Question: The table below has price data and daily
return data for Vanguard fund VB.
Calculate the arithmetic and geometric averages of the daily return
data. Show that the geometric average
accurately reflects the relationship between the initial and final stock price
and the arithmetic average does not accurately explain this relationship.
Daily Price and Returns
For Vanguard
Fund VB


Date

Adjusted Close

Daily Return

7/1/16

115.480674


7/5/16

113.99773

0.987158509

7/6/16

114.744179

1.006547929

7/7/16

114.913373

1.001474532

7/8/16

117.202487

1.019920345

7/11/16

118.128084

1.007897418

7/12/16

119.451781

1.011205608

7/13/16

119.10344

0.997083836

7/14/16

119.262686

1.001337039

7/15/16

119.402023

1.00116832

7/18/16

119.63093

1.001917112

7/19/16

119.202965

0.996422622

7/20/16

119.959369

1.006345513

7/21/16

119.481646

0.996017627

7/22/16

120.297763

1.00683048

7/25/16

120.019083

0.997683415

7/26/16

120.616248

1.004975584

7/27/16

120.347522

0.997772058

7/28/16

120.536625

1.001571308

7/29/16

120.894921

1.002972507

8/1/16

120.735675

0.998682773

8/2/16

119.12335

0.986645828

Analysis: The table below presents calculation of the
two averages and the count of return days.
The product of the initial value of the ETF, the pertinent average and
the count of return days is the estimate of the final value. Estimates of final ETF value are calculated
for both the arithmetic average and the geometric average and these estimates
are compared to the actual value of the stock on the final day in the period.
Understanding The
Difference Between Arithmetic Mean and Geometric Mean Returns


Statistic

Value

Note

Arithmetic Average of
Daily Stock Change Ratio

1.001506208

Average function

Geometric Average of
Daily Stock Change Ratio

1.001479966

Geomean function

Count of Return Days

21

Count Function

Estimate of final value
based on arithmetic average

119.1889153

Initial Value x
Arithmetic Return Average x Count Days

Estimate of final value
based on geometric average

119.12335

Initial Value X Geometric
Return Average x Count Days

Ending Value

119.12335

Copy from data table

There is another way to show that the daily return should be
modeled with the geometric mean rather than arithmetic mean. The average daily return of the stock is
(FV/IV)^{(1/n) }– 1 where FV is final value and IV is initial value and
n is the number of market days in the period, which for this problem is 21.
Using this formula we find the daily average holding period
return is 0.001479966. Note that 1 minus the geometric mean of the
daily stock price ratio is also 0.001479966.
The geometric mean gives us the correct holding period
return.
No comments:
Post a Comment