Portfolio allocation and borrowing constraints

ABSTRACT

Using the US Survey of Consumer Finances, we explore the empirical relationship between borrowing constraints and financial portfolio allocation by American households. To help motivate our empirical analysis we initially draw insights from a mean-variance model of optimal portfolio allocation with three tradable asset classes defined by increasing risk, and establish a link between borrowing restrictions and portfolio allocation in the presence of background risk. Our main contribution, however, lies in estimating the role that borrowing constraints play in shaping households' financial portfolio allocation. This is achieved using an ordered fractional probit model. In addition to revealing the significant empirical role played by household borrowing constraints in determining portfolio allocation, our analysis helps us to resolve ambiguity surrounding the behaviour of the medium-risk asset in our motivational theoretical framework. Further, the empirical distribution of medium-risk assets is found to be remarkably similar to that for high-risk assets, suggesting the presence of a more general ‘risk puzzle’, which our borrowing constraints measures partially explain.

KEYWORDS:

• Asset allocation
• Borrowing constraints
• Fractional models
• Background risk

JEL CLASSIFICATIONS:

• G11
• D11
• D14
• C35

Acknowledgments

We thank Alberto Montagnoli, Christoph Thoenissen and Mark Tippett for valuable comments and suggestions. The usual disclaimer applies. The authors have no declarations of interest to make.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 These authors also report that the number of Americans who cannot easily access loans may be twice as many as previously estimated, when people who cannot easily qualify for loans because of blemishes in their credit histories are taken into account.

2 This is of course, notwithstanding the detrimental effect that the inability to borrow plays in exacerbating social inequalities, and its implications for household consumption smoothing.

3 In this paper, our concept of background risk therefore corresponds to risk associated with the household wealth process, and not so-called ‘systemic risk’, which is a type of background risk that is present in financial markets. Previous theoretical and empirical research has indicated that the presence of background risk can have a significant impact on the allocation of household portfolios.

4 See for instance (Carroll 2002; Hurd 2002). Our motivational theoretical model may have implications for other empirical contributions where financial portfolios are treated as being comprised of low, medium, and high-risk assets. From an empirical perspective, many asset management companies describe the portfolio structure of their investment products using this taxonomy.

5 In addition to using a standard set of controls, our subsequent econometric analysis therefore includes variables that proxy for such risk when estimating the impact of borrowing constraints on a household's financial portfolio allocation.

6 As argued below, the exogenous and directly uninsurable nature of background risk amounts to the assumption of incomplete markets. The presence of such risk can lead a decision-maker to ‘reduce exposure to another risk even if the two risks are statistically independent’ (Kimball 1992), a concept described as ‘temperance’ (Kimball 1991). For related theoretical contributions see Pratt and Zeckhauser (1987), Kimball (1991), Gollier and Pratt (1996), and Heaton and Lucas (2000)

7 Canner, Mankiw, and Weil (1997) explore portfolio allocation among cash, bonds, and stocks, with a view to reconciling the household stockholding puzzle in light of the financial advice received by households.

8 We choose to use a Markowitz mean-variance framework due to its relative simplicity and tractability. In this regard, our model provides a useful starting point for motivating our subsequent empirical analysis. Nevertheless, since the publication of Markowitz (1952)–which uses variance as a measure of risk–more sophisticated techniques to inform portfolio selection have been developed. Although these techniques go beyond the scope of our contribution, they include conditional value at risk (Rockafellar and Uryasev 2000) and distortion risk measures (Wang 2000). We thank an anonymous referee for raising this issue.

9 Although ‘safe’ instruments such as government bonds may be riskless in nominal terms, inflation means that their return may be uncertain in real terms.

10 Eichner and Wagener (2012) consider the case of two tradable assets whose weights sum to unity. We extend this approach to the case of three tradable assets, which is uncommon in the literature. Unlike our contribution, Eichner and Wagener (2012) do not consider the impact of short-sale constraints on the tradable assets, which in the context of our contribution, are important model features.

11 Whilst this admits a natural, albeit simplistic, interpretation as a borrowing constraint, Kaplow (1994, 1505) notes that: ‘Individuals’ ability to borrow against human capital is limited, but some such borrowing occurs'. For instance, residential mortgages are based in part on predictions of future earnings; further, in the United States, the largest student loans are available to law and medical school students, due to the high and stable expected future income stream associated with jobs in the legal and medical professions.

12 Restricting the portfolio selection by only having positive weights of the assets limits the amounts of possible portfolios and introduces complexity that cannot be handled by closed-form mathematics, and requires computation of the corresponding Kuhn-Tucker conditions.

13 In the context of Italian households, Jappelli, Julliard, and Pagano (2010) also suggest that whereas institutional investors are able to borrow at low-risk to invest in high-risk assets with a greater expected return, households cannot act the same way, for example by funding stock market investments through borrowing from financial intermediaries such as banks. Using household data, the authors show that explicitly considering the no short-selling constraint helps significantly in reconciling individual portfolio choices with the efficient ones implied by portfolio theory (Markowitz 1952). Borrowing constraints which prevent households from short-selling are also explored in Bucciol, Miniaci, and Pastorello (2017), who also develop a model of household portfolio choice with borrowing restrictions in a mean-variance expected utility maximisation framework.

14 Under a Markowitz mean-variance approach, the presence of zero correlations between the tradable assets does not preclude a portfolio from benefitting from diversification. However, if assets are perfectly positively correlated, no benefit from diversification arises. In assuming zero correlations, households can be perceived as selecting uncorrelated tradable assets to help diversify risk.

15 In setting out our motivational theoretical framework, we are able to demonstrate that even in a relatively parsimonious model without correlated tradable assets, the impact of background risk on asset allocation is subject to some ambiguity. In our model, households exploit the information contained in background risk covariance structures, which has implications for portfolio diversification and risk reduction. We stress here that including correlations between the tradable assets introduces additional complexity to our findings as the predictions of the model become considerably more difficult to disentangle. Additional numerical experiments not presented here suggest that allowing for such correlations does not help to resolve the ambiguity described above. This leads us to suggest that an empirical approach such as that used in Section 5 can be usefully exploited to resolve such ambiguity.

16 Jiang, Ma, and An (2010) also explore the impact of background risk on the efficient frontier, and similarly find that an increase in background risk will shift the frontier to the right. Unlike our contribution, however, these authors do not focus on the implications of shifts in the frontier on household welfare or the optimal composition of a household's portfolio given mean-variance preferences.

17 As is standard in financial portfolio theory our interest is with the part of the efficient frontier in the ${\mu }_W$–${\upsilon }_W$ space such that ${\mu }_W^{\ast }\ge$ ${\mu }_W^{\mathrm{g}}$, where ${\mu }_W^{\mathrm{g}}$ denotes the value of ${\mu }_W$ at the global minimum portfolio and ${\mu }_W^{\ast }$ is a value of ${\mu }_W$ on the efficient fronter, as captured by expression (A6(A6) ${\upsilon }_W^{\ast }=\left[\begin{array}{c}{q}_L^{\ast 2}{\upsilon }_L+q_M^{\ast 2}{\upsilon }_M+q_H^{\ast 2}{\upsilon }_H+{\lambda }^2{\upsilon }_B\\ +2q_L^{\ast }{q}_M^{\ast }\mathrm{Cov}(L,M)+2q_L^{\ast }{q}_H^{\ast }\mathrm{Cov}(L,H)+2q_M^{\ast }{q}_H^{\ast }\mathrm{Cov}(M,H)\\ +2q_L^{\ast }\mathrm{\lambda Cov}(L,B)+2q_M^{\ast }\mathrm{\lambda Cov}(M,B)+2q_H^{\ast }\mathrm{\lambda Cov}(H,B)\end{array}\right].$(A6) ) in Appendix 1. A discussion of ${\mu }_W^{\mathrm{g}}$ is provided in Appendix 3. An important corollary of the results in expressions (4(4) $\frac{\mathrm{\partial }{q}_L^{\ast }}{\mathrm{\partial }{\mu }_W}<0;\frac{\mathrm{\partial }{q}_M^{\ast }}{\mathrm{\partial }{\mu }_W}⪋0;\frac{\mathrm{\partial }{q}_H^{\ast }}{\mathrm{\partial }{\mu }_W}>0,$(4) ) and (5(5) $\frac{\mathrm{\partial }{q}_M^{\ast }}{\mathrm{\partial }{\mu }_W}⪋0\mathrm{iff}\frac{{\upsilon }_L}{{\upsilon }_H}⪌\frac{{({\mu }_L-{\mu }_M)}^2}{[({\mu }_L-{\mu }_H)({\mu }_M-{\mu }_H)-{({\mu }_M-{\mu }_H)}^2]}.$(5) ) is that whilst in an unconstrained setting, increasing risk aversion is associated with a decrease (increase) in the share of assets in the high-risk (low-risk) category, for the medium-risk asset share $q_M$, the effect will also be a function of the ratio of ${\upsilon }_L$ and ${\upsilon }_H$, and its relationship to the expected returns of the three tradable assets. Under certain conditions, risk-loving households may hold a smaller share of their wealth in medium-risk assets relative to risk-averse households.

18 Although our empirical examples are highly stylised and tied to an explicit formal model, similarities in the profiles of the efficient frontiers for US and Italian households can be observed. See Figure 2 in Jappelli, Julliard, and Pagano (2010), and the associated discussion.

19 In practice, prohibitively high overdraft charges may act as an impediment to sustaining a negative balance for a considerable period of time.

20 We stress here that in the presence of a short-sale constraint, the non-binding return-risk space available to households satisfying $q_L^{\ast }\ge 0$, $q_M^{\ast }\ge 0$, and $q_H^{\ast }\ge 0$ may be limited, even when background risk is independent of the tradable assets. Under independent background risk, neither the optimal asset allocations associated with different points on the efficient frontier nor the regions of the non-binding risk-return space are functions of background risk.

21 Interest rates in US checking and savings accounts are closely tied to the fed funds rate, which is set by the Federal Open Market Committee (FOMC).

22 Our choice of mean-variance utility is commonplace in standard financial economics texts (Bailey 2005), and is appealing due to its analytical tractability. In our empirical analysis, we use measures of risk attitudes to capture a household's attitude towards risk. We stress here that these measures only proxy for risk preferences, and may not be tightly linked to the shape of a household's utility function.

23 Preference functions for which households are decreasingly absolute risk averse (DARA) do not lead to tractable results, and require solving by linear quadratic programming techniques. The expected log utility function given by $E\left(U\right(W\left)\right)=\ln {\mu }_W-{\upsilon }_W/2{\mu }_W^2$, where W denotes wealth, would fall into this class of function (see Pulley 1993, Eqn (2), 686 and the corresponding discussion for further details).

24 The extent to which a change in ${\sigma }_B$ impacts on the value of ${\mu }_W^{\ast }$ at the point of tangency is provided in Appendix 4, as is the corresponding impact of a change in risk-aversion γ. The Kuhn-Tucker conditions for our model are set out in Appendix 5.

25 This effect arises irrespective of the nature of correlation between background risk and the tradable assets.

26 Although not shown in Figure 1, the constraint becomes binding at approximately $\gamma =0.0761$. Under background risk, the constraint becomes binding at approximately $\gamma =0.0688$.

27 Under independent background risk an increase in ${\sigma }_B$ will, ceteris paribus, reduce household welfare. The composition of the household's optimal portfolio will be identical to the no background risk case.

28 Given the high rate of non-response associated with micro-data relating to wealth information, the SCF provides five imputations which give a distribution of outcomes. These multiple imputations increase the efficiency of the estimation, whilst also providing uncertainty surrounding the imputed values. Our summary statistics are based on taking the average of the corresponding five implicates for each cross-section. However, the econometric estimates are based on implementing the repeated imputation inference (RII) method, as described by Little and Rubin (1987), Rubin (1987) and Montalto and Sung (1996). The RII uses the average of the coefficients across the five imputations and adjusts the corresponding standard errors accordingly.

29 These measures are similar to those used by Fulford (2015). However, we allow a differential impact of being above or below a household's normal income by entering positive and negative values as separate variables.

30 The question changed the time period from 5 years to 12 months in the 2016 and 2019 waves of the survey. If we restrict our analysis to the consistent questions (i.e. between 1995 and 2013) we find no differences in the results.

31 We also use Tobit analysis (Tobin 1958) to model high-risk asset shares in line with the existing literature. The coefficients and associated marginal effects are discussed in Appendix 6. Generally these results are in line with those for the FRM approach.

32 In our dataset, this assumption is also reflected in the values of US households' observable asset shares.

33 It would be possible to model the effects of drivers on each of the $s_{\mathrm{ij}}$ shares as a linear system, such that $s_{\mathrm{ij}}={\mathbf{x}}_i^{\mathrm{\prime }}{\text{β}}_j+u_{\mathrm{ij}},$ where ${\mathbf{x}}_i$ is a matrix of (household) covariates and $u_{\mathrm{ij}}$ is a random error. This would effectively be an extension of the linear probability model, the shortcomings of which are well-known (Gujarati 1995). Indeed, such an approach would not be ideal: it would not guarantee that $0\le E\left(s_{\mathrm{ij}}\right|{\mathbf{x}}_i)={\mathbf{x}}_i^{\mathrm{\prime }}{\text{β}}_j\le 1$; would have repercussions on ensuring the adding-up constraint that ${\sum }_j=E\left(s_{\mathrm{ij}}\right|{\mathbf{x}}_i)\equiv 1$; it would also be unable to handle boundary observations of 0 or 1 shares; and would likely embody heteroskedasticity in $u_{\mathrm{ij}}$. We have also explored the use of a multi-nominal fractional response model, see for example, Becker (2014). In this setting, the inherent risk ordering of asset classes is not accounted for in the estimation strategy, instead the multi-nominal probit model is used as the foundation of the estimation strategy. We obtain similar results to those presented when we adopt this alternative modelling strategy.

34 We have also explored the robustness of our results subject to the inclusion of a dummy variable that captures if an individual falls in the bottom 25% of the income distribution. We find that the results corresponding to the borrowing constraint variables remain consistent with those presented in the paper, in terms of both the signs and statistical significance of the estimated parameters. This suggests that our borrowing constraint variables are not merely capturing non-linear income effects.

35 We thank an anonymous referee for making this suggestion, and further, for proposing a number of alternative asset classifications which are used in our estimations.

36 Deriving (A1(A1) $\begin{array}{rl}{q}_L& =\alpha +\text{β}\mathrm{\Gamma };\\ q_M& =1-\alpha -(1+\text{β})\mathrm{\Gamma };\mathrm{and}\\ q_H& =\mathrm{\Gamma },\end{array}$(A1) ) requires solving a two equation system with three unknowns. As there are an infinite number of solutions it is necessary to pin down the value of the scalar Γ. It is also possible to specify $q_M=1-\alpha -(1+\text{β})\mathrm{\Gamma }$ in (A1(A1) $\begin{array}{rl}{q}_L& =\alpha +\text{β}\mathrm{\Gamma };\\ q_M& =1-\alpha -(1+\text{β})\mathrm{\Gamma };\mathrm{and}\\ q_H& =\mathrm{\Gamma },\end{array}$(A1) ) as $1-\mathrm{\Gamma }-q_L$.

37 Alternatively, we can express (A10(A10) $\frac{\mathrm{\partial }{q}_M^{\ast }}{\mathrm{\partial }{\mu }_W}⪋0\mathrm{iff}\frac{{\upsilon }_L}{{\upsilon }_H}⪌\frac{{({\mu }_L-{\mu }_M)}^2}{[({\mu }_L-{\mu }_H)({\mu }_M-{\mu }_H)-{({\mu }_M-{\mu }_H)}^2]}.$(A10) ) as ${\upsilon }_L({\mu }_L-{\mu }_H)({\mu }_M-{\mu }_H)-{\upsilon }_L({\mu }_M-{\mu }_H{)}^2⪌{\upsilon }_H({\mu }_L-{\mu }_M{)}^2$. Once a household's optimal portfolio allocation is bound by the no short-sale constraint, this condition will not hold.

38 In our simple static model, borrowing restrictions assume the form of short-selling constraints. However, in a dynamic setting, Dybvig and Huang (1988) note that imposing a non-negativity constraint (or in fact any negative lower bound) on wealth is a plausible economic assumption, and admits a natural interpretation as a credit constraint. Another way is to prevent borrowing from future income. There are many institutional restrictions on the amount of credit an individual can obtain.

39 The impact on the efficient frontier when tradable assets are correlated is well explored in the literature. We therefore refrain from focusing on this area.

Additional information

Notes on contributors

Raslan Alzuabi

Raslan Alzuabi is a Lecturer in Economics at the University of Sheffield.

Sarah Brown

Sarah Brown is a Professor of Economics at the University of Sheffield.

Daniel Gray

Daniel Gray is a Lecturer in Economics at the University of Sheffield.

Mark N. Harris

Mark N. Harris is a Professor of Economics at Curtin University.

Christopher Spencer

Christopher Spencer is Senior Lecturer in Economics at Loughborough University.