The author provides a theoretical framework for long-horizon, dynamic, incomplete-market portfolio theory and shows that the familiar mean–variance characterization for one-period returns applies to the stream of payoffs in a dynamic intertemporal environment with incomplete markets and outside income. In equilibrium, long-run expected returns vary with long-run betas and outside-income betas.

## What’s Inside?

The author explores long-term portfolio problems and corresponding equilibriums. He attempts to derive the optimal payoff, the yield of the optimal payoff, and the expected long-term yield of an asset in a multiperiod, dynamic environment. His findings reveal that the optimal yield is on the long-term mean–variance frontier and that the long-term expected yield follows a long-term capital asset pricing model (CAPM). The optimal payoff does not include state-variable hedging demands, and pricing does not depend on state-variable pricing factors. The payoff description is static; it does not even require rebalancing.

## How Is This Research Useful to Practitioners?

Asset returns are not independent and identically distributed, which means that investors should hedge state variables. Calculating unknown value functions to implement state-variable hedging, however, is difficult and imprecise. The author provides a straightforward and intuitive benchmark characterization of optimal payoffs without deriving and characterizing the dynamic portfolio strategy.

He shows that the yield on the optimal payoff for the income investor without outside income is on the long-horizon mean–variance frontier. The investor does not have to rebalance in response to changes in state variables because the risk aversion that directs the payoff allocation does not vary over time.

When such nonmarketed outside income as wages and business income is available for investors, those investors can optimize their payoff by selling the outside income and investing the proceeds along with their wealth. A labor income and preference shock hedge payoff, together with an investment in a long-term mean–variance-efficient yield, make up the optimal total payoff.

In this case, time-varying investments and hedging demands for state variable shocks are still absent. Instead, the optimal asset yield of an investor is positively related to market asset yield, the average outside income hedge yield, and the share of outside wealth; it is negatively related to the level of risk aversion and the share of asset wealth. In other words, investors with stable outside income (e.g., a stable wage) and higher wealth tend to take higher asset risk.

In equilibrium, long-term expected yields obey a one-factor CAPM, using the aggregate total payoff as a reference. Finally, assets are expected to have a higher long-term yield when their cash flows have higher long-term covariance with aggregate outside income.

## How Did the Author Conduct This Research?

The author focuses on the optimal stream of final payoffs rather than on the composition and dynamics of portfolio returns. He begins by defining the asset stream payoffs, which can be coupons, dividends, or payouts from dynamic trading strategies. The author then defines the discount factor, risk-free yield, long-run mean, long-run variance, mean–variance frontier, expected yields, and expected betas in the dynamic and multiperiod environment.

To tackle the general portfolio problem, the author specifies that an investor has initial wealth and a stream of human or other nonmarketable income with quadratic utility (resulting in linear marginal utility). Investors maximize their utility of consumption subject to a wealth constraint.

After specifying a stochastic bliss consumption point, the author then solves the optimal payoff with algebra, showing that the optimal payoff is a combination of a labor income, a preference shock hedge payoff, and an investment in a long-run mean–variance-efficient yield.

The author starts with the standard case without outside income. In an environment where all investors have quadratic utility, a constant bliss point, and no outside income, the market payoff is long-run-variance efficient. In such cases, given investors are all of the same type but have different levels of risk aversion, equilibrium prices follow a long-term version of the CAPM. Mertonian state variables for time-varying investment opportunities vanish from both long-term expected yields and the optimal payoff.

Based on this framework, the optimal payoff is derived for investors with outside income and preference shocks. The author’s theoretical investor modifies his holding of the long-run mean–variance-efficient payoff to account for effective risk-aversion changes because of the market and perpetuity exposure of his stream of outside income. He shorts a payoff with no expected yield that hedges the idiosyncratic component of his outside income. With outside income, the expected long-run yield of assets follows a multifactor model, with the market payoff and the average outside income hedge payoff as factors.

## Abstractor’s Viewpoint

Dynamic hedging with state variables is technically complex and difficult to implement in practice. The author’s finding shows that the one-period CAPM is simple but practical in the dynamic environment.