Using linear and quadratic rent functions to discuss skewness of distributions.
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?
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(XE(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

(RENTE(RENT))2

(RENTE(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

(RENTE(RENT))2

(RENTE(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 (XE(X))3 and divide by the standard
deviation to the 3^{rd} 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
realworld 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.
http://www.sethlevine.com/archives/2014/08/ventureoutcomesareevenmoreskewedthanyouthink.html
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