Portfolio choice and insurance represent two fundamental markets for risk-sharing — one where individuals actively acquire risk exposure for potential gains, the other where they pay to transfer risk away.
In this post, I explore the mathematical equivalence between these choices, how both markets thrive on imperfection and share vulnerability to correlated risks, but also retain important differences that make each market unique. I have benefited from discussions with Xuelin Li.
At their core, portfolio choice and insurance decisions both represent risk-sharing mechanisms, though they operate through different market structures. This equivalence can be expressed mathematically, as described in Armantier, Foncel, and Treich (2023).
The Insurance Problem
In the canonical insurance model (Mossin, 1968), an individual with wealth $w$ faces a potential random loss $\tilde{L}$. The individual can purchase insurance coverage at level $\alpha$, where $\alpha$ represents the proportion of the loss covered by insurance. The insurance premium is $\alpha\pi$, where $\pi$ is the premium rate for full coverage.
The individual's objective is to maximize expected utility:
$$ \max_{\alpha} \mathbb{E}u[w - \alpha\pi - (1-\alpha)\tilde{L}] $$
This expression has a clear economic interpretation: the individual's final wealth is their initial wealth ($w$) minus the insurance premium paid ($\alpha\pi$) minus the portion of the loss they bear themselves ($(1-\alpha)\tilde{L}$). When $\alpha=0$, the individual has no insurance and bears the full loss; when $\alpha=1$, the individual has full insurance and pays no losses out of pocket (but pays the full premium π).
The Portfolio Choice Problem
In the standard portfolio choice problem (Pratt, 1964), an investor with initial wealth $w_0$ decides how much to invest in a risky asset offering a random excess return $\tilde{X}$.
The investor's objective is to maximize expected utility:
$$ \max_{a} \mathbb{E}u[w_0 + a\tilde{X}] $$