INTRODUCTION TO RISK ANALYSIS IN CAPITAL BUDGETING PRACTICAL PROBLEMS

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1 CHAPTER8 INTRODUCTION TO RISK ANALYSIS IN CAPITAL BUDGETING PRACTICAL PROBLEMS PROBABILISTIC APPROACH Question 1: A project under consideration is likely to cost `5 lakh by way of fixed assets and requires another `2 lakh for current assets. It is expected to have a life of 1 years, during which the returns are likely to be uniform, and at the end of which, it is likely to be have scrap value of `5 lakh. Various estimates of the gross income before depreciation and tax have been made. These are as follows: Annual amount (` in lakh) The rate of income tax is 35 percent. The cut-off rate is 12 percent. Assuming straight line method of depreciation is allowed for tax purposes, would you recommend acceptance of the project? Solution: Computation of NPV (` in lakhs) Mean CFBT - (1) (5 x.1) + (1 x.2) (2 x.5) + (3 x.1) + (4 x.1) 19.5 Less: Depreciation 5 5 (4.5) 1 PBT 15 Tax Liability (35%) (2) 5.25 Mean CFAT (1) (2) Cumulative PVAF (1 1) (12%) 5.65 P.V. of CFAT Add: PV of terminal value (1 th year) (2 lakhs + 5 lakhs) x PVCI Less: PVCO (5 + 2) (7) NPV Advise: Accept the Project. Question 2: Cyber Company is considering two mutually exclusive projects. Investment outlay of both the projects is `5,, and each is expected to have a life of 5 years. Under three possible situations their annual cash flows and probabilities are as under: Situation Probabilities Cash Flows (`) Project A Project B Good.3 6,, 5,, Normal.4 4,, 4,, Worse.3 2,, 3,, The cost of capital is 7 percent, which project should be accepted? Explain with workings. Solution: Computation of NPV of Project A: Situation NPV Expected NPV Good.3 (6,, x 4.1) - 5,, = 19,6, 5,88, Normal.4 (4,, x 4.1) - 5,, = 11,4, 4,56,

2 Worse.3 (2,, x 4.1) - 5,, = 3,2, 96, 11,4, Computation of NPV of Project B: Situation NPV Expected NPV Good.3 (5,, x 4.1) - 5,, = 15,5, 4,65, Normal.4 (4,, x 4.1) - 5,, = 11,4, 4,56, Worse.3 (3,, x 4.1) - 5,, = 7,3, 2,19, 11,4, Computation of Standard Deviation Project A: (19,6, 11,4,) 2 x.3 + (11,4, 11,4,) 2 x.4 + (3,2, 11,4,) 2 x.3 = 6,35,169 Project B: (15,5, 11,4,) 2 x.3 + (11,4, 11,4,) 2 x.4 + (7,3, 11,4,) 2 x.3 = 3,17,585 Since the both Projects have the same NPV but Project B has lesser standard deviation than Project A, hence Project B shall be accepted. Question 3: A company is considering Projects X and Y with following information: Project Expected NPV (`) Standard deviation X 1,22, 9, Y 2,25, 1,2, [ (i) Which project will you recommend based on the above data? (ii) Explain whether your opinion will change, if you use coefficient of variation as a measure of risk. (iii) Which measure is more appropriate in this situation and why? Solution: (i) On the basis of standard deviation project X be chosen because it is less risky than Project Y having higher standard deviation. (ii) CV = SD ENPV CVX = 9, =.738 1,22, CVY = 1,2, =.533 2,25, On the basis of Co-efficient of Variation (C.V.) Project Y appears to be less risky and hence should be accepted. (iii) However, the coefficient of variation as a measure of risk in such conflicting situation is best because it relates the risk and return. Question 4: Probabilities for net cash flows for 3 years of project (Project life) are as follows: Year 1 Year 2 Year 3 Cash Flow (`) Cash Flow (`) Cash Flow (`) 2,.1 2,.2 2,.3 4,.2 4,.3 4,.4 6,.3 6,.4 6,.2 8,.4 8,.1 8,.1 Calculate the expected net cash flows. Also calculate the present value of the expected cash flow, using 1 per cent discount rate. Initial Investment is `1,. Solution: Cash Flow (`) Year 1 Year 2 Year 3 Expected Value (`) Cash Flow (`) Expected Value (`) Cash Flow (`) Expected Value (`) 2,.1 2 2,.2 4 2,.3 6 4,.2 8 4,.3 1,2 4,.4 1,6 6,.3 1,8 6,.4 2,4 6,.2 1,2 8,.4 3,2 8,.1 8 8,.1 8 6, 4,8 4,2

3 Present value of cash flows: Years Cash flows PVF PV 1 6,.99 5, , , , ,154 PVCF 12,573 Expected Net Present Value = Present Value of cash flows Initial Investment = `12,573 - `1, = `2,573 CERTAINTY EQUIVALENT APPROACH Question 5: XYZ Ltd is considering the proposal of buying one of the two machines to manufacture a new product. Each of these machines requires an investment of `5,, and is expected to provide benefits over a period of 4 years. After the expiry of the useful life of the machines, the sellers of both the machines have guaranteed to buy back the machines at `5,. The management of the company uses CE approach to evaluate risky investments. The company's risk adjusted discount rate is 16 percent and the risk-free rate is 1 percent. The expected values of net cash flows (CFAT) with their respective CE are: Year Proposal A Proposal B CFAT CE CFAT CE 3, 3, 3, 3, , 36, 24, 32, Which machine, if either, should be purchased by the company? Solution: Computation of NPV Time PV factor Proposal A Proposal B Cash flow CE Certain cash flow PV Cash flow CE Certain cash flow ,.8 24, 21,816 18,.9 16,2 14, ,.7 21, 17,346 36,.8 28,8 23, ,.6 18, 13,518 24,.7 16,8 12, ,.5 15, 1,245 32,.4 12,8 8,742 (+) Terminal Value.683 5, 3,415 5, 3,415 PVCI 66,34 63,289 (-) PVCO (5,) (5,) NPV 16,34 13,289 Advise: Proposal A should be accepted. RISK ADJUSTED DISCOUNTING RATE Question 6: An enterprise is investing `1 lakhs in a project. The risk-free rate of return is 7%. Risk premium expected by the Management is 7%. The life of the project is 5 years. Following are the cash flows that are estimated over the life of the project. Year Cash flows (` in lakhs) Calculate Net Present Value of the project based on Risk free rate and also on the basis of Risks adjusted discount rate. Solution: The present Value of the Cash Flows for all the years by discounting the cash flow at 7% is calculated as below: PV

4 Year Cash flows (` in lakhs) Discounting 7% Present value of Cash Flows (` in lakhs) Total of present value of Cash flow Less: Initial Investment (1) Net Present Value (NPV) Now when the risk-free rate is 7% and the risk premium expected by the Management is 7%. So the risk adjusted discount rate is 7% + 7% = 14%. Discounting the above cash flows using the Risk Adjusted Discount Rate would be as below: Year Cash flows (` in lakhs) Discounting 14% Present value of Cash Flows (` in lakhs) Total of present value of Cash flows Less: Initial Investment (1) Net Present Value (NPV) Question 7: Determine the risk adjusted net present value of the following projects: X Y Z Net cash outlays (`) 2,1, 1,2, 1,, Project life 5 years 5 years 5 years Annual Cash inflow (`) 7, 42, 3, Coefficient of variation The Company selects the risk-adjusted rate of discount on the basis of the coefficient of variation: Coefficient of Variation Risk-Adjusted Rate of Return P.V. Factor 1 to 5 years At risk adjusted rate of discount. 1% % % % % % More than 2. 25% Solution: Statement showing the determination of the Risk Adjusted Net Present Value Project Net Cash Outlays Coefficient of Variation Risk Adjusted Discounted Rate Annual Cash Inflow PV Factor 1-5 years Discounted Cash Inflows Net Present Value (i) (`) (ii) (iii) (iv) (`) (v) (vi) (vii) = (v) x (vi) (viii)=(vii) (ii) X 2,1, % 7, ,29,18 19,18 Y 1,2,.8 14% 42, ,44,186 24,186 Z 1,,.4 12% 3, ,8,15 8,15 SCENARIO ANALYSIS Question 8: From the following information compute the net present value (NPVs) of the two projects for each of the possible cash flows, using scenario analysis: Project X( `) Project Y( `) Initial cash outflows (t = ) 3 3

5 Cash inflows estimates (t = 1 1) Worst 5 8 Most Likely 8 1 Best 15 2 Required Rate of Return 14% 14% Economic life (years) 1 1 Solution: The NPV of each project, assuming a 14% required rate of return, can be calculated for each of the possible cash flows. The present value interest factor annuity (PVIFA) of `1 for 1 years at 14% discount is Multiplying each possible cash flows by PVIFA, we get the following information. ( `) Determination of NPVs Project X Project Y Expected cash inflows PV NPV PV NPV Worst 26.8 (3.92) Most likely Best Conclusion: The aforesaid table shows that in case of Project X under worst circumstances there is negative NPV whereas in case of most likely and best circumstances there is positive NPV. However, in case of Project Y there is positive NPV under all the circumstances, at the same time cash inflows are more, than project X. Hence, Project Y is more profitable and hence, be accepted. DECISION TREE Question 9: A firm is required to choose between constructing a large or small factory to produce a new line of products. The large plant would be needed if the future brings a high demand for new products. But the large plant would have new cash inflows below the `1,, outlay, if demand is medium or low. The present value cash flows are `14,, with high demand, `9,, with medium demand, and `6,, with low demand. The smaller plant produces a lower return if demand is high but has positive net present values at medium demand. It would cost `2,, as a cash outlay and would return a present value inflow of `3,2, with high demand, `2,7, with medium demand, and `1,8, with low demand. Draw a Decision tree for the proposals under consideration. The decision is mutually exclusive. What is the net present value of each alternative if there is a 4% chance of high demand, 4% chance of medium demand and a 2% chance of low demand. Solution:.4 NPV = 4,, D1 NPV = 4, Construct a Large Factory NPV = 72, Construct a Small Factory.4 NPV = (1,,).2 NPV = (4,,).4 NPV = 1,2,.4 NPV = 7, Do Nothing NPV =.2 NPV = (2,) Advise: Construct a small factory. Question 1: X Ltd. is considering the purchase of a new plant requiring a cash outlay of `2,. The plant is expected to have a useful life of 2 years without any salvage value. The cash flows and their associated probabilities for the two years are as follows: 1st year Cash flows `8,.3 `11,.4 `15,.3 Depending upon the cash flows of year 1 to be `8,, or `11, or `15,, the cash flows for year 2 may be:

6 Year 1 - `8, Year 1 - `11, Year 1 - `15, Yr. 2 CF (`) Yr. 2 CF (`) Yr. 2 CF (`) 4,.2 13,.3 16,.1 1,.6 15,.4 2,.8 15,.2 16,.3 24,.1 Draw a Decision tree for the proposal under consideration. Presuming that 1% is the cost of capital, calculate the NPV of the project and suggest whether the project should be undertaken. Solution:.2 4, 8,.6.2 1,.3 15,.3 13, (2,).4 11, ,.3 16,.1 16, 15,.8.1 2, 24, Mean Cash Inflows (Year 1): (8, x.3) + (11, x.4) + (15, x.3) = 11,3 Mean Cash Inflows (year 2): (4, x.6) + (1, x.18) + (15, x.6) + (13, x.12) + (15, x.16) + (16, x.12) + (16, x.3) + (2, x.24) + (24, x.3) = 14,82 Mean NPV = (11,3 x.99) + (14,82 x.826) 2, = 2,513 Advise: Accept the proposal. Question 11: Pearson Publishing House, a small publisher is publishing economy edition of a new book on Financial Management. The cost of typesetting and other related cost will be `1,,. The cost of printing per book will be `2. If additional books are needed at a later time, the setup cost will be `5, per setup, however, cost of printing the book shall remain the same. The book will be sold for `14 per copy. Royalty to author, commission of agent and other related delivery charges shall be `4 per book. The future sale of book depends upon the review of the book. In case book gets good review. It is expected that sale of book will 5, copies per year for three years. On the other hand, if it gets bad review, the sale will be 2, copies in the first year and then sale will be ceased. for good review is.3. Mr. X, owner of publishing house faces a choice between ordering an immediate production of 15, copies or 5, copies, followed by additional production run at the end of the first year if the book is successful. The cost of capital is 1%. Using Decision Tree analysis recommend production schedule and decide whether book should be published or not. Solution:

7 NPV = 1,,31 D1 Print 15, copies NPV = 2,32,135 Print 5, copies.3.7 Success of book NPV = (1 x 5, x 2.487) 4,,* = 8,43,5 Failure of book NPV = (1 x 2, x.99) 4,,* = (-) 2,18,2 Success of book.3 NPV = (1 x 5, x 2.487) 2,,** 2,5,*** x.99 = 8,16,25.7 Failure of book NPV = (1 x 2, x.99) 2,,** = (-) 18,2 Do nothing * (1,, + 2 x 15,) = 4,, ** (1,, + 2 x 5,) = 2,, *** (5, + 2 x 1,) = 2,5, The expected net present value is larger and the potential loss is smaller if the initial run is 5, copies. The 5, copy choice is preferred since it produces both more expected value and less risk. * In above answer it has been assumed that to avoid again set up cost in the beginning of 2 nd year 1 copies will be printed in the end of first year. Question 12: A business man 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 on A is.7, while for B it is.4. Both investment require an initial capital outlay of `2,, and both return nothing if the venture is unsuccessful. Successful completion of A will return `3, (over cost), and successful completion of B will return `5, (over cost). Draw the decision tree and determine the best strategy. Solution: Accept A 2,6 (2,).3 Fail.7 Succeed 3, + 8 D2 8 Accept B Stop (2,).6 Fail.4 Succeed 5, D1A Do nothing Accept B 1,4.4 Succeed 5, D3 Stop 1,5.7 Succeed 3, + 1,5 Accept A.6 Fail (2,).3 Fail (2,) The required decision tree is as shown below: There are three decision points in this tree. These are indicated as 1, 2 and 3.

8 Evaluation of decision point 3: 1. Accept A (Amount in `) Outcome Conditional Values Expected Values Success.7 3, 2,1 Failure.3 (2,) (6) 1,5 2. Stop: Expected value = Evaluation of decision point 2: 1. Accept B (Amount in `) Outcome Conditional Values Expected Values Success.4 5, 2, Failure.6 (2,) (1,2) 8 2. Stop: Expected value = Evaluation of decision point 1: 1. Accept A (Amount in `) Outcome Conditional Values Expected Values Success.7 3, + 8 2,66 Failure.3 (2,) (6) 2,6 2. Accept B (Amount in `) Outcome Conditional Values Expected Values Success.4 5, + 1,5 2,6 Failure.6 (2,) (1,2) 1,4 3. Do Nothing: Expected value = Hence, the best strategy is to accept A first, and if it is successful, then accept B. Question 13: MCL Technologies is evaluating new software for ERP. The software will have a 3-year life and cost `1, thousands. Its impact on cash flows is subject to risk. Management estimates that there is a 5-5 chance that the software will either save the company `1, thousands in the first year or save it nothing at all. If nothing at all, savings in the last 2 years would be zero. Even worse, in the second year an additional outlay of `3 thousands may be required to convert back to the original process, for the new software may result in less efficiency. Management attaches a 4 percent probability to this occurrence, given the fact that the new software failed in the first year. If the software proves itself, second-year cash flows may be either `1,8 thousands, `1,4 thousands, or `1, thousands, with probabilities of.2,.6 and.2, respectively. In the third year, cash, inflows are expected to be `2 thousands greater or `2 thousands less than the cash flow in period 2, with an equal chance of occurrence. (Again, these cash flows depend on the cash flow in period 1 being `1, thousands.) All the cash flows are after taxes. (a) Set up a probability tree to depict the foregoing cash-flow possibilities. (b) Calculate a net present value for each three-year possibility, using a risk-free rate of 5 percent. (c) What is the risk of the project? Solution: (a)

9 First Year Second Year Third Year Invest in Software.5.5 1, 1, ,4.2 1, ,6.5 2, 1,6 1,2 1,2.5 8 D1.4 (3) Do not invest (b) Statement showing computation of NPV Case Time P.V. Factor Cash Inflows P.V.C.I P.V.C.O NPV Prob. Expected NPV I , ,8 1, , 1,728 4,312.6 (1,) 3, II , ,8 1, ,6 1, ,967 (1,) 2, III , ,4 1, ,6 1, ,64.2 (1,) 2, IV , ,4 1, ,2 1,36.8 3,258.6 (1,) 2, V , , ,2 1,36.8 2,895.8 (1,) 1, VI , , ,55.2 (1,) 1, VII (1,) (1,).3 (3) VIII 1.952

10 2.97 (3) (272.1) (1,) (1,272.1).2 (254.42) (c) Standard Deviation = (3, ) 2 x.5 + (2, ) 2 x.5 + (2, ) 2 x.15 + (2, ) 2 x.15 + (1, ) 2 x.5 + (1, ) 2 x.5 + (-1, ) 2 x.3 + (-1, ) 2 x.2 = 1,84.9 (272.1) SENSITIVITY ANALYSIS Question 14: Red Ltd. is considering a project with the following Cash flows: Amount in (`) Years Cost of Plant Recurring Cost Savings 1, 1 4, 12, 2 5, 14, The cost of capital is 9%. Measure the sensitivity of the project to changes in the levels of plant value, running cost and savings (considering each factor at a time) such that the NPV becomes zero. The P.V. factor at 9% are as under: Year Factor Which factor is the most sensitive to affect the acceptability of the project? Solution: P.V of Cash Flows: Amount in (`) Year 1 Running Cost 4, x.917 = (3,668) Savings 12 x.917 = 11,4 Year 2 Running Cost 5, x.842 = (4,21) Savings 14, x.842 = 11,788 14,914 Year Less: P.V of Cash outflow 1, x 1 (1,) NPV 4,914 Sensitivity Analysis: (i) Increase of Plant Value by `4,914 4,914 x 1 = 49.14% 1, (ii) Increase of Running Cost by `4,914 4,914 = 4,914 x 1 = 62.38% 3, ,21 7,878 (iii) Fall in saving by `4,914 4,914 = 4,914 x 1 = 21.56% 11,4 + 11,788 22,792 Hence, savings factor is the most sensitive to affect the acceptability of the project. Question 15: A company is considering investing in a new manufacturing project with following characteristics: A. Initial investment `35 lakhs scrap value nil B. Expected life 1 years C. Sales volume 2, unit per year D. Selling price `2, per unit E. Variable direct costs `1,5 per unit

11 F. Fixed costs excluding depreciation `25,, per year. The project shows an IRR of 17%. The MD is concerned about the viability of the investment as the return is close to company s threshold rate of 15%. He has requested a sensitivity analysis. You are required to: a) Re-calculate IRR assuming each of the characteristics A to F above in isolation varies by 1%. b) Advice the MD of the most vulnerable area likely to prevent the project meeting the company s hurdle rate. c) Revaluate the situation if another company, already manufacturing a similar product, offered to supply the units at `18 each; this would reduce the investment to `25 lakhs & fixed cost p.a. to `1 lakhs. Life of proposal = 1 years. Solution: (a) (A) Revised Initial Investment (35 lakh x 11%) = 385 lakh Cumulative PVAF (1 1) = 385 lakh = lakh Table B (1 th year) Project IRR = 14.5% (B) Revised expected life = 1 years x 9% = 9 years Cumulative PVAF (1 9) = 35 lakh = lakh Table B (9 th year) Project IRR = 15.5% (C) Revised Annual sales volume (2, units x 9%) 18, Units Annual contribution (18, units x `5) `9 lakh Less: Annual cash fixed cost (`25 lakh) Annual net cash inflows `65 lakh Cumulative PVAF (1 1) = 35 lakh = lakh Table B (1 th year) Project IRR = 13% (D) Revised S.P. per unit (`2, x 9%) `18 Less: Variable direct costs per unit (`15) Contribution per unit `3 Annual sales volume 2 Units Annual contribution `6 lakh Less: Cash Fixed Cost p.a. (`25 lakh) Annual net cash inflows `35 lakh Cumulative PVAF of IRR = 35 lakh = 1 35 lakh Project IRR = % (E) Selling price per unit `2, Less: Revised V.C. per unit (15 x 1%) (1,65) Revised contribution per unit `35 Annual sales volume 2, Units Annual contribution `7 lakh Less: Cash fixed cost p.a. (`25 lakh) Annual net cash inflows `45 lakh Cum PVAF of 1 yrs of IRR = 35 lakh = lakh Table B (1 th year) Project IRR = 4.5% (F) Existing annual contribution Less: Revised cash annual fixed cost `1 lakh (`27.5 lakh) Annual net cash inflows `72.5 lakh Cum. PVAF of IRR (1 1) = 35 lakh = lakh Table B (1 th year) Project IRR = 16% (b) Most vulnerable (sensitive) variable = selling price per unit (As a 1% change in selling price per unit will bring down the IRR to % level) (c) Selling price per unit `2, Less: Variable cost per unit (`1,8)

12 Contribution per unit `2 Annual sales volume 2, units Annual contribution `4,, Less: Annual cash fixed costs (`1,,) Annual net cash inflow `3,, Initial investment 25,, The project IRR in the above case is more than 1% & hence the project should be accepted. SIMULATION Question 16: The director of finance for a farm cooperative is concerned about the yield per acre he can expect from this year s corn crop. The probability distribution of the yields for the current weather conditions is: Yield kg per acre He would like to see a simulation of the yields he might expect over the next 1 years for weather conditions similar to those he is now experiencing. (i) Simulate the average yield he might expect per acre during the next 1 years using the following random numbers: 2, 72, 34, 54, 3, 22, 48, 74, 76, 2 (ii) He is also interested in the effect of market price fluctuations on the co-operative s farm revenue. He makes this estimate of per kg. Prices for corn. Price per kg (`) Simulate the revenues he might expect to observe over the next 1 years using the following random numbers for SP per kg: 82, 95, 18, 96, 2, 84, 56, 11, 52, 3. Solution: If the numbers -99 are allocated in proportion to the probabilities associated with each category of yield per acre, then various kinds of yields can be sampled using random number table: Yields in kg per acre Cumulative Random Numbers assigned (i) Let us simulate the yield per acre for the next 1 years based on the given 1 random numbers. Year Random Number Simulated Yield Total 148 The average yield is 148/1 = 148 kg/acre.

13 (ii) Let us now simulate the price he might expect in the next 1 years based on the random numbers given: Price per kg. Cumulative Random Numbers assigned This simulated prices are developed using the random numbers given for next 1 ten years. Year Random Number Simulated Price Per Kg Simulated Yield Revenue per acre Total 3,386 Question 17: Annual Net Cash Flows & Life of the project with their probability distribution are as follows: Annual Cash Flow Project Life Value (`) Value (Year) 1, , , , , , , Risk free rate is 1%, and Initial Investment is `1,3,. Various Random Number generated are as follows: 53,479 81,115 97,344 7,328 66,23 38,277 99,776 75,723 3,176 48,979 81,874 83,339 19,839 9,63 9,337 33,435 31,151 58,295 67,619 52,515 Calculate NPV in each Run. Solution: Correspondence between Values of Variables and two Digit Random Numbers: Annual Cash Flow Project Life Value (`) Cumulative Two Digit Random No. Value (Year) Cumulative Two Digit Random No. 1, , , ,

14 3, , , For the first simulation run we need two digit random number (1) For Annual Cash Flow (2) For Project Life. The numbers are 53 & 97 and corresponding value of Annual Cash Flow and Project Life are `3, and 9 years respectively and so on. Calculation of NPV through Simulation: Run Annual Cash Flow Random No. Annual Cash Flow (1) Random No. Project Life Project Life 1% (2) NPV (1) x (2) 1,3, , ,77* , , , (8,3) , (66,6) , (21,125) , , , , , , , , , *(3, x ,3,), Cumulative 1% for 9 years is 5.759, NPV of other runs are calculated similarly.

Final Course Paper 2 Strategic Financial Management Chapter 2 Part 8. CA. Anurag Singal

Final Course Paper 2 Strategic Financial Management Chapter 2 Part 8 CA. Anurag Singal Internal Rate of Return Miscellaneous Sums Internal Rate of Return (IRR) is the rate at which NPV = 0 XYZ Ltd., an