Comm 221 Extra NPV problems

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Apr 3, 2024

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1 Woodlot NPV problem considering sustainability A woodlot is for sale for a price of 15 million dollars. You estimate that next year, the lot will harvest 2 million board feet of good quality lumber. If you use optimized sustainable practices in managing the lot you believe that this amount can grow by 3% per year. Following a sustainable process requires a monthly investment of $85,000 in perpetuity. If instead, you follow an unsustainable process, you think the output will instead decline by 2% per year. If you do not follow a sustainable process, you can clear cut some sections and receive an immediate (time 0) cashflow of $200K for some lower value scrap lumber. You have forecasted the inflation adjusted value of the lumber at $1.25/board foot for the long foreseeable future. Similar forestry projects have been capitalized at a rate of 1% effective monthly. To simplify your analysis, you may assume that you receive all revenues in single lump sum at the end of each year. Should you buy the woodlot and if you do, should you use sustainable or non-sustainable practices? With sustainable practices: NPV = value of growing annuity of lumber – cost of wood lot – cost of sustainable practice Convert EMR to EAR to match cashflow, (1+EMR)^12-1 = EAR = 12.6825% D1 = first sale at end of year = 2 million feet * 1.25 a foot = $2.5M Value of annuity now, P0 = D1/(r-g) = 2.5/(0.126825-0.03) = $25.819M Cost of sustainable practice, perpetuity formula = monthly pmt/EMR = 85K/0.01 = $8.5M Total value = 25.819M-15M-8.5M = $2.319M > 0 , this is profitable, we could pursue this

2 Without sustainable practices: NPV = value of shrinking annuity of lumber + clear cut scrap lumber – cost of wood lot Value of shrinking annuity now, P0 = D1/(r-g) = 2.5/(0.126825-(-0.02)) = 2.5/0.146825 = 17.027M NPV = 17.027M + 0.2M – 15M = $2.227M > 0, this is profitable, we could pursue this So now we have a choice of do nothing (spend 0, get 0), do the sustainable for an estimated net benefit of 2.319M or do the unsustainable for a benefit of 2.227M Between these two choices, for the current situation, the sustainable project appears to be more profitable, we should do that.

3 NPV of a Delayed growing/shrinking/ending annuity project You are launching a new toy, the fidget-blinder, which should be a rage. You estimate that once you start production, in 3-months time, you will earn a revenue stream at the end of the 4 th month equal to $100K. From there, you expect sales to rise meteorically, at an effective rate of 6% per year for 2 full years. After that, once people realize that this toy is stupid and can leave them blind, sales are expected to plummet by 25% per month until the end of time. Projects like this are risky and so you estimate that the appropriate discount rate is 15% APR compounded monthly. Given that it will cost $750K for three consecutive months, at the START OF EACH month, to build the manufacturing facility, should you pursue this project? Approach: This problem requires many slow thought-out calculations to ensure everything is correct in terms of time. The rough sketch is: NPV = SUBTRACT annuity due for 3 payments to build facility ADD growing annuity from month 4 to eternity. SUBTRACT growing annuity paying in month 28 (2 years + 4 months, value in month 27) to eternity (cutting off the tail) ADD shrinking annuity paying from month 28 to eternity (2 years + 4 months, value in month 27)

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4 Cost of building facility (annuity due): Find the EMR. We have quoted rate of 15% APR co. monthly so just divided by 12, 15/12 = 1.25% EMR 750K now plus 2 payments like a standard annuity which is just PMT [(1-(1+r)^-n)/r] = 750K[(1-(1.0125)^-2 )/0.0125] = 750K*1.963115… = 1,472.33K or 1.47233M to build the facility in time 0 dollars (for the two standard payments). Add 750K for the annuity due for a total cost of 2.22233M If you rather, you might use a calculator key stroke as N=2, PMT = 750, FV = 0, I/Y = 1.25, <CPT> PV = -1472.33K …. same answer, also account for the 750 K for the immediate payment and get a total cost of 2.22233M First stage growth Now we need to know the value of the basic growing annuity from time 4 months (first payment) to eternity. For this, we need to turn the annual growth rate into an effective monthly growth rate. That is: (1+EAR)^1/12-1 = EMR = (1.06)^(1/12)-1= 0.4867551% or about 49 bps (basis points) D4 = 100K P3 = D4/(r-g) = 100K/(0.0125-0.004867) = 13.101M However, this needs to be discounted from month 3 to the present so P0 = P3/(1+EMR)^3 = 13.101M/(1.0125)^3 = 12.715M

5 Ending first stage growth (“cutting off the tail”) Unfortunately, this alone overstates the income because things will grow for 24 months (2 years) and then will start shrinking. We need to remove the value of the income flow from 24 months after the first sale (4+24 = 28) and substitute for a slower growth rate. The income flow 24 months after the first payment will be D28 = D4(1+g_monthlty)^24 = 100K(1.004867551)^24 = 112.354K Thus the value of this at P27 = D28/(r-g) = 112.354/(0.0125-0.004867) = 14.719576M but this is in time month 27 dollars, it needs to be brought to the present P0 = P27/(1+EMR)^27 = 14.719576M/(1+0.0125)^27 = 10.525177M We deduct this from the original value of the growth which would leave us the income flow from month 4 to month 28 Adding period of decline (Ah….I’m blind!) Next, we add back an alternative cash flow that is declining over time. The amount will be at the previous identified value, D28 = 112.354K, but instead of growing it will decline from that point to eternity. Thus P27 = D28/(r-g) = 112.354K/(0.0125-(-0.25)) = 428.0152K but this is in the future and needs to be discounted to the present P0=P27/(1+EMR)^27 = 428.0152K/(1+0.0125)^27 = 306.050K Altogether: NPV = -2.22233M + 12.715M-10.525177M+0.30605M = 273.543K > 0 This project is positive NPV but not by a wide margin. We’d probably go ahead and invest but it’s worth checking your analys is because if for example sales drop a little faster or don’t grow as quickly as expected then this project would probably lose val ue. Also, if the cost of financing were to climb, the value of this project would decline, and you’d probably reject it.