separate funders; on the right, the same four providers draw on a shared core. The total funding is identical.

5. What follows

Two implications are worth stating plainly. Reserve policy set per organisation is set against the wrong risk, because the correlated case is the one that matters and it is invisible from inside a single organisation. And intermediary funders — the regranting bodies that are often criticised as an extra layer — reduce $\kappa$ by construction, which is an argument for them that the efficiency critique does not address.

References

1. Okonjo, R. (2023). Portfolio overlap among regional grantmakers. Working paper, Institute for Regional Policy.
2. Bhattacharya, S., & Lindqvist, M. (2022). Threshold effects in service delivery organisations. Public Administration Quarterly, 46(3), 388–410.
3. Mackie, C. J. (2021). Coordination costs in distributed philanthropy. Regional Policy Press.
4. Vance, T. (2024). Correlated risk in charitable funding: a survey. Nonprofit Review, 9(1), 5–41.

Footnotes

1. This is a fabricated example written to exercise a publishing pipeline. The argument is coherent but the results are invented, and nothing here should be cited. See the notice at the top of every page on this site. ↩