Multiple Risky Assets Portfolio Choice

[alanlujan91]

Summary

HARK's PortfolioConsumerType supports portfolio choice between one risky asset and one riskless asset. Real-world portfolio decisions involve multiple asset classes (equities, bonds, international stocks, real estate, commodities, alternatives) whose correlations shift over time: stock-bond correlation was positive before 2000 and negative after. Extending HARK to $N$ risky assets would connect the toolkit to the life-cycle portfolio allocation literature and allow study of how correlation structure, diversification, and time-varying risk premia affect household portfolio composition.

Connection to Financial Services Practice

The LSEG partner plugin contains skills for multiple asset classes simultaneously: fixed-income-portfolio (partner-built/lseg/skills/fixed-income-portfolio/SKILL.md) analyzes bond portfolios with duration, convexity, and scenario analysis; fx-carry-trade (partner-built/lseg/skills/fx-carry-trade/SKILL.md) models foreign exchange carry strategies; option-vol-analysis (partner-built/lseg/skills/option-vol-analysis/SKILL.md) handles equity and FX volatility surfaces. The wealth-management portfolio-rebalance skill (wealth-management/skills/portfolio-rebalance/SKILL.md) manages drift across 9 asset classes (US Large Cap, US Small/Mid, International Developed, Emerging Markets, IG Bonds, High Yield, TIPS, Alternatives, Cash). A multi-asset HARK model would derive the optimal life-cycle allocation that these practitioner tools approximate through rebalancing bands and strategic targets.

Mathematical Model

Environment

Time is discrete, $t = 0, 1, \ldots, T$ (lifecycle). The agent allocates wealth across $N$ risky assets (e.g., domestic equity, bonds, international equity) and one riskless asset with gross return $R_f$. The agent also receives stochastic labor income.

State Variables

The individual state is $(w_t, z_t, t)$, where we denote by $w_t$ total financial wealth and by $z_t$ the persistent income component. In the most general version, the state includes the current portfolio composition (for tax or transaction cost reasons). In the frictionless benchmark with i.i.d. returns and no labor income, total wealth suffices; with persistent income, the state is $(w_t, z_t, t)$.

Asset Returns

The $N$ risky assets have excess returns $\mathbf{r}{t+1} = (r{1,t+1}, \ldots, r_{N,t+1})'$ where $r_{i,t+1} = R_{i,t+1} - R_f$. The joint distribution of excess returns is

$$\mathbf{r}_{t+1} \sim \mathcal{D}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$$

where the expected excess return vector is $\boldsymbol{\mu} = (\mu_1, \ldots, \mu_N)'$ and the variance-covariance matrix is $\boldsymbol{\Sigma}$ with elements $\sigma_{ij}$.

A standard specification uses the multivariate lognormal:

$$\log(1 + R_{i,t+1}) \sim \mathcal{N}(\tilde{\mu}_i, \tilde{\sigma}_i^2)$$

with correlation structure specified through the covariance matrix $\boldsymbol{\Sigma}$.

Preferences

$$u(c) = \frac{c^{1-\rho}}{1-\rho}$$

with relative risk aversion $\rho$ and discount factor $\beta$.

The age-dependent survival probability $\varsigma_t = \Pr(\text{alive at } t+1 \mid \text{alive at } t)$ abstracts from health-state dependence.

Choice Variables

Each period the agent chooses consumption $c_t$ and a vector of portfolio weights $\boldsymbol{\alpha}t = (\alpha{1,t}, \ldots, \alpha_{N,t})'$, where $\alpha_{i,t}$ is the fraction of post-consumption savings invested in risky asset $i$. The fraction in the riskless asset is $1 - \mathbf{1}'\boldsymbol{\alpha}_t$.

Wealth Evolution

$$w_{t+1} = (w_t - c_t), R_{t+1}^p + Y_{t+1}$$

where the portfolio gross return is

$$R_{t+1}^p = R_f + \boldsymbol{\alpha}_t' \mathbf{r}_{t+1} = R_f + \sum_{i=1}^{N} \alpha_{i,t}, r_{i,t+1}$$

and $Y_{t+1}$ is stochastic labor income (possibly correlated with asset returns). The income process follows the standard permanent-transitory decomposition: $Y_{t+1} = P_{t+1} \theta_{t+1}$, where the permanent component evolves as $\log P_{t+1} = \log P_t + \psi_{t+1}$ with $\psi_{t+1} \sim \mathcal{N}(-\sigma_\psi^2/2, \sigma_\psi^2)$, and the transitory shock is $\theta_{t+1}$ with $\mathbb{E}[\theta] = 1$. The correlation between permanent income shocks $\psi$ and asset returns $\mathbf{r}$ is a key parameter governing life-cycle portfolio allocation.

Recursive Formulation

$$V_t(w, z) = \max_{c, \boldsymbol{\alpha}} ; u(c) + \beta, \varsigma_t; \mathbb{E}\bigl[V_{t+1}(w', z')\bigr]$$

subject to:

1. Budget constraint: $w' = (w - c) R^p_{t+1} + Y_{t+1}(z')$
2. No-shorting constraints: $\alpha_{i,t} \geq 0$ for all $i = 1, \ldots, N$
3. No-leverage constraint: $\sum_{i=1}^{N} \alpha_{i,t} \leq 1$
4. Borrowing constraint: $w - c \geq 0$ (or some minimum savings level)

First-Order Conditions

For an interior solution, the Euler equations for each risky asset $i$ are:

$$\mathbb{E}_t\bigl[u'(c_{t+1}), r_{i,t+1}\bigr] = 0 \quad \text{for } i = 1, \ldots, N$$

combined with the standard consumption Euler equation:

$$u'(c_t) = \beta; \mathbb{E}_t\bigl[u'(c_{t+1}), R^p_{t+1}\bigr]$$

When shorting constraints bind ($\alpha_{i,t} = 0$ for some $i$), the corresponding Euler equation holds with inequality.

Analytical Benchmark: Merton-Samuelson

In the special case of no labor income ($Y = 0$), infinite horizon, and i.i.d. lognormal returns, the optimal portfolio is the myopic solution:

$$\boldsymbol{\alpha}^* = \frac{1}{\rho} \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}$$

This continuous-time result uses instantaneous expected excess returns $\boldsymbol{\mu}$ and the instantaneous covariance matrix $\boldsymbol{\Sigma}$. In the discrete-time lognormal specification, the mapping between log-return parameters $(\tilde{\mu}_i, \tilde{\sigma}_i^2)$ and the arithmetic excess return $\mu_i \approx \exp(\tilde{\mu}_i + \tilde{\sigma}_i^2/2) - R_f$ involves a Jensen's inequality correction that should be applied in calibration.

This mean-variance efficient portfolio, scaled by inverse risk aversion, provides a useful benchmark. With labor income (a bond-like asset), the agent tilts toward equities when young (human capital substitutes for bonds, provided labor income is uncorrelated with equity returns) and toward bonds when old, the central result of Cocco, Gomes, and Maenhout (2005). When income covaries positively with stock returns, the hedging motive pushes the young agent away from equities.

Practical Case: $N = 3$ (Stocks, Bonds, Housing/REITs)

For the most policy-relevant case, we consider three risky assets:

Asset	Symbol	Calibration target
Domestic equity	$R^{eq}$	S&P 500 total return
Long-term bonds	$R^{bd}$	10-year Treasury total return
Real estate / REITs	$R^{re}$	NCREIF or FTSE NAREIT

Stocks and bonds have time-varying correlation (positive pre-2000, negative post-2000), and real estate is moderately correlated with both. Over the life cycle, the optimal allocation shifts from equity-heavy (young, when labor income acts as a bond) to bond-heavy (retired, drawing down) with a moderate real estate allocation throughout.

Extension: Time-Varying Risk Premia

If the expected return vector $\boldsymbol{\mu}_t$ varies with an observable state variable $x_t$ (e.g., dividend-price ratio, term spread):

$$\boldsymbol{\mu}_t = \boldsymbol{\mu}_0 + \boldsymbol{B} x_t$$

the state space expands to $(w_t, z_t, x_t, t)$ and the optimal portfolio includes a hedging demand component in addition to the myopic demand. This is the Merton (1973) intertemporal hedging motive. The agent holds extra assets that pay off well when investment opportunities deteriorate.

Extension: Transaction Costs

With proportional transaction costs $\tau_i$ on trades in asset $i$, the state must track the current portfolio composition $\boldsymbol{\alpha}_{t}^{-}$ (before rebalancing). The agent faces a trade-off between rebalancing toward the optimal portfolio and paying transaction costs, leading to a no-trade region around the optimal weights.

Key References

• Cocco, J. F., Gomes, F. J., and Maenhout, P. J. (2005). "Consumption and Portfolio Choice over the Life Cycle." Review of Financial Studies, 18(2), pp. 491-533.
• Gomes, F. and Michaelides, A. (2005). "Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence." Journal of Finance, 60(2), pp. 869-904.
• Campbell, J. Y. and Viceira, L. M. (2002). Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press.
• Campbell, J. Y., Chan, Y. L., and Viceira, L. M. (2003). "A Multivariate Model of Strategic Asset Allocation." Journal of Financial Economics, 67(1), pp. 41-80.
• Merton, R. C. (1971). "Optimum Consumption and Portfolio Rules in a Continuous-Time Model." Journal of Economic Theory, 3(4), pp. 373-413.
• Merton, R. C. (1973). 'An Intertemporal Capital Asset Pricing Model.' Econometrica, 41(5), pp. 867-887.

Implementation Notes

• At each grid point, the $N$-dimensional portfolio optimization is the main computational challenge. For $N = 2$, a 2D grid search or root-finding on two Euler equations is tractable. For $N \geq 3$, the nested optimization becomes expensive, requiring an optimizer (e.g., scipy's constrained minimizer) within backward induction.
• Alternatively, the endogenous grid method for non-concave problems (Fella 2014; Iskhakov, Jørgensen, Rust, and Schjerning 2017) can smooth the portfolio choice.
• PortfolioConsumerType ($N = 1$) solves the portfolio FOC on a 1D grid. Extending to $N = 2$ requires solving a 2D FOC system at each wealth grid point, feasible with Newton's method.
• For $N = 3$ with transaction costs, the state $(w, \alpha_1^-, \alpha_2^-, z, t)$ is 5-dimensional, requiring sparse grid or machine learning approximation.
• HARK's DiscreteDistribution and combine_indep_dstns machinery already supports multivariate discrete approximations to the return distribution.
• Start with the frictionless $N = 2$ case (stocks + bonds), then extend to $N = 3$ and add transaction costs.
• Within-account portfolio choice in the tax-differentiated accounts model (Issue Tax-Differentiated Account Model (401k/IRA/Roth/Taxable) #1731) is a special case of this framework with account-type constraints on asset placement.