Modeling Non-profit Organisations’ Funding Diversification

Abstract

There is a rich literature connecting for profit financial portfolio theory to non-profit organisations’ funding diversification policies. This research typically ignores fundamental specificities of the non-profit context. The present paper integrates three of these specificities: non-profits’ performance being measured by other standards than their financial profitability, the non-proportionality between funding expenses and expected incoming funds due to varying marginal effects of the former, and crowding in/out effects between non-profit funding sources. It is shown that, contrary to what is the case in financial portfolio theory, a positive relation between expected performance and its variability cannot be taken for granted when increasing the number of non-profit’s funding sources. An important reason for this is the presence of crowding effects, which therefore cannot be ignored when looking for the best non-profit’s funding diversification strategy.

Introduction

Non-profit organisations are commonly defined as organisations founded to pursue non-financial objectives, not allowing the distribution of any financial surplus to owners or staff (Hansmann, 1987). Having other objectives than wealth maximising implies that their managerial behaviour (in all functional domains) can differ from that of the for profit organisations. Obviously, like all other kinds of organisations, they need financial resources to enable their operations. Those resources can be drawn from different sources, such as donors, governments, clients, and endowments, none of them necessitating the pursuit of financial returns, unlike the financial sources feeding for profit organisations.

It is generally accepted that non-profit organisations have more potential financial sources than for profit organisations (Tuckman, 1993). Their observed combination henceforward will be called the organisation’s funding portfolio. Young’s (2017, 41, 45) ‘benefits theory’ posits that the composition of this portfolio depends on the organisational mission and the ensuing activities and perceived benefits. This insight is consistent with the observation by Chang and Tuckman (1994, pp. 281–284) that industry affiliation, reflecting a specific category of missions, affects non-profit funding portfolio diversification (capturing both the number and distribution of financial resources in the funding portfolio). A link with institutions (and hence with institutional theory) is already found in results by Salamon and Anheier (1998, p. 219), who document marked differences in funding portfolio composition between countries (controling for industry).

Starting with Chang and Tuckman (1994), there is a long tradition of empirical research linking funding diversification to non-profit organisations’ financial health/vulnerability, ¹ mostly presented in a resource dependency framework. This has yielded qualitatively diverging conclusions as to the question whether more or less portfolio diversification is beneficial, balancing dependency considerations relevant in case of a few (or one) source(s) and rising fundraising costs together with possible mission-related conflicts in case of numerous sources.

As to theory on non-profit fundraising diversification, since the publication of Kingma’s (1993) seminal paper applying ‘modern’ financial portfolio theory (Markowitz, 1952) to select a portfolio of non-profit organisations’ funding sources (government and donations in his case), this theory attracted much research attention. Comprehensive literature reviews by Hung and Hager (2019) and Qu (2019) provide excellent and up-to-date states of the art, but also the warning that ‘one must be very cautious in directly transplanting portfolio theory into the nonprofit setting without considering … complications’ (Qu, 2019, p. 210). An obvious reason for this is the fact that non-profit performance differs from purely financial performance. Another problem, already alluded to by Jegers (1997, p. 68), is the observation that traditional portfolio theories discuss the composition of financial portfolios generating independent (albeit correlated) incoming streams of financial revenues (interests, dividends, gains realised when selling financial instruments), whereas the non-profit funding portfolio literature deals with the configuration of periodically incoming funds possibly directly affecting each other such as the abovementioned subsidies/grants and donations, or, among others, programme/earned revenues, all of which can be subdivided in finer-grained categories. Each of these categories is linked to a ‘funding source’, from which funds can be tapped as the consequence of specific expenses (e.g., fundraising expenses), which in cases of spontaneously incoming funds might be zero. In what follows these expenses will be called ‘funding expenses’.

This has at least two important implications, justifying the development of a financial portfolio model focused on non-profits. First, investors in financial portfolios can exactly determine the expected return of each instrument in their portfolio, the expected return being proportional to the amount invested in the instrument, a proportionality which cannot be expected when considering the relation between funding expenses and the amount of incoming funds, as one cannot a priori assume the marginal effect of funding expenses to be constant, each additional Euro of funding expenses not always raising the same amount of funds.

Second, financial portfolio theory does not consider crowding in/out effects known to characterise relations between non-profit organisations’ funding sources. One such effect is the extensively studied interaction between donations and subsidies assessing whether higher levels of one lead to higher or lower levels of the other. ² Such crowding mechanisms are fundamentally different from random co-movements, called hereafter stochastic correlations, implying that both should be explicitly distinguished, in theory as well as in empirical work.

The present paper presents a dedicated theory of non-profit organisations’ funding diversification accounting for the abovementioned differences between the for profit and non-profit funding contexts. ³ A novel taxonomy of conceivable non-profit organisations’ funding portfolio models is presented combining the absence/presence of crowding effects with the absence/presence of stochastic (co)variations. Original insights, contributing to the extant non-profit funding portfolio theory, are derived from the implications of the different models. Firstly, the traditional positive risk-performance association and the ensuing portfolio selection steered by risk attitudes will not automatically prevail, as ‘traditional’ diversification strategies will not always reduce the non-profit funding portfolio risk. Further, it will be shown that, as a consequence, optimal funding portfolios need not contain funds from all available funding sources, also contrasting with financial portfolio theory, in which by virtue of the separation theorem optimal portfolios are a combination of a risk-free asset and the market portfolio, by definition containing all financial assets.

In what follows the different non-profit funding portfolio models studied will be presented in ascending order of complexity and realism. For each model mathematical optimisation techniques will be used to characterise ‘optimal’ portfolios in terms of organisational performance and performance variability. Practical implications will be suggested. In the concluding section, the model’s main implications will be reiterated, followed by a discussion, an overview of limitations, and opportunities for further research.

Portfolio Models of Funding Diversification in Non-profit Organisations

We assume the non-profit organisation has an organisational objective function P, which is not financial profitability as in traditional portfolio theory. It is commonly known that in non-profit organisations, as elsewhere, managerial objectives need not coincide with board objectives (Van Puyvelde et al., 2012). Thus we model P, in line with the seminal work of Steinberg (1986), in a stylised way as the weighted average of service level (reflected by the amount of funds available for providing services) and budget level (gross funding amount). The weight k (∈[0,1]) is determined by the relative power, respectively, of board (assumed to prefer as much activity as possible) and management (assumed to strive for as high budgets as possible), and is therefore an indicator of management-board agency problems within the organisation. When the board (management) has all the power, k = 1 (k = 0). ⁴ The eventual objective pursued by the organisation (P) is the outcome of a balancing process between board and management.

Considering n funding sources, this leads to the following expression for the expected organisational performance, given the level of funding expenses, E being the expected value operator:

$\mathrm{E}\left(\mathrm{P}\right)=\mathrm{E}({\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}(\mathrm{k}\left({\mathrm{F}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}}\right)+\left(1-\mathrm{k}\right){\mathrm{F}}_{\mathrm{i}}))=\mathrm{E}\left({\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}}_{\mathrm{i}}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}\right)\right)=\mathrm{E}\left(\mathrm{F}-\text{kf}\right)$

where, in present value terms or in a one period model, f_i (≥0) are the funding expenses related to funding category i, and F_i the funding inflows from source i. In turn, F_i is shorthand for ${\mathrm{F}}_{\mathrm{i}}({\mathrm{f}}_{\mathrm{i}},{\mathbf{F}}_{-\mathbf{i}})+{\mathrm{\epsilon }}_{\mathrm{i}}$ , where ε_i is the stochastic fluctuation and F_-i the vector of the (n-1) values F_j (j ≠ i). ${\mathrm{F}}_{\mathrm{i}}({\mathrm{f}}_{\mathrm{i}},{\mathbf{F}}_{-\mathrm{i}})$ is assumed to be increasing at a decreasing rate in f_i, making the non-stochastic part of F_i concave in f_i. ⁵ ${\mathrm{F}}_{\mathrm{i}}({\mathrm{f}}_{\mathrm{i}},{\mathbf{F}}_{-\mathrm{i}})$ is assumed to be strictly positive, even if no funding efforts are made $({\mathrm{F}}_{\mathrm{i}}\left(0,{\mathbf{F}}_{-\mathrm{i}}\right)>0$ , but might be very small). F and f are, respectively, (the present value of the) total funds generated and (the present value of the) total funding expenses.

If there is no uncertainty as to the funds generated, striving for the highest attainable performance level (P*) is optimal.

If however the funds generated are stochastic, uncertainty comes into play. Then, optimality will be determined by the interplay among three factors: the expected level of P (called in what follows ‘expected performance’), its variance (‘performance variance’), and the level of organisational risk-aversion or, in other words, the expected performance the organisation is willing to sacrifice for a reduction in performance uncertainty. If risk-neutrality prevails, maximising P will determine the optimal funding portfolio as for the case without uncertainty.

Six different basic models can be considered, differentiated by the (non) inclusion of crowding effects, stochastic effects ε_i, ⁶ and correlations between funding sources ⁷ :

No crowding, no stochastic effects: $\forall \mathrm{i}$ : F_i = F_i(f_i)

No crowding, stochastic effects: $\forall \mathrm{i}$ : F_i = F_i(f_i) + ε_i with E(ε_i) = 0

No covariances between funding sources: cov(ε_i, ε_j) = 0 for i ≠ j

Covariances between funding sources: cov(ε_i, ε_j) ≠ 0 for i ≠ j

Crowding, no stochastic effects: $\forall \mathrm{i}$ : F_i = F_i(f_i, F _-i )

Crowding, stochastic effects: $\forall \mathrm{i}$ : F_i = F_i(f_i, F _-i ) + ε_i with E(ε_i) = 0

No covariances between funding sources: cov(ε_i, ε_j) = 0 for i ≠ j

Covariances between funding sources: cov(ε_i, ε_j) ≠ 0 for i ≠ j

Clearly, the more complex models are more realistic.

No Crowding, No Stochastic Effects

As this is a non-random, deterministic, case, no expectation operator is needed to derive the optimal funding portfolio (F₁, …, F_n):

$\underset{{\mathrm{f}}_{\mathrm{i}}}{\max }{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}\right)$

For each funding source i the amount invested, assuming interior solutions, should meet

$\frac{\mathrm{d}{\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\right)}{\mathrm{d}{\mathrm{f}}_{\mathrm{i}}}=\mathrm{k}$

Given the concavity of ${\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\right)$ higher values of k (the organisation being more inclined to pursue the board’s objectives) imply lower funding expenses.

No Crowding, Stochastic Effects, No Covariances Between Funding Sources

Introduction

Now, we consider not only the expected value of P, but also its variability, ⁸ which with no covariances is simply the sum of the variances related to each funding source:

$\mathrm{E}\left({{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\right)+{\mathrm{\epsilon }}_{\mathrm{i}}\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}\right)\right)$

${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}$

The conditions derived in the previous section also now characterise the maximally attainable level of P (P*), being the effect of optimal funding expenses ${\mathrm{f}}_{\mathrm{i}}^{*}$ . However, we must now take variability into consideration. Risk-aversion implies that organisations are willing to trade off P for reduced uncertainty. How this is done depends on the properties of $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}$ .

Equal Variances

In this case $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}=\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{j}}=\text{var}\hspace{0.17em}\mathrm{\epsilon }$ . Assuming funding from each of the funding sources (remember F_i(0) > 0), total variance of P is $\mathrm{n}.\text{var}\hspace{0.17em}\mathrm{\epsilon }$ . Starting at P*, increasing or decreasing one or more of the ${\mathrm{f}}_{\mathrm{i}}^{*}$ would decrease the expected P without any variance change. The only way to decrease total variance is to completely eliminate funding sources, ⁹ which is more drastic than setting funding expenses to zero as then some funds still can be expected (by the assumption that F_i(0) > 0). As we want to obtain the highest possible expected P for a given variance, the first funding source to be eliminated is the one for which $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)$ is minimal. Call the ensuing level ${\mathrm{P}}_{-1}^{*}$ . Of the remaining funding sources we again select the one for which $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)$ is minimal, leading to ${\mathrm{P}}_{-2}^{*}$ , continuing until no funding sources are left. The obtained performance variance-E(P) combinations then are $\left(\left(\mathrm{n}-\mathrm{j}\right)\text{var}\hspace{0.17em}\mathrm{\epsilon },{\mathrm{P}}_{-\mathrm{j}}^{*}\right)$ , with ${(\mathrm{P}}_{-\mathrm{j}}^{*}-{\mathrm{P}}_{-\left(\mathrm{j}+1\right)}^{*})>\left({\mathrm{P}}_{-\left(\mathrm{j}-1\right)}^{*}-{\mathrm{P}}_{-\mathrm{j}}^{*}\right)$ . Connecting these combinations in a performance variance-E(P) space results in a concave (kinked) curve starting at (0, 0) and ending in (n.var ε, P*) (see Figure 1 for n = 5 and var ε = 1). Note that the choice set consists only of these combinations, and not of the segments linking them. Which combination is optimal depends on the level of risk-aversion: under the given assumptions avoiding risk means limiting the number of funding sources and hence sacrificing an amount of expected performance, essentially because of the absense of covariance-based diversification effects. ¹⁰ The sources selected will be the ones with the highest values of $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)$ .

Figure 1.

Performance variance-E(P) curve (no crowding, no covariances, constant and equal variances).

Different Variances

In a more general case, we have $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}\ne \text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{j}}$ . Also in this case, when starting at P*, increasing or decreasing one or more of the ${\mathrm{f}}_{\mathrm{i}}^{*}$ would decrease E(P) without any variance change. Hence, again, total variance can only be reduced by reducing the number of funding sources. It is straightforward to see that, covariances being zero, the effect of eliminating a funding source i will have a fixed effect on E(P) and performance variance, whatever the number of sources left: E(P) will decrease by (F_if_i*−kf_i*) and total variance by $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}$ . The performance variance-E(P) combinations leading, once connected, to a concave (kinked) curve also starting at (0, 0) and going to ( ${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}$ , P*) can only be obtained by eliminating the funding sources in ascending order of $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)/\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}$ . As in the case of equal variances, more risk-aversion will imply (at least weakly) fewer funding sources and less expected performance. The sources selected in the eventual portfolio will be the ones with the highest values of. Note that the rule derived above for the case $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}=\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{j}}=\text{var}\hspace{0.17em}\mathrm{\epsilon }$ is just a special case of this one.

No Crowding, Stochastic Effects, Covariances Between Funding Sources

Introduction

Now, expected performance and performance variance are:

$\mathrm{E}\left({{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\right)+{\mathrm{\epsilon }}_{\mathrm{i}}\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}\right)\right)$

${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}+{\displaystyle \sum }_{\mathrm{i}}{\displaystyle \sum }_{\mathrm{j}\ne \mathrm{i}}\text{cov}\left({\mathrm{\epsilon }}_{\mathrm{i}},{\mathrm{\epsilon }}_{\mathrm{j}}\right)$

Equal Variances and Correlations

We start again with the simplest case: equal variances ( $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}=\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{j}}=\text{var}\hspace{0.17em}\mathrm{\epsilon }$ ), and equal correlations (−1 ≤ ρ ≤ 1) between each pair of two funding sources $(\text{cov}({\mathrm{\epsilon }}_{\mathrm{i}},{\mathrm{\epsilon }}_{\mathrm{j}})=\rho \text{var}\hspace{0.17em}\mathrm{\epsilon })$ . Performance variance with a portfolio of all n funding sources then is

$\mathrm{n}.\text{var}\hspace{0.17em}\mathrm{\epsilon }+\rho \mathrm{n}\left(\mathrm{n}-1\right)\text{var}\hspace{0.17em}\mathrm{\epsilon }$

For a portfolio of s (≤n) sources this becomes

$\mathrm{s}.\text{var}\hspace{0.17em}\mathrm{\epsilon }+\rho \mathrm{s}\left(\mathrm{s}-1\right)\text{var}\hspace{0.17em}\mathrm{\epsilon }$

Variance by definition being positive implies the following condition for ρ:

$\mathrm{\rho }(\mathrm{n}\hspace{0.17em}–\hspace{0.17em}1)\ge -1\hspace{0.17em}\text{or}\hspace{0.17em}\mathrm{\rho }\ge \frac{-1}{\left(\mathrm{n}\hspace{0.17em}–\hspace{0.17em}1\right)\hspace{0.17em}}$

In the limiting case $\mathrm{\rho }\left(\mathrm{n}-1\right)=-1$ performance variance of a portfolio maximising E(P) is zero, hence under that condition this portfolio is optimal.

Going from s sources in the portfolio to (s + 1) sources changes the performance variance independently from the specific funding sources involved by

$\text{var}\hspace{0.17em}\mathrm{\epsilon }+2\mathrm{\rho }\text{svar}\hspace{0.17em}\mathrm{\epsilon }$

The conditions for reaching the highest possible E(P) still are the same, as is the observation that performance variance can only be affected by increasing/decreasing the number of funding sources, but not by changing funding expenses.

If $\mathrm{\rho }>0$ increasing the number of funding sources increases performance variance at an increasing rate (due to the impact of s in the last formula). So, we can obtain the performance variance-E(P) curve by starting with a one source portfolio selected by looking at the funding source for which $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)$ is maximal, then selecting the remaining source for which $\left({\mathrm{F}}_{\mathrm{j}}\left({\mathrm{f}}_{\mathrm{j}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{j}}^{*}\right)$ again is maximal, and so on. Performance variance increasing at an accelerated rate and E(P) decreasingly increasing step by step results in performance variance-E(P) combinations forming, when connected, a concave kinked curve. As in the case without covariances, risk-aversion will determine the optimal number of funding sources.

Diversification benefits, in the sense that performance variance is smaller due to correlations between funding sources than without correlations, can only occur for a negative value of $\mathrm{\rho }$ . Three situations are then possible: adding a new funding source always makes the performance variance increase, adding a new funding source always makes performance variance decrease, or adding a new funding source only up to a given number of sources increases performance variance, after which it starts decreasing.

If $2\mathrm{\rho }(\mathrm{n}\hspace{0.17em}–\hspace{0.17em}1)>–1$ we are in the first situation, ¹¹ as from s < n this implies $2\rho \mathrm{s}>-1$ . The difference with the case in which ρ > 0 is that by adding new funding sources performance variance increases at a decreasing rate. Consider the kinked curve connecting the performance variance-E(P) combinations of the portfolio constructed by sequentially adding the remaining source for which, given the number of already included sources, $\left({\mathrm{F}}_{\mathrm{j}}\left({\mathrm{f}}_{\mathrm{j}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{j}}^{*}\right)/\left(\text{var}\hspace{0.17em}\mathrm{\epsilon }+2\mathrm{\rho }\text{svar}\hspace{0.17em}\mathrm{\epsilon }\right)$ is maximal (or equivalently $\left({\mathrm{F}}_{\mathrm{j}}\left({\mathrm{f}}_{\mathrm{j}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{j}}^{*}\right)$ is maximal, as the denominator is the same for all funding sources considered). This curve is concave only if each $\left({\mathrm{F}}_{\mathrm{j}}\left({\mathrm{f}}_{\mathrm{j}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{j}}^{*}\right)/\left(\text{var}\hspace{0.17em}\mathrm{\epsilon }+2\mathrm{\rho }\text{svar}\hspace{0.17em}\mathrm{\epsilon }\right)$ is smaller than the previous one despite the denominator decreasing with increasing s. Stated otherwise, the curve connecting the optimal performance variance-E(P) combinations is not necessarily concave (and in an extreme case even convex). As before, risk attitude will determine the optimal portfolio, but not so elegantly as when the performance variance-E(P) curve would be concave.

The second situation will arise when $\mathrm{\rho }<-\hspace{0.17em}½$ . This could be considered diversification at its best: increasing the number of funding sources increases E(P) and decreases performance variance. The optimal portfolio then contains all funding sources, for which optimal funding efforts are made. Note however that this situation is only possible when there are two funding sources. In that case $\mathrm{\rho }\left(\mathrm{n}–1\right)=\mathrm{\rho }\ge -1$ is met. For n ≥ 3, ρ must at least equal - ½, contradicting $\mathrm{\rho }<-\hspace{0.17em}½$ . All in all, this signifies that in practice such a situation will rarely be observed.

The third situation occurs when $2\mathrm{\rho }\left(\mathrm{n}\hspace{0.17em}–\hspace{0.17em}1\right)<-1$ (but of course $\mathrm{\rho }\left(\mathrm{n}\hspace{0.17em}–\hspace{0.17em}1\right)\ge -1$ ) and $2\mathrm{\rho }>-1$ . Say that from the ( $\stackrel{\sim }{\mathrm{s}}+1$ )^th source onwards the performance variance starts decreasing. As in the previous case, the performance variance-E(P) combinations result in a (not necessarily concave) kinked curve up to the ${\stackrel{\sim }{\mathrm{s}}}^{th}$ funding source and the point ( $\mathrm{n}.\text{var}\hspace{0.17em}\mathrm{\epsilon }+\rho \mathrm{n}\left(\mathrm{n}-1\right)\text{var}\hspace{0.17em}\mathrm{\epsilon }$ , P^*) ¹² . Generally speaking, the smaller the difference between the E(P) of the first $\stackrel{\sim }{\mathrm{s}}$ funding sources and P^*, and the lower risk aversion, the higher the chances that a funding portfolio consisting of all funding sources would be optimal (see Figure 2 for an example).

Figure 2.

Example of a performance variance-E(P) curve without crowding and with constant and equal (co)variances.

Different Variances and Correlations

As in the section ‘Different Variances’ when analysing the situation without crowding but with stochastic effects and no covariances between funding sources, we now have $\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}\ne \text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{j}}$ .

A first case is the one in which all covariances are positive. As in that same section, and as shown in Appendix 2, we start from the P^* portfolio and sequentially drop from the remaining s funding sources the source i for which $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)/\left(\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}+2{{\displaystyle \sum }}_{\begin{array}{l}\mathrm{j}=1\\ \mathrm{j}\ne \mathrm{i}\end{array}}^{\mathrm{s}}\text{cov}\left({\mathrm{\epsilon }}_{\mathrm{i}},{\mathrm{\epsilon }}_{\mathrm{j}}\right)\right)$ is minimal. Connecting the n performance variance-E(P) points results in a kinked concave function. The connecting segments’ slopes are equal to $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)/\left(\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}+2{{\displaystyle \sum }}_{\begin{array}{l}\mathrm{j}=1\\ \mathrm{j}\ne \mathrm{i}\end{array}}^{\mathrm{s}}\text{cov}\left({\mathrm{\epsilon }}_{\mathrm{i}},{\mathrm{\epsilon }}_{\mathrm{j}}\right)\right)$ and increasing when starting from the P* portfolio. So, in this case, more risk-aversion will imply less funding sources.

The next case ¹³ is that when all covariances are negative. Dropping a funding source (i) from a portfolio of s sources results in the following reduction (which can be negative, then reflecting an increase) of the performance variance:

$2\sqrt{\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}}\left({\mathrm{\rho }}_{1\mathrm{i}}\sqrt{\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_1}+{\mathrm{\rho }}_{2\mathrm{i}}\sqrt{\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_2}+\dots +\frac{1}{2}\sqrt{\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{i}}}+\dots +{\mathrm{\rho }}_{\text{si}}\sqrt{\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_{\mathrm{s}}}\right)$

The correlations being negative, three subcases are a priori possible: the reductions (call them, with a common symbol, Δ) are all positive for all s, they are all negative for all s, and some are negative while other are positive. The first subcase (implying the correlations are (very) small in absolute terms) can be analysed in the same way as the case of positive correlations. The second subcase implies that adding a funding source would always both increase E(P) and decrease the ensuing performance variance, automatically making the portfolio combining all sources at their values $\left({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)\hspace{0.17em}-\mathrm{k}{\mathrm{f}}_{\mathrm{i}}^{*}\right)$ optimal. The third subcase leads to no general conclusion. A numerical example shows that even in a very simple case entirely different situations can arise: consider three funding sources (labeled 1, 2, and 3) each with the same maximal $\mathrm{E}\left(\mathrm{P}\right)={\mathrm{F}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}^{*}\right)-\mathrm{k}\hspace{0.17em}{\mathrm{f}}_{\mathrm{i}}^{*}$ and variances 1, 4, and 9 respectively. We construct examples with all correlations being negative, meeting the condition derived by Rousseeuw and Molenberghs (1994) and Langford et al. (2001):

$1+2{\mathrm{\rho }}_{12}{\mathrm{\rho }}_{13}{\mathrm{\rho }}_{23}-{\mathrm{\rho }}_{12}^2-{\mathrm{\rho }}_{13}^2-{\mathrm{\rho }}_{23}^2\ge 0$

The first example (Table 1) shows a situation in which reducing the number of sources in the funding portfolio from the maximum number 3 always increases performance variance. Going from 2 sources to 1 source makes Δ positive or negative, depending on the funding sources involved. Note that the variance of source 1 is smaller than that resulting from the portfolio of all sources and all portfolios of 2 sources. Hence, it is impossible to construct a concave curve connecting the relevant performance variance-E(P) combinations.

Table 1.

Example 1.

ρ12 = −.2 ρ13 = −.8 ρ23 = −.4
Variance portfolio (123)	3.6
Variance portfolio (12)	4.2
Variance portfolio (13)	5.2
Variance portfolio (23)	8.2
Variance 1	1
Variance 2	4
Variance 3	9

The second example (Table 2) is one in which such a connecting curve can be drawn: going from the portfolio (123) to (12) reduces performance variance by 6.6, and then further leaving out source 2 further reduces it to 1 (decrease of 3.5). Note that performance variance would increase if going from portfolio (13) to (3), so we again have positive and negative values of Δ.

Table 2.

Example 2.

ρ12 = −.1 ρ13 = −.2 ρ23 = −.1
Variance portfolio (123)	11.2
Variance portfolio (12)	4.6
Variance portfolio (13)	8.8
Variance portfolio (23)	11.8
Variance 1	1
Variance 2	4
Variance 3	9

The last example (Table 3), also with positive and negative values of Δ, describes a case where there is only one possibility to go from 3 to 2 sources (12) of funds in the portfolio while decreasing performance variance (by .6). Further decreasing the number of sources leads to a more pronounced variance decrease (.8 or 3.8), again making the curve connecting the relevant variance-E(P) combinations not concave.

Table 3.

Example 3.

ρ12 = −.05 ρ13 = −.20 ρ23 = −.60
Variance portfolio (123)	5.4
Variance portfolio (12)	4.8
Variance portfolio (13)	8.8
Variance portfolio (23)	5.8
Variance 1	1
Variance 2	4
Variance 3	9

The last case, in which some correlations are positive and other ones negative does not differ fundamentally from the previous one. Also here some numerical examples support the statement that all kinds of performance variance-E(P) connecting curves can emerge, making it impossible to derive a general statement as to the optimal portfolio of funding sources. We work with the same funding sources as above, leading to comparable cases: all portfolios of 2 funding sources exhibit more performance variance than the portfolio of all 3 funding sources (Table 4), a concave curve linking the relevant performance variance-E(P) combinations can be constructed (Table 5), and, as the last case, this curve cannot be concave (Table 6).

Table 4.

Example 1’.

ρ12 = .10 ρ13 = −.75 ρ23 = −.40
Variance portfolio (123)	5.1
Variance portfolio (12)	5.4
Variance portfolio (13)	5.5
Variance portfolio (23)	8.2
Variance 1	1
Variance 2	4
Variance 3	9

Table 5.

Example 2’.

ρ12 = .10 ρ13 = −.20 ρ23 = −.40
Variance portfolio (123)	8.4
Variance portfolio (12)	5.4
Variance portfolio (13)	8.8
Variance portfolio (23)	8.2
Variance 1	1
Variance 2	4
Variance 3	9

Table 6.

Example 3’.

ρ12 = .10 ρ13 = −.20 ρ23 = −.60
Variance portfolio (123)	6.0
Variance portfolio (12)	5.4
Variance portfolio (13)	8.8
Variance portfolio (23)	5.8
Variance 1	1
Variance 2	4
Variance 3	9

Crowding, No Stochastic Effects

This is basically a simple case, even though its solution might be numerically complex, depending on the articulation of the relationships involved. For given values of f_i we indeed have a system of n equations in n variables F_i, which, if it meets the regularity conditions, can be solved. The value of P obtained then is function of the f_i, so it can be maximised by solving for the optimal values of the f_i.

Suppose, in a simple algebraic example, two funding sources (n = 2). The crowding in/out effects are reflected by the parameters a and b in the equations below. Positive (negative) values reflect crowding in (out). To avoid pathological outcomes, we assume the absolute values of a and b to be below 1. In a simple additive model we assume the level of funding F_i equals the funds raised by the funding efforts f_i to which the crowding effect of F_j (j ≠ i) is added:

${\mathrm{F}}_1={\mathrm{F}}_1\left({\mathrm{f}}_1\right)+{\text{aF}}_2$

${\mathrm{F}}_2={\mathrm{F}}_2\left({\mathrm{f}}_2\right)+{\text{bF}}_1$

from which

${\mathrm{F}}_1=\frac{1}{1-\text{ab}}\left({\mathrm{F}}_1\left({\mathrm{f}}_1\right)+\mathrm{a}{\mathrm{F}}_2\left({\mathrm{f}}_2\right)\right)$

${\mathrm{F}}_2=\frac{1}{1-\text{ab}}(\mathrm{b}{\mathrm{F}}_1\left({\mathrm{f}}_1\right)+{\mathrm{F}}_2\left({\mathrm{f}}_2\right))$

Acknowledging the absence of stochastic variability, the ensuing value of P then is

${\mathrm{F}}_1+{\mathrm{F}}_2–\mathrm{k}\left({\mathrm{f}}_1+{\mathrm{f}}_2\right)=\frac{1}{1-\text{ab}}(\left(1+\mathrm{b}\right){\mathrm{F}}_1\left({\mathrm{f}}_1\right)+\left(1+\mathrm{a}\right){\mathrm{F}}_2\left({\mathrm{f}}_2\right))-\mathrm{k}\left({\mathrm{f}}_1+{\mathrm{f}}_2\right)$

If there is an interior maximum, the concavity of F_i(f_i) guarantees its uniqueness when the following conditions are met:

$\frac{1+\mathrm{b}}{1-\text{ab}}\frac{\partial {\mathrm{F}}_1}{\partial {\mathrm{f}}_1}-\mathrm{k}=0$

$\frac{1+\mathrm{a}}{1-\text{ab}}\frac{\partial {\mathrm{F}}_2}{\partial {\mathrm{f}}_2}-\mathrm{k}=0$

Crowding and Stochastic Effects

The model discussed in the section ‘No Crowding, Stochastic Effects, Covariances Between Funding Sources’, is a special case of the more general model in which crowding, stochastic effects, and covariances between funding sources are considered. It was established that even in this special case one cannot be sure the curve linking the relevant performance variance-E(P) pairs is concave. Hence, this also applies to the case with crowding effects and non-zero correlations.

The model with crowding effects, but no correlations between funding sources, generalises the model described in the section ‘No Crowding, Stochastic Effects, No Covariances Between Funding Sources’, where it was established that the curve linking the relevant performance variance-E(P) combinations was concave. We now show that this can not longer be guaranteed any more. To that end, we generalise the algebraic example of the previous section:

${\mathrm{F}}_1={\mathrm{F}}_1\left({\mathrm{f}}_1\right)+{\text{aF}}_2+{\mathrm{\epsilon }}_1$

${\mathrm{F}}_2={\mathrm{F}}_2\left({\mathrm{f}}_2\right)+{\text{bF}}_1+{\mathrm{\epsilon }}_2$

The maximum E(P) is the same as described in Section 2.4, but now comes with performance variance

$\text{var}({\mathrm{F}}_1+{\mathrm{F}}_2–\mathrm{k}({\mathrm{f}}_1+{\mathrm{f}}_2))=\frac{1}{{\left(1-\text{ab}\right)}^2}\left({\left(1+\mathrm{b}\right)}^2\text{var}\hspace{0.17em}{\mathrm{\epsilon }}_1+{\left(1+\mathrm{a}\right)}^2\text{var}{\hspace{0.17em}\mathrm{\epsilon }}_2\right)$

In the simplest case where variances of both funding sources are equal this becomes

$\text{var}\left(F_1+F_2-\mathrm{k}\left(F_1+F_2\right)\right)=\frac{1}{{\left(1-\text{ab}\right)}^2}\left({\left(1+\mathrm{b}\right)}^2+{\left(1+\mathrm{a}\right)}^2\right)\text{var}\hspace{0.17em}\mathrm{\epsilon }$

Performance variance reduction by eliminating one of both funding sources has the sign of

${\left(1+\mathrm{b}\right)}^2+{\left(1+\mathrm{a}\right)}^2-{\left(1-\text{ab}\right)}^2$

$={\left(1+\mathrm{a}+\mathrm{b}\right)}^2-{\mathrm{a}}^2{\mathrm{b}}^2$

Considering the even more special case in which a = b this becomes

${\left(1+2\mathrm{a}\right)}^2-{\mathrm{a}}^4$

which is smaller than 1 in absolute value, and negative for values of a being lower than $1-\sqrt{2}\approx -.414$ , not an exceptionally low crowding coefficient. It can also easily be seen that if a differs from b (also with an absolute value smaller than 1), higher values of the one together with values below −.41 for the other still make this difference negative, implying that the performance variance of the smaller portfolio is larger than that of the optimal larger one. Hence, we cannot ensure that the curve linking the relevant performance variance-E(P) combinations is concave, even in this specific case, let alone in more general versions of this model.

Practical Implications

Despite the present paper being theoretical in nature, there are practical implications, especially from the more realistic models. The first is that it need not be true that increasing the number of funding sources automatically leads to ‘better’ portfolios. The second is that the traditional (financial) diversification logic is not always valid regarding performance variability, necessitating consideration of crowding effects among potential financing sources. Executives (or consultants) of non-profit organisations obviously cannot be expected to engage in calculations implied by this paper, but they might benefit from at least considering intuitively the underlying mechanisms.

Conclusions, Implications, Limitations, and Suggestions for Further Research

Apart from suggesting the abovementioned practical implications, this paper adds to non-profit funding portfolio theory by combining elements of financial portfolio theory and the idiosyncrasies of non-profit organisations’ funding streams, leading to theoretical results not always similar to the ones obtained in classical portfolio theory or the extant non-profit funding diversification literature. Fundamental characteristics, including the performance indicator and the role of crowding in/out effects, markedly differ between for profit and non-profit funding contexts. These elements are now for the first time integrated in a more comprehensive theoretical approach to non-profit funding portfolios. The main insight obtained is that, contrary to financial portfolio theory, general statements about optimal funding portfolios of non-profit organisations cannot be made, which might explain the ‘diverging conclusions’ of the empircal literature referred to in the Introduction. The composition of optimal non-profit funding portfolios is shown to be determined by the severity of agency problems between board and management, the specific values of performance variances and covariances and the crowding structure linking the available funding sources. Crowding effects indeed can even invert diversification effects resulting from stochastic relations between funding streams: in these cases, portfolio risk might increase by adding funding from additional sources, instead of being reduced.

Models like the ones elaborated in the present paper are characterised by limitations and omissions, which in turn might inspire further extensions and refinements. A number of them are enumerated above (organisation level, (sub)sector level, and ecosystem-related effects on expected funding streams, specific non-profit regulations as to fundraising, dynamic mechanisms including potential ‘starvation cycle’ effects (Schubert & Boenigk, 2019) ¹⁴ ). Further, apart from amending the traditional Steinberg (1986) approach of reflecting the organisational objective in monetary units and only considering activity level (board) and budget (management) determining the eventual organisational objectives (ignoring other potential stakeholders such as the funders themselves: large donors, governments, …), ¹⁵ the implications of more articulated specifications of the stochastic and crowding effects might be established. Finally, the models developed in the present paper make abstraction of the institutional context. Their potential usefulness could be enhanced by taking this context in consideration, theoretically or (in particular) empirically by ‘incorporat[ing] various institutional contexts to test the model’s applicability’. ¹⁶ Examples of elements that might be considered are fiscal rules on donations and non-profit surplus, and subsidy regulations, ¹⁷ to which more general constructs at the jurisdictional level might be added.

Also concerning empirical research, the reviews referred to above show that the distinct literature streams on non-profit funding portfolios and crowding effects are isolated from one another: empirical non-profit funding portfolio work ignores crowding effects, and the empirical crowding literature abstracts from any funding portfolio consideration. The results obtained in the present paper imply that both should be integrated.