Return on Investment (RoI) is the profit divided by the amount of investment so if I invest $1000 and get $1080 back it is an RoI of 8%. But that does not account for the length of time it took to earn that 8% so a better evaluation of how good the investment was is the rate of return which looks at the RoI per annum.

The obvious way is to divide the RoI by the length of the investment so if it took me 2 years to earn that 8% then my rate of return would be the average of 2% per year. But that is not the standard way of doing it. The standard rate of return is CAGR which assumes annual compounding. My question is why do we assume annual compounding? My investment may compound quarterly (dividends) or not at all (bonds) so annual compounding rate seems a bit of an artificial construction whereas the average rate of return doesn’t.

You want to be able to compare things. 4% compounded quarterly is better than 4% compounded annually. If you actually have something that returns 4% compounded quarterly, you should say so or report it as as an effective (1.01)[sup]4[/sup] = 1.0406 So state this as 4.06% effective annual rate (as if compounded annually).

We wouldn’t have to use the annual convention, but there needs to be some standard convention for comparison.

Because, theoretically, you could have have cashed out at any point and re-invested in something else. In reality, neither CAGR or ROI (or IRR for that matter) all that good in measuring investment success. This is because they all fail to consider risk or volatility. In reality, one should use NPV since the cash flows are discounted by the inherent risk of the investment. For funds, the Sharpe and Information ratios have far more explanatory power.

The choice of 1 year as a time period is arbitrary; in that sense it is an artificial construction. But some kind of simple mean of ROI wouldn’t cut it as this does not reflect the way investment value changes; investment value changes geometrically, not arithmetically. Another way to think about it is that each time-period’s worth of gain/loss is not independent; the gain/loss you obtain this time period depends on the principal value at the beginning of the time period, which in turn is dependent upon the gain/loss obtained during the prior time period.

An investment that returns 4% per year for two years is not equivalent to an investment that returns 8% over two years; the former investment is worth more due to compounding. The fact that various investment may not actually compound annually is irrelevant since their returns can always be re-expressed in terms of CAGR.

No. The best model is continuous compounding; this is the way growth is expected to work. The return can be normalized to reflect annual compounding or quarterly compounding, but you need compounding. In your example, the annual returns would be stated as

4.00% – your average method
3.92% – annual compounding
3.87% – quarterly compounding

It doesn’t seem to make a big difference which way you calculate.
But next suppose that instead of 8% net after 2 years, you have 80% net after 20 years. Now you’ll see that failing to assume compounding exaggerates your return hugely:

4.00% – your average method
2.98% – annual compounding
2.95% – quarterly compounding

If the purpose is just to compare like for like, all that matters is that the period be the same between the two numbers you are comparing.

I suspect annual is the standard because few financial instruments actually compound daily or weekly, and if you pick monthly the fact that months vary significantly in length make the numbers weird (e.g. a bond that pays a constant monthly coupon would have a higher rate of return in February than August because you earn the same amount in fewer days).

Annual is standard when trying to explain things to high-school students. Or for comparable consumer finance contracts.

For people dealing with financial instruments, there are all kinds of numbers, and (when I was involved), annual rate of return would have been used mostly when selling to retail clients. For everything else, overnight rates and 90 day rates and all kinds of simple-interest rates.

If you’re stating that an investment is returning 4% a year, you’re going to be assuming yearly compounding, because that’s how you’re stating the rate of growth. Almost always rates of return are based on a year, not on any other period of time, just for consistency’s sake. It’s possible to use some other period, but if you wanted to use continuous compounding, you’d have to come up with the rate that’s in the exponential function defining the constant compounding, a number that wouldn’t mean very much to the people trying to understand how fast the investment grows. A year is generally a good period of time for financial reporting because of the fact that many businesses are seasonal. There are very few businesses that have periodic business cycles longer than a year; perhaps convention centers get rented out more in Presidential election years, but that might just be a small blip in the grand scheme of things. So we’re using a year period so that we are comparing numbers that are actually comparable and don’t have a good reason to be different. And so we’re assuming all compounding is annual because we have to assume some method of compounding. We could assume continuous compounding despite using annual rates, but given there is a fixed relationship between them, it makes more sense to use an annul compounding period when describing the rate.

This is why you see a lot of interest rate offers that have two rates: one is the nominal Annual Percentage Rate, the rate used to calculate each amount of interest due by dividing the rate by the fraction of the year the interest covers, and the other is the the Annual Percentage Yield, the “official” guide to how much interest there is over the year, which is the total interest paid assuming that no payments on the principal + interest balance were made over the course of the year. I think there was a time when a company could publish the APR and not state the compounding period, which made it impossible to perfectly compare two offers unless you had knowledge not advertised; at some point they were required to provide the APY, and generally will provide the APR as well since that’s what’s lower (assuming we’re talking about consumer debt interest rates).