## Wednesday, July 11, 2018

### A Probabilistic Model of ETF Return and Risk

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 25th percentile, 50th percentile, 75th 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 year-to-date, 1-year, 5-year and 10-year 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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