A note on the difference between geometric and arithmetic averages

13 Oct

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.


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