Survival Analysis and the Cox Proportional Hazards Model

You want to know what predicts how long a hedge fund survives before shutting down, or how long until a loan defaults. This looks like an ordinary regression problem — regress "time to event" on fund or loan characteristics — except many funds in your dataset haven't closed yet by the time you're analyzing the data. You know they've survived at least this long, but not their eventual lifespan. Throwing those funds out wastes information and biases the sample toward short-lived funds; treating their unknown future lifespan as if it were their true lifespan is simply wrong. Survival analysis is built specifically to use this partial information — called censoring — correctly.

An analogy: a race that's still running

Imagine timing a marathon, but some runners haven't crossed the finish line by the time you have to file your report. You know each of those runners has been running for at least, say, 3 hours 40 minutes — that's real, usable information about how slow or fast they are — but you don't know their actual finish time. Ignoring these runners (only reporting on people who've finished) skews your report toward the fast ones. Guessing their exact finish time and treating it as known also introduces fake precision. Survival analysis treats "still running, known to have lasted at least this long" as its own distinct, legitimate piece of data, called a censored observation.

The model, one piece at a time

Survival analysis centers on the hazard function $h(t)$: the instantaneous rate at which the event happens at time $t$, given that it hasn't happened yet. The Cox proportional hazards model lets covariates shift this hazard multiplicatively without needing to specify the hazard's overall shape over time:

$$h(t \mid x_i) = h_0(t) \exp(\beta_1 x_{1i} + \beta_2 x_{2i} + \cdots),$$

where $h_0(t)$ is a baseline hazard — the risk over time for a hypothetical observation with all covariates at zero — left completely unspecified and never directly estimated, and $\exp(\beta_1 x_{1i} + \cdots)$ is a multiplier that scales that baseline risk up or down for observation $i$ based on its characteristics. In plain English: the model doesn't try to say when the average fund closes, only how much more or less likely one fund is to close at any given moment relative to another, based on their characteristics — which is why it's called "proportional hazards" and why it can be estimated from a mix of observed and censored lifetimes without ever pinning down $h_0(t)$'s shape.

The key quantity for interpretation is the hazard ratio, $e^{\beta_1}$: if $\beta_1 = 0.5$ for "uses leverage," then $e^{0.5}\approx 1.65$ means a levered fund faces 65% higher instantaneous closure risk than an otherwise identical unlevered fund, at every point in time.

Worked example 1: fund closure hazard by strategy

Fitting a Cox model to 300 hedge funds, with strategy type as a covariate, gives $\hat\beta_{\text{L/S equity}} = -0.4$ relative to the baseline (multi-strategy) group. The hazard ratio is $e^{-0.4} \approx 0.67$: long/short equity funds close at about 67% of the rate of multi-strategy funds at any given tenure, i.e., roughly a 33% lower instantaneous closure risk, holding other characteristics fixed. This says nothing about the average lifespan directly — a fund with a lower hazard throughout still has, on average, a longer survival time, but the Cox model quantifies the risk ratio, not the lifespan ratio, directly.

Worked example 2: censoring in a loan book

A bank tracks 1,000 loans issued over the past 5 years for time to default. 220 have already defaulted (observed event times, e.g., default at month 14, month 31, etc.), and 780 are still performing as of the report date — each contributes a censored observation ("survived at least $T$ months, true default time unknown"). A naive analysis using only the 220 defaulted loans would overstate default risk (it silently drops every loan that's performing well), while an analysis that pretends the 780 performing loans defaulted "now" would understate typical loan lifespan. The Cox partial likelihood correctly uses all 1,000 loans, letting each censored loan contribute the information "still alive at time $T$" without requiring a guess at its ultimate outcome.

What this means in practice

Survival analysis is the right tool wherever the outcome is "time until X" and some observations haven't experienced X yet: fund and firm closures, loan default timing, time-to-fill for resting orders, time until a corporate bond is called or matures early, or customer churn. Ignoring censoring — dropping still-active observations, or treating "hasn't happened yet" as "never will" — is one of the most common silent biases in financial time-to-event data.