DYNAMIC CASH FLOW ANALYSIS HANDOUTS

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DYNAMIC CASH FLOW ANALYSIS I. Natural Resource Analysis A company has the option to acquire a ten-year lease to extract gold from a gold mine. The lease should be acquired if its cost is less than the value of the lease. Here are relevant data:

• Price of gold is constant over the life of the lease. It is currently P = $400 per ounce.

• The mine’s capacity is 50,000 ounces.

• Operating costs reflect inherent nonlinearities in mining, namely, decreasing returns-to-scale and increasing costs as reserves are depleted. Specifically, when the begin-of-year mine inventory is I, the cost to mine Q ounces for the upcoming year is C*Q2/I, where C = 500.

• The annual cost of capital is 10%.

The following assumptions are made:

• Cash flows will be estimated on an annual basis.

• Taxes will be ignored.

• Production occurs instantaneously.

• Production quantities will be multiples of the lot size, in this case 1000 ounces.

• Profits each year accrue at the beginning of the year.

A production policy specifies the quantity and timing of production. Each production policy yields a corresponding profit cash flow stream, that is, yearly profits over time. The present value of yearly profits (at the appropriate cost of capital) represents the value of the leave given this production policy. A reasonable objective is to find the production policy that maximizes the present value of yearly profits. The resulting present value is the value of this lease.

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II. Fishing Example You own both a lake and a fishing boat as an investment package. You plan to profit by taking fish from the lake. Each season you decide either to fish or not fish. You have only three seasons to fish. Here are the relevant data:

• If you do not fish, the fish population will double by the start of next season.

• If you do fish, you will extract 70% of the fish population at the beginning of the season. At the start of the next season, the fish population will be the same as it was at the beginning of the current season.

• The initial fish population is 10 tons.

• The profit per ton is $1 (unit of money), which is constant.

• The cost of capital is 25%.

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III. Oil Pump Example You have purchased a lease for an oil well. The lease duration is three years. Here are the relevant data:

• In each year, you have three choices of how to operate the well: o Not pump. No operating cost and no change in oil reserves. o Normal pump. Operating cost is $50 thousand and 20% of current reserves will be

extracted. o Enhance pump. Operating cost is $120 thousand and 36% of current reserves will be

extracted.

• The price of oil is constant at $10 per barrel.

• This well has initial reserves of 100 thousand barrels of oil.

• The cost of capital is 10%.

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IV. Machine Replacement Analysis A company has to purchase a machine to produce a new product within its manufacturing plant. The product’s life is 10 years. Here are the relevant data obtained from the vendor who sells the machine:

• Purchase price is $15 million.

• The market value depreciation is 20% per year.

• The maintenance and operations cost in the first year is $0.50 million. Each year thereafter this cost grows at 65% per year.

• The equipment can be replaced at any time for a new machine. No technological improvements

are expected, and all of the current machine’s characteristics listed above will apply in the future.

• The annual cost of capital is 18%.

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NRA_OptimalAnalysis.01.hava r 0.0953102 p 400 dt 1 planningHorizon 2 I0 10 numPeriods 2 initialState State(2, 10) ProjectValue 1256.49 DDP_AO ddp_Result state value n I policy information action value 0 1 IGNORE 0.0 0 2 IGNORE 0.0 0 3 IGNORE 0.0 0 4 IGNORE 0.0 0 5 IGNORE 0.0 0 6 IGNORE 0.0 0 7 IGNORE 0.0 0 8 IGNORE 0.0 0 9 IGNORE 0.0 0 10 IGNORE 0.0 1 1 0 0.0 1 1 2 1 150.0 1 1 3 1 233.333 2 1 4 2 300.0 2 1 5 2 400.0 3 1 6 2 466.667 4 1 7 3 557.143 4 1 8 3 637.5 5 1 9 4 711.111 5 1 10 4 800.0 6 2 10 3 1256.49 7 DDP_AO ddp_TerminalResult state value n I policy information action value 0 1 IGNORE 0.0 0 2 IGNORE 0.0 0 3 IGNORE 0.0 0 4 IGNORE 0.0 0 5 IGNORE 0.0 0 6 IGNORE 0.0 0 7 IGNORE 0.0 0 8 IGNORE 0.0 0 9 IGNORE 0.0 0 10 IGNORE 0.0

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DDP_AO ddp_Value state action value n I 1 1 0 0.0 1 2 0 0.0 1 2 1 150.0 1 3 0 0.0 1 3 1 233.333 1 3 2 133.333 1 4 0 0.0 1 4 1 275.0 1 4 2 300.0 1 4 3 75.0 1 5 0 0.0 1 5 1 300.0 1 5 2 400.0 1 5 3 300.0 1 5 4 0.0 1 6 0 0.0 1 6 1 316.667 1 6 2 466.667 1 6 3 450.0 1 6 4 266.667 1 6 5 -83.3333 1 7 0 0.0 1 7 1 328.571 1 7 2 514.286 1 7 3 557.143 1 7 4 457.143 1 7 5 214.286 1 7 6 -171.429 1 8 0 0.0 1 8 1 337.5 1 8 2 550.0 1 8 3 637.5 1 8 4 600.0 1 8 5 437.5 1 8 6 150.0 1 8 7 -262.5 1 9 0 0.0 1 9 1 344.444 1 9 2 577.778 1 9 3 700.0 1 9 4 711.111 1 9 5 611.111 1 9 6 400.0 1 9 7 77.7778 1 9 8 -355.556 1 10 0 0.0 1 10 1 350.0 1 10 2 600.0 1 10 3 750.0 1 10 4 800.0 1 10 5 750.0 1 10 6 600.0 1 10 7 350.0 1 10 8 0.0 1 10 9 -450.0 2 10 0 727.273 2 10 1 996.465 2 10 2 1179.55 2 10 3 1256.49 2 10 4 1224.24 2 10 5 1113.64 2 10 6 872.727 2 10 7 562.121 2 10 8 136.364 2 10 9 -450.0

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DDP_AO ddp_PresentValue state action value n I 1 1 0 0.0 1 2 0 0.0 1 2 1 0.0 1 3 0 0.0 1 3 1 0.0 1 3 2 0.0 1 4 0 0.0 1 4 1 0.0 1 4 2 0.0 1 4 3 0.0 1 5 0 0.0 1 5 1 0.0 1 5 2 0.0 1 5 3 0.0 1 5 4 0.0 1 6 0 0.0 1 6 1 0.0 1 6 2 0.0 1 6 3 0.0 1 6 4 0.0 1 6 5 0.0 1 7 0 0.0 1 7 1 0.0 1 7 2 0.0 1 7 3 0.0 1 7 4 0.0 1 7 5 0.0 1 7 6 0.0 1 8 0 0.0 1 8 1 0.0 1 8 2 0.0 1 8 3 0.0 1 8 4 0.0 1 8 5 0.0 1 8 6 0.0 1 8 7 0.0 1 9 0 0.0 1 9 1 0.0 1 9 2 0.0 1 9 3 0.0 1 9 4 0.0 1 9 5 0.0 1 9 6 0.0 1 9 7 0.0 1 9 8 0.0 1 10 0 0.0 1 10 1 0.0 1 10 2 0.0 1 10 3 0.0 1 10 4 0.0 1 10 5 0.0 1 10 6 0.0 1 10 7 0.0 1 10 8 0.0 1 10 9 0.0 2 10 0 727.273 2 10 1 646.465 2 10 2 579.545 2 10 3 506.494 2 10 4 424.242 2 10 5 363.636 2 10 6 272.727 2 10 7 212.121 2 10 8 136.364 2 10 9 0.0

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ddp_CashFlow state action value n I 1 1 0 0.0 1 2 0 0.0 1 2 1 150 1 3 0 0.0 1 3 1 233.333 1 3 2 133.333 1 4 0 0.0 1 4 1 275 1 4 2 300 1 4 3 75 1 5 0 0.0 1 5 1 300 1 5 2 400 1 5 3 300 1 5 4 0 1 6 0 0.0 1 6 1 316.667 1 6 2 466.667 1 6 3 450 1 6 4 266.667 1 6 5 -83.3333 1 7 0 0.0 1 7 1 328.571 1 7 2 514.286 1 7 3 557.143 1 7 4 457.143 1 7 5 214.286 1 7 6 -171.429 1 8 0 0.0 1 8 1 337.5 1 8 2 550 1 8 3 637.5 1 8 4 600 1 8 5 437.5 1 8 6 150 1 8 7 -262.5 1 9 0 0.0 1 9 1 344.444 1 9 2 577.778 1 9 3 700 1 9 4 711.111 1 9 5 611.111 1 9 6 400 1 9 7 77.7778 1 9 8 -355.556 1 10 0 0.0 1 10 1 350 1 10 2 600 1 10 3 750 1 10 4 800 1 10 5 750 1 10 6 600 1 10 7 350 1 10 8 0 1 10 9 -450 2 10 0 0.0 2 10 1 350 2 10 2 600 2 10 3 750 2 10 4 800 2 10 5 750 2 10 6 600 2 10 7 350 2 10 8 0 2 10 9 -450

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DCFA_FishingExample.hava r 0.25 planningHorizon 3 I0 10 P 1 Actions (NONE, FISH) prodPercentage(NONE) 0 prodPercentage(FISH) 0.7 newInventoryFactor(NONE) 2 newInventoryFactor(FISH) 1 initialState State(3, 10) ProjectValue 20.16 DDP_AO ddp_Result state value n I policy information action value 0 10 IGNORE 0.0 0 20 IGNORE 0.0 0 40 IGNORE 0.0 0 80 IGNORE 0.0 1 10 FISH 7.0 10 1 20 FISH 14.0 20 1 40 FISH 28.0 40 2 10 FISH 12.6 10 2 20 FISH 25.2 20 3 10 NONE 20.16 20 DDP_AO ddp_Value state action value n I 1 10 NONE 0.0 1 10 FISH 7.0 1 20 NONE 0.0 1 20 FISH 14.0 1 40 NONE 0.0 1 40 FISH 28.0 2 10 NONE 11.2 2 10 FISH 12.6 2 20 NONE 22.4 2 20 FISH 25.2 3 10 NONE 20.16 3 10 FISH 17.08 DDP_AO ddp_PresentValue state action value n I 1 10 NONE 0.0 1 10 FISH 0.0 1 20 NONE 0.0 1 20 FISH 0.0 1 40 NONE 0.0 1 40 FISH 0.0 2 10 NONE 11.2 2 10 FISH 5.6 2 20 NONE 22.4 2 20 FISH 11.2 3 10 NONE 20.16 3 10 FISH 10.08 ddp_CashFlow state action value n I 1 10 NONE 0 1 10 FISH 7.0 1 20 NONE 0 1 20 FISH 14.0 1 40 NONE 0 1 40 FISH 28.0 2 10 NONE 0 2 10 FISH 7.0 2 20 NONE 0 2 20 FISH 14.0 3 10 NONE 0 3 10 FISH 7.0

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DCFA_OilPumpExample.hava r 0.1 planningHorizon 3 I0 100000 P 10 Actions (NONE, NORMAL, ENHANCE) prodPercentage(NONE) 0 prodPercentage(NORMAL) 0.2 prodPercentage(ENHANCE) 0.36 cost(NONE) 0 cost(NORMAL) 50000 cost(ENHANCE) 120000 initialState State(3, 100000) ProjectValue 366744 DDP_AO ddp_Result state value n I policy information action value 0 26214.4 IGNORE 0 0 32768.0 IGNORE 0 0 40960.0 IGNORE 0 0 51200.0 IGNORE 0 0 64000.0 IGNORE 0 0 80000.0 IGNORE 0 0 100000 IGNORE 0 1 40960.0 NORMAL 31920.0 32768.0 1 51200.0 ENHANCE 64320.0 32768.0 1 64000.0 ENHANCE 110400 40960.0 1 80000.0 ENHANCE 168000 51200.0 1 100000 ENHANCE 240000 64000.0 2 64000.0 ENHANCE 139418 40960.0 2 80000.0 ENHANCE 226473 51200.0 2 100000 ENHANCE 340364 64000.0 3 100000 ENHANCE 366744 64000.0 DDP_AO ddp_Value state action value n I 1 40960.0 NONE 0.0 1 40960.0 NORMAL 31920.0 1 40960.0 ENHANCE 27456.0 1 51200.0 NONE 0.0 1 51200.0 NORMAL 52400.0 1 51200.0 ENHANCE 64320.0 1 64000.0 NONE 0.0 1 64000.0 NORMAL 78000.0 1 64000.0 ENHANCE 110400 1 80000.0 NONE 0.0 1 80000.0 NORMAL 110000 1 80000.0 ENHANCE 168000 1 100000 NONE 0.0 1 100000 NORMAL 150000 1 100000 ENHANCE 240000 2 64000.0 NONE 100364 2 64000.0 NORMAL 136473 2 64000.0 ENHANCE 139418 2 80000.0 NONE 152727 2 80000.0 NORMAL 210364 2 80000.0 ENHANCE 226473 2 100000 NONE 218182 2 100000 NORMAL 302727 2 100000 ENHANCE 340364 3 100000 NONE 309422 3 100000 NORMAL 355885 3 100000 ENHANCE 366744

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DDP_AO ddp_PresentValue state action value n I 1 40960.0 NONE 0.0 1 40960.0 NORMAL 0.0 1 40960.0 ENHANCE 0.0 1 51200.0 NONE 0.0 1 51200.0 NORMAL 0.0 1 51200.0 ENHANCE 0.0 1 64000.0 NONE 0.0 1 64000.0 NORMAL 0.0 1 64000.0 ENHANCE 0.0 1 80000.0 NONE 0.0 1 80000.0 NORMAL 0.0 1 80000.0 ENHANCE 0.0 1 100000 NONE 0.0 1 100000 NORMAL 0.0 1 100000 ENHANCE 0.0 2 64000.0 NONE 100364 2 64000.0 NORMAL 58472.7 2 64000.0 ENHANCE 29018.2 2 80000.0 NONE 152727 2 80000.0 NORMAL 100364 2 80000.0 ENHANCE 58472.7 2 100000 NONE 218182 2 100000 NORMAL 152727 2 100000 ENHANCE 100364 3 100000 NONE 309422 3 100000 NORMAL 205885 3 100000 ENHANCE 126744 ddp_CashFlow state action value n I 1 40960.0 NONE 0.0 1 40960.0 NORMAL 31920.0 1 40960.0 ENHANCE 27456.0 1 51200.0 NONE 0.0 1 51200.0 NORMAL 52400.0 1 51200.0 ENHANCE 64320.0 1 64000.0 NONE 0.0 1 64000.0 NORMAL 78000.0 1 64000.0 ENHANCE 110400 1 80000.0 NONE 0.0 1 80000.0 NORMAL 110000 1 80000.0 ENHANCE 168000 1 100000 NONE 0 1 100000 NORMAL 150000 1 100000 ENHANCE 240000 2 64000.0 NONE 0.0 2 64000.0 NORMAL 78000.0 2 64000.0 ENHANCE 110400 2 80000.0 NONE 0.0 2 80000.0 NORMAL 110000 2 80000.0 ENHANCE 168000 2 100000 NONE 0 2 100000 NORMAL 150000 2 100000 ENHANCE 240000 3 100000 NONE 0 3 100000 NORMAL 150000 3 100000 ENHANCE 240000

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MRA_NewMachineAnalysis.hava r 0.18 P 15.0 mvd 0.2 mc1 0.5 mcg 0.65 PH 5 initialState State(5) ProjectValue 17.4747 ddp_Obj ddp_MIN DDP_AO ddp_Result state value n policy information action value n 0 IGNORE 0.0 1 1 5.2542 0 2 2 9.1217 0 3 3 12.1704 0 4 4 14.8342 0 5 5 17.4747 0 DDP_AO ddp_TerminalResult state value n policy information action value 0 IGNORE 0.0 DDP_AO ddp_Value state action value n 1 1 5.2542 2 1 9.7069 2 2 9.1217 3 1 12.9845 3 2 12.8951 3 3 12.1704 4 1 15.5681 4 2 15.6727 4 3 15.3683 4 4 14.8342 5 1 17.8256 5 2 17.8623 5 3 17.7222 5 4 17.5443 5 5 17.4747

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DDP_AO ddp_PresentValue state action value n 1 1 0.0 2 1 4.45271 2 2 0.0 3 1 7.73025 3 2 3.77348 3 3 0.0 4 1 10.3139 4 2 6.55106 4 3 3.19787 4 4 0.0 5 1 12.5714 5 2 8.74059 5 3 5.55175 5 4 2.71006 5 5 0.0 ddp_CashFlow state action value n 1 1 5.25424 2 1 5.25424 2 2 9.12166 3 1 5.25424 3 2 9.12166 3 3 12.1704 4 1 5.25424 4 2 9.12166 4 3 12.1704 4 4 14.8342 5 1 5.25424 5 2 9.12166 5 3 12.1704 5 4 14.8342 5 5 17.4747 pvMCost action value 1 0.423729 2 1.01623 3 1.84473 4 3.00322 5 4.62315 pvSValue action value 1 10.1695 2 6.89457 3 4.67429 4 3.16901 5 2.14848 ANNUAL EQUIVALENT METHOD AE n value 1 6.2 2 5.82615 3 5.59748 4 5.51445 5 5.58801 nStar 4 infHorizonPolicyCost 17.5443

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MRA_ExistingMachineAnalysis.hava r 0.18 P 15.0 mvd 0.2 mc1 0.5 mcg 0.65 PH 5 age 3 initialState State(5, 3) ProjectValue 9.268 ddp_Obj ddp_MIN DDP_AO ddp_Result state value n age policy information action value n age 0 0 IGNORE 0.0 1 0 1 5.2542 0 0 2 0 2 9.1217 0 0 3 0 3 12.1704 0 0 4 0 4 14.8342 0 0 5 0 5 17.4747 0 0 5 3 1 9.268 4 0 DDP_AO ddp_Value state action value n age 1 0 1 5.2542 2 0 1 9.7069 2 0 2 9.1217 3 0 1 12.9845 3 0 2 12.8951 3 0 3 12.1704 4 0 1 15.5681 4 0 2 15.6727 4 0 3 15.3683 4 0 4 14.8342 5 0 1 17.8256 5 0 2 17.8623 5 0 3 17.7222 5 0 4 17.5443 5 0 5 17.4747 5 3 0 9.7947 5 3 1 9.268 5 3 2 9.7756 5 3 3 11.4453 5 3 4 14.5784 5 3 5 19.6678 DDP_AO ddp_PresentValue state action value n age 1 0 1 0.0 2 0 1 4.45271 2 0 2 0.0 3 0 1 7.73025 3 0 2 3.77348 3 0 3 0.0 4 0 1 10.3139 4 0 2 6.55106 4 0 3 3.19787 4 0 4 0.0 5 0 1 12.5714 5 0 2 8.74059 5 0 3 5.55175 5 0 4 2.71006 5 0 5 0.0 5 3 0 17.4747 5 3 1 12.5714 5 3 2 8.74059 5 3 3 5.55175 5 3 4 2.71006 5 3 5 0.0

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ddp_CashFlow state action value n age 1 0 1 5.25424 2 0 1 5.25424 2 0 2 9.12166 3 0 1 5.25424 3 0 2 9.12166 3 0 3 12.1704 4 0 1 5.25424 4 0 2 9.12166 4 0 3 12.1704 4 0 4 14.8342 5 0 1 5.25424 5 0 2 9.12166 5 0 3 12.1704 5 0 4 14.8342 5 0 5 17.4747 5 3 0 -7.68 5 3 1 -3.30334 5 3 2 1.03502 5 3 3 5.89352 5 3 4 11.8683 5 3 5 19.6678 pvMCost state action value n age 1 0 1 0.423729 2 0 1 0.423729 2 0 2 1.01623 3 0 1 0.423729 3 0 2 1.01623 3 0 3 1.84473 4 0 1 0.423729 4 0 2 1.01623 4 0 3 1.84473 4 0 4 3.00322 5 0 1 0.423729 5 0 2 1.01623 5 0 3 1.84473 5 0 4 3.00322 5 0 5 4.62315 5 3 0 0 5 3 1 1.90344 5 3 2 4.56504 5 3 3 8.28676 5 3 4 13.4909 5 3 5 20.7678 pvSValue state action value n age 1 0 1 10.1695 2 0 1 10.1695 2 0 2 6.89457 3 0 1 10.1695 3 0 2 6.89457 3 0 3 4.67429 4 0 1 10.1695 4 0 2 6.89457 4 0 3 4.67429 4 0 4 3.16901 5 0 1 10.1695 5 0 2 6.89457 5 0 3 4.67429 5 0 4 3.16901 5 0 5 2.14848 5 3 0 7.68 5 3 1 5.20678 5 3 2 3.53002 5 3 3 2.39323 5 3 4 1.62253 5 3 5 1.10002

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DYNAMIC CASH FLOW ANALYSIS HOMEWORK PROBLEMS 1. Consider the gold mining problem analyzed in class. Relevant data pertaining to the optimal policy are provided in the following table:

Optimal Production Schedule 0 1 2 3 4 5 6 7 8 9 K values 213.8149 211.454 208.17 203.58 197.126 187.965 174.79 155.47 126.281 80 Percentage 0.207769 0.21075 0.21493 0.22079 0.22912 0.2411 0.25866 0.2852 0.32727 0.4 Inventory 50000 39611.5 31263.2 24543.9 19124.7 14742.8 11188.3 8294.31 5928.78 3988.45 Production 10388.45 8348.32 6719.31 5419.17 4381.92 3554.5 2894.01 2365.53 1940.33 1595.38 Profit 3076182 2459604 1965644 1569406 1250769 993304 783317 608889 458623 319076 PV 3076182 2236004 1624499 1179118 854292 616763 442162 312456 213951 135320 Value 10690747

a. What is the inventory at the end of the 10-year lease? b. Suppose the gold mine has been in operation for exactly 7 years. The inventory left is 15,000 ounces.

What is the value of the gold mine? c. Your answer to part b assumes what? d. Given the situation in part b, what is the optimal production plan to the end of the lease?

2. Produce the Optimal Production Schedule table shown in problem #1 for the gold mine lease when the profit flow is π(I, z) = z*(I-z) instead of g*z – Cz2/I. (Take I0 = 50, i.e., measure in units of thousands.) 3. Recall the fishing problem described in the handout. You own a lake. Each season you decide either to fish or not to fish. If you do not fish, the fishing population will increase by 50% by the start of the next season. (In the example, the population doubled if you did not fish.) If you do fish, you will extract 70% of the fish that were in the lake in the beginning of the season, and the fish population at the beginning of the next season will be the same as at the beginning of the season. The initial fish population is 10 tons. Your profit is $1 per ton. The discount factor is d = 0.80. You have three seasons to fish. Determine the value of your lake, and the optimal fishing strategy over time. 4. A company has purchased a lease for an oil well. Here are the relevant data: • In each year, the company has three choices of how to operate the well:

o Not pump. No operating cost and no change in oil reserves. o Normal pump. Operating cost is $75 thousand and 30% of current reserves will be extracted. o Enhance pump. Operating cost is $200 thousand and 51% of current reserves will be

extracted. • The price of oil is constant at $10 per barrel. • This well has initial reserves of 100 thousand barrels of oil. • The cost of capital is 15%. • Profits accrue at the beginning of the year. a. Determine the value of a two-year lease and the optimal pumping strategy using the method of dynamic programming. b. The company’s actual lease duration was 10 years. It is the end of the 4th year (i.e. t = 4) and the company is deciding what its pumping strategy should be for year 5. The company’s production strategy for the first four years had been to pump normal, no pump, pump normal, no pump. What is the optimal pumping strategy for year 5?

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DYNAMIC CASH FLOW ANALYSIS HOMEWORK PROBLEM SOLUTIONS 1. (a) x10 = x9 – z9 = 3,988.45 – 1,595.38 = 2,393. (b) V7(x7) = K7x7 = 155.47(15,000) = 2,322,050. (c) Assumes the gold mine will be operated in the optimal way until the end of the lease. (d) z7 = 0.2852(15,000) = 4,278. x8 = x7 – z7 = 15,000 – 4,278 = 10,722. z8 = 0.32727(10,722) = 3,509. x9 = x8 – z8 = 10,722 – 3,509 = 7,213. z9 = 0.40(7,213) = 2,885. 2. Use GoldMineHmkSoln.hava. 3. Use DCFA_FishingExample.hava. 4. Use DCFA_OilPumpExample.hava.

V(1, 100,000) = max{0, 10(30,000) – 75,000, 10(51,000) – 200,000} = 310,000. Enhance. V(1, 70,000) = max{0, 10(21,000) – 75,000, 10(35,700) – 200,000} = 157,000. Enhance.

V(1, 49,000) = max{0, 10(14,700) – 75,000, 10(24,990) – 200,000} = 72,000. Normal. V(0, 100,000) = max{0, [10(30,000)-75,000] + (1/1.15)[157,000],

[10(51,000)-200,000] + (1/1.15)[72,000]} = 372,609. Enhance. Optimal pumping strategy is to enhance pump first year, then pump normal in second year. Value of the two-year lease is 372,609.

The amount of inventory at the beginning of year 5 is 49,000. The cash flow using normal pump is 10(0.3)(49,000) - 75,000 = 72,000 and the remaining inventory will be 34,300. The cash flow using enhance pump is 10(0.51)(49,000) - 200,000 = 49,900, and the remaining inventory will be 24,010. As both cash flows are positive, it is optimal to pump at some level. Since the cash flow using normal pump is HIGHER than the cash flow using enhance pump, and since the remaining inventory will obviously be higher when using normal pump, the optimal pumping strategy is to pump normal.

NOTE: You cannot use your reasoning in part (a) to answer part (b). This is because the optimal

decision for the last period of part (a) may not be the optimal solution for year 5 in a 10-year horizon. The future value of all cash flow obtained from years 6-10 as a result of the decision in year 5 must be considered when deciding the optimal decision in year 5.

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