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# Remember Compounding When Computing Annualized Returns

You are offered the choice between two investments. They could be anything from merger arbitrage opportunities to certificates of deposit, doesn't really matter, but we'll assume their rates of return are known. Your options are:

• commit capital for one year to receive a return of 12%, or
• commit capital for one month to receive a return of 1%.

Assume you get the original capital back plus the return at the end of the investment period in one lump sum.

Which of these offers is better?

If you answered that they are equal, you'd be using an approach that appears in many good quality Seeking Alpha articles on special situations, such as these by Maudes Capital, Lukas Neely and Jim Sloan. Alas, you would also be wrong.

Many assumptions are implicit to the setup of the question above. For example, if you assume opportunities of this type to dry up in the future, you'll be more keen to opt for the secure return of the longer option. But let's make the simple (and dangerous) assumption of a static universe: interest rates (or other returns) remain as they are and these opportunities will be available with similar conditions also in the future. In that case the answer is simple. Let's say you have \$100,000 at the beginning of the year.

• For the one-year investment, you end up with \$112,000.
• For the short investment, you get \$101,000 at the end of January. In February, you have \$101,000 to invest and hence end the month with \$102,010, which you invest in March to receive \$103,030.10 and so on. You end the year with \$112,682.50.

Thus, the second option is the better one. The return over the year on your original capital is 12.68%, or to put it another way, your return is about 5.7% higher than the simple formula suggests.

All of this is of course well known: it is simply another manifestation of the power of compound interest. The annualized return that accounts for the compounding effect is known as CAGR (Compound Annual Growth Rate).

Now here's the key question: how much do annualized returns computed using the simple approach differ from the compound rate?

If the investment period is a year, the two results coincide perfectly. The shorter the investment horizon, the larger the understatement from the naive formula. For example, a return of 1% per week produces a CAGR of 67.8% vs. 52% simple expectation (a 30% higher total return). A return of 1% per day for the entire 365-day year would generate a CAGR of 3678% -- multiplying the inital capital over 37-fold instead of the 3.65 times of the simple formula.

On the other hand, the error is also proportional to the rate of return. The return in our 1% per month scenario was 5.7% higher that the naive formula. At 2% per month, the annualized return is 26.8%, almost 12% higher than expectation. At 5% per month, the error is over 30%.

Granted, these examples are rather forced, and if you ever actually find a horn of plenty that spits out a 1% return every single day, you probably don't need a lot of math to see that it's a keeper. Nevertheless, it's good to keep the compounding effect in mind especially when evaluating a situation with a short investment horizon. If something ties up capital for less than a month and you consider it without taking compounding into account, you may be significantly underestimating the opportunity in front of you.