Topic 2 – Investment Appraisal: Uncertainty and Capital Rationing

1. Topic 2 – Investment Appraisal: Uncertainty and Capital Rationing

2. Learning Objectives After studying this topic you will: • understand why uncertainty must be considered in investment appraisal; • know the time based methods of incorporating inflation into the appraisal; • be able to use Probability to assess uncertainty; • know how to calculate and use Expected Value; • be able to use Discrete and Continuous Probabilistic Analysis; • know how to conduct a Sensitivity analysis of a project; • be able to evaluate a Portfolio’s Risk; • understand the way projects are selected when Capital Rationing exists for Single and Multi-Periods.

3. Uncertainty in investment appraisal • The risk and uncertainty associated with the individual project • The effect on the overall risk and uncertainty of the firm when the project being considered is combined with the rest of the firm’s operations – the portfolio effect • The decision maker’s attitude to risk and its effect on the final decision

4. Methods of considering the uncertainty • Time based • Probability based • Sensitivity analysis and simulation.

5. Time based • Payback • Risk Premium • Finite Horizon

6. Payback • The number of periods cash flows required to recoup the original investment. • Advantages: • Simplicity of calculation • General acceptability and ease of understanding • Disadvantages: • Assumes that uncertainty relates only to the time elapsed • Assumes that cash flows within the calculated payback period are certain • Makes a single blanket assumption and does not attempt to consider the variabilities of the cash flows estimated for the particular project being appraised.

7. Risk Premium • The discount rate is raised above the cost of capital in an attempt to allow for the riskiness of projects. The extra percentage being known as the risk premium.

8. The cash flows for a project are shown below. The cost of capital is 10% and as the project is considered to be risky, a risk premium of 5% is to be added to the basic rate. The effects of the two discount rates are shown . Example 1

9. Risk Premium • The discount rate is raised above the cost of capital in an attempt to allow for the riskiness of projects. The extra percentage being known as the risk premium. • Advantages: • Simple to use • Disadvantages: • Makes the implicit assumption that uncertainty is a function of time. • Does not consider the individual project characteristics and the variability of the project cash flows. • Problems of deciding upon a suitable risk premium

10. Finite Horizon • Project results beyond a certain period were ignored. All pojects are thus appraised over the same time period. • Advantages: • Simplicity • Disadvantages: • Projects do vary in length. Appraisal should reflects it. • Considering certain in project cash flows • Dose not consider the variabilities of cash flows.

11. Probability based • Expected value • Discrete probabilistic analysis • Continuous probabilistic analysis

12. Expected Value

13. The cash flows and probability estimates for a project are shown below. Calculate: • expected value of the cash flows in each period; and • expected value of the NPV when the initial project outlay is $11,000 and the cost of capital is 15%. ENPV = -11,000+(3,890 x 0.870) + (4,520 x 0.756) + (3,870 x 0.658) + (4,550 x 0.572) = $950 Example 2

14. Discrete Probability Analysis (DPA) • Instead of merely averaging these estimates it uses the component parts of the estimates to show the various outcomes and probabilities possible.

15. The NPV of Example 2 was $950 and management consider this somewhat marginal and wish to explore the range of outcomes possible. Further investigation reveals that two capital costs are possible: the $11,000 as stated with a probability of 0.8 and $15,000 with a probability of 0.2. This results in a new expected NPV of $150 using the new expected capital cost of $11,800. Present value of most likely cash flows = 11,143 Less most likely capital cost = 11,000 NPV = 143 The combination has a probability of 0.5 x 0.8 = 0.4 • The outcome range from -$5,167 to $3,707 • The probability of making a loss is 0.16 + 0.06 + 0.1+ 0.04 = 0.36 Example 3

16. Discrete Probability Analysis (DPA) • Advantages • Simple to apply and understand • Gives some indication of the range of possible outcomes and their probabilities • Considers the detailed variations in the cash flows and investment required for a project rather than merely making one overall assumption • Disadvantages • Uses discrete estimates • Increases the amount of subjective estimation

17. Continuous Probabilistic Analysis • Continuous distributions and aspects of statistical theory are used instead of the discrete estimates which are a feature of DPA

18. Continuous Probabilistic Analysis • “Given that the most likely value of the cash flow is $30,000 within what limits would you expect the cash flow to be 50% of the time?” • Assume the answer is: “25,000-35,000”. • From Normal Area Table 50% lies between ±⅔σ • 35,000 - 25,000 = ⅔σ, therefore, ŝ ≈ 7,500 • The total range of cash flow expected is 7,500 - 52,500, the whole of a normal distribution is within the range of ±3σ • σ = (52,500 - 7,500) / 6 = 7,500

19. 95% 99.74% -3 -2 -1 =0 1 2 3 The Normal Distribution

20. Combining the means and standard deviations of the cash flows

21. The means and standard deviations of the cash flows of a project are shown below and it is required to calculate: • The project NPV (i.e. the mean) • The variability of the project NPV (i.e. σNPV) • The probability of obtaining • A negative NPV • An NPV of at least $20,000 Example 4

22. The project NPV = -200,000 + (55,000 x 0.909) + (48,000 x 0.826) + (65,000 x 0.751) + (70,000 x 0.683) + (40,000 x 0.621) = 11,108 The standard deviation of the NPV, σNPV = From table, Z = 1.622, probability = 0.9474, Therefore, there is (1-0.9474)=5.3% chance of being –ve NPV Example 4

23. | 8 + 8 z 0 Normal Distribution .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7828 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 . 9 .8159 .8186 .212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

24. The project NPV = -200,000 + (55,000 x 0.909) + (48,000 x 0.826) + (65,000 x 0.751) + (70,000 x 0.683) + (40,000 x 0.621) = 11,108 The standard deviation of the NPV, σNPV = From table, Z = 1.299, probability = 0.9032, therefore, there is (1-0.9032) = 9.68% chance of obtaining at least 20,000 Example 4

25. | 8 + 8 z 0 Normal Distribution .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7828 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 . 9 .8159 .8186 .212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

26. x 11,108 y Point x is 11,108 – (3 x 6,847) = -9,433 Point y is 11,108 + (3 x 6,847) = 31,649 Example 4

27. Comparing of projects using CPA Project A: mean = 80,000 , s.d. = 12,500 Project B: mean = 130,000 , s.d. = 17,500 Project B is relatively less risky

28. Sensitivity Analysis • Showing the effects of uncertainty by varying the values of the key factors (e.g. sales volume, price and rates of inflation, cost per unit) and showing the resulting effect on the project. • Objective: to establish which of the factor affect the project most.

29. Assume that a project (using single valued estimates) has a positive NPV of $25,000 at a 10% discounting rate. This value would be calculated by the normal methods using particular values for sales volume, sales price, cost per unit, inflation rate, length of life, etc. Once the basic value (i.e. the NPV of $25,000) has been obtained the sensitivity analysis is carried out by flexing, both upwards and downwards, each of the factors in turn. Example 5

31. Risk and Portfolio Effect

32. A firm with $100,000 to invest is considering two projects, X and Y each requiring an investment of $100,000. The returns from the proposed projects and from existing operations under three possible views of expected market conditions are shown in Table 4 together with the calculated standard deviations of returns, i.e. the measure of riskiness used by the company. The firm considers that the risk and return of their existing operations are similar to the market as a whole and that a reasonable estimate of a risk free interest rate is 8%. Which, if either, of the two proposed investments should be initiated and why? Example 6

33. Covariance (x,0) σxσ0 -0.009745 0.22 x0.18 Project X: Rx = (0.20 x 0.3) + (0.20 x 0.4) + (-0.01667 x 0.3) = 0.135 = 13.5% Project Y: RY = (-0.02 x 0.3) + (0.15 x 0.4) + (0.27 x 0.3) = 0.135 = 13.5% Existing operations: Ro = (-0.09 x 0.3) + (0.1825 x 0.4) + (0.28 x 0.3) = 0.13 = 13% Covariance between Project X return (Rx) and Company Return (Ro) Correlation coefficient between X and 0 = = Covariance (x,0) = -0.246

34. Covariance (y,0) σyσ0 0.01662 0.15 x0.18 Covariance between Project Y return (Rx) and Company Return (Ro) Correlation coefficient between Y and 0 = = Covariance (y,0) = -0.615

35. RF + RF + σy σx X X X X Correlation (y,0) Correlation (x,0) 0.13-0.08 0.13-0.08 0.08 0.08 = = - 0.615 - 0.246 + + 0.22 0.15 X X X X 0.18 0.18 Ro-RF Ro-RF 6.5% 0.08 0.08 = = = - 0.0256 - 0.0015 σ0 σ0 10.65% = Required return of Project X = Required return of Project Y =

36. Attitude to Risk Investment A: • Return of $100,000 with a probability of 1, i.e. certainty Investment B: • Return of $300,000 with a probability of 0.5 • Return of 0 with a probability of 0.5 The expected returns are: Investment A: $100,000 Investment B: ($300,000 x 0.5) + ($0 x 0.5) = $150,000

37. a b Utility c Monetary returns a = Convex curve Risk aversion b = Straight line at 45% Risk neutral c = Concave curve Risk seeking

38. Capital Rationing • To maximise the return from the batch of projects selected having regard to capital limitation.

39. Single Period Capital Rationing – Divisible Projects CR Ltd has a cost of capital of 15% and has a limit of $100,000 available for investment in the current period It is expected that capital will be freely available in the future. The investment required, the NPV at 15% and the EVPI for each of the 6 projects currently being considered are shown below. What projects should be initiated? Ranking by EVPI is C, B, D, A and E. Example 7

40. Optima Investment Plan Example 7

41. Single Period Capital Rationing with mutually exclusive divisible projects Assume the same data as Example 7 except that projects B and D are mutually exclusive. What projects should be initiated? Group I Group II Example 8

42. Single Period Capital Rationing – Indivisible Projects Lloyds Ltd has a cost of capital of 10% and has a limit of $100,000 available for investment in the current period. Capital is expected to be freely available in future periods. The following indivisible projects are being considered. • It is required to calculate the optimal investment plan • Where there are no alternative investments available for any surplus funds. • Where surplus funds can be invested to produce 12% in perpetuity. Example 9

43. Single Period Capital Rationing – Indivisible Projects No alternative investments available for any surplus funds Example 9

44. Single Period Capital Rationing – Indivisible Projects Surplus funds can be invested to produce 12% in perpetuity Example 9

45. Multi-period Period Capital Rationing Trend Ltd has a cost of capital of 10% and is considering which project or projects it should initiate. The following projects are being considered: Estimated cash flows Capital is limited to $40,000 now and $35,000 in Year 1. The projects are divisible. Example 10

46. Multi-period Period Capital Rationing A = 23,245 B = 28,414 C = 37,939 Maximise 23,245XA + 28,414XB + 37,939XC where XA = the proportion of Project A to be initiated XB = the proportion of Project B to be initiated XC = the proportion of Project C to be initiated Capital at time 0: 15,000XA + 25,000XB + 25,000XC ≤ 40,000 Capital at time 1: 25,000XA + 15,000XB + 15,000XC ≤ 35,000 XA, XB, XC ≤ 1 XA, XB, XC ≥ 0 Example 10

47. Learning Objectives After studying this topic you will: • understand why uncertainty must be considered in investment appraisal; • know the time based methods of incorporating inflation into the appraisal; • be able to use Probability to assess uncertainty; • know how to calculate and use Expected Value; • be able to use Discrete and Continuous Probabilistic Analysis; • know how to conduct a Sensitivity analysis of a project; • be able to evaluate a Portfolio’s Risk; • understand the way projects are selected when Capital Rationing exists for Single and Multi-Periods.