## Thursday, January 17, 2013

### Understanding Annualized Percentage Yield (APY)

Ulmer Scientific frequently compares the historic performance of actively traded securities like stocks, bonds, and exchange trade funds using an “Annualized Percentage Yield-to-Date” methodology.  While this might sound like a string of complicated finance jargon, understanding the basics of this relatively simple method is key to performing any financial or investing comparison.

Most financial websites like Yahoo! Finance and Google Finance will often provide a table of annualized returns (1 year, 3 year, 5 year) which make for great stock-to-stock comparisons.  However in depth analysis between multiple investments or analysis of multi-year trends demands the ability to calculate these returns from historical prices.

The following is intended to provide the basics for calculating annualized returns and should help explain the APY-to-Date charts used on Ulmer Scientific:

First the Math:

The premise behind calculating returns is that all the trades, dividends, distributions and on-goings with market securities generate some net change in value.  Typically, this change is represented by a percentage; or more appropriately, since investors are out to make money, “percentage yield”.

Market analysts frequently rely on the exponential growth formula:

Where P(t) is the present value, P(0) is the initial value, e is Euler's Number, r is the exponential growth rate (usually a decimal), and t is time.  Note: r is not the same as the percent change (i.e. r of 0.07 ≠ 7% growth), but we’ll take care of that.

In finance, the most common basis for comparison is the year; its quick, easy, and gives the investor a pretty clear idea of how their investment would have faired if they picked this security 365 days ago.  The standard year also allows for easy comparisons between multiple securities.  Rather than trying to relate the S&P 500’s gain of 142 points to the DOW Jones’ 12, investors just convert the change into a percent.

To calculate this annual percentage change, or yield, we rearrange the exponential growth formula to solve for r (and convert r to an actual percentage):

You’ll notice, if you plug in the prices of your favorite stock the percentage change is just the same if you were to simply divided the two.  Why the extra math?

In short, investors want to know how their investments perform over multiple years.  20% in one year is far more productive than 20% in 100 years.  So to keep things simple, investors prefer to keep everything on the same time scale, aka one year.  Thus we need to convert 20% in 100 years to some percentage per year

(Hint: it won’t work with simple division)

Give up?  We need to modify the annual percent yield formula to average (or annualize) our change over multiple years:

Now whither we use stock prices from last year, or last decade, we can easily compare multiple investments with a single %.

What Ulmer Scientific Does

This blog uses the APY method to calculate investments’ yield from any time (or value) in the past to the time of publishing; essentially producing an APY-to-Date Chart.  These charts can be used to compare similar investment products (GLD vs IAU), investment classes (bonds, equities, etc.), or anything we can think of.  Here’s an example:

Annualized Percent Yield on the y-axis, dates to the present on the x-axis.  The true power of these charts comes into play when you can see the effects of major financial events, like the market crash of late 2008, or the difference dividends and distributions make over the long run.