Tuesday, May 10, 2016

Monopoly PR4: Skewness of Returns from Different Rent Functions

Question:  What is the skewness of rent from electric company when rent is $4.00 times the sum of the values from two dice rolls?

Would the skewness number change if rent was $10 times the sum of the two dice rolls?

What is the skewness number if rent was equal to the square of the sum of the two dice rolls?

Background:   The formula for Skewness is

SK=E(X-E(X))3 /  STD(X)3




Analysis:


Below I have the data needed to calculate skewness for the situation where rent is $4.00 times the sum of the values on two dice.



Data for Variance and Skewness Calculations
on Electric Company
Dice Sum
Rent
Probability
Expected Rent
(RENT-E(RENT))2
(RENT-E(RENT))3
2
8
0.0278
28
400
-8000
3
12
0.0556
28
256
-4096
4
16
0.0833
28
144
-1728
5
20
0.1111
28
64
-512
6
24
0.1389
28
16
-64
7
28
0.1667
28
0
0
8
32
0.1389
28
16
64
9
36
0.1111
28
64
512
10
40
0.0833
28
144
1728
11
44
0.0556
28
256
4096
12
48
0.0278
28
400
8000
Rent equals $4 times sum of two dice. 

Note the probabilities and the returns are symmetrically distributed.

The perfect symmetry leads to a skewness of 0.

When the player owns Waterworks rent goes to $10 times the sum of the two dice rolls.   I will leave it to the reader to show that the higher rent does not change the shape of the distribution.

The table below is set up for the calculation of the skewness of rents when rent is the square of the sum of the values on the two dice.

Data for Variance and Skewness When Rent is the Square of
the Sum of Two Dice Rolls
Dice Sum
Rent
Probability
Expected Rent
(RENT-E(RENT))2
(RENT-E(RENT))3
2
4
0.0278
28
576
-13824
3
9
0.0556
28
361
-6859
4
16
0.0833
28
144
-1728
5
25
0.1111
28
9
-27
6
36
0.1389
28
64
512
7
49
0.1667
28
441
9261
8
64
0.1389
28
1296
46656
9
81
0.1111
28
2809
148877
10
100
0.0833
28
5184
373248
11
121
0.0556
28
8649
804357
12
144
0.0278
28
13456
1560896

Note the rent distribution shape from 4 times the sum of dice is very different from the rent distribution shape when rent is the square of the sum of two dice rolls.


The difference between high and low rents is much larger when rent is determined by a quadratic function of dice sums than a linear function of dice sums.


The values of the skewness statistics for the two situations are presented below.

How did I get the skewness figures?

·      Take the SUMPRODUCT of  (X-E(X))3 and divide by the standard deviation to the 3rd power.



Skewness Results for the Linear
 and Quadratic Rents
Rent $4 times
dice sum
Rent dice sum
squared
Var(Rent)
93.3
1909.8
Std (Rent)
9.7
43.7
Skewness (Rent)
0.0
1.7

The quadratic rent function provides higher returns when the sum of the dice is high.  This results in positive skewness.


Skewness is an important factor in monopoly and in real-world financial decisions.


Other Readings:  Most financial analysts emphasize expected returns and risk of returns but the skewness of stock prices or interest rates is also an important factor.  Venture capitalists often make most of their money from a small number of stocks (Google or Facebook most recently) that do very well.  Some investors who take profits too early miss out on the larger returns.    In the current macro environment, bond investors are naturally wary because interest rates are low and a large interest rate hike could lead to spectacular losses.   Future returns in the bond market are likely negatively skewed at this time.









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