*By Joel Redmond, CFP, United States*

Life doesn’t always go according to formula, but formulas play a much bigger part in our lives than we might think. For example, one popular formula is used in every love story: boy meets girl, boy loses girl, boy wins girl back. Another formula characterizes the action genre: world-class action hero who quit the game is pushed over the edge and back on the scene by arch-nemesis preparing to take over the world. Still another formula characterizes many marriages today: happy spouse, happy house.

What formula is the typical client’s financial life expressed by? For some, it would be the simple time value of money formulas: the future value of a single sum (what they have saved already) and the future value of periodic payments (the amount they contribute each period to their savings).*

It’s common for financial planners to meet with clients and show them how much they will have at period end assuming certain assumptions. It’s also not uncommon to show clients the impact of various scenarios in financial markets: Monte Carlo simulations do this fairly well. These are theoretical calculations. But what about the *en masse*, big-picture connections we always suspect between major economic factors: new housing starts and GDP growth, interest rates and mortgage refinancings, and so on? What can we use to find out if one thing is generally related to another thing, so that given a change in the first, we can predict the value of the second? For something like this, we can use *linear regression*.

**Linear Regressions**

Linear regression is a method of statistical inference that expresses the relationship between two or more variables using a straight line. The line is the best approximation of this relationship, and it takes the form

Y = aX + b

Here, Y is the dependent variable, or the variable we’re trying to figure out. X is the independent variable: we’re trying to figure out how to predict a change in Y, given a specific change in X. A and b are constants – if we know them, and we’re given a value of X, we can find out the value of Y.**

What could X and Y be? Almost any quantity with enough data to suggest a relationship: snowfall and ski equipment sales for a particular region, for example.

For our case, we can use linear regression to measure something more interesting: we can make X the percentage of a nation’s citizens with a comprehensive financial plan, and Y the average family’s net worth at retirement, for example. (This will be a fictional nation.)

Often, linear regression merely confirms a suspicion we already have: for example, it seems reasonable to expect that more people with completed financial plans points to a higher average net worth. To find out if this is true, what would we need to do? Well, since we’re looking for the relationship, we look at our equation again. It was:

Y = aX + b

Where X = percentage of families with a financial plan, and Y = net worth at retirement. Instead of doing algebra the way we learned it in school, where we’re given a, X, and b and told to solve for Y, linear regression is like Jeopardy: we have to begin with the answers (X & Y) and solve for the question (a & b). We do this by collecting data points on X and Y.

X, the percentage of families with a financial plan, could be collected as an annual survey over a ten-year period (the number of periods we use in this example). Y, the average net worth at retirement of the families of our fictional nation, could be collected from government data archives or websites. Here is what a sample with ten observation periods might look like Figure 1:***

Once we’ve found out all of these data points for X and Y, we take the average of them. We then use a special formula to determine b****, and then plug in X and Y into the basic equation to solve for a. In the end, we find that:

Y = 11.388 + .178X (See Figure 2)

In this example. This means that, for every additional 1 percent of families with a financial plan, the average net worth at retirement of a family rises $115,660. How did we get this? Simply by plugging 1 into the X in the above equation.

**Uses of Linear Regressions**

Linear regressions are tremendously useful in several respects. First, they help us determine if there really is a relationship between two variables. The example we used was fairly obvious: if people pay more attention to their finances and engage a professional to help them, it means they’re likely to get richer. But many variables have a much subtler interaction with each other, and it isn’t always easy to find out whether they have a linear relationship at all. Linear regression can immediately help us visualize whether or not the data support a linear relationship.

**Technically, this is incorrect – the true form adds a random error term ε. The real expression is Y = aX + b + ε. But we don’t need to go into that in this post.

***This is a somewhat modest amount of observations for a linear regression – often the observations extend into the thousands and higher. Also – keep in mind these numbers are completely arbitrary and aren’t intended to represent an actual nation.

****The formula is b = Cov(x,y)/Var(x) – we take the covariance of X and Y and divide it by the variance of X. You can use investment texts to refresh your memory on these formulas.

*****This is the general principle behind the speculative (i.e. non-hedging) aspect of world derivative markets.

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