Question

How does compounding of returns play a role in calculating returns to an investor? Explain how you would calculate how much an investor would need to save for X years if they desired Y annual returns? How would the number of years to retirement, the projected life expectancy, and the level of risk tolerance, affect your answer? How do taxable vs. non-taxable returns affect the answer?

Answer 1

Understanding Compound Return

• Compound return is viewed as a much more accurate measure of performance of an investment's return over time than the average return. This is because the average annual return does not take compounding into effect, which results in a gross misstatement of an investor's actual returns. Average returns either overestimate or underestimate growth or decline in returns. In effect, compound returns ensure that volatility, which can inflate or deflate returns, is accounted for in calculations.

  KEY TAKEAWAYS

• Compound return is the rate of return for capital over a cumulative series of time.
• Compound returns are a more accurate measure as compared to average returns to calculate growth or decline in an investment over a period of time.

• Example of Compound Return

  For example, suppose you started with an initial investment of $1,000. If you multiply 1,000 by 1.1 five times, that is, $1,000 x (1.1)5, you will end up with about $1,611. If an investment of $1,000 ended up being worth $1,611 by the end of five years, the investment could be said to have generated a 10% annual compound return over that five-year period.

  Here is the math:

• Year 1: $1,000 x 10% = $1,100
• Year 2: $1,100 x 10% = $1,210
• Year 3: $1,210 x 10% = $1,331
• Year 4: $1,331 x 10% = $1,464.10
• Year 5: $1,464 x 10% = $1,610.51

• Among the choices, the geometric average (also known as the "compound average") does the best job of describing investment return reality. To illustrate, imagine that you have an investment that provides the following total returns over a three-year period:

  Year 1: 15%
  Year 2: -10%
  Year 3: 5%

  To calculate the compound average return, we first add 1 to each annual return, which gives us 1.15, 0.9, and 1.05, respectively. We then multiply those figures together and raise the product to the power of one-third to adjust for the fact that we have combined returns from three periods.

  (1.15)*(0.9)*(1.05)^1/3 = 1.0281

  Finally, to convert to a percentage, we subtract the 1 and multiply by 100. In doing so, we find that we earned 2.81% annually over the three-year period.