Friday, November 27, 2015

Using the geometric distribution to model myopic investment behavior

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 kth trial after k-1 failures.   In the following expression, p is the probability of a success on each trial and (1-p) is the probability of a failure on each trial.   The probability of the first success on the kth trial is:


P (X=k) = (1-p)k-1   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 24-month period is the sumproduct of the probability vector and the actual price vector.

The dispersion measure is the E(P2) – 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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