Question

The price of shares of ABC stock was

• $60 on January 1, 2018
• $70 on January 1, 2019; and
• $50 on January 1, 2020.

Paul bought one share of ABC on January 1, 2018, four more on January 1, 2019, and sold all five shares on January 1, 2020.

• 1. What is the annual arithmetic (time-weighted) average return on ABC stock during the two-year period?
  2. What is the annual geometric (time-weighted) average return on ABC stock during the two-year period?
  3. What is Paul's annual dollar-weighted average return during the two-year period? (To find the return, use Excel, a financial calculator, or trial and error. Using approximation is okay. Hint - Look for an integer between -7% and -12%).
  4. Why is Paul's annual dollar-weighted average return lower than the arithmetic and geometric time-weighted annual average return of ABC?

Answer 1

a. Time-weighted average return = [(1+HP1​)×(1+HP2​)×⋯×(1+HPn​)]−1

where: n= Number of sub-periods, HP= (End Value - Initial Value+Cash Flow)/(Initial Value+Cash Flow​) HPn​= Return for sub-period n​

Time weighted average return of Paul = [(1 + (50 - 60)/60)] x [1 + (50-70)/70] - 1

= [1-0.1667] x [1-0.286] - 1 = [0.833 x 0.714] - 1 = 0.5949 - 1 = 0.4050 (or) 40%

b. Geometric Time weighted return is same as Arithmetic Time rated return. Hence it is also 40%

c. Dollar weighted average return =

PVO=PVI=CF0​+CF1​​/(1+IRR)+CF2​​/(1+IRR)^2+....+CFn/(1+IRR)^n

​where:PVO=PV Outflows; PVI=PV Inflows; CF0​=Initial cash outlay or investment; CF1​,CF2​,CF3​,...CFn​=Cash flows; N=Each period; IRR=Initial rate of return​

Substituting values in above forumla, 50 = -60/(1+IRR) - 70/(1+IRR)^2

IRR = - 8.5% upon solving the same

x. Dollar weighted return is lower than arithmetic average return because of Lower IRR