A Probabilistic Model of ETF
Return and Risk
Situation: An investor considers investing in Vanguard
Consumer Staples Fund VDC on four dates ranging from 7/1/13 to 4/1/14 and selling
all shares on four dates ranging from 7/1/17 to 4/1/18.
The date and adjusted close
stock price data for the four funds are presented in the table below.
Adjusted Close Price for Vanguard Consumer Staples Fund VDC
on Four Purchase Dates and Four Sale Dates


Purchase Date

Adjusted Close Price
on Purchase Date

Sale Date

Adjusted Close Price on Sale Date

7/1/13

93.55

7/1/17

10.3%

10/1/13

96.38

10/1/17

8.8%

1/1/14

93.50

1/1/18

10.5%

4/1/14

101.93

4/1/18

7.1%

Questions on this investment opportunity
How many possible outcomes
are there for an investor who purchases on one of the four possible purchase
dates and sells on one of the four possible sale dates?
What is the annual return for
each of these transactions?
What is the average annual
return and the standard deviation of returns over all possible outcomes defined
by the four purchase dates and the four sale dates?
What is minimum 25^{th}
percentile, 50^{th} percentile, 75^{th} percentile, and max
return for an investor considering an investment in VDC on the four purchase
dates coupled with the four transaction dates?
Questions on this approach to measuring risk and
return
Typically, mutual funds and
ETFs publish return statistics based on holding periods with a single purchase
date and a single sale date? What is
the advantage of the use of return statistics based on multiple purchase and
sale dates?
This procedure examines
average annual return. Why are
statistics based on average annual return more meaningful than statistics based
on total returns in a setting where holding periods differ?
Analysis of Investment Opportunity: There are 16
possible holding periods based on the four possible purchase dates and the four
possible sale dates. The table
summarizing the calculation of returns for these 16 outcomes is presented
below.
Returns for Four
Purchase Dates and Four Sale Dates of Fund VDC


Purchase Date

Sale Date

Years

Adjusted Close on Purchase Date

Adjusted Close on Sale Date

Annualized Return


1

7/1/13

7/1/17

4.00

93.55

138.29

10.3%

2

10/1/13

7/1/17

3.75

96.38

138.29

10.1%

3

1/1/14

7/1/17

3.50

93.50

138.29

11.8%

4

4/1/14

7/1/17

3.25

101.93

138.29

9.8%

5

7/1/13

10/1/17

4.25

93.55

133.78

8.8%

6

10/1/13

10/1/17

4.00

96.38

133.78

8.5%

7

1/1/14

10/1/17

3.75

93.50

133.78

10.0%

8

4/1/14

10/1/17

3.50

101.93

133.78

8.1%

9

7/1/13

1/1/18

4.50

93.55

146.60

10.5%

10

10/1/13

1/1/18

4.25

96.38

146.60

10.4%

11

1/1/14

1/1/18

4.00

93.50

146.60

11.9%

12

4/1/14

1/1/18

3.75

101.93

146.60

10.2%

13

7/1/13

4/1/18

4.75

93.55

129.86

7.1%

14

10/1/13

4/1/18

4.50

96.38

129.86

6.9%

15

1/1/14

4/1/18

4.25

93.50

129.86

8.0%

16

4/1/14

4/1/18

4.00

101.93

129.86

6.2%

Technical Note: Years is simply difference between sale date
and purchase date divided by 365.25.
The annualized return is (PSALE/PURCHASE^{)(1/YEARS)}
Below are the return
statistics for the 16 possible outcomes.
Return Statistics for 16 Possible Return Outcomes


Average

9.3%

STD

1.7%

Percentile


0

6.2%

25

8.1%

50

9.9%

75

10.3%

100

11.9%

Discussion of issues pertaining to the return and risk
measures:
Typically, mutual funds and ETFs publish return
statistics based on holding periods with a single purchase date and a single
sale date? What is the advantage of the
use of return statistics based on multiple purchase dates and multiple sale
dates?
Very few investors buy and
sell stocks on an arbitrarily chosen holding period. The average and dispersion of returns from
multiple holding periods (in this case 16) provides a more realistic portrayal of
possible outcomes than a return statistic based on a single holding period.
It is impossible to measure
risk based on one holding period. The
use of multiple holding periods allows us to measure dispersion of returns,
which is one measure of risk.
Investor are also concerned
about the correlation of returns between different investment opportunities. We could calculate returns for these 16 holding
periods for some other ETF and measure correlation with 16 returns from this
ETF.
ETFs and funds often report
yeartodate, 1year, 5year and 10year holding periods. However, the end date on all these return
measures is usually identical – often the most recent date or the date at the
end of the most recent month. An
abnormally high or low closing price affects all reported traditional return
measures. By contrast, the probabilistic
return measure is less sensitive to returns on a specific date.
This procedure examines average annual return. Why are statistics based on average annual
return more meaningful than statistics based on total returns?
Total returns depend on the
length of the holding period and expected total return should increase for
longer holding periods. However,
comparisons based on total return measure are misleading. What return would you prefer total returns
of 9.5% for 3.5 years or total returns of 9.6% for 4.5 years.
The variance of total returns
should also be larger for longer time periods.
Heteroskedastic variance estimates will reduce the validity of
statistical tests based on the return data.
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