Question

The returns on a stock for the last 5 years have been 32%, 18%, -20%, 16%, and -16%. Assume that you purchased the stock 5 years ago for $34.25 and that all returns have come in the form of either capital gains or losses (i.e., there have been no dividends). (4 points) a. What is the price of the stock today? b. Compute the average (arithmetic) annual return. c. Compute the geometric average annual return.

Answer 1

Part a:

Given that the stock returns for last 5 years are: 32%, 18%,-20%,16%, and -16%.
Stock price 5 years ago=$34.25
The stock value changed as:
Initial stock price*(1+Percentage change)

Year 1:$34.25*(1+32%)=$34.25*(1.32)=$45.21

Year 2:$45.21*(1+18%)=$45.21*(1.18)=$53.3478

Year 3:$53.3478*(1-20%)=$53.3478*(0.8)=$42.67824

Year 4:$42.67824*(1+16%)=$42.67824*(1.16)=$49.5067584

Year 5:$49.5067584*(1-16%)=$49.5067584*(0.84)=$41.58567706

Price of the stock today=$41.58567706 or $41.59 (Rounded up to two decimal places)

Part b:

Average arithmetic annual return=Sum of all the observations/Number of observations
Sum of all observations=32%+18%+(-20%)+16%+(-16%)=30%
Number of observations=5
Average arithmetic annual return=30%/5=6%

Part c:

Geometric average annual return=[(1+Rate of return in year 1)*(1+Rate of return in year 2)*(1+Rate of return in year 3)*(1+Rate of return in year 4)*(1+Rate of return in year 5)]^(1/Number of years) -1

Rate of return in year 1=32%
Rate of return in year 2=18%
Rate of return in year 3=-20%
Rate of return in year 4=16%
Rate of return in year 5=-16%

Number of years=5
Geometric average annual return=[(1+32%)*(1+18%)*(1-20%)*(1+16%)*(1-16%)]^1/5 -1
=[(1.32)*(1.18)*(0.8)*(1.16)*(0.84)]^1/5 -1
=1.214180352^(1/5) -1
=1.039576947-1
=0.039576947 or 3.96% (Rounded up to two decimal places)