Strategic Investment & Finance Solutions to Exercises Exercise 1 Question a 40 30 30 20 20 0 1 2 3 4 5-100 With a discount rate equal to 10%: NPV 0 = 100 +40 1.1 1 +30 1.1 2 +30 1.1 3 +20 1.1 4 + 20 1.1 5 = 100 + 36.36 + 24.79 + 22.54 + 13.66 + 12.42 = 9.77 With a discount rate equal to 20%: NPV 0 = 100 +40 1.2 1 +30 1.2 2 +30 1.2 3 +20 1.2 4 +20 1.2 5 = 100 + 33.33 + 20.83 + 17.36 + 9.65 + 8.04 = 10.79 Question b Discount rate: 10% p.a. Discount rate after tax: 10 (1 0.3) = 7% p.a. Depreciation: Straight line Tax: 30% Periods: 5 years We make use of an after-tax discount rate. 1
The cash flow is calculated on an after-tax basis each year, and the results are discounted back to period 0. 0 1 2 3 4 5 Operational Net Cash Flow -100 40 30 30 20 20 Depreciation 20 20 20 20 20 Net taxable income / tax base 20 10 10 0 0 Tax payment 6 3 3 0 0 Cash Flow after tax -100 34 27 27 20 20 The annual net cash after tax in each year is discounted using the after-tax discount rate: NPV 0 = 100 + 34 1.07 1 + 27 1.07 2 + 27 1.07 3 + 20 1.07 4 + 20 1.07 5 = 6.92 Discount rate: 20% After-tax discount rate: 20 (1 0.3) = 14% Depreciation: Straight line Tax: 30% Periods: 5 years NPV 0 = 100 + 34 1.14 1 + 27 1.14 2 + 27 1.14 3 + 20 1.14 4 + 20 1.14 5 = 8.95 Exercise 2 Investment: 200,000 Lifetime: 5 years Operating costs: 158,000/year Revenue: 208,000/year Net cash: Scrap value: 208,000 158,000 = 50,000/year 15,000 after 5 years 2
208,000 15,000 0 1 2 3 4 5 200,000 158,000 Question a The internal rate of return is found by setting NPV 0 equal to zero. Internal rate of return: NPV 0 = 0 0 = 200 + 50 PVIFA IRR;5 + 15 (1+IRR) 5 The internal rate of return is either found by trial and error, Excel or by using a financial calculator. Using any of the three methods yields an internal rate of return = 9.78% p.a. Question b An investment is profitable if the internal rate of return is larger than the applied discount rate. We can conclude that this investment is attractive as it satisfies the company s requirement of a return on investment larger than 8%, as 9.78% > 8%. 3
Question c Discount rate: 8% p.a. NPV 0 = 200,000 + 50,000 5 8% + 15,000 (1,08) 5 = 200,000 + 50,000 3.99271 + 15,000 0.680583 = 200,000 + 209,844.25 = 9,844.25 Expressed as an average annual cash flow (annuity): A = 9,844.25 PVIFA 8%:5 1 = 9,844.25 0.250456 = 2,465.55 Exercise 3 Discount rate 7% p.a. Periods: 8 Annual savings: 27,460 27,460 0 1 2 3 4 5 6 7 8? NPV 0 = 27,460 PVIFA 7%; 8 = 27,460 5.9713 = 163,971.86 The maximum price that the company on these assumptions should be willing to pay is equal to the discounted value of the future annual savings, i.e. 163,971.86 4
Exercise 4 Question a Discount rate: 12% p.a. Alternative A: The company establishes the production facility on its own Initial investment: 5 million Expenses: Year 1: 1 million Year 2-4: 500,000 Marginal Contribution: Year 1: 0 Year 2: 2 million Year 3: 4 million Year 4: 6 million 4,000,000 6,000,000 2,000,000 0 1 2 3 4 1,000,000 500,000 5,000,000 NPV 0 = 5,000,000 1,000,000 1.12 1 + 1,500,000 1.12 2 + 3,500,000 1.12 3 + 5,500,000 1.12 4 = 5,000,000 1,000,000 0.892857 + 1,500,000 0.797194 + 3,500,000 0.71178 + 5,500,000 0.635518 = 1,289,514 Alternative B: The company acquires an existing manufacturer Investments:?? 5
Marketing expenses: Year 1: 1 million (It is assumed that the marketing expenses are equal for both alternatives) Year 2-4: 500,000 Costs: 400,000/Year Marginal Contribution: Year 1: 1.2 million Year 2: Year 3: Year 4: 2.0 million 1.2 = 2.4 million 4.0 million 1.2 = 4.8 million 6.0 million 1.2 = 7.2 million 4,800,000 7,200,000 1,200,000 2,400,000 0 1 2 3 4? 1,400,000 900,000 The NPV 0 of making the acquisition must be at least equal to the NPV of establishing the production on our own, i.e. 1,289,514 in order to make the acquisition as attractive as alternative 1: NPV 0 = 1,289,514 = I 0 200,000 1.12 1 + 1,500,000 1.12 2 + 3,900,000 1.12 3 + 6,300,000 1.12 4 NPV 0 = 1,289,514 = I 0 200,000 0.892857 + 1,500,000 0.797194 + 3,900,000 0.71178 + 6,300,000 0.635518 1,289,514 = I 0 + 7,796,926 I 0 = 6,507,412 This means that the maximum acquisition price is 6,507,412 when benchmarked against the alternative of establishing the production on our own. If the price exceeds this limit, NPV 0,BUY < NPV 0,PROD, then alternative 1 should be chosen. 6
Exercise 5 Question a Discount rate: Method 1 Investment: Lifetime: Variable costs: 12% p.a. 5 million 10 years 500 per unit 0 1 2 3 4 5 6 7 8 9 10 500 Q 5 million Method 2 Investment: Lifetime: Variable costs: 10 million 15 year 400 per unit 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 400 Q 10 million It is assumed that identical replacement of both facilities will take place. Due to the different lifetimes, the net present value method is not used. Instead we use the annuity method. One could also find the common multiple, under the presumption that identical replacement is undertaken. Method 1 A = 5,000,000 PVIFA 12%; 10 1 = 5,000,000 0.176984 = 884,920 7
Cost-function: 884,920 + 500 Q Method 2 A = 10,000,000 PVIFA 12%; 15 1 = 10,000,000 0.146824 = 1,468,240 Cost-function: 1,468,240 + 400 Q In order to find the production level at which the two investments are considered equally desirable from a financial point of view we set the cost-functions equal to each other. TC 7.000.000 6.000.000 5.000.000 4.000.000 3.000.000 2.000.000 1.000.000 0 Method 2 Method 1 Q 0 2.000 4.000 6.000 8.000 10.000 12.000 8
This equation is solved with one unknown variable the production quantity. 884,920 + 500 Q = 1,468,240 + 400 Q 100 Q = 583,320 Q = 5,833 If the company expects to produce a quantity less than 5,833 units, it should invest in method 1, whereas it should invest in method 2 if the expected quantity produced is larger than 5,833 units. At a production level of the abovementioned quantity or above, the lower variable costs in method 2 have outweighed the relatively large investment in the fixed assets; hence the two methods breakeven at a quantity equal to 5,833 units. Exercise 6 Question a Project X Discount rate: 12% p.a. Investment: 10,000 Cash flow: Year 1: 6,500 Year 2: 3,000 Year 3: 3,000 Year 4: 1,000 6,500 3,000 3,000 1,000 0 1 2 3 4 10,000 9
NPV 0 = 10,000 + 6,500 1.12 1 + 3,000 1.12 2 + 3,000 1.12 3 + 1,000 1.12 4 = 10,000 + 10,966.01 = 966.01 In order to find the internal rate of return, we set NPV 0 = 0: 0 = 10,000 + 6,500 (1+IRR) 1 + 3,000 (1+IRR) 2 + 3,000 (1+IRR) 3 + 1,000 (1+IRR) 4 The internal rate of return is calculated using trial and error, Excel, or a financial calculator. The internal rate of return is calculated to be 18.03% per year. Pay Back-method (static and dynamic) Period 0 1 2 3 4 Net CF 10,000 6,500 3,000 3,000 1,000 Net CF discounted 10,000 5,804 2,392 2,135 636 Acc. CF un-discounted 10,000 3,500 500 2,500 3,500 Acc. CF discounted 10,000 4,196 1,805 331 966 Note: CF = Cash Flow Both methods yield a pay-back period between 2 and 3 years. If we use linear interpolation, we have: Pay Back period, static method = 2.2 years Pay Back period, dynamic method = 2.9 years Project Y Discount rate: 12% p.a. Investment: 10,000 Cash flow: 3,500/year 3,500 0 1 2 3 4 10,000 10
NPV 0 = 10,000 + 3,500 PVIFA 12%; 4 = 10,000 + 3,500 3,037349 = 10,000 + 10,630.72 = 630.72 To find the internal rate of return, we set NPV 0 = 0 0= 10,000 + 3,500 PVIFA R; 4 PVIFA R;4 = 2.85714 The internal rate of return is calculated using trial and error, Excel, or a financial calculator. The internal rate of return is 14.96% per year. Pay Back-method (static and dynamic) Period 0 1 2 3 4 Net CF 10,000 3,500 3,500 3,500 3,500 Net CF discounted 10,000 3,125 2,790 2,491 2,224 Acc. CF un-discounted 10,000 6,500 3,000 500 4,000 Acc. CF discounted 10,000 6,875 4,085 1,594 631 Note: CF = Cash Flow Using linear interpolation we have: Pay Back period, static method Pay Back period, dynamic method = 2.9 years = 3.7 years Question b If the two investments are mutually independent, the criteria is that NPV 0 > 0. As NPV 0 > 0 for both projects, both projects should/could be initiated. Question c If the two projects are mutually exclusive, the project with the largest net present value should be preferred. As NPV 0,X > NPV 0,Y project X should be the preferred one. Question d Yes - maybe!! 11
The discount rate may affect the ranking of the projects because the net present value is influenced by changes to the discount rate. In general we can say: A high interest rate favours short-term investments because cash flows further into the future have a comparatively lower net present value A low interest rate favours long-term investments as cash flows further into the future maintain a comparatively higher net present value Below figure illustrates how the ranking of the two projects is altered when changing the discount rate. Project Y is the most favourable, when the interest rate is low (below 6.22% p.a.). When the interest rate is higher (above 6.22% p.a.), project X becomes the most favourable. NPV 0 5.000 Project Y 4.000 3.000 Project X 2.000 1.000 Discount rate 0-1.000 0 10 20 30 40-2.000-3.000-4.000 Question e The internal rate of return method is only applicable, when we are performing absolute evaluations of the investments. If the point in case is two investments (which are mutually exclusive), it is not possible to use this method, as: 1. The internal rate of return is a relative number (Is 50% if 100 worth more than 2% of 5 million?). 2. The time dimension is lacking. 3. When comparing investment alternatives, the ranking may change (compare with the graph above). 12
In the graph above it is shown that if one chooses to make use of the internal rate of return method, investment X will be perceived as being the most profitable. However, this is not true for every interest rate, which the investor may apply. Exercise 7 Question a: Investment: Operating expenses: Inflow 21,000, lifetime of 15 years 210/year 34.100 kwh of 0.075, with a utilisation rate of 80%, yields: 34.100 0.075 0,8 = 2,046/year Annual net: 2,046 210 = 1,836 Real rate of interest: 5% per year 1,836 0 1 2 3 13 14 15 21,000 NPV 0 = 21,000 + 1,836 PVIFA 5; 15 = 21,000 + 1,836 10.3797 = 1,943 which expressed as an annual expense yields: A = 1,943 PVIFA 5; 15-1 = 1,943 0,096342 = 187 As the net present value is negative, the project is not profitable. 13
Determination of the internal rate of return: NPV 0 = 21,000 + 1,836 PVIFA?; 15 = 0 yields PVIFA?; 15 = 21,000 1,836 = 11,437908 By looking in an annuity table we see that this corresponds to an interest rate between 3.5% and 3.75%. Using linear interpolation we can determine that the internal rate of return is 3.6%, which is less than the interest rate of calculation. This confirms that the project is not profitable. Questions b and c: If we assume that the windmill already has been acquired, we must now evaluate the profitability of the two alternative add-on investments. The two alternatives have the same lifetimes, i.e. 25 years, and we can therefore use the net present value method. However, there is one problem the add-on investments have a longer lifetime than the windmill does, so we must either consider a total lifetime of 15 years, or assume that we in 15 years re-invest in a new windmill. 1. Total lifetime of 15 years Alternative 1: Selling the electricity Investment: 1,500 20% of 34.100 kwh at a price of 0.032 per kwh yields 34.100 0,2 0.032 = 218/year NPV 0 = 1,500 + 218 PVIFA 5; 15 = 1,500 + 218 10.3797 = 767 which, expressed as an annual inflow yields A = 767 PVIFA 5; 15 1 = 767 0.096342 = 74 As the net present value is positive, the project is profitable. 14
Alternative 2: Hot water Investment: 3,000 20% of 34,100 kwh = 6,820 kwh 1 kwh corresponds to 0.2 litres of oil at a price of 0.26 per litre Annual savings: 6,820 0.2 0.26 = 355 NPV 0 = 3,000 + 355 PVIFA 5; 15 = -3,000 + 355 10.3797 = 681 which expressed as an annual inflow yields: A = 681 PVIFA 5; 15 1 = 681 0.096342 = 66 As the net present value is positive, the project is profitable. Under the assumption that the windmill has been acquired, and that we only consider a period of 15 years, both add-on investments are profitable. However, alternative 1 is a little bit better. Although this is the case, none of the two alternative add-on investments are able to make the entire project profitable. 2. Re-investment in a mill every 15 years plus best alternative every 25 years. The point in case is a series of chain investments. Therefore, the annuity method is the only method, which will lead to the correct result. Alternative 1: Selling electricity Investment: 1,500 20% of 34,100 kwh at a price of 0.032 per KWh gives 34,100 0.2 0.032 = 218 /year A = 1,500 PVIFA 5; 15 1 + 218 = 1,500 0.070952 + 218 15
= 112 /year As the annuity is positive, the project is profitable Alternative 2: Heated water Investment: 3,000 20% of 34,100 kwh = 6,820 kwh 1 kwh corresponds to 0.2 litre of oil at a price of 0.26 per litre Annual savings: 6.820 0.2 0.26 = 355 A = 3,000 PVIFA 5; 15 1 + 355 = 3,000 0.070952 + 355 = 142 /year As the annuity is positive, the project is profitable. Under the assumption that the mill and the add-on investment are repeated infinitely, both alternative add-on investments are profitable. However, alternative 2 is marginally better. Although this is the case, none of the two add-on investments are able to make the total project profitable, as the mill has an annual average cost of 1,871. Question d: We are now considering a total investment consisting of both the windmill and the best alternative add-on investment. 1. Total lifetime 15 years. Best add-on investment: Selling electricity NPV 0 = 21,000 + (2,046 210) PVIFA 5; 15 1,500 + 218 PVIFA 5; 15 If the price of energy changes with x%, we get: NPV 0 = 21,000 + (2,046 (1+x) 210) PVIFA 5; 15 1,500 + 218 (1+x) PVIFA 5; 15 In order for the total project to be profitable, NPV 0 must be larger than zero. Therefore: 16
0 = 21,000 + (2,046 (1+x) 210) PVIFA 5; 15 1,500 + 218 (1+x) PVIFA 5; 15 = 21,000 + (1,836 + 2,046 x) 10,3797 1,500 + (218 + 218 x) 10,3797 = 11.759,45 + 235.037,92 x 11759., 45 x = = 0.05 234. 037, 92 which means that the total project is in balance, if the energy prices increase by 5%. 2. Re-investment in a windmill every 15 years plus the best alternative every 25 years. Best add-on investment: Heated water A = 21,000 PVIFA 5; 15-1 +2,046 210 3,000 PVIFA 5; 15-1 + 355 If the energy prices change by x%, we get: A = 21,000 PVIFA 5; 15 1 + 2,046 (1+x) 210 3,000 PVIFA 5; 25 1 + 355 (1+x) In order for the total project to be profitable, the annuity must be equal to or greater than 0. 0 = 21,000 PVIFA 5; 15 1 +2,046 (1+x) 210 3,000 PVIFA 5; 15 1 + 355 (1+x) = 21,000 0.096342 + 2,046 + 2,046 x 210 3,000 0.070952 + 355 + 355*x = 454,38 + 24.006 x 454, 38 x = = 0.0189 24. 006 which means that the total project is in balance, if the energy prices increase by 1.89%. Exercise 8 Question a Both aircraft types have an expected lifetime of 15 years. Therefore, we can use the net present value method to evaluate the two alternatives. Type I: Investment: Annual energy consumption: Other annual operating costs: 12 million 2.5 million 1.8 million 17
Expected scrap value: 2 million 2 0 1 2 3 13 14 15 12 4.3 NPV 0 = - 12 4.3 PVIFA 9; 15 + 2 (1 + 0.09) 15 = - 12 4.3 8.06069 + 2 0.274538 = - 46.1 Type II: Investment: Annual energy consumption: Other annual operating costs: Expected scrap value: 16 million 2.25 million 1.64 million 2.4 million 2.4 Time 0 1 2 3 13 14 15 3.89 16 NPV 0 = - 16 3.89 PVIFA 9; 15 + 2.4 (1+ 0.09) 15 = - 16 3.89 8.06069 + 2.4 0.274538 = - 46.7 From a financial point of view, Type I should be chosen as it has the lowest costs. Question b The assumptions are changed, such that type II now has a lifetime of 20 years. This being the case, we must assume that the airline carrier expects to continue its operations in excess of the 20 years, and, in lack of better knowledge, it is natural to assume identical re-investment. Therefore, the net present value method is no longer applicable, and we must evaluate the project by use of the annuity method. 18
Type I NPV 0 for 15 years has been calculated to 46.1 million in question 1. This, distributed as an annuity over 15 years, gives us: Type II A I = - 46.1 PVIFA 1 9; 15 = - 46.1 0.124059 = - 5.7 million /year NPV 0 for 20 years is calculated to be: NPV 0 = - 16 3.89 PVIFA 9; 20 + 2.4 (1+0.09) 20 = - 16 3.89 9.12855 + 2.4 0.178431 = - 51.1 Distributed as an annuity over 20 years gives us: A II = - 51.1 PVIFA 1 9; 20 = - 51.1 0.109546 = - 5.6 million /year Given the changes in circumstances, type II is now the most attractive investment. Question c Aircraft type I consumes what corresponds to 2.5 million in fuel costs per year, whilst type II consumes what corresponds to 2.25 million in fuel costs per year. If we change the price of fuel by x%, the costs of type I can be written as: 2.5 (1+x), and for type II we get: 2.25 (1+x). Type I NPV 0,I = ( - 12 - (2.5 (1+x) + 1.8) PVIFA 9; 15 + 2 (1+0.09) 15 ) Average annual costs: A I A I = NPV 0 PVIFA 9; 15 1 = (- 12 - (2.5 (1+x) + 1.8) PVIFA 9; 15 + 2 (1+0.09) 15 ) PVIFA 9; 15 1 19
Type II NPV 0,II = (- 16 - (2.25 (1+x) + 1.64) PVIFA 9; 20 + 2.4 (1+0.09) 20 ) Average annual costs: A II A II = NPV 0 PVIFA 9; 20 1 = (-16-(2.25 (1+x) + 1.64) PVIFA 9; 20 + 2.4 (1+0.09) 20 ) PVIFA 9; 20 1 By setting A I = A II, we have an equation in which we can solve for the value of x. However, this may be done much more easily. The annual costs for types I and II, have been calculated, in question 2, to take on the value of 5.7 million/year and 5.6 million/year, respectively. These costs consist in part of fuel costs, and of other costs, including the investment in itself, distributed as an annuity. If we express the annual costs as: Annual costs = fuel costs + other costs we get: Type I A I = 2.5 + 3.22 Type II A II = 2.25 + 3.35 If the cost of fuel is changed by x%, we get: Type I A I = 2.5 (1+x) + 3.22 Type II A II = 2.25 (1+x) + 3.35 If we set A I = A II, we get one equation with one unknown. 2.5 (1+x) + 3.22 = 2.25 (1+x) + 3.35 2.5 + 2.5 x + 3.22 = 2.25 + 2.25 x + 3.35 2.5 x = 1.25 x = 0.5 From the calculations above we see that the cost of fuel must decrease by 50%, in order for type I to be as profitable as type II. Hence, the calculus is rather insensitive towards changes in the fuel price. Exercise 9 20
Assumptions: Interest rate: 10% per year Initial investment: 1,000,000 Depreciation: 30% fixed percentage rate per year, reducingbalance method Operating costs: 50,000 year 1 increases with 20% per year from here on Retail price: 300 AVC: 210 Contribution Margin/unit 90 Sales: 9,000 units Total Contribution 90 9,000 = Margin: 810,000 Wages: 2 men of 180,000 each= 360,000 Profit: 450,000 Year Machine Operating Marketing expenses Profit Net NPV 0 Acc. NPV 0 0 1,000,000 1,000,000 1,000,000 1,000,000 1 50,000 400,000 450,000 0 0 1,000,000 2 60,000 100,000 450,000 290,000 239,669 760,331 3 72,000 100,000 450,000 278,000 208,866 551,465 4 86,400 100,000 450,000 263,600 180,042 371,423 5 168,070 103,680 100,000 450,000 414,390 48,587-114,120 On the basis of the calculations above, the company should not initiate the production, as the net present value is negative with -114.120 over a 5-year period. Identical re-investment: Now, only the costs, which are directly associated with operating the machine, are of interest i.e. wages and marketing expenses are not relevant. 21
Year Scrap Depre. Interest Operating Total npv 0 NPV 0 A 0 1.000.000 1 700.000 300.000 100.000 50.000 450.000 409.091 409.091 450.000 2 490.000 210.000 70.000 60.000 340.000 280.992 690.083 397.619 3 343.000 147.000 49.000 72.000 268.000 201.352 891.435 358.459 4 240.100 102.900 34.300 86.400 223.600 152.722 1.044.157 329.401 5 168.070 72.030 24.010 103.680 199.720 124.010 1.168.167 308.160 6 117.649 50.421 16.807 124.416 191.644 108.178 1.276.345 293.058 7 82.354 35.295 11.765 149.299 196.359 100.763 1.377.108 282.866 8 57.648 24.706 8.235 179.159 212.101 98.947 1.476.055 276.678 9 40.354 17.294 5.765 214.991 238.050 100.956 1.577.011 273.833 10 28.248 12.106 4.035 257.989 274.130 105.689 1.682.701 273.852 11 19.773 8.474 2.825 309.587 320.886 112.469 1.795.169 276.390 12 13.841 5.932 1.977 371.504 379.413 120.893 1.916.062 281.208 As the average annual costs, A, are at their minimum in the ninth year, it will be optimal to replace the machine every 9-10 years. From the seventh year and onwards, the sales revenue exceeds the average annual costs. Hence, the project is profitable, if the time horizon is 7 or more years. 500.000 450.000 400.000 350.000 300.000 250.000 200.000 150.000 100.000 50.000 0 Average annual cos ts Year 0 5 10 15 Exercise 10 Assumptions: 22
Interest rate of calculation: 15 % per year Monthly Contribution Margin: 300,000 Annual Contribution Margin: 3,600,000 Annual depreciation: 10% of the initial investment Operating costs: 1,000,000 in year 1 from thereon an increase of 25% per year The production should continue as long as the annual inflows exceed the annual costs. Therefore, it is necessary to construct a table showing the annual costs. Year Scrap Depre. Interest Operating Total Con.Mar. Net 0 3.500.000 1 3.150.000 350.000 525.000 1.000.000 1.875.000 3.600.000 1.725.000 2 2.800.000 350.000 472.500 1.250.000 2.072.500 3.600.000 1.527.500 3 2.450.000 350.000 420.000 1.562.500 2.332.500 3.600.000 1.267.500 4 2.100.000 350.000 367.500 1.953.125 2.670.625 3.600.000 929.375 5 1.750.000 350.000 315.000 2.441.406 3.106.406 3.600.000 493.594 6 1.400.000 350.000 262.500 3.051.758 3.664.258 3.600.000 64.258 7 1.050.000 350.000 210.000 3.814.697 4.374.697 3.600.000 777.697 8 700.000 350.000 157.500 4.768.372 5.275.872 3.600.000 1.675.872 9 350.000 350.000 105.000 5.960.464 6.415.464 3.600.000 2.815.464 10 0 350.000 52.500 7.450.581 7.853.081 3.600.000 4.253.081 Note: Depr. = Depreciation Con.Mar. = Contribution Margin As we can see in the table above, the total costs in year 6 will exceed 3,600,000. Therefore, it is only profitable to continue the production including the fifth year. 23
9000000 8000000 7000000 6000000 5000000 4000000 3000000 2000000 1000000 0 Costs Annual Con.Mar. Year 0 2 4 6 8 10 12 24
Exercise 11 Assumptions: Interest rate of calculation: 15% per year Annual depreciation: 30% reducing-balance Annual MCON: 450,000 Annual operating costs: 45,000 year 1 increase with 45,000 per year afterwards Initial investment: 1,000,000 Question a: Year Scrap value Depreciation Interest Operating Total 0 1.000.000 0 1 700.000 300.000 150.000 45.000 495.000 2 490.000 210.000 105.000 90.000 405.000 3 343.000 147.000 73.500 135.000 355.500 4 240.100 102.900 51.450 180.000 334.350 5 168.070 72.030 36.015 225.000 333.045 6 117.649 50.421 25.211 270.000 345.632 7 82.354 35.295 17.647 315.000 367.942 8 57.648 24.706 12.353 360.000 397.059 9 40.354 17.294 8.647 405.000 430.942 10 28.248 12.106 6.053 450.000 468.159 The production should be continued as long as the annual costs are lower than the annual marginal contribution. In year 10 the costs exceed the marginal contribution. Therefore, the production should be stopped after the ninth year. 25
600000 500000 400000 300000 200000 100000 0 Con.M ar. per year Costs Year 0 2 4 6 8 10 12 Question b: Year Scrap value Depre. Interest Operating Total npv 0 NPV 0 A 0 1.000.000 0 1 700.000 300.000 150.000 45.000 495.000 430.435 430.435 495.000 2 490.000 210.000 105.000 90.000 405.000 306.238 736.673 453.140 3 343.000 147.000 73.500 135.000 355.500 233.747 970.420 425.022 4 240.100 102.900 51.450 180.000 334.350 191.166 1.161.586 406.863 5 168.070 72.030 36.015 225.000 333.045 165.582 1.327.168 395.915 6 117.649 50.421 25.211 270.000 345.632 149.426 1.476.594 390.171 7 82.354 35.295 17.647 315.000 367.942 138.323 1.614.917 388.162 8 57.648 24.706 12.353 360.000 397.059 129.799 1.744.716 388.810 9 40.354 17.294 8.647 405.000 430.942 122.500 1.867.217 391.320 10 28.248 12.106 6.053 450.000 468.159 115.722 1.982.939 395.105 As the average annual costs are at their minimum in the seventh year, it is optimal to replace the equipment every seven years. 26
600000 500000 Average annual costs 400000 300000 200000 100000 0 Year 0 2 4 6 8 10 12 Question c: The question may be solved using Excel, or performing the calculations by hand. Excel: Year Initial investment Scrap value Con.Mar. Operating costs Net Cash NPV 0 0 1.197.833 0 1.197.833 1.197.833 1 838.483 450.000 45.000 405.000 352.173,9 2 586.938,1 450.000 90.000 360.000 272.211,7 3 410.856,7 450.000 135.000 315.000 207.117,6 4 287.599,7 450.000 180.000 270.000 154.373,4 5 201.319,8 450.000 225.000 426.319,8 211.956,3 k 0 = 0 The cell, which can be changed, is the initial investment. Note that the scrap value in year 5 is included in the Net Cash amount. The target cell includes NPV 0, which must equal 0. By hand: The balance equation for the initial investment: 27
I = 405,000 1.15 1 + 360,000 1.15 2 + 315,000 1.15 3 + 270,000 1.15 4 + 225,000 1.15 5 + I 0.7 5 1.15 5 I = 352,173.9 + 272,211.7 + 207,117.6 + 154,373.1 + 111,864.8 + 0.08356 I 0.91644 I = 1,097,741.4 I = 1,197,832 Question d: Again, this question may be solved using the solver function in Excel, or manually by construction the balance equation. Excel: Annual Con.Mar. 395,914.8 Year Initial Investment Scrap value Contribution Margin Operating costs Net cash NPV 0 0 1.000.000 0 1.000.000 1.000.000 1 700.000 395.914,8 45.000 350.914,8 305.143,3 2 490.000 395.914,8 90.000 305.914,8 231.315,6 3 343.000 395.914,8 135.000 260.914,8 171.555,7 4 240.100 395.914,8 180.000 215.914,8 123.450,0 5 168.070 395.914,8 225.000 338.984,8 168.535,4 k 0 = 0 The cell which varies is now the annual contribution margin. Other than that, everything is analogous to question c. By hand: Balance equation: 1.000.000 = Con.Mar. PVIFA 15; 5 45,000 1.15 1 90,000 1.15 2 135,000 1.15 3 180,000 1.15 4 225,000 1.15 5 + 1,000,000 0.7 5 1.15 5 1,000,000 = 3.35216 Con.Mar. 39,130.43 68,052.96 88,764.66 102,915.54 111,864.82 + 83,560 28
3.35216 Con.Mar. = 1,327,167.51 Contribution Margin = 395,914 Exercise 12 Question a In order for the investment to be profitable, NPVo > 0. It is assumed that the same number of units is sold each year. In the following, let X express the number of printed circuit boards sold. Interest rate of calculation: 15% per year Investment: 10 million Expenses: Year 1: 3 million Years 2-5: 2 million Retail salesprice: Years 1-3: 700 /unit Years 4-5: 350 /unit Variable costs: 450 /unit Con.Mar. per unit: years 1-3: 700 450 = 250 /unit years 4-5: 350 450 = 100 /unit Net income: year 1: X 250 3 million Years 2-3: X 250 2 million Years 4-5: X ( 100) 2 million X 250 X 100 0 1 2 3 4 5 3.000.000 2.000.000 10.000.000 29
We must construct an equation with X as the unknown variable, and where NPV 0 is set equal to zero: NPV 0 = 0 = 10.000.000 3.000.000 1,15 1 2.000.000 PVIFA 15%; 4 1,15 1 + 250 X PVIFA 15%; 3 + ( 100 X PVIFA 15%;2 1,15 3 ) 0 = 10.000.000 3.000.000 0,8696 2.000.000 2,4826 + 250 X 2,2832 100 X 1,0689 0 = 17.573.875,41 + 570.8063 X 106,8930 X 463,9133 X = 17.573.875,41 X = 37.881,8 This means that in order for the project to become profitable, the company must sell 37.882 printed circuit boards every year! As the contribution margin is negative the last two years, it would be best to end the production and the marketing after 3 years. Doing so will yield the following result: NPV 0 = 0 = 10.000.000 3.000.000 1,1) 1 (2.000.000 PVIFA 15%; 2 1,15 1 ) + 250 X Question b PVIFA 15%; 3 0 = 10.000.000 3.000.000 0,8696 2.000.000 1,4137 + 250 X 2,2832 570,8 X = 15.436.200 x = 27.043 units sold every year Now Digiprint is contemplating whether or not it should lower its prices to 600 (lower than that of its competitors), and at the same time reduce its marketing costs by 1/3 per year. In order for the investment to be profitable, NPV 0 > 0. In the following, let X express the number of sold printed circuit boards. Interest rate: 15% per year Investment: 10 million Expenses: Year 1: 2 million Years 2-5: 1,333 million 30
Retail salesprice: Years 1-3: 600 /unit Years 4-5: 300 /unit Variable costs: 450 /unit Con.Mar. per unit: Years 1-3: 600 450 = 150 /unit Years 4-5: 300 450 = 150 /unit Net income: Year 1: X 150 2 million Years 2-3: X 150 1,333 million Years 4-5: X ( 150) 1,333 million Again, we construct an equation with X as the unknown variable, and where NPV 0 is set equal to zero. NPV 0 = 0 = 10.000.000 2.000.000 1,15 1 (1.333.333,33 PVIFA 15%;4 1,15 1 )+ 150 X PVIFA 15% ;3 + ( 150 X PVIFA 15%; 2 1,15 3 ) 0 = 10.000.000 2.000.000 0,8696 1.333.333,33 2,4826 + 150 X 2,2832 150 X 1,0689 0 = 15.049.333.33 + 342,48 x -160,34 x 182,15 X = 15.049.333.33 X = 82.620,55 This means that a drop in price to 600 per unit will lead to a sale of 82.621 units each and every year. Again, it will be best to stop the production and marketing after the third year. Doing so gives us the following result: NPV 0 = 0 = 10.000.000 2.000.000 1,15 1 (1.333.333,33 PVIFA 15%;2 ) 1,15 1 ) +150*X*PVIFA 15%;3 0 = 10.000.000 2.000.000 0,8696 1.333.333,33 1,4137 + 150 X 2,2832 342,48 X = 13.624.133 X = 39.781 units sold every year 31
Exercise 13 Question a Initial investment: X Depreciation: 25% straight-line method Scrap value: Accounted value Increased Con.Mar.: 500.000 per quarter Taxes: 40% at the end of the following year Interest rate of calculation: 25% nominal per year before taxes Interest, quarterly calculated, before taxes: 25%/4 = 6,25% Interest, quarterly calculated, after taxes: 0,0625 (1-0,40) = 3,75%. 500.000 X 0,25 0 1 2 3 4 5 6 7 8 9 10 11 12 16 X Annual taxes = (500.000 4 X 0,25) 0,40 = 800.000 0,1 X NPV 0 of taxes = (800.000 0,1 X) 1,0375 8 + (800.000 0,1 X) 1,0375 12 + (800.000 0,1 X) 1,0375 16 = 1.554.130 0,194266 X NPV 0 of scrap = X 0,25 1,0375 12 = 0,160725 X NPV 0 of Con.Mar. = 500.000 PVIFA 3,75% ; 12 = 4.761.345 NPV 0 total = investment + NPV 0 of Con.Mar. + NPV 0 of scrap NPV 0 of taxes = X +4.761.345 + 0,160725 X 1.554.130 + 0,194266 X = 3.207.215 0,645009 X The average quarterly profit = (3.207.215 0,645009 X) PVIFA 3,75%; 12 1 = 336.796 0,0677337 X 32
Question b The maximum price for the production facility is attained, when the average quarterly earning capacity is 0. 0 = 336.796 0,0677337 X X = 336. 796 0, 0677337 = 4.972.355 Exercise 14 In order to determine if it will be profitable to expand the production facility, we must first examine, what the optimal production level is with the current production facility Given the assumptions that The sales of the three products are not interrelated That the stated sales numbers represent the maximum quantity, which may be sold of each product. Ranking of the products in accordance with their contribution margin per machine minute: Single Multiple Future Price per unit 200,00 300,00 170,00 Variable cost 170,00 240,00 110,00 Con.Mar. per unit 30,00 60,00 60,00 Time consumption 6 min. 30 min. 15 min. Con.Mar. per minute 5,00 2,00 4,00 As the table shows, the product Single has the highest contribution margin per minute, followed by Future, and lastly Multiple. If we fill the production capacity in accordance with this ranking, the optimal production on the existing facility will look like this: Quantity Time per unit Total time consumption Single 40.000 units 6 min. 4.000 hours Future 60.000 units 15 min. 15.000 hours Multiple 4.240 units 30 min. 2.120 hours Hence, there is a market for additionally selling 25.760 units of the product Multiple. 33
An additional injection moulding machine has a capacity of 5.280 hours per year, which corresponds to a production of 10.560 units of the product Multiple. Therefore the company has 3 possibilities: 1. Do not expand capacity 2. Expand capacity with two facilities 3. Expand capacity with three facilities Each expansion of production and sales with one more unit of Multiple will increase the annual contribution margin with 60,00. But, doing so will also entail an increase in invested capital, in the form of increased raw material stocks and debtor accounts. On average, the raw materials are in stock for two months at a time, which corresponds to 1/6 of a year. As half of the raw materials are bought on two months credit, this half does not require further investment of capital, whereas the other half, which is paid in cash, requires an increase in invested capital of 240/2/6 = 20,00. On top of this, the invested capital in the debtors of the firm must be added, and as the company offers a 30-day credit account, this amounts to a total of 240/12 = 20,00. Here, only the costs of the raw materials related to the additional capital investment in the debtor accounts, have been included. We now have the necessary information for constructing an investment calculus for an additional facility, which is to be utilised at its maximum capacity: Machine investment: 1.750.000 Increased capital expenditure the first year: 10.560 (20 + 20) = 422.400 Annual increase in contribution margin: 10.560 60 = 633.600 633.000 422.400 1.750.000 0 1 2 3 4 422.400 34
NPV 0 = NPV 0 = 161.029,71 1.750.000 422.400 1,1 1 + 633.000 PVIFA 10; 4 + 422.400 1,1 4 The corresponding annuity value is calculated as 161.029,71 PVIFA 10; 4-1 = 13.405 The internal rate of return is found based on the following equation: 1.750.000 = 422.400 (1+R) 1 + 633.000 PVIFA R;4 + 422.400 (1+R) 4 IRR = 13,4% per year All three methods show that investing in an additional injection moulding machine, which is able to be utilised 100% for the production of the product Multiple, is profitable. In other words, the production facility must be expanded by at least two new facilities. Whether or not the facility should be expanded by 3 facilities is dependent upon the fact, if the remaining sales possibilities of 4.640 units for the product Multiple is enough, for the third facility to become profitable. Machine investment: 1.750.000 Increased capital investment the first year: 4.640 (20 + 20) = 185.600 Annual increase in contribution margin the first year: 4.640 60 = 278.400 NPV 0 = 1.750.000 185.600 1,1 1 + 278.400 PVIFA 10;4 + 185.600 1,1 4 NPV 0 = 909.469 As a result, the third facility is not profitable. Exercise 15 Interest rate of calculation: Initial investment: 12% per year 800,000, of which 400,000 is paid upon delivery, and, following this, 200,000 is paid each of the two following years. Lifetime: 5 years Scrap value: 0, in the fifth year Retail price per unit: 150 35
VC per unit: 100 Con.Mar. per unit: 50 Sales per year: Year 1: 5,000 units Years 2-5: 7,000 units 250,000 350,000 0 1 2 3 4 5 200,000 Question a 400,000 The criterion for initiating the proposed project is that the net present value must be greater than zero. Year Initial inv Revenue VC Con.Mar. Net Cash npv 0 NPV 0 0 400,000 400,000 400,000 434,376 1 200,000 750,000 500,000 250,000 50,000 44,643 2 200,000 1,050,000 700,000 350,000 150,000 119,579 3 1,050,000 700,000 350,000 350,000 249,123 4 1,050,000 700,000 350,000 350,000 222,431 5 1,050,000 700,000 350,000 350,000 198,599 As NPV 0 is greater than zero, the manager should recommend that the production be commenced. Question b In the following it is assumed in diametrical opposition to generally accepted theory that the sales follow the listed values from question a, completely independent of the pricing of the products. In order to find that retail price, which on the margin exactly makes the project profitable, we set NPV 0 = 0, and afterwards the equation is solved with respect to the unknown variable, P. NPV 0 = 0 = 400,000 200,000 1.12 1 200,000 1.12 2 + (P 100) 5,000 1.12 1 + 36
(P 100) 7,000 PVIFA 12%; 4 1.12 1 = 400,000 178,571.43 159,438.78 + (P 100) 4,464.29 + (P 100) 18,983.43 = 738,010.21 + 4,464.29 P 446,428.57 + 18,983.43 P 1,898,343.34 3,082,782.12 = 23,447.72 P P = 131.47 If the retail price is set at 131.47, the proposed project will attain an exact net present value of 0. Exercise 16 Retail price: 8,100 VC, wages: 2,500 VC, materials: 3,500 Credit account to customers: 2 months Sales forecast: Quarter Q1 Q2 Q3 Q4-8 Units 7 27 37 40 Question a Cash budget In the cash budget it is assumed that there are no stocks with materials. In other words, the quantity sold is equal to the purchased quantity in each of the quarters. Furthermore, it is assumed that the sales in each quarter is equally distributed between the months, as it is not possible to calculate the customer credits, unless this assumption is made. Moreover, it is also assumed, that all outstanding claims at the end of the 8. quarter, are paid in the ninth quarter. Lastly, sales in the 9. quarter and further on, are ignored. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Sales in units 7 27 37 40 40 40 40 40 Price per unit 8.100 8.100 8.100 8.100 8.100 8.100 8.100 8.100 Materials 3.500 3.500 3.500 3.500 3.500 3.500 3.500 3.500 37
Wages 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 Revenue 56.700 218.700 299.700 324.000 324.000 324.000 324.000 324.000 TVC 42.000 162.000 222.000 240.000 240.000 240.000 240.000 240.000 Con.Mar. 14.700 56.700 77.700 84.000 84.000 84.000 84.000 84.000 Deb. End 1) 37.800 145.800 199.800 216.000 216.000 216.000 216.000 216.000 Deb. Begin 0 37.800 145.800 199.800 216.000 216.000 216.000 216.000 debtors 37.800 108.000 54.000 16.200 0 0 0 0 Con.Mar. 14.700 56.700 77.700 84.000 84.000 84.000 84.000 84.000 debtors 37.800 108.000 54.000 16.200 0 0 0 0 Funds available 23.100 51.300 23.700 67.800 84.000 84.000 84.000 84.000 1) The revenue in Q1 is 56.700, which corresponds to a turnover of 18.900 per month. Debtors end is 2 18.900 = 37.800, as the company offers two months of credit to its customers. Question b Interest rate of calculation: 12% per year Per quarter: 3% Investment: 100,000 Sunk costs: 250,000 84,000 216,000 23,700 67,800 0 1 2 3 4 5 6 7 8 9 100,000 23,100 51,300 NPV 0 = 100,000 23,100 1.03 1 51,300 1.03 2 + 23,700 1.03 3 + 67,800 1.03 4 + 84,000 PVIFA 3%; 4 1.03 4 + 216,000 1.03 9 NPV 0 = 100,000 22,427 48,355 + 21,689 + 60,239 + 277,418 + 165,546 NPV 0 = 354,110 The project is profitable, as it generates a positive net present value. The project will, when considering the already invested 250.000, yield a profit of 104.110 in the first two years. 38
Exercise 17 Question a The first thing to do is to calculate the optimal lifespan: Period Scrap value Value loss 0 1,000,000 Lost interest Operational costs Maintenance costs Total MC NPV Acc. NPV Annuity 1 0 1,000,000 80,000 230,000 90,000 1,400,000 1,296,296 1,296,296 1,400,000 2 230,000 126,000 356,000 305,213 1,601,509 898,077 3 230,000 176,400 406,400 322,613 1,924,122 746,624 4 230,000 246,960 476,960 350,580 2,274,702 686,780 5 230,000 345,744 575,744 391,842 2,666,544 667,853 6 230,000 484,042 714,042 449,967 3,116,511 674,149 7 230,000 500,000 730,000 425,948 3,542,459 680,409 The optimal lifespan of the equipment is 5 years. The NPV of marginal revenue (MR) minus NPV of annual costs is to be equal to a net profit of 200,000 annually over the 5 years. NPV of net profit = NPV of MR NPV of costs: 5 (1 8%) 1 200,000 5 (1 8%) 8% = 2,000 Price 5 (1 8%) 1 5 2,666,544 (1 8%) 8% Price = 200,000 3.99271 2,666,544 433.93 2,000 3.99271 Question b One year less: Net profit per year 4 (1 8%) 1 4 (1 8%) 8% = 2,000 433.93 4 (1 8%) 1 4 (1 8%) 8% 2,274,702 2,000 433.93 3.312127 2,274,702 Netprofitp eryear 181,080 3.312127 One year more: 6 (1 8%) -1 Net profit per year 6 (1 8%) 8% 6 (1 8%) -1 = 2,000 433.93 6 (1 8%) 8% 3,116,511 2,000 433.93 4.62288 3,116. 511 Netprofit pryear 193,710 4,62288 39
Exercise 18 Interest rate: 11% per year Machine A Investment: 6 million Lifetime: 7 years Scrap value: 300,000 Savings: 2 million per year Operating costs: 200,000 the first year, increasing with 10,000 per year. 2,000,000 300,000 0 1 2 3 4 5 6 7 200,000 260,000 6,000,000 Year Costs Savings Scrap value Net cash npv 0 0 6,000,000-6,000,000-6,000,000 1 200,000 2,000,000 1,800,000 1,621,622 2 210,000 2,000,000 1,790,000 1,452,804 3 220,000 2,000,000 1,780,000 1,301,521 4 230,000 2,000,000 1,770,000 1,165,954 5 240,000 2,000,000 1,760,000 1,044,474 6 250,000 2,000,000 1,750,000 935,622 7 260,000 2,000,000 300,000 2,040,000 982,583 2,504,579 Machine B Investment: 5 million 40
Lifetime: 4 years Scrap value: 0 Savings: 2.5 million per year Operating costs: 300,000 the first year, increasing by 10% per year 2,500,000 0 1 2 3 4 300,000 399,300 5,000,000 Year Costs Savings Scrap value Net Cash NPV 0 0 5,000,000-5,000,000 1 300,000 2,500,000 2,200,000 1,981,982 2 330,000 2,500,000 2,170,000 1,761,221 3 363,000 2,500,000 2,137,000 1,562,556 4 399,300 2,500,000 2,100,700 1,383,796 1,689,555 As the two alternatives have different lifetimes, the profitability cannot be determined by comparing net present values. Therefore, we transform the net present values into the corresponding annual average net payments. Alternative A Alternative B A = 2,504,579 PVIFA 11%; 7-1 A = 531,510 A = 1,689,555 PVIFA 11%; 4-1 A = 544,588 Alternative B is the most profitable, as it generates the largest value of the annuity. 41
Exercise 19 Question a In order to find the highest price, which the company is willing to pay for the production facility, we must set NPV 0 = 0. This is where the entire investment breaks even, and, as such, this is where we have the maximum price for the equipment. 4,750,000 0 1 2 3 4 750,000 I 0.1 I First of all, we calculate the present value, PV 0,, of the contribution margin for the 4 years: Year Costs Additional Con.Mar. Net Cash PV 0 1 750,000 4,750,000 4,000,000 3,636,364 2 750,000 4,750,000 4,000,000 3,305,785 3 750,000 4,750,000 4,000,000 3,005,259 4 750,000 4,750,000 4,000,000 2,732,054 12,679,462 Secondly, we must make the following assumptions, before the NPV 0 may be set equal to zero: The additional contribution margin, which the facility generates, must be able to cover the recovery and the interest of the facility, which may be interpreted as the company requiring a return on investment of 10% per year on the facility. Moreover, there is a cost of 10% of the initial investment, in order to dismantle and dispose of the facility at the end of year 4. Given the above assumptions, the expression for NPV 0 will look like this: 0 = NPV 0 = I + 12,679,462 (I 0.10 ) (1.10) 4 42
I + 0.0683 I = 12,679,462 I = 11,868,821.5 This means that the company is willing to pay a maximum of 11,868,821.5 for the facility, when the additional marginal contribution must be able to cover the costs of dismantling and disposing of the facility. Question b Principal: Term: 8 million 4 years Annuity loan: Principal 8,000,000 - Capital loss (8%) 640,000 - Service dee (0.1%) 8,000 - Document fee (0.5%) 40,000 - Stamp duty (1.5%) 120,000 Net cash 7,192,000 Annual nominal interest rate: 7 % Semi-annual interest rate: 3.5% Term: 8 periods Semi-annual commission: 40,000 (0.5% of the principal) Annuity = 8,000,000 PVIFA 3,5%; 8-1 + 40,000 Annuity = 8,000,000 0.145476 + 40,000 Annuity = 1,203,808 semi-annually 7,192,000 0 1 2 3 4 5 6 7 8 Time 1,203,808 Semi-annual rate of interest: 43
7,192,000 = 1,203,808 PVIFA R; 8 PVIFA R; 8 = 5.97437 R = 6.9868 % semi-annually, which is an approximate rate, found by looking in a table. The precise value may be found by using the solver-function in Excel, or a financial calculator. Effective interest rate per year: 1.069868 2 1 = 0.1446 = 14.46% Fixed loan Principal 8,000,000 Capital Loss (4%) 320,000 Service fee (0,1%) 8,000 Document fee (0,4%) 32,000 Stamp duty (1,5%) 120,000 Net cash 7,520,000 Annual interest rate: 12 % Quarterly interest rate: 3% Term: 16 periods Interest is paid quarterly. 7,520,000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 8,000,000 Semi-annual rate of interest: 7,520,000 = (0.03 8,000,000) PVIFA R; 16 + 8,000,000 (1 + R) -16 R = 3.496 % per quarter (solver) Effective interest per year: 1.03496 4 1 = 0.14734 = 14.734 % 44
As we can see, the effective interest rate in the annuity loan is a little bit smaller than that of the bullet loan. Therefore, in accordance with general financial theory, we should prefer this financing option. Question c Due to cash preferences it is imaginable that we should choose the bullet loan anyways as we do not pay out that much on an annual basis, as we would with an annuity loan. The argument for this is that the new project may have larger requirements to the liquidity in the beginning of the project, and, therefore, the liquidity may be upheld the first difficult years of the project. However, we must still repay the 8,000,000, when the 4 years have passed, and, as such, we must, of course, pay attention to possible fluctuations in the revenue from the facility. What this means is that we must have the liquidity available in the fourth year, when the principal of the loan is to be redeemed. Exercise 20 Question a 9,25 Interest per quarter is calculated as = 2.3125 % per quarter. 4 Nominal yearly interest = (1+2.3125 %) 4 1 = 9.5758 % Question b Interest per quarter 2.3125 % of 50,000 = 1,156.25 Total yearly interest payment = 1,156.25 4 = 4,625. 45
Question c 50,000 0 1 2 3 4 750 1,156.25 50,000 So: 49,250 = 1,156.25 (1 R) (1 R) N N 1 R + 50,000 (1+R) 4 Effective interest per quarter: 2.713277 % Effective interest per year = ((1 + 0.02713277) 4 1) 100 = 11.3 % p.a. 46
Exercise 21 Question a Interest rate of calculation: Investment: 10 % per year 2.5 million 3 alternative ways of investment: Option 1: 1. Cash purchase 2. Supplier financed 3. Raising a serial loan 0 1 2 3 4 5 6 2,500,000 Down payment: I 0 = 2,500,000 This amount is already given to us at its present value, wherefore we do not need to calculate the present value for this outflow. Option 2: 0 1 2 3 4 5 6 Time 1,000,000 900,000 Payment at signing: Repayments: 1 million 900,000 in year 1 and 2, respectively I 0 = 1,000,000 + (900,000 1.1-1 ) + (900,000 1.1-2 ) I 0 = 2,561,983 This means that this financing form entails an expense calculated here and now of 2,561,983. 47
Option 3: Quotation rate: 98 Term: 5 years with annual settling periods Interest rate: 6 % per year The principal is calculated by dividing (net proceeds + other costs) with 0.98 as is the case with VAT-calculations Principal 2,560,000 Capital loss 2% 51,200 Other costs 8,800 Net cash 2,500,000 Now we are able to construct a table for the cash flow covering the repayment, interest and instalments on the loan: Year Repayment Interest Installment Remaining debt 1 512,000 153,600 665,600 2,048,000 2 512,000 122,880 634,880 1,536,000 3 512,000 92,160 604,160 1,024,000 4 512,000 61,440 573,440 512,000 5 512,000 30,720 542,720 0 Now the payments in each of the years, respectively, must be discounted with the subjective interest rate of the company: I 0 = 665,600 1.10 1 + 634,880 1.10 2 + 604,160 1.10 3 + 573,440 1.10 4 + 542,720 1.10 5 = 605,090.9 + 524,694.2 + 453,914.4 + 391,667.2 + 336,986.4 = 2,312,353 As option 3 has the lowest initial investment, the company should choose to raise a serial loan. It is possible to argument that, although there is a capital loss of 2%, the company achieves a kind of interest mark-up with this loan, as the company s interest rate is 10%. 48
Exercise 22 Loan amount: 100,000 Term: 5 years with quarterly equally sized installments Number of periods: 20 settlement periods Interest rate: 2% per quarter As we are dealing with equally sized installments each period, the exercise is solved as an annuity loan. Question a Quarterly installment Annuity = 100,000 PVIFA 1 2%; 20 Annuity = 100,000 0.06116 Annuity = 6,116 per quarter 100,000 0 1 2 19 20 6,116 Question b In order to find the quarterly effective interest rate, we must construct the balance equation. Doing so entails discounting the quarterly installments back to period 0, and comparing the discounted values with the net proceeds. As there are no additional costs in connection apart from nominal interest, the net proceeds equal the principal. Therefore, the nominal interest rate per quarter is equal to the effective interest rate per quarter. Effective interest rate per year: 1.02 4 = 1.0824 = 8.24% Question c The scenario now changes, as we now have a capital loss due to the quotation rate of 95. Principal 100,000 Capital loss (5%) 5,000 49
Net cash 95,000 The quarterly payment is still the same, at the principal does not change. 95,000 0 1 2 19 20 6,116 Time Effective interest rate per quarter: 95,000 = 6,116 PVIFA R; 20 PVIFA R; 20 = 15.5229 R = 0.02538 per quarter Effective interest rate per year: 1.02538 4 = 1.1055 = 10.54% Question d Quotation rate: 95 Term: 10 years with equal quarterly installments Number of periods: 40 periods Annuity per quarter Annuity = 100,000 PVIFA 2%; 40 1 Annuity = 100,000 0.03656 Annuity = 3,656 per quarter 95,000 0 1 2 39 40 3,656 Effective interest rate per quarter: 95,000 = 3,656 PVIFA R; 40 PVIFA R; 40 = 25.9847 R = 0.02296 per quarter 50
Effective interest rate per year: 1.02296 4 = 1.0951 = 9.51% 51
Exercise 23 Question a Cash discount when payments take place within 5 days: 1.5% Credit opportunity: current month + 20 days Average credit time: 30 days ((15 + 20) 5) Assume an invoice value of 100 with a cash discount equal to 1.5: 98.50 0 1 2 3 4 5 6 33 34 35 Time 100 R = 1, 5 100 1, 5 R = 0.01523 per 30 days There are 12 credit periods per year Effective interest rate per year: 1.01523 12 1 = 0.1989 = 19.89% Exercise 24 Question a Principal: US$ 200,000 Interest: 8.5 % p.a. Time horison 5 years 1st year interest accru to principal Monthly payments year 2 5 US$ 5,348.68 1st year s interest: 8.5 % of 200,000 = US$ 17,000 Outstanding end of year 1 = US$ 200,000 + US$ 17,000 = US$ 217,000 Outstanding debt is repaid over following 48 months with monthly payment of US$ 5,348.68. 52
8, 5 An annual interest of 8,5 % with monthly payments gives an interest per month of 12 = 0.7083 % per month. Monthly payment: 217,000 48 (1 0,70833%) 0,70833% = US$ 5,348.68 48 (1 0,70833%) 1 Question b 196,000 13 US$ 5,348.68 48 Balance: (1 R) 196,000 = (5,348.68 (1 R) 48 48 1 ) (1 + R) 12 R = 0.75735 % R Effective interest p.a. = ((1 + 0.75735) 12 1) 100 = 9.47 % p.a. Exercise 25 Question a Bullet loan Quotation rate: 95.95 Serial loan: Quotation rate: 99.5 Annuity loan: Principal 1,000,000 Capital loss x%? Quotation rate 100 x As the nominal interest rate per quarter is solely used when determining the amount of the payment, and the effective interest rate is used in the balance equation, we can now calculate the following: 53
Annuity = 1,000,000 PVIFA 1,875%; 20 1 = 60,421.48 Balance equation: Question b Bullet loan 1,000,000 x = 60,421.48 PVIFA 2%; 20 1,000,000 x = 60,421.48 16.351433 Quotation price = 98.7978 Principal 1,000,000 Capital loss (4.05%) 40,500 Net cash 959,500 The semi-annual interest rate is a simple interest rate, as the annual interest rate is nominal; hence: 7%/2 =3.5% Normally, interest on a bullet is paid at the end of each period. Therefore, the cash flow looks like this: 959,500 0 1 2 7 8 9 10 35,000 1,000,000 959,500 = 35,000 PVIFA R; 10 + 1,000,000 (1 + R) 10 R = 3.9993% semi-annually 8.15860 % per year 54
If we accrue interest to the principal, we get: 959,500 0 1 2 9 10 Time 1,410,599 959,500 = 1,000,000 (1 + 0.035) 10 (1+R) 10 959,500 = 1,410,598.76 (1 + R) 10 (1 + R) 10 = 0.6802 R = 3.9288% semi-annually 8.0119% per year Serial loan Principal 1,000,000 Capital loss (0.5%) 5,000 Net cash 995,000 995,000 0 1 2 3 4 5 Balance equation: 995.000 5 t 1 PMT (1 R) t t 995,000 =280,000 (1 + R) 1 + 264,000 (1 + R) 2 + 248,000 (1 + R) 3 + 232,000 (1 + R) 4 + 216,000 (1 + R) 5 The payment fluctuates from year to year, because we are dealing with a serial loan. Therefore, we must find the effective interest rate either by using a financial calculator, or by using the solver function in Excel. 55
Annual real rate of return: 8.1994% Annuity loan Quarterly real rate of return: 2% Annual real rate of return: 1.02 4 = 1.0824 Annual real rate of return: 8.24% Question c The company ought to prioritize the loans in accordance with the following ranking: Bullet loan with interest accrued to the principal Bullet loan with continuously paid interest Serial loan Annuity loan Question d Now accrued interest payments and debt discounts of the loans are tax-deductable, and the company is able to make use of this in the periods in question. Bullet loan: Interest: 3.5% semi-annual Principal: 1,000,000 Periods: 10 After taxes 973,270 0 1 2 8 9 10 After taxes 23,100 973,270 = 23,100 PVIFA R; 10 + 1,000,000 (1 + R at ) 10 R at = 2.6173% semi-annually 5.303% annually 1,000,000 Even though the interest is accrued, and is not paid on a continuous basis, the company can deducti tax-wise. The first time the company accrues an interest expense on 1 million. The next time interest on 1,035,000 (1 million plus interest from the first period) is accrued. Here the interest 56
accrued is 0.035 1,035,000 = 36,225. The tax-effect of the interest-deduction is therefore 0.34 36,225, which may be perceived as an inflow of cash. This generates the following cash flow: After taxes 973,270 11,900 12,316.5 15,670 0 1 2 7 8 9 10 1,394,380 Now we are able to construct the following balance equation, in order to find the effective after-tax interest rate: 973,270 = 11,900 (1 + R at ) 1 12,316.5 (1 + R at ) 2.. Serial loan: + 1,394,380 (1 + R at ) 10 R at = 2.7532% semi-annually 5.2126% per year Now we construct a table for the serial loan, which shows the annual after-tax installments in the respective years. Year Interest Repayment Payments Tax After-tax Payment 0 995.000 1.700 996.700 Remaining debt 1 80.000 200.000 280.000 27.200 252.800 800.000 2 64.000 200.000 264.000 21.760 242.240 600.000 3 48.000 200.000 248.000 16.320 231.680 400.000 4 32.000 200.000 232.000 10.880 221.120 200.000 5 16.000 200.000 216.000 5.440 210.560 0 In accordance with the table above, the balance sheet assessment is as follows: 996,700 = 252,800 (1 + R at ) 1 + 242,240 (1 + R at ) 2 + 231,680 (1 + R at ) 3 + 221,120 (1 + R at ) 4 + 210,560 (1 + R at ) 5 The effective interest rate after taxes = 5.4041% per year 57
Annuity loan Calculating the real rate of return after taxes for an annuity loan is very complicated. In principle, we must construct a table, as we did with the serial loan, but given the fact that the annuity loan has quarterly installments, the term horizon is 20 periods. Hence, without the use of Excel, one would not get a plausible result. If we use Excel, the real rate of return after taxes is calculated to be 5.4041% per year. Question e After the changes in the tax conditions, the annuity loan becomes a little bit better than the serial loan. However, there is only a very little difference, and it will not require many changes, before the ranking of the alternatives will change. In any case, the bullet loan is the cheapest, and it offers a cash advantage. Exercise 26 Alternative 1 Principal: 6,000,000 Quotation price: 96 Interest: 6 % per year, quarterly applied with a rate of 1.5% per quarter Service Fee 0.1 % Document fee: 0.4 % Stamp duty: 1.5 % Year 0 Down payment 6,000,000 0.96 = 5,760,000 Fee loan application 6,000,000 0.001 = 6,000 Commitment commission 6,000,000 0.004 = 24,000 Stamp duty 6,000,000 0.015 = 90,000 Net proceeds 5,640,000 The exercise may be solved by constructing the balance equation with the quarterly interest rate as the unknown variable, and from there on using the goal-seek /solver function in Excel. Manual calculation: Quarterly interest payment: 6,000,000 0.015 = 90,000 Balance equation: 58
5,640,000 = 90,000 PVIFA R; 20 + 6,000,000 (1+R) 20 By trial and error, we find R to be = 1.862074 % per quarter Excel: In Excel we construct the cash flow with net proceeds, interest payments and redemption of the loan. All amounts are discounted back to time 0 with the unknown quarterly interest rate. The quarterly interest rate is changed, until the total discounted value equals zero. Quarterly interest rate = 1.862074 % Annual interest rate = 7.658932 % Cash flow Q Net Cash Interest Repayment Payment Present value 0 5.640.000 5.640.000 5.640.000 1 90.000-90.000-88.354 2 90.000-90.000-86.739 3 90.000-90.000-85.154 4 90.000-90.000-83.597 5 90.000-90.000-82.069 6 90.000-90.000-80.568 7 90.000-90.000-79.096 8 90.000-90.000-77.650 9 90.000-90.000-76.230 10 90.000-90.000-74.837 11 90.000-90.000-73.469 12 90.000-90.000-72.126 13 90.000-90.000-70.807 14 90.000-90.000-69.513 15 90.000-90.000-68.242 16 90.000-90.000-66.995 17 90.000-90.000-65.770 18 90.000-90.000-64.568 19 90.000-90.000-63.387 20 90.000 6.000.000-6.090.000-4.210.823 Sum of present values: -0,0006 59
Alternative 2: Alternative 2 may also be solved using the goal-seek function in Excel, or by construction of the balance sheet assessment. Principal 1,500,000 SGD Exchange rate 3.86 Increase in exchange rate 1.5% per year Interest 3% per year prepaid Stamp duty & fee application 1% Underwriting commission 2% per year prepaid Repayment 500,000 SGD after 2 years 500,000 SGD after 4 years 500,000 SGD after 5 years Outflow Service Fee Net cash 1,500,000 SGD 15,000 SGD 1,485,000 SGD Year 0 1 2 3 4 5 Repayment 500,000 500,000 500,000 Remaining debt 1,500,000 1,500,000 1.000,000 1,000,000 500,000 0 Underwriting com. 30,000 30,000 20,000 20,000 10,000 0 Interest 45,000 45,000 30,000 30,000 15,000 0 Net payment SGD 1,410,000 75,000 550,000 50,000 525,000 500,000 SGD exch.rate 3.86 3.9179 3.976669 4.036319 4.096863 4.158316 Net payment 5,442,600 293,843 2,187,168 201,816 2,150,853 2,079,158 Present value 5,442,600 274,168 1,904,090 163,932 1,630,127 1,470,283 NPV = 0 Effective interest rate: 7.175946% per year Balance equation: 5,442,600 = 293,843 (1+R) 1 + 2,187,168 (1+) 2 + 201,816 (1+R) 3 + 2,150,853 (1+R) 4 + 2,079,158 (1+R) 5 R = 7.175946% per year Alternative 2 has the lowest effective interest rate. Therefore, it should be chosen. 60
Exercise 27 Question a Offer 1: Offer 2: 100,000 in cash Annuity loan with the following terms: Principal = 110,000 R = 5% n = 5 Annuity = PVIFA 5%; 5 1 110,000 = 0.230974 110,000 = 25,407.23 Now this annual annuity must be discounted using the company s subjective interest rate: I 0 = 25,407.23 PVIFA 7%; 5 I 0 = 104,174.66 Offer 3: Lease contract, with annual lease payment of 26,000 I 0 = PVIFA 7%; 5 26,000 = 106,605 From an initial investment point of view, the company should choose the alternative, which generates the lowest I 0. Hence, the company should choose offer 1. Question b When a company decides to lease its equipment, it is often times under the conditions that the lessor pays for the maintenance of the equipment. In this case, there is not that big a difference between the leasing price and the purchase, it may very quickly become profitable to lease the machine instead of buying it, as the company will then save money on maintenance of the equipment. Furthermore, as there is no scrap value tied to the machine, i.e. there is no possibility of getting additional cash inflow in year 5, this also promotes the choice of leasing instead of buying. It might even be the case that there may be additional expenses in connection with dismantling and disposing of the machine, which also promotes leasing instead of buying. 61
Exercise 28 Question a Alternative A, Bullet loan Net proceeds 12,500,000 / 1.25 = 10,000,000 Annual interest 12,500,000 0,04 = 500,000 Cash flow: 0 1 2 3 Loan in USD 12,500,000-12,500,000 Interest in USD -500,000-500,000-500,000-500,000 Cash-flow in USD 12,000,000-500.000-500.000-12,000,000 Exchange rate 1.25 1.23 1.20 1.18 Cash-flow in 9,600,000-406,504-416,667-10,169,492 Balance equation: NPV 0 = 9,600,000 406,504 (1+R) 1 416,667 (1+R) 2 10,169,492 (1+R) 3 NPV 0 = 0 IRR = 4.8% p.a. 62
Alternative B, Annuity loan: Calculation of net proceeds: Principal 10,000,000 Issuing loss (bond price 98) 200,000 Handling fee 100,000 Document fee 50,000 Stamp duty 150,000 Net proceeds 9,500,000 Quarterly annuity : 1 10,000,000 PVIFA = 902,580 1,25;12 Balance assessment: 9,500,000 = 902,580 PVIFA?; 12 PVIFA?; 12 = 10.53 2.25% per quarter PVIFA 2,25%; 12 = 10.41 2.00% per quarter PVIFA 2%; 12 = 10.58 10.58 10.53 2.0 0,25 2.07 % per quarter 10.58 10.41 Annual real rate of interest: (1 + 0.0207) 4 1 = 0.0854 8.5 % per year 63
. Question b Even though the borrower gets a lower real rate of interest, when choosing the loan in foreign currency, he must remain critical towards the risk, which is directly related to the loss due to the changes in the exchange rate. Therefore, the interest savings achieved when choosing the loan in USD, must be higher than the loss due to the exchange rate changes. Therefore, there is not a great difference between the two real rates of interest for the two alternatives, and it will not take a lot of changes on the foreign exchange market, before alternative 2 loses its profitability. The two alternative financing forms also have a remarkable difference in the cash flow, when it comes to the repayment of the loans. Exercise 29 Question a Increase in exchange rate: 1% per year Period 1 2 3 4 Exchange rate 1.25 1.26 1.27 1.28 Principal 3,000,000 Repayment, end (1/3 of principal) -1,000,000-1,000,000-1,000,000 Borrowing costs (1%) -30,000 Interest (5% of remaining debt, prepaid) -150,000-100,000-50,000 0 Net Payment in USD 2,820,000-1,100,000-1,050,000-1,000,000 Net Payment in 2,256,000-873,016-826,772-781,250 64
Balance assessment: 2.256,000 = 873,016 (1+R) 1 + 826,772 (1+R) 2 + 781,250 (1+R) 3 The real of interest may now be calculated to 5.05% per year Question b The following conditions should be considered before the loan is raised: Worst-case with regards to the exchange rate risk, if the price of the USD should become sky high, The possibilities of refinancing, when the three years have expired. The connections to the Danish day-to-day financing sources, if one makes use of new financing sources. Pros and cons of a fixed interest rate for 3 years. The use of the USD as a risk coverage of corresponding inflows in USD. A number of other factors could, of course, also be included in the considerations. Exercise 30 Question a Annuity loan The total payment is constant during the entire loan period. This provides for an easy overview of payment schedule The share of interest payments is high in the beginning of the loan. This may be an advantage, as interest payments are typically tax deductible. Contrarily, reduction of outstanding amount is relatively slow. Loan with grace period/installment-free period in the beginning This can be an advantage particularly during start-ups where liquidity is tight in the beginning. Reduction of outstanding amount is relatively faster with this type of loan. Serial loan The installment is constant whereas interest, and consequently total payments, will reduce over time. Assures a constant repayment schedule. Leasing Leasing is a flexible form of financing giving access to use of the asset against leasing payments. The asset kan be returned/replaced, which gives flexibility. Question b: 65
A total financing of 8 mill. is to be sourced. This is a start-up project in an industrial sector with a sizeable risk. The cost structure contains both fixed and variable costs in a relatively balanced ratio. It would be sensible to have a not-too-high financial risk, for instance an equity share of 40% (3.2 mill.) and the residual (4.8 mill.) financed with debt. Depending on the sponsor s ability to inject share capital, the remaining financing will have to be sourced through external investors and financing institutions. 66
Question c: Discounting assures that cash flows spread over time are made comparable. The discount rate is fixed by taking into account: - Time preference - Opportunity cost - Market rate - Risk - Tax - Inflation In real life we fix the discount rate by weighing together the price of equity (opportunity cost) and the price of debt (capital cost) by using the WACC. If we assume that financing is structured with 40% equity and 60% debt, and that the price of equity fixed by the investors is 20% p.a. and that debt financing has a capital cost of 10% p.a. then WACC will be: 40% x 20% p.a. + 60% x 10% p.a. = 14% p.a. Question d: Cash flows are estimated based on strategic analysis and on the basis of qualified estimates of investment costs (Capex), sales, costs and assumptions (not financing). 67
Question e: See Strategic Investment and Finance. Question f: Quantity Price Dir.wages El/water Contrib.Marg. Mainten. Sale/ marketing Salaries Terminal value Cash Flow -8.000.000 85.000 30 8 2 1.700.000 200.000 1.500.000 1.000.000-1.000.000 140.000 30 8 2 2.800.000 200.000 1.000.000 1.000.000 600.000 154.000 30 8 2 3.080.000 200.000 1.000.000 1.000.000 880.000 169.400 30 8 2 3.388.000 200.000 1.000.000 1.200.000 988.000 186.340 30 8 2 3.726.800 200.000 1.000.000 1.500.000 1.026.800 204.974 30 8 2 4.099.480 200.000 1.000.000 1.500.000 1.399.480 225.471 30 8 2 4.509.428 200.000 1.000.000 1.500.000 1.809.428 248.019 30 8 2 4.960.371 200.000 1.000.000 1.500.000 2.260.371 272.820 30 8 2 5.456.408 200.000 1.000.000 1.500.000 2.756.408 300.102 30 8 2 6.002.049 200.000 1.000.000 1.500.000 1.500.000 4.802.049 IRR: 8,81% Disc.rate: 14% NPV -2.407.242 With the assumptions made the investment gives a positive cash flow, but a financial evaluation with a discount rate of 14% p.a. shows that it is not a good investment as the NPV is negative. The IRR is 8.81% p.a. which is not sufficient compared to the required return on invested capital of 14% p.a. 68
Question g: The sales price should in the base case increase to 32.6 before NPV=0 is reached. Capital expenditures should reduce to 5.6 mill. to reach a NPV=0. These numbers provide input to considerations as to the project. Investors should evaluate the realism and consider ways to improve the project s viability, or whether they all-in-all consider the risk too large in comparison to expected return. Question h: The financial analysis above is based on the assumptions of the base case scenario, which is a static picture. We can envision that the project may develop in more directions; it may turn out very poorly if we are not successful at all, and there is the potential for establishing a substantial business if things develop positively and we understand how to adapt. Real options are a very useful tool for developing these scenarios and at the same time thing through how we as management can adapt and lead the company. Question i: - Base case as we know from above. NPV is -2.4 mill. 69
- A best case scenario where we are very successful; we have the option to increase sales price to 36, the yearly growth rate is 20% instead of 10%, we allocate more money for maintenance, sale, marketing, and salaries, and the terminal value of the company is assumed to reach 3 mill. after 10 years. NPV in this scenario = 10,258,724 Sales price 36 Sales growth Ingredients and El Unchanged 20% p.a. Maintenance 400,000 From year 6 Sale and marketing 1,500,000 In all years Terminal value 3,000,000 Discount rate 14% Year Volume Price Ingred. EL+water Cont.Marg. Mainten. Sale/mark. Salaries Scrap value Cash Flow 0-8.000,000 1 85,000 36 8.00 2.00 2,210,000 200,000 1,500,000 1,000,000-490,000 2 140,000 36 8.00 2.00 3,640,000 200,000 1,500,000 1,000,000 940,000 3 168,000 36 8.00 2.00 4,368,000 200,000 1,500,000 1,000,000 1,668,000 4 201,600 36 8.00 2.00 5,241,600 200,000 1,500,000 1,200,000 2,341,600 5 241,920 36 8.00 2.00 6,289,920 200,000 1,500,000 1,500,000 3,089,920 6 290,304 36 8.00 2.00 7,547,904 400,000 1,500,000 1,500,000 4,147,904 7 348,365 36 8.00 2.00 9,057,485 400,000 1,500,000 1,500,000 5,657,485 8 418,038 36 8.00 2.00 10,868,982 400,000 1,500,000 1,500,000 7,468,982 9 501,645 36 8.00 2.00 13,042,778 400,000 1,500,000 1,500,000 9,642,778 10 601,974 36 8.00 2.00 15,651,334 400,000 1,500,000 1,500,000 3,000,000 15,251,334 NPV 10,258,724 70
- A worst case scenario where we have to reduce price to 28, and sales volumes fall by 10% annually, and in parallel we have to reduce expenditures to maintenance, sale, marketing and salaries, and after 4 years we decide to close the company and sell buildings and equipment at an estimated 3 mill. NPV = 6,969,286. Sales price 28 Sales growth Ingred. and El Unchanged -10% p.a. Maintenance 100,000 From year 2 Sale and marketing 500,000 From year 2 Close in year 4 Terminal value 3,000,000 Discount rate 14% Year Volume Price Ingred. EL+water Contr. Marg. Mainten. Sale/mark. Salaries Scap value Cash Flow 0-8,000,000 1 85,000 28 8 2 1,530,000 200,000 1,500,000 1,000,000-1,700,000 2 76,500 28 8 2 1,377,000 100,000 500,000 700,000 77,000 3 68,850 28 8 2 1,239,300 100,000 500,000 500,000 139,300 4 61,965 28 8 2 1,115,370 100,000 500,000 300,000 3,000,000 3,215,370 NPV -6,969,286 Assuming a 30% probability for the best case scenario, 50% probability for the base case and 20% probability for the worst case, then the total expected NPV of the investment positive with 480,139. What we see is that by taking into account real options into the investment analysis we can add value. We now have a solid and valuable overview of the possibilities and risks. Whether we consider the potential return worthwhile given the risks is a concrete and subjective decision. 71
Exercise 31 Question a Set A 0-500,000 1 130,000 2 130,000 3 130,000 4 130,000 5 130,000 6 130,000 7 130,000 8 130,000 NPV 193,540.41 Set B 0-650,000 1 213,000 2 200,000 3 184,400 4 165,680 5 143,216 6 116,259 7 83,911 8 45,093 NPV 179,276.86 After having looked closer at option Set B you realize that realistically there is probably an optimal economic lifespan for this investment Question b 72
1 2 3 4 5 6 7 8 650000 568.750 487.500 406.250 325.000 243.750 162.500 81.250 - Value loss 81.250 81.250 81.250 81.250 81.250 81.250 81.250 81.250 Interest loss 65.000 56.875 48.750 40.625 32.500 24.375 16.250 8.125 Op costs 65.000 78.000 93.600 112.320 134.784 161.741 194.089 232.907 Opp. Rent cost 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 Total MC 233.250 238.125 245.600 256.195 270.534 289.366 313.589 344.282 MR 300000 300000 300000 300000 300000 300000 300000 300000 MR-MC 66.750 61.875 54.400 43.805 29.466 10.634-13.589-44.282 NPV 60.682 51.136 40.872 29.919 18.296 6.003-6.973-20.658 Acc NPV 60.682 111.818 152.690 182.609 200.905 206.908 199.935 179.277 After year 6 the marginal contribution becomes negative, i.e. optimal lifespan is 6 years. Question c NPV (year 6) = 206,908 Question d Annuity over 8 years Annuity over 6 years 33,604 47,507 6 years is the optimal lifespan and therefore the annuity over 6 years becomes higher than over 8 years. Question e We compare the expected NPV with and without an analysis. If the viewer base is large (2 million persons), then: NPV = 8 million 3 million = 5 million. If the viewer base is small (0.4 million persons), then: NPV = 2 million 3 million = 1 million Without testing the expected P is a weighted average of the possible P s, weighted by the given probabilities (50%). Expected NPV (no test) = (0.5 5 million) + [0.5 ( 1 million)] = 2 million If the analysis is undertaken and the market is found to be too small, the project will be abandoned implying a loss of NPV = -0.1 million. If the analysis proves that there is a large market, the project is pursued creating a NPV = 5 million - 0.1 million. Consequently: Expected NPV (with market analysis) = 0.5 (5 million - 0.1 million) + 0.5 (-0.1) = 2.4 million 73
Therefore, it pays to undertake the market analysis! Exercise 32 Question a The business risk is not of a fixed, given size. The following elements should be part of the evaluation: - Demand uncertainty (product life cycle): this is a new market with already proof tested products that are to some extent necessary for buyers the uncertainty can be assessed as medium. 74
- Price volatility of input factors (supplier strength): there is no dependency on few, key input factors/suppliers this uncertainty can be assessed as relatively low - Growth drivers: particularly environmental and technological drivers, though still associated with uncertainty as transition into new techniques and technologies is typically slow within agriculture. Medium risk. - Sale price volatility: pricing is assessed to be relatively stable, and therefore relatively low uncertainty - Competitive positioning: BioProduction s competitive position as a small niche player is assessed to be neither strong nor weak - Opportunities and threats: opportunities are particularly related to the expected growth in the market, the threats could be competitior s patents - Management capacity: depends heavily on possible joint venture partner s capacity, and the risk on this parameter can be assessed as high or possibly high-medium. - Cost structure: assessed to be medium risk as there is a certain need for capital investments, although not very large investments. Contribution margin is relatively high which helps in reducing the risk. On top of this comes business risk associated with business operations in the Middle East; the political coutry risk, legal risks, cultural differences, infrastructure etc. A number of factors point in the direction of medium risk. However, establishing on an entirely new marked points in the direction of high risk, particularly with regards to management capacity, which can be a determinant factor. Depending on how investor assess the risk of establishing on a new marked, then business risk could be assumed to be mediumhigh. It is important to note that the assessment is subjective. Question b: The discount rate is set according to the Weighted Average Cost of Capital (WACC) methods which combines the price of equity and the price of debt with the weight of equity with the weight of debt. The text does not specify weighs or price, so this has to be developed. We could for instance assume that 50 % of total financing is debt and that the price of debt is 10 % p.a. And let us consequently assume that the residual 50 % of financing is then equity and that the price of equity is set at 20% p.a. based on the above analysis of business risk. 75
WACC will then be 15 % p.a. If for instance the price of equity is set at 30 % p.a. then WACC will be 20 % p.a. The key point here is that some assumptions will have to be made as to cost of capital because the final balance between debt and equity is typically not known at the time of the investment analysis. Question c: Given the assumptions on sales prices, sales volume, costs and investment amounts, then the following cash flow can be estimated: Year 0 1 2 3 4 5 6 7 8 9 10 11 Turnover 945,000 1.890,000 2,835,000 2,835,000 2,835,000 2,835,000 2,835,000 2,835,000 2,835,000 2,835,000 Variable costs 117,600 235,200 352,800 352,800 352,800 352,800 352,800 352,800 352,800 352,800 Water, electr,, materials 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 Salaries 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 Repair and maintenance 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 Sale and marketing 28,350 56,700 85,050 85,050 85,050 85,050 85,050 85,050 85,050 85,050 Administration 18,900 37,800 56,700 56,700 56,700 56,700 56,700 56,700 56,700 56,700 Net cash flow -6,000,000 0 540,150 1,320,300 2,100,450 2,100,450 2,100,450 2,100,450 2,100,450 2,100,450 2,100,450 2,100,450 15% 1,473,900 20% -196,619 IRR 19.32% It is assumed that the 6 mill. capital investment takes place at time 0 and that cash flow during year 1 is 0. Other assumptions are of course possible. Calculation of the NPV to 1,473,900 is based on a discount rate of 15 % p.a. If a 20% p.a. is applied, the NPV is -196,619. The IRR is 19.3% p.a. Based on the assumptions made, the investment is attractive as long as the discount rate, WACC, is below 19.3% p.a. It is of course ultimately up to the individual investor to judge if the business case is strong and solid enough for it to be worthwhile doing the investment. 76
Question d: Taking point of departure in a discount rate of 15% p.a. then the following critical values can be calculated (goal seek/solver): Investment: Max. 7,473,900 Price: Min. 127.50 Costs: Can increase with totally 337,729. The key point is not the exact numbers but the assessment of how risky the project is, when for instance the budgeted sales price of 150 can fall to minimum 127.50 in all years before the NPV turns negative. Again, the assessment is individual and ultimately based upon the investors understanding and perception of the business case. Question e Answers will vary and the key point here is to maintain logic, structure, creativity and realism in the scenarios being developed. Below is a suggestion for a future scenario: - Initial capital expenditure 6 mill. as above - A best case scenario could be to expand production capacity for instance a doubling of production capacity and a calculation could yield a NPV at time 0 of 3 mill. (based on a number of assumptions regarding sales, costs, investment, financing) - A base case scenario could for instance be an alliance with a strategic partner or expansion with a new product. If a strategic alliance is pursued the NPV could for instance be calculated at 2.2 mill. Expansion with a new product could lead to a 2 mill. NPV. - A worst case scenario might lead to a sell off where NPV = -3 mill. or restructuring to a simpler production lay out where NVP = 0.1 mill. Investment: 6 mill. Best Base Worst Expand - -NPV 0 = 3 mill. Add new product -- NPV 0 = 2 mill. Strategic partner -- NPV 0 = 2.2 mill. Restructure to simple production -- NPV 0 = 0,1 m mio. kr. Sell off -- NPV 0 = -3 mill. All scenarios have is this way been thought through strategically and NPVs have been calculated. 77
If for simplicity we assume equal probability for each of the options, then the total NPV will be: NPV 0 = 20% 3 mill. + 20% 2 mill. + 20% 2.2 mill. + 20% 0.1 mill. + 20% -3 mill. = 0.86 mill. So all in all an attractive investment strategy, however with a lower NPV than calculated in question c given the assumptions we have made here. The focal point of the scenario analysis is again not the exact number for the NPV but the strategic analysis of possible scenarios and their probabilities and the associated options. Question f Year Scrap value 0 1,200,000 Based on data given in the text the following calculations can be made as to marginal contribution margin in each period and thereby optimal life span. Value loss Interest loss Operational costs MC MR MR-MC NPV Accumulated of MR-MC NPV 1 1,100,000 100,000 180,000 100,000 380,000 600,000 220,000 191,304 191,304 2 1,000,000 100,000 165,000 105,000 370,000 540,000 170,000 128,544 319,849 3 900,000 100,000 150,000 110,250 360,250 486,000 125,750 82,683 402,531 4 800,000 100,000 135,000 115,763 350,763 437,400 86,638 49,535 452,067 5 700,000 100,000 120,000 121,551 341,551 393,660 52,109 25,908 477,974 6 600,000 100,000 105,000 127,628 332,628 354,294 21,666 9,367 487,341 7 500,000 100,000 90,000 134,010 324,010 318,865-5,145-1,934 485,407 8 400,000 100,000 75,000 140,710 315,710 286,978-28,732-9,393 476,014 9 300,000 100,000 60,000 147,746 307,746 258,280-49,465-14,061 461,953 10 200,000 100,000 45,000 155,133 300,133 232,452-67,681-16,730 445,224 11 100,000 100,000 30,000 162,889 292,889 209,207-83,682-17,987 427,237 12-100,000 15,000 171,034 286,034 188,286-97,748-18,270 408,967 The numbers show that the optimal life span is somewhere between year 6 and year 7. It the investor continues beyond this then marginal costs will exceed marginal revenue which is unfeasible.. The maximal NPV is achieved in year 6 at 487,341. 78
Question g: Yes, the investment is feasible. In effect, the investment is feasible over all time spans but with an optimal life span around 6-7 years. Question h: Cash flow for the loan and effective interest rate: Period Outstanding debt Instalment Interest Admin. fee Cash flow, USD Cash flow, DKK 0 800.000 784.000 4.312.000 1 720.000 80.000 32.000-112.000-616.000 2 640.000 80.000 28.800 2.000-110.800-609.400 3 560.000 80.000 25.600-105.600-580.800 4 480.000 80.000 22.400 2.000-104.400-574.200 5 400.000 80.000 19.200-99.200-545.600 6 320.000 80.000 16.000 2.000-98.000-539.000 7 240.000 80.000 12.800-92.800-510.400 8 160.000 80.000 9.600 2.000-91.600-503.800 9 80.000 80.000 6.400-86.400-475.200 10-80.000 3.200 2.000-85.200-468.600 Effective interest, semi annual 4.64% Annual effective interest: 9.50% The effective interest rate in a semi-annual basis is calculated to 4.64%. Converted into annual effective interest rate this is 9.50% p.a. Question i: If this loan is taken in year 0 to finance the 6 mill investment, there will not be enough liquidity generated from the project itself to honour payments on the loan during the first 2 years. If we convert cash flow on the loan to DKK at an exchange rate of DKK/USD 5,5 (last column above) then we can see that in year 2 there will not be enough cash flow from the project (540,150) in order to honour the two payments during the two semi-annual periods. Only in year 3 will there be enough liquidity (1,320,300) to honour payments on the loan. It is therefore necessary either to re-negotiate the repayment profile of the loan or secure some other additional financing (bridge financing) which can make sure the above loan can be repaid according to schedule. 79
In any case, the above loan is not sufficient to secure closure of the total financing needs of teh 6 mill. investment. It will have to be studied closely how total financing looks when a comprehensive financial package is on the table. The above analysis shows that the existing proposal for a loan does not fit the liquidity needs optimally. Question j: The required security needs to be considered, and if these requirements might hinder access to additional financing. The project s liquidity is of course uncertain. It is therefore important to consider the conditions and consequences of running into difficulties of servicing the debt. Flexibility as to repayment in full, restructuring etc of the loan should be considered. The loan is in USD. The text does not indicate in which currency business is transacted, but if revenues and costs are transacted in e.g. and/or Egyptian Pounds, then there are potentially significant currency/economic risks. Exercise 33 Question a A = 2,500,000 A = 292,073.86 (1 8%) (1 8%) 15 15 8% 1 Question b: 80
Question c: The effective interest rate is: 8.5 % p.a. Question d: Outstanding amount on the annuity loan after 3 years is: 2,201,091. To be able to repay this loan in full, a new loan will have to be taken with a principal of 2,269,166. Annual payments on a 7 years serial loan will be: 2,269,166 / 7 = 324,167 Question e: 81
Exercise 34 Question a Question b A new issue of shares can be: 82
- Bonus shares to existing shareholders this doesn t add new fresh money - New stock with privileged subscription rights for existing shareholders - New stock without privileged subscription rights Can be either through public stock market or private equity in unlisted shares. Question c - Weather - Environment - Bad infrastructure - Political instability - Consumers standards/certifications etc - Regulations - Skills - Change in taste/income Question d Map the investment strategy over time through a decision tree. The approach will vary according to which options the students apply. Question e The risk element is particularly important here. A discount rate for equity of around 20%-30% p.a. could indicatively be considered reasonable. Question f Question g - Repayment schedule - Currency - Financial gearing 83
- Grace periods Exercise 35 Question a - Political instability - Terrorism - Growing industry globally - Demand uncertainty - Change in taste holiday preferences - Consumer income - Competition - Degree of Operational Leverage high for industry (high FC) High/medium-high risk seems a fair evaluation. Question b An approach could be to start with WACC. 50% is debt, 50% equity. The cost of debt can be taken from the effective interest rate in question 22 = 16.23% p.a. The cost of equity could then be set at for instance 30% taking into account the risk of this investment. All in all the discount rate would then be: WACC = 50% * 16.23% p.a. + 50% * 30% p.a. = 23% p.a. Question c Net cash flow is 15,550,557 in year 10 and expected to remain constant in following years, i.e. growth rate = 0%. Terminal value in year 10 can then be calculated as: 15,550,557/0.23 = 67,611,117. Question d If using a discount rate of 23% p.a. and following cash flows, then NPV will be negative as IRR is only 19% p.a. 84
Question e Four fundamental methods can be applied: critical values, scenario analysis, real options analysis and stochastic simulation. The results will differ according to the student s decision regarding discount rate, and scenarios applied. Examples of critical values for the investment analysis as defined in this draft solutions would be that in order for the investment to become viable: - Occupancy rates in all years should be increase with 19.5% - Total costs in all years should be decreased with 27.1% - Investment cost reduced by 8.98 million These are just examples. The key here is to get an overview of the uncertainty and reflect upon this. Question f Outstanding principal Instalment Interest Admin.fee Front fee CF 0 3200000-48000 3152000 1 3200000 112000 5500-117500 2 3200000 112000 5500-117500 3 3200000 112000 5500-117500 4 3200000 112000 5500-117500 5 3200000 112000 5500-117500 6 3200000 112000 5500-117500 7 3200000 112000 5500-117500 8 3200000 112000 5500-117500 9 3100000 100000 112000 5500-217500 10 3000000 100000 108500 5500-214000 11 2900000 100000 105000 5500-210500 12 2800000 100000 101500 5500-207000 13 2700000 100000 98000 5500-203500 14 2600000 100000 94500 5500-200000 15 2500000 100000 91000 5500-196500 16 2400000 100000 87500 5500-193000 17 2300000 100000 84000 5500-189500 18 2200000 100000 80500 5500-186000 19 2100000 100000 77000 5500-182500 20 2000000 100000 73500 5500-179000 21 1900000 100000 70000 5500-175500 22 1800000 100000 66500 5500-172000 85
23 1700000 100000 63000 5500-168500 24 1600000 100000 59500 5500-165000 25 1500000 100000 56000 5500-161500 26 1400000 100000 52500 5500-158000 27 1300000 100000 49000 5500-154500 28 1200000 100000 45500 5500-151000 29 1100000 100000 42000 5500-147500 30 1000000 100000 38500 5500-144000 31 900000 100000 35000 5500-140500 32 800000 100000 31500 5500-137000 33 700000 100000 28000 5500-133500 34 600000 100000 24500 5500-130000 35 500000 100000 21000 5500-126500 36 400000 100000 17500 5500-123000 37 300000 100000 14000 5500-119500 38 200000 100000 10500 5500-116000 39 100000 100000 7000 5500-112500 40 0 100000 3500 5500-109000 3,83% Question g Annual eff. Rate: 16,23% Question h - Liquidity/cash flow possibly longer grace period - Security - Flexibility - Currency - Stability of bank Exercise 36 Question a The ultimate discount rate will vary, the key point is reflections on WACC, capital structure chosen, and reasoned thoughts on opportunity cost and risk. Reflections also on business risk and financial risk. The discount rate should of course be reasonable. 86
In the following a discount rate of 15% p.a. has been chosen. Question b Scenario analysis and real options may vary significantly the important thing is that arguments are provided and the structure is logical. Question c E.g. critical value for annual maintenance costs are 654,061 as calculated above with goal seek. Price can be reduced by 20% in all years as illustrated above. 87
Question d Question e 17.76% p.a. Question f - Liquidity/cash flow possibly longer grace period - Security - Flexibility - Currency - Stability of bank - Reporting requirements Exercise 37 Question a The business risk checklist may be used commenting on: Demand uncertainty (product life cycle) Price fluctuations and availability of input factors (supplier strength) Growth drivers (expansion opportunities/-ability) Sale price volatility (price setting/market forms) Competitive rivalry/market desirability (Porters 5 forces) Company s position among competitors Opportunities and threaths (PESTEL) Degree of Operational Leverage 88
Question b Scenario planning/real options/phasing of investments. Concrete measures as for example: Secure pricing and reduce competition by having strong products and patents. Keep strong relations with buyers. Use BCG matrix to balance portfolio. Joint entures. Strategic alliance. Question c WACC should be used here arguing for debt/equity ratio and cost of capital. Discount rates in the range of 10-25 % p.a. should be expected. Question d Will vary -the key here is structure, clarity and clear assumption plus a final assessment of investment viability. Below a guideline proposal: This is a very simple version where the first 7 years are lined up according to data provided. What happens thereafter is of course based on everyone s sound judgement: how long will the patent (and the price) maintain, what happens thereafter, etc. Question e Again, answers will vary; it should be expected that a number of critical values are calculated and commented, and also that best case, base case and worst case scenarios are calculated and described. Question f The 34 million should now be regarded as a sunk cost and the investment regarded as a phased investment where the results (scenarios) of Phase II can lead to a number of options it is important to realize that there are now options and these should be described. 89
Below a simple calculation where a good result leads to continued investments as originally planned, a mediocre result makes PharmaGroup do a joint venture with another company thereby reducing their part of NPV to 50% and finally a worst case leading to a stop of further investments. Question g+h 90
Question i - Currency - Could it be a more suitable currency? - Security what security is required by lender? - Horizontal Balance Structure (cash flow stream) does the cash flow stream of investment and financing fit? - Vertical balance structure Debt to equity - Bank - what kind of bank is it? Is it solid? 91