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# Survival Analysis

Updated on January 8, 2024
Article byKumar Rahul
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

## What Is Survival Analysis?

Survival analysis, also known as time-to-event analysis or event history analysis, is a statistical and analytical method used to study the time it takes for an event of interest to occur. This event can be any event with a temporal dimension, such as the failure of a machine, the onset of a disease, or various financial events like default on a loan, bankruptcy, or a stock price hitting a certain level.

For eg:
Source: Survival Analysis (wallstreetmojo.com)

It predicts the likelihood and timing of default events for loans, bonds, or other financial instruments. Financial institutions can develop models to estimate the probability of default for individual borrowers or issuers. Financial institutions and credit rating agencies use survival analysis to assess credit risk.

### Key Takeaways

• Survival analysis is a statistical method tailored to model the timing of financial events. It is especially valuable where timing is often as crucial as the event itself.
• It accommodates censored data, where events have not occurred for all observations. It allows for a more realistic analysis of financial events.
• It is essential for credit risk assessment, helping financial institutions predict and manage the timing of loan defaults. It also guides lending decisions and risk pricing.
• It accounts for time-varying covariates, making it suitable for assessing changing risk factors over time, an essential consideration.

### Survival Analysis Explained

Survival analysis is a specialized statistical technique that explores and predicts the timing of crucial financial events and outcomes. Outcomes involve loan defaults, bond defaults, market crashes, or investment returns.

The origins of survival analysis can be attributed to actuarial science and the insurance industry. Historically, it was initially developed to calculate the longevity of insurance policyholders and estimate payouts. Over time, financial experts recognized its broader applicability and adapted it to assess financial risks.

From this perspective, it seeks to achieve several goals:

1. Risk Quantification: It enables financial institutions and investors to quantify and manage risks. It models the probability of financial events occurring within specific timeframes.
2. Default Prediction: For lenders, it is a powerful tool for predicting loan defaults and credit risk. By analyzing historical data, they can estimate when borrowers are likely to default, helping make sound lending decisions.
3. Portfolio Optimization: It assists financial institutions in optimizing portfolios by evaluating the expected duration of assets and investments.
4. Investment Insights: Investors use it to assess the timing of market events and make informed decisions regarding investments, factoring in the probability of reaching specific financial goals.
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### Assumptions

Survival analysis, like in any other application, is based on certain assumptions to make meaningful predictions and inferences about the timing of financial events. Here are the key assumptions when applying it in a financial context:

1. Non-Informative Censoring: It assumes that censoring (i.e., the event of interest not occurring during the study period) is non-informative. This means that the reasons for censoring (e.g., data on a loan default being unavailable because the loan is still active) are not related to the likelihood of the event occurring.
2. Constant Hazard: It typically assumes that the hazard rate, which represents the instantaneous risk of the event occurring, given that it has not occurred yet, remains relatively constant over time.
3. Independence: It is assumed that the survival times (times to financial events) of different individuals or financial instruments are independent of each other. In reality, correlations and dependencies might exist in financial markets or portfolios, but for simplicity, survival analysis often assumes independence.
4. Proportional Hazards (Cox Model): When using the Cox proportional hazards model, an essential assumption is that the hazard ratio between two groups remains constant over time.
5. No Competing Risks: It assumes that there is only one type of event of interest. This might only sometimes hold. For example, a financial instrument might be subject to both default and prepayment. In such cases, the assumption of no competing risks may not be satisfactory, and more complex models are required.
6. Exogeneity: Financial events should not affect the covariates or predictors used in the survival analysis. If an event itself impacts the variables used for modeling, it can violate this assumption.
7. Homogeneous Populations: It often assumes that the population under study is homogeneous, meaning that all individuals or financial instruments share the same characteristics or risk factors.

### Types

Survival analysis encompasses several types or variations, each tailored to address specific financial applications and challenges. Here are some common types of it from a financial perspective:

1. Kaplan-Meier Survival Analysis: The Kaplan-Meier estimator is a fundamental tool for estimating the survival function, which represents the probability of a financial event not occurring up to a specific time. This analyzes the survival or default rates of financial instruments, such as bonds or loans, and compares different groups of financial assets.
2. Cox Proportional Hazards Model: The Cox model analyzes the impact of various factors (covariates) on the timing of financial events, like loan defaults or bankruptcy. It assumes that the hazard ratio between different groups remains constant over time, making it suitable for modeling risk factors and predicting default probabilities.
3. Parametric Survival Models: Parametric models, like the Weibull or exponential distributions, make explicit assumptions about the functional form of the hazard rate. They estimate the time of financial events, incorporating specific assumptions about the event’s distribution.
4. Censored Data Handling: It often deals with censored data, where events haven’t occurred for all observations in the dataset. Specialized techniques like the Turnbull and Nelson-Aalen estimators handle censored data and provide more accurate survival estimates.
5. Accelerated Failure Time Models: These models assume that the logarithm of the survival time is linearly related to covariates. They quantify how financial events affect other factors, helping assess the impact of different variables on the timing of the event.
6. Frailty Models: These models account for unobserved heterogeneity among financial instruments or individuals. Frailty models help in understanding how unobserved factors may influence the timing of events like loan defaults.

### Examples

Let us understand it better via the following examples.

#### Example #1

Suppose a fictional bank, “FuturaBank,” wants to predict loan defaults more accurately using survival analysis. They collect data on various loans, including loan amounts, interest rates, credit scores, and employment histories. The event of interest is the default on a loan.

FuturaBank employs the Cox Proportional Hazards model to analyze the data. After setting up the model, they found that higher interest rates and lower credit scores significantly increase the risk of loan default. The model also reveals that the risk of default is relatively stable over time.

By applying survival analysis, FuturaBank can estimate the probability of loan default for individual borrowers throughout the life of their loans. This helps the bank make better-informed lending decisions, set interest rates more accurately, and manage credit risk more effectively.

#### Example #2

In a recent development of 2023, Tune Insight, an AI-driven financial technology company, has successfully secured \$3.4 million in funding. The company specializes in credit risk analysis and offers a unique approach to the world of finance. Using advanced algorithms, including survival analysis, Tune Insight provides an innovative solution for predicting credit risk and analyzing loan defaults. This new funding round, led by prominent investors, will further strengthen the company’s mission to revolutionize credit risk assessment.

Survival analysis, a statistical method crucial in financial risk assessment, is instrumental in Tune Insight’s predictive capabilities. By modeling the timing of financial events, such as loan defaults, Tune Insight aims to empower financial institutions with more accurate tools for credit scoring and portfolio management. This recent funding injection bolsters their innovative approach, helping to refine the art of credit risk assessment in the financial industry.

### Applications

Survival analysis has various applications, providing valuable insights and predictive capabilities for understanding and managing financial risks. Some of its critical applications  include:

1. Credit Risk Assessment: Financial institutions use it to assess and predict the risk of loan defaults and credit events. By modeling the time to default for borrowers, they can make more informed lending decisions and set appropriate interest rates based on individual creditworthiness.
2. Default Prediction: It predicts the likelihood and timing of bond and loan defaults. This is essential for both creditors and investors in assessing the creditworthiness of issuers and making investment decisions.
3. Loan Portfolio Management: Banks and other lending institutions apply such analysis to optimize their loan portfolios. They can analyze the expected time to default for different types of loans, helping them diversify their portfolios and allocate capital efficiently to manage risk.
4. Risk Assessment for Investments: Investors use this analysis to assess the risks associated with various financial investments, such as stocks and bonds. It allows them to model the time-to-market events, such as reaching a specific price level, and make more informed investment choices.
5. Market Event Analysis: It analyzes the timing of significant market events like stock market crashes or changes in asset prices. Understanding the timing of such events is essential for traders, investors, and risk managers to respond effectively.
6. Customer Lifetime Value: Financial institutions use this analysis to estimate the lifetime value of a customer. By modeling the duration of customer relationships and predicting churn, they can develop strategies to retain profitable customers and acquire new ones.
7. Insurance Risk Assessment: Insurance companies utilize this analysis to assess the timing of insurance claims, payouts, and policy lapses. This is critical for pricing insurance products and managing their financial liabilities.

Following is a representation of the advantages and disadvantages of survival analysis:

### Survival Analysis vs Logistic Regression

Below is a comparison of survival analysis and logistic regression, focusing on their application:

1. How does survival analysis differ from logistic regression?

Survival analysis models the timing of events, focusing on time-to-event data, while logistic regression is used for binary classification without considering the timing aspect. The choice depends on whether timing is essential in the analysis.

2. How is survival analysis applied to credit risk assessment?

It is applied to assess credit risk by modeling the time to loan default. By analyzing historical data and borrower characteristics, it estimates the probability and timing of loan defaults, aiding in credit scoring and lending decisions.

3. Can survival analysis be used for investment decision-making?

Yes, it can be applied to assess the timing of investment events, such as reaching a specific return threshold or market events. It helps investors make more informed decisions about their investment strategies.

This article has been a guide to what is Survival Analysis. We explain its examples, assumptions, applications, comparison with logistic regression, and types. You may also find some useful articles here –