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Abstract

We construct a life-cycle model that delivers realistic behavior for both equity holdings and borrowing. The key model ingredient is a wedge between the cost of borrowing and the risk-free investment return. Borrowing can either raise or lower equity demand, depending on the cost of borrowing. A borrowing rate equal to the expected return on equity--which we show roughly matches the data--minimizes the demand for equity. Alternative models with no borrowing or limited borrowing at the risk-free rate cannot simultaneously fit empirical evidence on borrowing and equity holdings.

Introduction

Borrowing presents a problem for life-cycle models of consumption and portfolio choice. In the classic Merton-Samuelson model, modified to include a realistic process for labor income, unsecured borrowing leads to huge, highly levered equity positions. For example, with relative risk aversion of 2 and standard specifications for income and asset returns, the model yields average equity holdings more than 20 times bigger than average annual income. To be sure, life-cycle models that preclude borrowing can generate realistic equity holdings, but they fly in the face of evidence that unsecured consumer credit is widely available and widely used. In fact, unsecured debt is much more prevalent than equity in the portfolios of younger households.

In this paper, we construct a life-cycle model that resolves the tension between borrowing and equity holdings. Households can borrow in our setup--but at rates that exceed the risk-free investment return. Given realistic borrowing costs, the model yields both debt positions and equity holdings that fit the main features of the data.

Except for its treatment of borrowing, our preferred model is entirely standard. Agents have time-separable, isoelastic preferences with moderate risk aversion. They face realistic income processes and can invest in risky and risk-free assets. We do not rely on habit formation, self-control problems, myopia or costs of participating and trading in equity markets to obtain sensible life-cycle profiles for borrowing and equity holdings. Neither do we rely on informational barriers, time-varying asset returns or enforcement problems in loan markets. Instead, the key elements of our analysis are realistic borrowing costs and the life-cycle structure. But as we explain, realistic borrowing costs magnify the impact of certain other frictions--such as fixed costs of participating in equity markets or liquidity benefits from bond holdings--on participation rates and portfolio shares.

Table 1 reports data on the size of the wedge between borrowing costs and the risk-free return. The bottom two rows show that household borrowing costs on unsecured loans exceed the risk-free return by about six to nine percentage points on an annual basis, after adjusting for tax considerations and charge-offs for uncollected loan obligations. Since 1987, roughly two percentage points arise from the asymmetric income tax treatment of household interest receipts and payments. However, the bulk of the wedge arises from transactions costs in the loan market. Despite the evident size of these costs, they have been largely ignored in theoretical analyses of life-cycle consumption and portfolio behavior. They have also been ignored in most empirical studies of asset-pricing behavior.

The relationship between equity holdings and the cost of borrowing is non-monotonic in our model. To see why, suppose initially that the borrowing rate equals the expected return on equity. No one borrows to buy equity in this case, because the net return is zero and the investment would increase risk exposure. At a slightly lower borrowing rate, however, the net return is positive and the household adopts a small debt-financed equity position. Further reductions in the borrowing rate lead to greater leverage and further increases in equity demand. Now move in the other direction and consider a borrowing rate that slightly exceeds the equity return. In this case, households with debt hold no equity (because debt repayment offers a better return), so the borrowing rate has no immediate impact on their equity demand. But higher borrowing rates discourage borrowing for consumption-smoothing purposes. As a result, households borrow less at each age, achieve a positive financial position earlier in life, invest in equity at an earlier age and hold more equity at later ages. Further increases in the borrowing rate imply a further upward shift in the life-cycle equity profile, and sufficiently high borrowing costs choke off all borrowing. Hence, equity holdings and participation rates are minimized when the borrowing rate equals the expected return on equity--a scenario consistent with Table 1.

We also develop several other points. First, our model implies high non-participation rates in equity markets, much higher than in otherwise identical models with no borrowing and much closer to the data. Second, even a small wedge between borrowing rates and the risk-free return dramatically reduces the demand for equity. Third, greater income uncertainty raises equity demand in our model with realistic borrowing costs, contrary to its effect in the standard model with no wedge. Fourth, equity demand is a non-monotonic function of relative risk aversion with realistic borrowing costs, again contrary to the standard model. Fifth, and not surprisingly in light of our previous remarks, equity demand is sensitive to the shape of the life-cycle income profile in our preferred model. Finally, we also consider a model with limited borrowing at the risk-free rate and show that it does a poor job of resolving the tension between borrowing and equity holdings. The limited-borrowing model implies that households borrow to finance equity holdings and always exhaust borrowing capacity. Both implications are sharply at odds with observed behavior.

We reiterate that our main goal is to construct a model that delivers realistic life-cycle behavior for both equity holdings and unsecured borrowing. We largely meet that goal, but gaps between theory and data remain. When fit to the evidence on unsecured borrowing and the historical equity premium, equity holdings in our baseline specification are somewhat larger than in the data. And, like other lifecycle models with no liquidity motive for bond holdings, our model does not generate realistic bond portfolio shares with moderately risk averse investors.

The paper proceeds as follows. The balance of the introduction discusses related research and reviews some important facts about borrowing and equity holdings over the life cycle. Sections 2 and 3 describe the model and choice of parameters. Section 4 considers life-cycle behavior in our preferred model and alternatives, and section 5 compares model implications with empirical evidence. Where the models fail to fit the facts, we assess the significance of the failures. Section 6 offers some concluding remarks, and an appendix describes our numerical solution method.

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