# Hazard Rate: Definition, How to Calculate, and Example

> **Definition**
>
> The hazard rate measures how likely a non-repairable item is to fail at a specific age. It's part of the hazard function, which predicts survival chances over time. The hazard rate is calculated with a mathematical formula for use in industries like engineering, finance, and insurance. This information can be practically applied in asset management and risk assessment.

## What Is the Hazard Rate?

The hazard rate refers to the rate of death for an item of a given age (x). It is part of a larger equation called the hazard function, which analyzes the likelihood that an item will survive to a certain point in time, based on its survival to an earlier time (t). In other words, it is the likelihood that if something survives to one moment, it will also survive to the next.

The hazard rate only applies to items that cannot be repaired and is sometimes referred to as the failure rate. It is fundamental to the design of safe systems in applications and is often relied on in [commerce](https://www.investopedia.com/terms/c/commerce.asp), engineering, [finance](https://www.investopedia.com/ask/answers/what-is-finance/), [insurance](https://www.investopedia.com/terms/i/insurance.asp), and regulatory industries.

### Key Takeaways

*   The hazard rate measures the likelihood of an item's failure or death over time.  
    
*   Calculating the hazard rate involves the probability density and survival functions.  
    
*   The hazard rate curve often resembles a bathtub shape, indicating changes in failure likelihood over an item's lifespan.  
    
*   The hazard rate concept is crucial in fields like engineering, medicine, and insurance to predict system reliability and longevity.  
    
*   The failure rate is essentially another term for the hazard rate.

## Deep Dive into Hazard Rate Analysis 

The hazard rate shows how likely an item is to fail based on its age. It is part of a wider branch of [statistics](https://www.investopedia.com/terms/s/statistics.asp) called [survival analysis](https://www.investopedia.com/terms/s/survival-analysis.asp), which is a set of methods for predicting the amount of time until a certain event occurs, such as the death or failure of an engineering system or component.

The concept is also used in other fields with different names like reliability analysis in engineering, [duration analysis](https://www.investopedia.com/terms/d/duration.asp) in [economics](https://www.investopedia.com/terms/e/economics.asp), and event history analysis in sociology.

## How to Calculate the Hazard Rate

The hazard rate for any time can be determined using the following equation:

$h(t) = f(t) / R(t)$

F(t) is the [probability density function (PDF)](https://www.investopedia.com/terms/p/pdf.asp), or the probability that the value (failure or death) will fall in a specified interval, such as a specific year. R(t), on the other hand, represents the survival function, or the probability that something will survive past a certain time (t).

> **Important**
>
> The hazard rate cannot be negative.

## Real-World Example: Understanding Hazard Rate Applications

The probability density calculates the probability of failure at any given time. For instance, a person is certain to die eventually. As you get older, you have a greater chance of dying at a specific age, since the average failure rate is calculated as a fraction of the number of units that exist in a specific interval, divided by the number of total units at the beginning of the interval.

If we were to calculate a person's chances of dying at a certain age, we would divide one year by the number of years that person potentially has left to live. This number would grow larger each year.

A person aged 60 would have a higher probability of dying at age 65 than a person aged 30 because the person aged 30 still has many more units of time (years) left in their life, and the probability that the person will die during one specific unit of time is lower.

## Analyzing the Hazard Rate Curve: A Visual Perspective 

In many instances, the hazard rate can resemble the shape of a bathtub. The curve slopes downwards at the beginning, indicating a decreasing hazard rate, then levels out to be constant, before moving upwards as the item in question ages.

Imagine a car: its parts usually don't fail in the first few years after it's made. Still, a small number of cars might fail quickly because of factory defects. Once a new car has shown that it is free of such defects, it is not expected to fail for many years. As the car ages, the probability of malfunction increases. When the curve slopes up, the product's [useful life](https://www.investopedia.com/terms/u/usefullife.asp) is over, and unexpected failures become more likely.

## What Is the Difference Between the Failure Rate and Hazard Rate?

The hazard rate and failure rate are essentially the same thing. Failure rate is basically another way to say hazard rate.

## What Is the Hazard Rate Used for?

The hazard rate seeks to determine the chances of survival of its subject at a certain time point. It can effectively be applied to almost any item, and is commonly used in engineering, medicine, and insurance.

## What Is the Bathtub Hazard Rate Curve?

The bathtub curve is a visual representation of the typical failure rate of a product or group of products over time. The bathtub image maps out three distinct periods. The initial downward slope, also known as “infant mortality,” represents the timeframe when the product is first used and goes from a high failure occurrence to a low one. Next, the line levels out and becomes flat, depicting the useful life period when assets are expected to have a relatively constant failure risk. Then finally the slope moves upward, taking the form of a bathtub, as the product in question enters its wear-out period and degrades at an accelerated pace.

## The Bottom Line

The hazard rate effectively predicts how likely an item is to survive to a certain time point. That might not sound particularly useful or groundbreaking. However, in certain industries, the concept is used to make key decisions.

> **Article Sources**
>
> 1.  Boston University. "[Survival Analysis](https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_survival/BS704_Survival_print.html)."