## 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.