## Thursday, December 3, 2015

### Average stock price for two stock-purchase strategies

Average stock price for two stock-purchase strategies

Question Four:  Consider two people who choose to invest in FUSVX over the 2008 to 2009 time period.   One person spends a fixed amount of money per month  (say \$1,000) on the first day of the month for this investment.    The other person buys a fixed number of shares per month (say 25 shares) on the first day of each month.

Calculate the average price per share for the two investors.  Calculate the variance of the price per shares purchased by the two investors.

Which of these two investment strategy results in the lowest average price for securities purchased?

Is this result specific to this time period for this security or does it generalize to other securities and other time periods/

What does this result say about the relative return from the two share-purchase strategies?

 Monthly Share Price of FUSVX for 2008 and 2009 1/2/08 48.8 2/1/08 47.2 3/3/08 47.0 4/1/08 49.0 5/1/08 49.7 6/2/08 45.5 7/1/08 44.8 8/1/08 45.5 9/2/08 41.4 10/1/08 34.3 11/3/08 31.8 12/1/08 31.9 1/2/09 29.2 2/2/09 26.1 3/2/09 28.4 4/1/09 30.9 5/1/09 32.6 6/1/09 32.7 7/1/09 35.0 8/3/09 36.2 9/1/09 37.6 10/1/09 36.7 11/2/09 38.9 12/1/09 39.4

Answer:  The average price per share purchased over the 24 months is the weighted average of monthly share prices where the weights are shares purchased in a month as a percent of total shares in the entire time period.

The number of stocks purchased in a month is the amount spent divided by the price.  The number of shares purchased is higher when the stock price is low than when the stock price is high.  Hence,  a person who spends a fixed amount per month will buy more shares when the stock price is low.

By contrast, the number of shares purchased is the same each month when the person purposely purchases a fixed number of shares each month.

Below are the weights for the two stock purchase rules.

 Weights for constant expenditures and constant shares Date Stock Price Number of shares purchased when \$1,000 spent each month Weights at constant expenditure per month Weights at constant shares per month 1/2/08 48.8 20.5 0.032 0.042 2/1/08 47.2 21.2 0.033 0.042 3/3/08 47.0 21.3 0.033 0.042 4/1/08 49.0 20.4 0.031 0.042 5/1/08 49.7 20.1 0.031 0.042 6/2/08 45.5 22.0 0.034 0.042 7/1/08 44.8 22.3 0.034 0.042 8/1/08 45.5 22.0 0.034 0.042 9/2/08 41.4 24.1 0.037 0.042 10/1/08 34.3 29.2 0.045 0.042 11/3/08 31.8 31.4 0.048 0.042 12/1/08 31.9 31.3 0.048 0.042 1/2/09 29.2 34.2 0.053 0.042 2/2/09 26.1 38.3 0.059 0.042 3/2/09 28.4 35.2 0.054 0.042 4/1/09 30.9 32.4 0.050 0.042 5/1/09 32.6 30.6 0.047 0.042 6/1/09 32.7 30.6 0.047 0.042 7/1/09 35.0 28.6 0.044 0.042 8/3/09 36.2 27.6 0.043 0.042 9/1/09 37.6 26.6 0.041 0.042 10/1/09 36.7 27.3 0.042 0.042 11/2/09 38.9 25.7 0.040 0.042 12/1/09 39.4 25.4 0.039 0.042 Total 648.4 1.0 1.0

The average share price is the dot product of the weights and the monthly share price.

 Expected Share Prices from Two Purchase Strategies Expected Price Constant Expenditures 37.0 Expected Price Constant Share Purchases 38.4

The strategy of purchasing a constant dollar amount of shares each month results in a lower purchase price than a strategy of purchasing a constant number of shares each month.

This makes sense because by spending a constant dollar amount each month you purchase more shares when the stock price is low and fewer shares when the stock price is high

Note:   This problem reminds me of a classic SAT problem.  A person drives 60 miles from his house to the beach and 50 miles back.  The trip to the beach takes 1 hour.   The trip back takes 2 hours.   What is the average speed?

One of the options 45 mph is wrong because the person spends more time in the car on the slow leg.

Note:   The constant-expenditure strategy provides a lower average price than a constant-share strategy.   Does this mean that the constant expenditure strategy will result in better investment returns?   Not in all circumstances.    Investment returns are based on both purchase price and sale price.

When a stock soars perpetually like Apple did a few years ago the thing to do is buy.  In this example, FSUVX fell over much of the time period.  Hence a person who purchased fewer shares at the beginning of the period and more at the end did much better than a person who bought a constant amount of the entire period.

People who liked this post may also be interested in some work on my new finance blog.