### Agonizing Over Annualizing

Annualized returns allow for assets with different time horizons to be more easily compared. The uncertainty of returns causes two major problems: 1) multiple years of return is better than a single-year of the same annualized return and 2) return path influences preference. In most of my posts, I have a preformed opinion and solution. For this topic, I’m looking for input from readers.

**Multi-Year Returns**

Imagine three assets that are all at $100. One you believe will grow by $40 in a year, another $80 in two years, and a third will grow $120 in three years (see table below). I’ve laid out the Arithmetic (no compounding), Geometric (compounding), and Total returns. If I can find a brand new investment, identical to Stock #1, at the end of every year, I’m indifferent to the three assets (assuming a straight path – see next section about return path preference). It is a flawed assumption to assume I can find another Stock #1 at the end of Year 1 and Year 2, so, arguably, I’d prefer Stock #3 which gives me 40% arithmetic returns every year.

Here’s the problem. The Arithmetic return measures them all identically and the Geometric^{(1)} return suggests that Stock #3 is the worst. I’ve historically defaulted to geometric return because fund returns are geometric, but working with clients that have long-dated return streams has made me question my assumption. In fact, I believe that Stock #3 may actually deserve a boost compared to other 40% returns because of its duration (assuming a straight-line path which I’ll discuss below).

**Path Dependency in Annualized Returns**

If the total return was realized in a straight path for stocks one through three, then all three stocks would be at $140 dollars at the end of year one and both Stock #2 and Stock #3 would be trading at $180 at the end of year two. If that is the case, I have a strong preference for Stock #3. But if all of Stock #3s returns are realized at the end of Year #3, I may not be indifferent (see below).

If my return happens all at the end of year 3, the total return is the same but my reported performance results aren’t as great at the end of year one and two. In this case, I may prefer Stock #1, which I expect to return 40% this year.

Clearly, the multi-year and path-dependence nuances make choosing the right annualization method, geometric or arithmetic, confusing. Maybe the best plan is to use geometric to calculate the return but give the position size a boost for the number of years. If you have thoughts, I’m all ears. Please post comments to the blog or email me directly.

^{(1)}Geometric returns finds the compound annual growth rate (CAGR) that equals the total return for the time period. If you could find a brand new investment with a 30% return for 3 years and you invested the proceeds from year one into each subsequent year, you would end up with a 120% return. To make the Geometric return constant, returns would need to be reinvested at a 40% rate (see below).