XYZ's stock price and dividend history are as follows:

Year Beginning-of-Year Price Dividend Paid at Year-End

2016 $ 110 $ 4

2017 120 4

2018 100 4

2019 110 4

An investor buys three shares of XYZ at the beginning of 2016, buys another two shares at the beginning of 2017, sells one share at the beginning of 2018, and sells all four remaining shares at the beginning of 2019.

a. What are the arithmetic and geometric average time-weighted rates of return for the investor?

Arithmetic time-weighted rates of return: 4.47%

Geometric time weighted rates of returns: 3.66%

Explanation:

Year-by-year time-weighted average rate of returns are:

2016-2017: (Beginning price of 2017 + Dividend at year end 2016 - Beginning price of 2016) / Beginning price of 2016 = (120+4-110) / 110 = 12.73%

2017-2018: (Beginning price of 2018 + Dividend at year end 2017 - Beginning price of 2017) / Beginning price of 2017 = (100+4-120) / 120 = - 13.33%

2018-2019: (Beginning price of 2019 + Dividend at year end 2018 - Beginning price of 2018) / Beginning price of 2018 = (110+4-100) / 100 = 14.00%

=> Arithmetic time-weighted rates of return = (12.73% - 13.33% + 14.00%) / 3 = 4.47%

=> Geometric time weighted rates of returns = [ (1 + 0.1273) x (1 - 0.1333) x (1 + 0.1400) ] ^ (1/3) - 1 = 3.66%