To meet a 99% service objective, you would need to set the inventory at roughly 2 or 3 standard deviations above the mean. (Mean and standard deviation here refer to the demand for the product.) The standard deviation of the total demand is given by "total = \\"walk-in + phone + 20 walk-in phoneP, where walk-in, phone are the standard deviations for the individual types of demands and p is their correlation. If the demand streams are positively correlated, the variability of total demand will be greater, resulting in a higher inventory requirement. If the demands are negatively correlated, total variability and required inventory will be lower. (a) E[3X + 2Y] = 3E[X] +2E[Y] = 3(.09) + 2(.12) = .51. (b) Var [3X + 2Y] = 903 + 40} + 2(3)(2)oxoyp = 1.04. The standard deviation is the square root, 1.02. The answer is (e). In more detail: (i) False, because ox could be larger than oz. (ii) False, because we don't know anything about the covariances. (iii) True, because standard deviations are positive so the correlation and covariance always have the same sign. (a) E[Y] = E[X2 - Xi] = E[X2] - E[X1] = 85 - 60 = 25. (b) Var[Y] = Var[X2+(-1)Xi] = ox, tox, -20x,ox,p = 58.4. So, oy = v58.4 = 7.64. (a) The standard deviation of the price change for one gallon is 0.032 so the standard deviation for -1, 000, 000 gallons is 32,000. (b) After buying 24 contracts, the company in effect has a position of -1, 000, 000.X + 24 . 42, 000) , using the notation in the hint. The resulting standard deviation is to hedge against a possible price increase. The company chooses to hedge by buying futures contracts on heating oil. Suppose the standard deviation of changes in the price per gallon of jet fuel over 3 months is 0.032, the standard deviation of changes in the futures price per gallon of heating oil is 0.040, and the correlation between the two is 0.8. Also, each heating oil futures contract is for 42,000 gallons. (a) What is the standard deviation of the company's unhedged exposure? (Think of the company as holding -1, 000, 000 gallons of jet fuel, because of its anticipated purchase.) (b) A simple gallon-for-gallon hedging rule would suggest that the company should buy 1, 000, 000/42, 000 = 23.8 ~ 24 contracts. What is the standard deviation of the com- pany's exposure under this strategy? (Hint: Let X = change in price of jet fuel and Y = change in futures price (per gallon) of heating oil. Write the exposure in terms of X and Y. ) (c) Find the number of contract that minimizes the standard deviation of the company's exposure