# Determinants of Housing Price

## What matters when determining the price of a house? Is the value of the components of a home quantifiable? What changes can you make to your house to increase value the most?

— Aidan Padala

Last year, in preparation for selling their house, my Grandparents dug a hole and built a koi pond in their yard in hopes of increasing house value and improving their odds of selling the house. Little did they know, an open body of water often reduces house value as it attracts mosquitos and reduces interest from families with pets and young children. With decisions like this, there is always money on the line, and making assumptions can be costly. Instead, I am falling back on data, and relying on it to show what determines the rises and falls in the value of a home.

**Square Footage**

There is a clear correlation here between living space and the value of a home. The market places a high value on usable property, and nothing is a better proxy for usable property than sq ft. of living space in the house. Between 1000 and 5000 square feet (the vast majority of the data), we see that the distribution is essentially linear, and by calculating the slope we see that on average, adding 150 sq ft. of living space could increase your house value by around $48,000.

Consider the houses above the red reference line in comparison to those below it. For any given property size, the points get more lighter in shade as living space increases (price goes up), however for any given living space, increases in property size don’t go hand in hand with increases in price. This is a clear indication that empty space is wasted space, and that house value will rise with renovations and extensions.

This leads to the big question: What do you add to a home to increase value?

# Bedrooms

Here we clearly see that the number of bedrooms in a house is a key determinant of house price, as there is a clear relationship between the number of bedrooms and the value of a house.

As we can see above, the relationship very closely mirrors

Avg. House Price ≈ 100,000 + 100,000 * Bedrooms

This gives us a rough indication that building a new bedroom onto a house can increase its value by about $100,000. According to homeguide.com,

The average bedroom is around 150 square feet, and as we saw earlier increasing your living space by 150 sq ft. can on average add around $48,000 in value to your home. Since bedrooms on average can increase your house value by around $100,000, we see that bedrooms are a very high value extension for a house, and make the most of the square footage, providing a higher than average return.

# Bathrooms: A Cautionary Tale

Here’s the catch: from any point on this plot, go over one and up one to simulate adding a bedroom without adding a bathroom as well. Typically, house value does not go up all that much, showing that the ratio of bathrooms to bedrooms is an important factor for consumers when deciding the value of a home. Houses with many bathrooms relative to the number of bedrooms are typically much higher value than those with fewer bathrooms. For example, according to this plot, a house with 4 bedrooms and 4 bathrooms is likely to be more expensive than a house with 5 bedrooms and 3 bathrooms.

In these plots, we see that there is an exponential relationship between the number of bathrooms and the average house value. Using the graph on the right, where the Y-axis is the log of the house price, we see a roughly linear relationship that indicates a relationship of

log(P/100,000) = 0.75 + 0.5(B) → P = 100,000*e^(0.75 + 0.5B)

where P = House Price and B = # of Bathrooms. This relationship is very closely mirrored on the graph on the left, where a plot of average house value and Number of Bathrooms almost identically mirrors the relationship derived from the graph on the right. Whereas adding a bedroom to a house is mostly linear, we see that the value added by adding a bathroom depends on how many bathrooms you already have.

# Crime Rates

In looking for a home, future homeowners want a place where they will feel safe, so logically housing prices and crime rates should be correlated.

Here we have a plot of house prices on a map of Seattle. The darker the red fill of the dot, the more expensive the house was. Right off the bat we see trends, where some areas (such as waterfront areas) are more expensive, whereas others (such as houses close to airports) are less expensive, showcasing how important location is to the price of a house.

As we can see from these figures, crime rates relative to population can be a great approximation of the value of a home. For example, the East Precinct boasts the highest population density, and likely the second highest population (after the North precinct), however it has roughly the same number of crimes as the South(Southeast) precinct, which is much lower in population and population density. As we would expect, we see that on average, points in the East precinct are much darker than those in the SE precinct, showing that house prices are lower in areas with higher crime rates.

Using data on Housing Price Indexes and Crime Rates in major cities, we are able to create this plot, where we see that there is a clear negative relationship between violent crime rates and house prices.

# Code

See the original code in our R Notebook