San Antonio is considering an option for providing water, including desalting. O

Question

San Antonio is considering an option for providing water, including desalting. One brackish aquifer is expected to yield desalted water that will generate revenue of $4.1 million per year for the first 4 years, after which less production will decrease revenue each year by 5%. If the aquifer will be totally depleted in 10 years, what is the present worth of the desalting option at an interest rate of 12% per year, with quarterly compounding the first 4 years and annually compounding there after?

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Explanation / Answer

This soln has 3 parts :- 1. We find PV of annuity of $4.1M for 4 Yrs @12% with Qtryly compounding ie. PMT=$4.1M/4 = $1.025M, i=12%/4 = 3%, period n = 4Yr*4 = 16 2. We find PV at end of 4 yrs of a Decreasing anunity of $4.1M, decreas factor g = -5%, i=12%, period n = 6 yrs 3. We find PV of decreasing annuity at no 2 above at Y0, dicouting by i=12% Part 1: Here we need to calculate PV of Annuity. PMT = $1.025M,n=16, INT i=3% Present value of annuity PVA = PMT(PVIFAi,n). So PVA = PMT*(1/i - 1/(i(1+i)^n)) So we get PVA = PMT*(1/3% -1/(3%*(1+3%)^16)) ie PVA = PMT*(33.3333-20.7723) = $1.025M*12.561 = $12.875M So PVA1 = $12.875M Part 2: Here we have to find the PV of a growing Anuity = PVA Decrease is g=-5%. i=12%, PMT=A =$4.1M, n=6 PVA = [A/(i-g)]*[1-{(1+g)/(1+i)}^n] ie PVA = [PMT/(12%+5%)]*[(1-{(1-5%)/(1+12%)}^6] SOllving, weget PVA = [PMT/17%]*0.6276 = 3.6918*PMT = 3.6918*$4.1M = $15.14M So PVA2 = $15.14M Part 3: Present value PV = FV/(1+r)^n. We have FV = $15.14M, r=12%, n=4 Yrs. Here we are just findg the PV & hence we are taking r=12% & not qtryly compunding int rate So we have PV = $15.14M/(1+12%)^4 = $9.62M From Part 1 & Part 3, we have PV = $12.875M + $9.62M =$22.495M So Present worth of desilting option is $22.495M

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