Question Three: A broker calls a person once a month to urge
him to invest in the Spartan 500 Index fund FUSVX The likelihood that the person will make
this purchase is geometrically distributed with the probability 0.20 that he
will make the investment and 0.80 that he will not make this investment.
Once
the investor purchases shares the broker moves on and stops calling.
If after two years a person does not make an
investment the broker stops calling.
What
percent of people make an initial purchase of FUSVX? What is the likelihood a person purchases
FUSVX for months 1 to 24?
The table below has monthly data on the share price for FUSVX
from 2008 and 2009. This is a period
where the market and this ETF are tanking.
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

What is the expected price conditional on actual share
prices for the investor with share purchase behavior described by the geometric
distribution? What is the variance of
the share purchase price?
Answer: The geometric distribution gives the
probability of a first success on the k^{th }trial after k1 failures. In the following expression, p is the
probability of a success on each trial and (1p) is the probability of a
failure on each trial. The probability
of the first success on the k^{th} trial is:
P (X=k) = (1p)^{k1} p
Success in this problem is the probability the broker
persuades the person to purchase FUSVX.
The probabilities for each month are given in the table below.
Month

Probability of First
Success

Beginning of month Stock
Price

1

0.2000

48.8

2

0.1600

47.2

3

0.1280

47.0

4

0.1024

49.0

5

0.0819

49.7

6

0.0655

45.5

7

0.0524

44.8

8

0.0419

45.5

9

0.0336

41.4

10

0.0268

34.3

11

0.0215

31.8

12

0.0172

31.9

13

0.0137

29.2

14

0.0110

26.1

15

0.0088

28.4

16

0.0070

30.9

17

0.0056

32.6

18

0.0045

32.7

19

0.0036

35.0

20

0.0029

36.2

21

0.0023

37.6

22

0.0018

36.7

23

0.0015

38.9

24

0.0012

39.4

My measure or expected stock price conditional on actual
price for people who buy at some point in the 24month period is the sumproduct
of the probability vector and the actual price vector.
The dispersion measure is the E(P^{2}) – E(P)^{2}
The results are presented below.
Conditional Expected
stock Price and Dispersion of Expected Stock Price From Truncated Geometric
Distribution


E(Stock Price)

E(Stock Price2)

E(Stock Price2) E(Stock
Price)2

45.2

2081.2

41.0

Notes on this problem:
Note One: The assumption that the geometric
distribution guides the stock purchase decision is not one that would be made
by a rational investor. Rational
investors would have an unbiased estimate of the future stock price. Note that rational investors can be
wrong. Rationality or an unbiased
expectation differs from having perfect foresight, another assumption often
made by economists. Observe the stock price fell by a lot during this time
period (2008 through 2009). I would expect a person who had the
expectation that the stock price of this ETF would fall from 48 to 39 would not
buy this ETF.
Were most people who purchased stocks in this period
irrational? This claim seems way too
strong. We can, however, state that
these purchasers lacked perfect foresight and may have been guided by
myopia.
The assumption that people buy
based on a geometric distribution is one way to model myopic investment
decision process.
Note Two: I have assumed the solicitation to purchase
the stock price ends after 24 months.
The probabilities sum to 0.9953 not 1.0 because some people do not buy prior
to the end of the two year period. The
actual geometric distribution continues forever. This distribution is censored because of the
stipulation that solicitations end at month 24.
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