Calculating Returns

Calculating Simple Returns

Here is one of my favorite little results in Finacial Math, because I use it constantly.

I tend to calculate simple returns in the following way:

$$r_{t} \equiv \frac{p_{t}}{p_{t-1}} - 1$$

Where:

$r_{t} :=$ % return at time $t$

$p_{t} :=$ $ price at time $t$

You can read it:

Today’s price, over yesterday’s price, minus one.

It is completely equivalent to the standard textbook formula:

$$r_{t} \equiv \frac{p_{t} - p_{t-1}}{p_{t-1}}$$

Proof:

$$\frac{p_{t} - p_{t-1}}{p_{t-1}} = \frac{p_{t}}{p_{t-1}} - \frac{p_{t-1}}{p_{t-1}} = \boxed{\frac{p_{t}}{p_{t-1}} - 1}$$

If there is any intermitent cash-flows its not much more complicated:

$$r_{t} \equiv \frac{p_{t} + cf_{t}}{p_{t-1}}$$

Why?

It is more immediate, and less buttons to press on a calculator, or values to track for mental math, for quick returns. Less buttons a human has to press to compute, the less chance for errors to occur.

A financial return is the ratio of an asset’s price at different points of time.

Also there is a subtle mindset difference with using this formula that fosters a perspective on capital appreciation.