Question

Handi Inc., a cell phone manufacturer, procures a standard display from LCD Inc. via an options contract. At the start of quarter 1 (Q1), Handi pays LCD $4.00 per option. At that time, Handi’s forecast of demand in Q2 is normally distributed with a mean of 30,000 and a standard deviation of 9,000. At the start of Q2, Handi learns exact demand for Q2 and then exercises options at the fee of $2.70 per option, (for every exercised option, LCD delivers one display to Handi). Assume Handi starts Q2 with no display inventory and displays owned at the end of Q2 are worthless. Should Handi’s demand in Q2 be larger than the number of options held, Handi purchases additional displays on the spot market for $8.00 per unit. For example, suppose Handi purchases 37,000 options at the start of Q1, but at the start of Q2 Handi realizes that demand will be 40,000 units. Then Handi exercises all of its options and purchases 6,000 additional units on the spot market. If, on the other hand, Handi realizes demand is only 35,000 units, then Handi merely exercises 35,000 options.

a. Suppose Handi purchases 37,000 options. What is the expected number of options that Handi will exercise?

b. Suppose Handi purchases 37,000 options. What is the expected number of displays Handi will buy on the spot market?

c. Suppose Handi purchases 37,000 options. What is Handi’s expected total procurement cost?

d. How many options should Handi purchase from LCD?

e. What is Handi’s expected total procurement cost given the number of purchased options from part d?

Answer 1

Given,

Mean Demand (D) = 30000

Standard Deviation () = 9000

Option Price = $4

Exercise Price = $2.7

Spot market price = $8

(a) Options Purchased = Q = 37000

z = Q - D / = 37000 - 30000 / 9000 = 0.78

Loss function : L(0.78) = 0.125

Expected loss demand = L(0.78) x

= 0.125 x 9000 = 1125

Expected number of options exercised = D - Lost Demand = 30,000 - 1125 = 28,875

(b)

Q = 37,000

Units bought from the spot market = Expected loss in demand = 1125 (As calculated in the part (a)

(c)

Total Procurement cost = Cost of option + Cost of exercise + Cost of buying from spot market

= 4 x 37,000 + 2.7 x 28875 + 8 x 1125

= $ 234962.5

(d)

Cu = Cost per unit of demand Underestimated

Co = Cost per unit of demand Overestimated

Cu = Spot Market - Exercise Price = 8 - 2.7 = $ 5.3

Co = Option price = $ 4

Critical ratio = Cu / Co + Cu = 5.3 / 5.3+4 = 0.57

Optimal point will be obtained when P(D<=Q) = Critical ratio

Hence,

P(D<=Q) = 0.57

Hence, F(z) = 0.57

From the z-table,

z = 0.1764

Order Quantity (Q') = Mean Demand + (z * ) = 30,000 + 0.1764 x 9000 = 31587.6

Hence, the optimal number of Options to be purchased = 31588 units.

(e)

Now, we know,

z = 0.1764

Corresponding Loss function will be : L(0.1764) = 0.315

Expected lost demand = 0.315 x 9000 = 2835

Expected exercise options = 30000 - 2835 = 27165

Total Procurement cost = 4 x 31588 + 2.7 x 27165 + 8 x 2835 = $ 222377.5