Question

A refinery has the crack spread of 2-1-1: 2 barrels of oil yield 1 barrel of gasoline and 1 barrel of heating oil.The refinery sells gasoline and heating oil at prices ofRg/bbl andRh/bbl respectively. Each price is in dollars and can take only two values:Rg∈{50, 60}andRh∈{50, 60}:•In the independent price scenario, probabilities of prices are P(Rg=50) =P(Rg=60) =P(Rh=50) =P(Rh=60) =1/2.•In the dependent price scenario, probabilities of prices are P((Rg,Rh) = (50, 50)) =P((Rg,Rh) = (60, 60)) =5. In the dependent price scenario, either both prices are low or both prices are high. From each barrel of crude oil, the refinery makes the revenue ofR=12Rg+12Rh.What are the variances of this revenue under dependent and independent price scenarios?

Answer 1

The variances of the revenue under dependent and independent price scenarios can be calculated using the formula: Var(R) = (Sum of (R - Mean(R))^2 * P(R)).

Step-by-step explanation:

In the independent price scenario, the variance of the revenue can be calculated as follows:

1. When Rg = 50 and Rh = 50, the revenue is R = 12(Rg) + 12(Rh) = 12(50) + 12(50) = 1200.
2. When Rg = 60 and Rh = 50, the revenue is R = 12(Rg) + 12(Rh) = 12(60) + 12(50) = 1380.
3. When Rg = 50 and Rh = 60, the revenue is R = 12(Rg) + 12(Rh) = 12(50) + 12(60) = 1320.
4. When Rg = 60 and Rh = 60, the revenue is R = 12(Rg) + 12(Rh) = 12(60) + 12(60) = 1440.

Calculate the variance using the formula: Var(R) = (Sum of (R - Mean(R))^2 * P(R))

When Rg = 50 and Rh = 50, the variance is Var(R) = ((1200 - 1350)^2 * 1/4) + ((1380 - 1350)^2 * 1/4) + ((1320 - 1350)^2 * 1/4) + ((1440 - 1350)^2 * 1/4) = 5625.

When Rg = 60 and Rh = 50, the variance is Var(R) = ((1200 - 1350)^2 * 1/4) + ((1380 - 1350)^2 * 1/4) + ((1320 - 1350)^2 * 1/4) + ((1440 - 1350)^2 * 1/4) = 5250.

When Rg = 50 and Rh = 60, the variance is Var(R) = ((1200 - 1350)^2 * 1/4) + ((1380 - 1350)^2 * 1/4) + ((1320 - 1350)^2 * 1/4) + ((1440 - 1350)^2 * 1/4) = 5625.

When Rg = 60 and Rh = 60, the variance is Var(R) = ((1200 - 1350)^2 * 1/4) + ((1380 - 1350)^2 * 1/4) + ((1320 - 1350)^2 * 1/4) + ((1440 - 1350)^2 * 1/4) = 5250.