loans

Time-varying rates of return create a new complication, that is best explained by an analogy. Is a car traveling 258,720 yards in 93 minutes fast or slow? It is not easy to say, because you are used to thinking in “miles per sixty minutes,” not in “yards per ninety-three minutes.” It makes sense to translate speeds into miles per hour for the purpose of comparing speeds. You can even do this for sprinters, who cannot run a whole hour. Speeds are just a standard measure of the rate of accumulation of distance per unit of time.
The same issue applies to rates of return: a rate of return of 58.6% over 8.32 years is notas easy to compare to other rates of return as a rate of return per year. So, most rates of return are quoted as average annualized rates. Of course, when you compute such an average annualized rate of return, you do not mean that the investment earned the same annualized rate of return of, say, 5.7% each year—just as the car need not have traveled at 94.8 mph (258,720 yards in 93 minutes) each instant. The average annualized rate of return is just a convenient unit of measurement for the rate at which money accumulates, a “sort-of-average measure of performance.”
So, if you were earning a total three-year holding return of 173% over the three year period, what would your average annualized rate of return be? The answer is not 173%/3 ≈ 57.7%,
because if you earned 57.7% per year, you would have ended up with (1 + 57.7%) · (1 + 57.7%) · (1 + 57.7%) − 1 = 287%, not 173%. This incorrect answer of 57.7% ignores the compounded interest on the interest that you would earn after the first year and second year. Instead, you need to find a single hypothetical rate of return which, if you received it each and every year, would give you a three-year rate of return of 173%.