Over the course of the last few months, we have heard a lot about modelling, as epidemiologists tried to forecast the spread of Covid-19. While some have criticised the validity of these models, it is easy to see their importance. A basic exponential model was a very good predictor before lockdown measures began.

Models are not limited to pandemic responses and complicated experimental research. I would argue they guide how we make decisions in our personal and professional lives. When we buy a house or car, decide on a new job, or pick a place for lunch, subconsciously we go through a checklist of pros and cons.

A statistical model is more precise, allowing us to guess outputs from a few given inputs. While not perfect, these models go a long way in describing many facets in our day-to-day lives.

How we use models when buying cars

What do you look for in a new car? What do you consider when deciding on a fair price?

For used cars at least, the process usually starts the same for everyone. First you go to trademe, look at a few models you like and check the prices. Then you compare the prices against each other.

This one is a 2010, but it has low k's. This one seems like a good deal, but I'm not fond of neon green. This one is almost perfect, but I'd rather it had the flat-screen in the dash.

All of these features influence what you would be willing to pay for each car. This is done mostly subconsciously; instead of discounting a few cents for each k on the car, we tend to factor it into our ballpark figures.

But this is similar to what a statistical model would do. With a perfect model, you would put in all the factors which affect the price and, voila, you get the value of the car.

Unfortunately, perfect models don't exist for practical applications, which is why weather forecasters aren't always on the money. Good models still explain a lot. The better the model, the more it explains.

When it comes to used cars, some statisticians have thrown together models to estimate prices. I actually had to study this myself in my university stats class. I found one example online where the analysis only included four variables:

1. Mileage

2. Year

3. Whether it came with the tech upgrade

4. Whether it included a mechanics report

These four factors explained 90% of the price, with the other 10% unaccounted for. This is possibly explained by condition, colour, location etc. However there is also going to be some variation as people negotiate on price. For a ballpark figure, this is a helpful result.

For the cars included in their sample, they estimated a car with zero miles, from the current year, without the tech upgrade or mechanics report costs about US$27,500. Each mile on the clock discounted the price by about 6 cents. Each additional year discounted it by about $1,300. And including the tech and report added about $1,500.

So a 5 year old car with 200,000 miles, no bells and whistles? Expect to pay around US$9,000.

Share price modelling

This same modelling process can be applied to share prices and returns, if we have the factors needed.

The most widely known model is the CAPM (Capital Asset Pricing Model). Essentially it only considers a share's volatility compared to the market as a whole. You start with a risk-free return, like that of cash, and add on the return implied by the share's volatility.

This is like the example above, where we started with the value of a current year car with no miles, then discounted the price based on the number of miles.

The CAPM model explains about 70% of returns, pretty good with only one factor! Obviously, with something as complicated as the share market, we need a few more variables to explain returns.

In 1992, Eugene Fama and Ken French developed the Fama-French Three-Factor Model, adding two more factors to the CAPM. From the Wikipedia page:

They noticed small company shares do better over time, along with those of value companies. Value companies are those with a high value of assets on the books when compared to the value of their shares, hence a high book-to-market.

Adding these two extra factors meant the model now explained more than 90% of a share portfolio's returns. Not perfect, but incredibly informative nonetheless. Eugene Fama went on to share the 2013 Nobel Price for Economics because of this important work.

Why is this important?

The first advantage is clear. If we know key factors driving share returns, we can build portfolios around these factors. This is the foundation of Dimensional's philosophy, building evidence-based investment strategies.

But this model doesn't just explain the returns of Dimensional's portfolios, it explains the returns of all share portfolios.

In our car example, 90% of a car's price was explained by mileage, year, tech and availability of a mechanic's report. The remaining 10% of the price could be due to unknown factors, like condition or colour. There will always be a bit of random variation on top of this.

If we know how exposed a portfolio is to small companies and value companies, we can explain 90% of the returns received. The remaining 10% is due to unknown factors, the decisions of the manager and some random variation.

Once we have accounted for small and value, we can see whether an active portfolio manager is adding to performance through their decisions. Unsurprisingly, we find the absence of any such outperformance.

Bart Frijns, AUT professor and director of the Auckland Centre for Financial Research, investigated KiwiSaver growth funds for a 2016 paper. He used the three-factor model to see if any funds outperformed after accounting for the 90% of returns the model explained. His findings:

Put simply, there was no evidence active management added any value, once you factor in the 90% of returns explained by the model. The 10% that isn't explained is supposedly where active management can make the difference, but there was no evidence to suggest they did.

This implies the returns of KiwiSaver funds in New Zealand is explained by the risks they take, the fees they charge and a bit of chance. There may be a few more factors which are not yet known, but again, there is no evidence active managers have been taking advantage of these unknown factors.

To add to this, another Bart Frijns paper, published in 2018, found active managers were ignoring these factors when making investment decisions:

We choose to ignore the unexplained 10% in favour of the 90% explained by known factors, avoiding the high fees of active management. This can be done with index-like strategies, based around the proven factors of higher returns.