Solutions to Practice Problems – Chapter 10

 

1 a)  The spreadsheet for cashflows and expected NPV is as follows.  Based on the expected values for each variable, NPV is $530,235.13.

 

 

 

 

 

cost

    6,000,000.00

 

 

 

 

 

total depreciation

    5,500,000.00

 

 

 

 

 

book value

       500,000.00

 

 

 

 

 

salvage value

    1,200,000.00

 

 

 

 

 

tax

       245,000.00

 

 

 

 

 

after-tax salvage value

       955,000.00

 

 

 

 

 

 

 

 

 

 

1

2

3

4

5

revenue

 

  16,316,437.50

  21,211,368.75

 25,453,642.50

  29,271,688.88

  33,662,442.21

variable cost

 

  10,302,187.50

  13,392,843.75

 16,071,412.50

  18,482,124.38

  21,254,443.03

fixed costs

 

    6,000,000.00

    6,300,000.00

   6,615,000.00

    6,945,750.00

    7,293,037.50

depreciation

 

    1,815,000.00

    1,375,000.00

   1,100,000.00

       880,000.00

       330,000.00

EBT

 

   (1,800,750.00)

       143,525.00

   1,667,230.00

    2,963,814.50

    4,784,961.67

tax @ 35%

 

      (630,262.50)

         50,233.75

      583,530.50

    1,037,335.08

    1,674,736.59

NI

 

   (1,170,487.50)

         93,291.25

   1,083,699.50

    1,926,479.43

    3,110,225.09

depreciation

 

    1,815,000.00

    1,375,000.00

   1,100,000.00

       880,000.00

       330,000.00

increase in WC

 

       489,493.13

       424,227.38

      381,804.64

       439,075.33

 

OCF

 

       155,019.38

    1,044,063.88

   1,801,894.86

    2,367,404.09

    3,440,225.09

capital inv

  (6,000,000.00)

 

 

 

 

 

initial wc

  (1,631,643.75)

 

 

 

 

 

recovery

 

 

 

 

 

    3,366,244.22

after tax salvage value

 

 

 

 

 

       955,000.00

Net cashflows

  (7,631,643.75)

       155,019.38

    1,044,063.88

   1,801,894.86

    2,367,404.09

    7,761,469.31

PV

  (7,631,643.75)

       138,410.16

       832,321.33

   1,282,553.17

    1,504,528.10

    4,404,066.13

NPV

      530,235.13

 

 

 

 

 

The results of the sensitivity analysis  will be as follows:

Pessimistic NPV Optimistic NPV
Market size (units)

162,301.96

898,168.31

Market Share 48,577.16 1,132,307.60
Sale price/unit 150,052.97 1,100,508.38
Variable Cost/unit -244,430.72 1,304,900.98
Fixed Costs -110,023.15 1,042,441.75

The critical variables are variable cost and fixed costs.  Both variables have a large enough impact on NPV that our calculated NPV could be positive simply due to estimation errors in these variables.

 

 

2 a)

           Operate plant  PV1 = 55,000/.1 = 550,000

High demand

    Prob 0.5                  Sell plant  PV1 = 475,000

Invest today

        Operate plant  PV1 = 20,000/.1 = 200,000

               Low demand

 Prob 0.5                 Sell plant  PV1 = 350,000

 

Don’t invest

 

o       If demand is high, operate the plant; if demand is low, sell it

o       Expected PV1 from investing = 0.5 * 550,000 + 0.5 * 350,000 = 450,000

o       NPV of project at time 0 = 450,000/1.1 – 395,000 = 14,090.91

 

b)    We already know that if demand is high, operating the plant is better than selling it, and if demand is low, selling is better than operating.  We can use that in the decision tree below:

               Invest => operate => NPV = 550,000/1.1 - 395,000 = 105,000

High demand

    Prob 0.5                     Don’t invest  NPV =0

Get info

               Invest => sell => NPV = 350,000/1.1 – 395,000 = -76,818.18

                    Low demand

    Prob 0.5                     Don’t invest  NPV = 0

 

So if demand is high we should invest; if demand is low we shouldn't.  

NPV after buying information = 0.5*105,000 + 0.5*0 - 25,000 = 27,500

Buying info increases NPV by  27,500 - 14,090.91 = 13,409.09.  So we should buy the information, and 13,409.09 is the NPV of buying the info.

 

c) The value of info = cost of info + NPV of info = 25,000 + 13,409.09 = 38,409.09.

 

(The value of info comes from the fact that with a probability of 0.5 you save the loss you otherwise make when demand turns out to be low.  After investing $395,000 today, when demand is low you end up with $350,000 at time 1, or a PV of $318,181.82.  You lose a PV of 395,000 – 318,181.82 = 76,818.18.  The expected saving is 0.5 * 76,818.18 = 38,409.09)

 

 

3)    

DO NOT DRILL  PV of water costs = 3750/.1 = 37,500

                                         

                                   FIND WATER  cost = 26,250

                                /    prob 0.65                            FIND WATER  33,250

            DRILL TO 150’/                             DRILL ON  /     prob 0.3

                                    \                           /                   \ DON’T FIND    59,500

                                      \  DON’T FIND /                           prob 0.7

                                            prob 0.35   \

                \ STOP DRILLING  52,500  

If you find water at 150', the PV of water costs = 15,000 for drilling + 1125/.1 for operating/maintenance costs = 26,250

If you don't find water and you stop drilling, PV of costs = 15,000 for drilling + 37,500 for buying water = 52,500

If you drill on and find water, PV of water costs = 22,000 for drilling + 1125/.1 for operating/maintenance costs = 33,250

If you drill on and don't find water, PV of costs = 22,000 for drilling + 37,500 for buying water = 59,500

Taking the second decision first, if you drill on at 150', the expected cost of water = .3*33,250 + .7*59,500 = 51,625

That's cheaper than not drilling on, so if you don't find water at 150' you will continue to drill, and end up with a cost of 51,625.

For the first decision, then, the expected cost from drilling = .65*26,250 + .35*51,625 = 35,131.25

That's cheaper than buying water, so the farmer should drill the well.  (NPV of drilling = the reduction in water cost = cost if buy - cost of drill = 37,500 - 35,131.25 = 2,368.75)

The decision tree for drilling the test well is:

 

    SKIP TEST WELL    cost 35,131.25

        /

/                                       FIND WATER  => drill => PV of costs = 6,000 + 15,000 + 1125/.1 = 32,250

\                                     /     prob 0.65

        \ DRILL TEST WELL /                          

                                                 \                                              

               \ DON’T FIND   => buy => PV of costs = 6,000 + 37,500 = 43,500

                                                        Prob 0.35

Expected cost if you drill the test well = .65 * 32,250 + .35*43,500 = 36,187.50.

Since this is more than the expected cost if you skip the test well, they should not drill the test well.  

 

Drilling the test well increases cost by 36,187.50 - 35,131.25 = 1,056.25.  This means that when you pay $6,000 for the information you get from the test well, you are overpaying by 1,056.25.  The information is worth 6,000 - 1,056.25 = 4,943.75.  That's the maximum you should be prepared to pay.

 

[This problem is only slightly different from what we did in class.  However, it is not at all easy now to get the value of information directly by computing the expected saving.  We can't say that with the information there is a 0.35 probability of saving the $15,000 cost of drilling a dry well, for an expected saving of .35*15,000 = 5,250.  The effect of getting information is now more complicated.  Without information, if the well turns out to be dry, the PV of water costs are $51,625.  With information, if the well is dry, the PV of water costs are $37,500.  So with a probability of 0.35 the amount we save with the information is $14,125.  The expected saving is then 0.35*14,125 = 4,943.75.

    What's happening here is that without information, when we spend $15,000 to drill to 150', all of that $15,000 is not wasted.  By drilling on to 200', we end up paying $51,625 for water, which is $875 less than not drilling on and just buying water for $52,500.  So out of the 15,000 we spend to drill to 150', only 15,000 - 875 = 14,125 is wasted.  Hence the value of information is 0.35 times 14,125 and not 0.35 times 15,000.

    The point of all this?  It's not a good strategy to plan to figure out the value of information directly by computing the expected saving.  There are all kinds of treacherous pitfalls.]

 

 

4 a) We really have three separate decisions here:

a) whether to make the initial $1,250,000 R&D investment at time 0

b) whether to invest the additional $500,000 R&D investment at time 1 if a working prototype has not resulted

c) whether to make the $10,000,000 project investment once a working prototype is developed

Clearly, if the answer to the third decision is "No", then we can forget about the earlier decisions.  If the project is not worth taking even when the R&D efforts succeed, the R&D expenditure clearly makes no sense.

                                                                           

                                                              LAUNCH PROJECT

                                   PROTOTYPE  /

                                /       prob 0.7       \ DON'T                                                           LAUNCH PROJECT

                              /                                                                           PROTOTYPE /  

             INVEST IN    /                                                 ADDITIONAL      /     prob 0.2      \ DON'T

          / R&D TODAY \                                      /  R&D INVESTMENT  \                         

       /                       \                             /                                     \NO PROTOTYPE => ABANDON

      /                                \ NO PROTOTYPE /                                                 prob 0.8

    /                                           prob 0.3          \

                         \  ABANDON  

 DON'T INVEST

 

Consider first whether it is worth investing $10,000,000 once you have a working prototype:

Expected cashflows each year, starting next year = .6*1,500,00 + .4*800,000 = 1,220,000

PV today of future cashflows = 1,220,000/.1 = 12,200,000

NPV of the investment today = 12,200,000 - 10,000,000 = 2,200,000

=> whenever you have a working prototype, you should invest in the project, and it will generate an NPV today of $2,200,000

 

Now consider if it's worth making the additional R&D investment at time 1:

With a probability of 0.2, you will have a working prototype at time 2.  Investing $10,000,000 at time 2 will generate a time 2 NPV of $2,200,000

With a probability of 0.8, there is no prototype.  You abandon the project and end up with zero

=> the NPV at time 1 of making the additional R&D investment = (0.2*2,200,000)/1.1 - 500,000 = -100,000

=> if there is no working prototype at time 1, you should abandon the project instead of investing in more R&D

 

So is it worth launching the R&D project today?

With a probability of 0.7, you will have a working prototype at time 1.  Investing $10,000,000 at time 1 will generate a time 1 NPV of $2,200,000

With a probability of 0.3, there is no prototype at time 1.  You abandon the project and end up with zero

=> the NPV at time 0 of making the initial R&D investment = (0.7*2,200,000)/1.1 - 1,250,000 = 150,000

=> ARC should invest in the R&D project

 

b) The information will affect your time 1 decision.  If the information says there will be a working prototype at time 2, you will make the additional R&D investment at time 1.  If the information is negative, you won't.  Also, if you don't get the information at time 1, then you will do the same thing as before -- abandon the project.  So the decision tree changes to:

                                                                           

                                                                LAUNCH PROJECT

                                      PROTOTYPE  /

                                 /        prob 0.7        \ DON'T                                                        

                               /                                                    PROTOTYPE  => MAKE ADDITIONAL 

              INVEST IN    /                                          GET  /     prob 0.2             R&D INVESTMENT

            R&D TODAY  \                                       /  INFO  \                         

                                     \                                   /                 NO PROTOTYPE => ABANDON

                                       \ NO PROTOTYPE /                      prob 0.8

                                                prob 0.3          \

                         \  DON'T  => ABANDON

If you get the information at time 1:

with probability 0.2, the information is positive; you make the time 1 R&D investment of $500,000, and get a time 2 NPV of $2,200,000

with a probability of 0.8, you abandon the project and get nothing

=> the NPV at time 1 of getting the information = 0.2*(2,200,000/1.1 - 500,00)  - 175,000 = 125,000

(We subtract away the time 1 information cost from the expected time 1 "cashflows" resulting from getting the information.) 

 

So when we consider the initial R&D investment:

with a probability of 0.7 you end up with a time 1 NPV (from investing in the project) of 2,200,000

with a probability of 0.3 you end up with a time 1 NPV (from buying the information) of 125,000

=> the NPV at time 0 of making the initial R&D investment = (0.7*2,200,000 + 0.3*125,000)/1.1 - 1,250,000 = 184,090.91

 

Bottom line:  ARC should buy the information.  After buying the information, the NPV of the project increases from 150,000 to 184,090.919.

(The time 1 NPV of buying the information = 125,000.  To get the contribution of the information to the time 0 NPV of the overall project, we have to factor in the 0.3 probability that we will end up buying the information, and also discount it one period.  This gives us the increase in the project's time 0 NPV: 0.3*125,000/1.1 = 34,090.91.)

 

 

5 a) There is a decision at time 0: whether to continue operations or liquidate.  There are two decisions at time 1: whether to continue operations or liquidate, both when the new Japanese technology works and when it doesn't.

LIQUIDATE                                                     LIQUIDATE                       

    /AT TIME 0                                                            /

  /                               NEW TECHNOLOGY WORKS /

/                              /                    prob 0.7                    \   

\                            /                                                         \ CONTINUE   

        \   CONTINUE  /                                       

            AT TIME 0   \                                                            LIQUIDATE

                                   \                                                        /

                                     \                DOESN'T WORK         /                                           

                                                            prob 0.3                  \

                                             \ CONTINUE

 

If the new technology works

NPV at time 1 of current and future cashflows if you continue = 6,000,000 + 4,500,000/[.12 - (-.05)] = 32,470,588.24

Net cashflow at time 1 if you liquidate = 6,000,000 + 30,000,000 = 36,000,000

=> if the new technology works you should liquidate

 

If the new technology fails

NPV at time 1 of current and future cashflows if you continue = 6,000,000 + 6,800,000/[.12 - .04)] = 91,000,000

Net cashflow at time 1 if you liquidate = 6,000,000 + 75,000,000 = 81,000,000

=> if the new technology fails you should continue operations

 

Time 0 decision:

With a probability of 0.6, you will end up with 36,000,000 at time 1; with a probability of 0.4, you will end up with 91,000,000 at time 1

PV at time 0 of expected time 1 value if you continue = (0.6*36,000,000 + 0.4*91,000,000)/1.12 = 51,785,714.29

Net cashflow at time 1 if you liquidate = 50,000,000

=> SAYY should continue operations at time 0 rather than liquidate, and end up with PV0 of $51,785,714.29

 

b) Once they have Miss Cleo's information, it is clear what SAYY should do:

if the new technology is going to succeed, they should just liquidate at time 0 instead of time 1 (50,000,000 today is better than 36,000,000 tomorrow)

if the new technology is going to fail, they should just continue in business (don't liquidate either today or tomorrow)

The decision tree can be represented as:

                                                                              LIQUIDATE AT 0  PV0 = 50

                                                               /           

                                                                                / 

                           NEW TECHNOLOGY WORKS  /    LIQUIDATE AT 1    PV1 = 36

                      /                    prob 0.6                       \ 

                    /                                                            \

              BUY   /                                                                DON'T LIQUIDATE    PV1 = 32.5

              INFO  \                                                               

                          \                                   

                            \                                                              LIQUIDATE AT 0  PV0 = 50

                              \                                                         / 

                                \               DOESN'T WORK          /    LIQUIDATE AT 1    PV1 = 81

      prob 0.4                   \ 

                                                                                  \

                                                                                           \ CONTINUE       PV1 = 91

 

PV0 after buying the information = .6*50,000,000 + .4*(91,000,000/1.12) - 5,000,000 = 57,500,000

Bottom line:  SAYY should buy the information.  After buying the information, their time 0 value  increases from $51,785,714.29 to $57,500,000

(The NPV of buying the information = 57,500,000 - 51,785,714.29 = 5,714,285.71.

Or the value of the information = NPV + cost = 10,714,285.71.

The benefit of the information is that is allows the firm to get $50 million today instead of $36 million tomorrow if the Japanese technology is going to succeed.  In PV terms this is a benefit of 50,000,000 - 36,000,000/1.12 = 17,857,142.86.  This benefit will be realized with a probability of 0.6, so the expected benefit = 0.6*17,857,142.86 = 10,714,285.71, and this is the value of the information.)

 

 

6 a) We have three separate decisions here:

a) whether to make the initial $6,666,666 R&D investment at time 0

b) whether to make the additional $4,000,000 R&D investment at time 1 if containment shield problems exist

c) whether to make the $125,000,000 project investment once they have a working technology

                                                                         

                                                              LAUNCH PROJECT

                                   READY TO GO /

                                /       prob 0.4        \ DON'T                                                               LAUNCH PROJECT

                              /                                                                            READY TO GO  /  

           INVEST IN    /                                                 ADDITIONAL      /     prob 0.25         \ DON'T

         / R&D TODAY \                                      / R&D INVESTMENT   \                         

       /                       \                            /                                 \    PROBLEMS    ABANDON

      /                                 \     PROBLEMS    /                                                 prob 0.75

    /                                           prob 0.6           \

                         \  ABANDON  

 DON'T INVEST

 

Consider first whether it is worth investing $125,000,000 once you have a working technology:

PV today of future cashflows = 10,000,000/(.13 - 0.06) = 142,857,142.86

NPV of the investment  = 142,857,142.86 - 125,000,000 = 17,857,142.86

=> whenever you have a working technology, you should invest in the project, and it will generate an NPV of 17,857,142.86 at that time

 

Now consider if it's worth making the additional R&D investment at time 1:

With a probability of 0.25, a working technology will result at time 2.  You would then invest $125 million at time 2, and generate a time 2 NPV of 17,857,142.86

With a probability of 0.75, there is no working technology and you abandon the project.

=> the NPV at time 1 of making the additional R&D investment = (0.25*17,857,142.86)/1.13 - 4,000,000 = -49,304.68

=> if there is no working technology at time 1, it is not worth making the additional investment.

 

So is it worth launching the R&D project today?

With a probability of 0.4, you will have a working technology at time 1, and generate a time 1 NPV of 17,857,142.86

With a probability of 0.6, you won't have a working technology at time 1.  Since you will abandon the project, the cahflows will then be zero.

=> the NPV at time 0 of making the initial R&D investment = (0.4*17,857,142.86 + 0.6*0)/1.13 - 6,666,666 = -345,553.48

=> NCC should not invest in the R&D project

 

b) The information will help you to avoid needlessly making the additional R&D investment if the containment shield problems are insurmountable.  The decision tree changes to:

                                                                           

                                                              LAUNCH PROJECT

                                   READY TO GO /

                                /       prob 0.4        \ DON'T                                    

                              /                                                        SOLVABLE   => MAKE R&D INV & LAUNCH PROJECT

           INVEST IN    /                                         GET       /     prob 0.25   

         / R&D TODAY \                                   /    INFO       \                         

       /                       \                          /                   \PROBLEMS => ABANDON

      /                                 \   PROBLEMS    /                            prob 0.75

    /                                           prob 0.6         \

                       \   DON'T    

 DON'T INVEST

 

If you get the information at time 1:

with probability 0.25, the information is positive; you make the time 1 R&D investment of $4,000,000, and go on to make the project investment of $125,000,000 at time 2

with a probability of 0.75, you abandon the project and avoid the $4,000,000 investment

=> the NPV at time 1 of getting the information = 0.25*(17,857,142.8/1.13 - 4,000,000)  - 2,500,000 = 450,695.32

(The information cost is paid for sure.  The additional R&D investment is made with a probability of 0.25, generating the time 2 NPV.) 

If you don't get the information at time 1, then you would have the same 0 you ended up with before without information.

=>

Clearly, the information is worth getting.  The time 1 NPV of getting the information is $450,695.32.  Or the value of the information is $2,950,695.32

 

So when we consider the initial R&D investment:

with a probability of 0.4 you end up with a time 1 NPV (from investing in the project) of 17,857,142.86

with a probability of 0.6 you end up with a time 1 NPV (from buying the information) of 450,695.32

=> the NPV at time 0 of making the initial R&D investment = (0.4*17,857,142.86 + 0.6*450,695.32/1.13 - 6,666,666 = -106,246.23

=> Once though buying the information improves the NPV of the project, it still remains negative.  NCC should stil reject the R&D project, even with information.

 

 

7 a) When we perform a sensitivity analysis, we compute three different versions of the NPV: pessimistic NPV (where all variables are set to their pessimistic value), expected NPV (where all variables are set to their expected value), and optimistic NPV (where all variables are set to their optimistic value).  F If we are looking at 10 different variables, we will end up doing 21 different NPV computations.  The base computation uses the expected value for each variable.  Then we take each variable one at a time, and change its value, once to the pessimistic value, once to the optimistic value, and re-compute NPV each time.  In these computations, all other variables remain at their expected values.

b)    Sensitivity analysis considers the impact of different variables on the NPV estimate, taking each variable one at a time.  T

c)    A decision tree typically involves an alternating sequence of decisions you make, followed by learning the random outcome of those decisions, followed by future decisions you make, etc. T  

d)    If the NPV of a project without information is $3,000 and information is available at a cost of $5,000 we will not consider buying the information, since the NPV of the project after buying the information would be negative.  F The NPV of the project after buying the information = NPV without info - cost of info + benefit of info, and could still be positive.

e)    Decision trees provide a simple way to take the value of real options into account when computing a project's NPV, but it yields only an approximate estimate of the NPV.  T An exact computation requires using option pricing theory.  Using decision trees takes into account the impact of real options on cashflows, but not the impact of real options on risk.  See the last paragraph of the section called "Pro and Con Decision Trees" in the book (p. 277) for more details.

f)    When a project has real options, the options may either increase or decrease the NPV of the project.  F An option can only have a positive value; i.e., real options can only increase the NPV of a project.  The worst case is that you choose not to do what you have the option of doing.  Having it as an extra choice though certainly cannot hurt.

g)    A decision tree problem always has to be solved backwards, starting with the last decision to be made.  T  Till you make the last decision you don't know what the relevant cashflows are for the previous decision, and so on.  That's why the problems must be solved backwards.

 

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