Decision Trees 1. Grow Growfa fast st Comp Compan any y is eval evalua uati ting ng four four alte altern rnat ativ ivee sing single le-p -per erio iod d inve invest stme ment nt opportunities whose returns are based on the state of the economy. The possible states of the economy and the associated probability distribution is as follows : State Probability
Fair 0 .2
Good 0. 5
Great 0 .3
The returns for each investment opportunity and each state of the economy are as follows :
Alternative W X Y Z
Fair (Rs.) 1000 500 0 (-) 4000
State of the economy Good (Rs.) 3000 4500 5000 6000
Great (Rs.) 6000 6800 8000 8500
Usin Using g the the deci decisi sionon-tr tree ee appro approach ach,, dete determ rmin inee the the expec expecte ted d retu return rn for for each each alternative. Which alternative investment proposal would you recommend if the expected monetary value criterion is to be employed ? Ans. : Invest in Y; Value Rs. 4,900
2. The investm investment ent staff staff of TNC Bank Bank is conside considerin ring g four four investme investment nt proposal proposalss for a client : Shares, Bonds, Real estate and Savings Certificates. The investments will be held for one year. The past data regarding the four proposals are given below : SHARES
There is a 25 % chance that shares will decline by 10 %, a 30 % chance that they will remain stable and a 45 % chance that they will increase in value by 15 %. Also the shares under consideration do not pay any dividends.
BONDS
These bonds stand a 40 % chance of increase in value by 5 % and 60 % chance of remaining stable and they yield 12 %.
REAL ESTATE
This pr proposal ha has a 20 20 % ch chance of of in increasing 30 30 % in in value, a 25 % chance of increasing 20 % in value, a 40 % chance of increasing 10 % in value, a 10 % chance of remaining stable and a 5 % chance of losing 5 % of its value.
SAVINGS CERTIFICATES
These certificates yield 8.5 % with certainity.
Use a Decision tree structure for the alternatives available to the investment staff, and using the expected value criterion, choose the alternative with the highest expected value. Ans. : Invest in Real Estate, Rs. 114.75 3. A finance manager is considering drilling a well. In the past only 70 % of wells drilled were successful at 20 metres depth in that area, moreover on finding no water at 20 metres some persons in that area drilled it further upto 25 metres but only 20 % struck water at that level. The prevailing cost of drilling is Rs. 500/- per metre. The finance manager estimated that in case he does not get water in his own well, he will have to pay Rs.15000/- to buy water from outside for the same period of getting water from the well. The following decisions are considered : a) do not drill any well b) drill upto 20 metres and c) if no water is found at 20 metres, drill further upto 25 metres. Draw an appropriate decision tree and determine the finance manager’s optimal strategy. Ans. : Drill upto 20M, if there is no water Drill upto 25M; Value Rs. 14,350 4. A company is contemplating whether to produce a new product. If it decides to produce the product it must either install a new division which needs a cash outlay of Rs.4,00,000/- or work overtime with OT expenses of Rs.1,50,000/-. If the company decides to install a new division it needs the approval of the Government and the company feels that there is a 70 % chance of getting the approval. A market survey has revealed the following facts regarding the magnitude of sales for the new products Magnitude of Sales Probability Resulting Profits in Lakhs High 0.45 15 Medium 0.30 7 Low 0.20 3 Nil 0.05 -5 (Loss) However by resorting to overtime the company will not be in a position to meet the high magnitude of sales, it will be able to satisfy upto the level of medium magnitude only, even if high magnitude of sales results. Solve the problem to suggest which option should be selected. Ans. : Produce New Product by resortingto Over Time; Rs. 4.1 lacs 5. Mr. X is trying to decide whether to travel to Sri Lanka from Delhi to negotiate sale of a shipment of Chinese novelties. He holds the novelties stock and is fairly confident but by no means sure that if he makes the trip he will sell the novelties at a price that will give him a profit of Rs.30,000/-. He puts the probability of obtaining the order at 0.6. If he does not make the trip he will certainly not get the order. If the
novelties are not sold in Sri Lanka there is an Indian customer who will certainly buy them at a price that leaves him a profit of Rs.15,000/- and this offer will be open atleast till Mr.X returns from Sri Lanka. Mr.X estimates the expenses of the trip to Sri Lanka at Rs.2500/-, he is however concerned that his absence even only for 3 days will lead to production inefficiencies in the factory. These could cause him to miss the deadline on another contract, with the consequence that a late penalty of Rs.10,000/- will be invoked. Mr.X assesses the probability of missing the deadline under these circumstances at 0.40 further he believes that in his absence there would be a lower standard of house keeping in the factory. If the raw material and labour costs on the other contract will rise by Rs.2000 above the budgeted figure. Draw an appropriate decision tree for Mr.X’s problem and using EMV as the appropriate criterion for the decision find the appropriate initial decision. Ans. : X should proceed Srilanka; Value Rs. 15,500 6. M/s B and Company is currently working with a process which after paying for materials, labour etc., brings a profit of Rs.10,000/-. The following alternatives are made available to the company : a) The company can conduct research (R 1) which is expected to cost Rs.10,000/and having 90% probability of success, the company gets a gross income of Rs.25,000/-. b) The company can conduct research (R 2) expected to cost Rs.5,000/- and having a probability of 60% success. If successful, the gross income will be Rs.25,000/-. c). The company can pay Rs.6,000/- as royalty of a new process which will bring a gross income of Rs.20,000/-. d) The company continues the current process. Because of limited resources, it is assumed that only 1 of the 2 types of research can be carried out at a time. Ans. :R2 is followed by Royalty; Value Rs.15,600 7. A businessman has two independent investments A and B available to him, but he lacks the capital to undertake both of them simultaneously, he can choose to take A first and then stop or if A is successful then take B or vice-versa. The probability of success of A is 0.70 while for B it is 0.40. Both investments require an initial capital outlay of Rs.2,000/- and both return nothing if the venture is unsuccessful. Successful completion of A will return Rs.3000/- (over cost) and successful completion of B will return Rs.5000/- (over cost). Draw the decision tree and determine the best strategy. Ans. :Accept A, if it is successful Accept B; Value Rs. 2,060 8. Oil India Corporation is considering whether to go for an offshore oil drilling contract to be awarded in Bombay. If they bid value would be Rs.600 million with a 65% chance of gaining the contract. They may set up a new drilling operation or move
already existing operation, which has proved successful to the new site. The probability of success and expected returns are as follows : Outcome
New Drilling Probability
Operation Existing Operation expected expected Revenue probability Revenue (in Rs. (in Rs. Million) Million) Success 0.75 800 0.85 700 Failure 0.25 200 0.15 350 If the corporation does not bid or lose the contract, they can use the 600 million to modernise their operations, this would result in a return of either 5% or 8% on the sum invested with probabilities of 0.45 and 0.55 (Assume that all costs and Revenue have been discounted to present value). a. Construct a decision tree for the problem showing clearly the course of action. b. By applying an appropriate decision criteria recommend whether or not Oil India Corporation should bid for the contract. Ans. :Bid the contract and go for New Drilling operation ; Vlaue Rs. 46.46 Million
9. The Indian Yacht Company has developed a new Cabin Cruiser, which they have earmarked for the medium to large boat market. The market analysis has a 30% probability of Annual sales being 5000 boats, 40% probability of 4000, and 30% probability of 3000 boats. The company can go into limited production where variable costs are Rs.10,000/- per boat and fixed costs are Rs.8,00,000/- annually. Alternatively they can go into full scale production where variable costs are Rs.9,000/- per boat and fixed costs are Rs.50 Lakhs annually. If the new boat is to be sold for Rs.11,000/- should the company go into limited of full scale production when their objective is to maximise the expected profits. Ans. : Limited Production ; Rs. 32 Lacs
10. An oil company has recently acquired rights in a certain are to conduct surveys and test drilling to lead to lifting oil. The area is considered to have good potential for finding oil. At the outset the company has a choice to conduct further geological tests or to carry out a drilling programme immediately. On the known conditions, the company estimates that there is a 70 : 30 chance of further tests showing a “Success”. Whether the tests show the possibility of ultimate success or not or even if no tests are undertaken at all, the company could still pursue its drilling programme or alternatively consider selling its rights to drill the area. Thereafter, however, if it carries out the drilling programme, the likelihood of final success or failure is considered dependent on its foregoing stages. Thus if successful tests have been carried out the expectation of success in drilling is given as 80 : 20. If the tests indicate “Failure” then the expectation of success in drilling is given as 20 : 80. If no tests have been carried out at all, the expectation of success in drilling is given as 55 : 45. Costs and Revenues have been estimated for all possible outcomes and the net present value of each is given below :
Outcomes
Net Present Value (Rs. Millions)
Success With Prior tests Without Prior tests Failure With Prior test s Without Prior tests Sale of exploitation rights Prior tests show success Prior tests show failure 15 Without prior tests
100 120 (-) (-)
50 40 65
45
Draw a decision tree diagram to represent the above information. Evaluate the tree in order to advise the management of the company on its best course of action. Ans. : Test, if it is +ve Drill, If it is -ve Sell; Value Rs. 53.50 11. A client asks an estate agent to sell 3 properties A, B, & C for him and agrees to pay him a 5 % commission on each sale. He specifies certain conditions. The estate agent must sell A first and this he must do within 60 days. If and when A is sold the agent receives his 5 % commission on that sale. He can either back out at this stage or nominate and try to sell one of the two remaining properties within 60 days. If he does not succeed in selling the nominated property in that period, he is not given the opportunity to sell the other. If he does sell it in the period, he is given the opportunity to sell the third property on the same conditions. The following table summarises the prices, selling costs (incurred whenever a sale is attempted) and the agent’s estimated probability of making a sale.
Property A B C
Data for Property Sales Price Rs. Selling Costs Rs. 12000 400 25000 225 50000 450
Probability 0.7 0.6 0.5
(i) draw an appropriate decision tree for the estate agent. (ii) what is the estate agent’s best strategy under EMV approach ? Ans. : Accept offer to try to sell A-if able to sell A, Then try to sell C-if able to sell C, try to sell B.
