Portfolio Optimization with Censored Return Distributions

The problem

Investors typically want the highest possible return for the lowest possible risk. The classical Markowitz model (1952) chooses portfolio weights that minimize variance for a target expected return. However, it offers no way to limit losses on a bad day. Put options, insurance-like instruments for equities, provide this capability. We ask: if an investor is allowed to buy protective puts on each stock, at what level should they buy, and how should that change the portfolio mix?

The main idea

A protective put truncates a stock's return distribution: any loss worse than a chosen floor is paid back by the option. Mathematically, the stock's returns are no longer normally distributed, they become a censored normal distribution (also called a Tobit distribution), with the left tail collapsed onto a single point. We derive closed-form expressions for the mean, variance, and covariance of this censored distribution, then plug them into a Markowitz-style optimization model where the level of protection is a decision variable.

We re-parameterize the floor as \(\Delta_i\), the probability that the insurance actually pays out. \(\Delta\) is thus bounded between 0 and 1, which makes the model numerically well-behaved. The cost of each put is priced automatically using the Black–Scholes (1973) formula.

Methods

Censored-moment derivations. Closed-form first and second moments for each protected asset, plus a covariance approximation that preserves positive definiteness of the portfolio covariance matrix.
Polynomial approximations. The normal CDF, PDF, and inverse CDF (probit) have no closed form, so we replace them with low-order Taylor / linear approximations to keep the model tractable.
Covariate conditioning. Stock means and variances are conditioned on macro covariates (core PCE inflation, the Effective Fed Funds rate), the same inputs analysts use in discounted-cash-flow valuation models.
Solver. The resulting nonconvex nonlinear program is solved in Gurobi on a 10-stock universe spanning tech, communications, industrial, and consumer sectors.

Takeaways

Adding protective puts reduces portfolio variance with only minimal loss of expected return in typical market regimes.
The optimal amount of insurance depends strongly on the macro environment. In high-return regimes (e.g. 2023-2025), buying puts can actually hurt the Sharpe ratio, downside protection is not always worth its cost.
Jointly optimizing portfolio weights and option levels is, to our knowledge, new to the literature; previous work treated option choices as exogenous or fixed.
The censored-distribution approach generalizes to any setting where outcomes are truncated. This includes left-side truncation such as downtime with service-level guarantees, or on the right-side for an insurance company with a reinsurance policy.