Average stock price for two stockpurchase 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 sharepurchase 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 constantexpenditure strategy provides a
lower average price than a constantshare 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.
This page describes several essays on personal financial
choices.
This page outlines several financial math problems solved in
Excel.
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