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"5to replace 7 of his watering troughs approximately 108years from now at a total cost of about $3,400. At a 7%9discount rate, what would be the present value (cost) of the troughs?.Present value of the troughs at 7% & 10 years:8Assuming he plans to farm for the forseable future, then5the farmer is committed to replacement of the troughs9life of his fencing, and from this example, $1,728.39 set
9,000lbs.
5,000lbs.YearsLbs.U(7) Present Value of an Increasing Annuity (capitalization of an increasing annuity):ProducedP 1 2 3 4 5 6 7 8 9 107Thus the investment appears to be economically feasible8with C/S, but isn't without cost/share. However, a more7complete analysis would require consideration of annual&operation and maintenance (O&M) costs.(1+I)n+1  (1+I)  n(I)
(1+I)n(I)2
? ?&(2) Future Value of One (compounding):7 Recovery Factor, Monthly Payment, Partial Payment): $ $ $ $:Farmer Gene Crabcakes currently rents farm land at $40 per;acre. He's considering the purchase of an adjacent farm of?comparable value to his rental land. The land is for sale at a>period and 7% interest to capitalize the $40/acre rental rate.9price of $750/acre. From this limited information, is he0Capitalized or PV of $40/acre/year for 30 years:;better off renting or purchasing? Use a 30 year evaluationThe purchase price is $750/acre<is reasonably certain the land will continue to be available;to him and that rent will not increase substantially in the<future. The rental rate would have to be around $60/acre to<have the capitalized value of his rental payments be approx.9the same as the purchase price. A more complete analysis6would include consideration of annual ownership costs ;(taxes) as well as other monetary and nonmonetary factors.9(12.409 = the PV of an annuity of one for 30 years at 7%)?Therefore, Mr. Crabcakes is better off continuing to rent if he ? ?7(4) Present Value of an Annuity of One (capitalization)3(5) Future Value of an Annuity of One (amount of an annuity of one):
(1+i)n  1
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$ $ $ $4Colonel Harvey is thinking about installing 2 spring7developments and gravityfed troughs at an approx. cost7of $2,500 each. They will allow his cattle to properly9graze 32 acres that are vastly underuitilized now. They6even go ungrazed at all in some years due to the long9distance to existing water sources. The increased forage:utilization and production response are expected to result6in an added animal unit month (AUM) per acre valued at9improved forage utilization and increased forage produced4at the end of 10 years, using a 6% rate of interest?2$10/AUM. (a) What would be the ending value of the8(b) Would Col. Harvey be better off investing the $5,0003required for the water systems in a CD earning 5% a
available?6year over the same time period? (c) What if 75% C/S isper year,(a) 32 acres * 1 AUM/acre * $10/AUM =4(13.181 = FV of an annuity of one for 10 years @ 6%)'(1.791 = FV of one 10 years hence @ 6%)6Thus the Col. Would be better off investing in the CD.+$5,000 total cost * 25% to get the farmer'scost =/Thus w/ C/S the investment in watering pays off5handsomely, i.e., it is expected to yield $1,979 more7than investing the $1,250 cost in a 6% CD for 10 years.:(6) Amount of an Annuity for a Future Value (sinking fund) $
? ? ?? ? ?9Renowned GIS specialist and parttime farmer Fred, a.k.a.8"Sugar Bear" wants to set aside some money every year to<be able to build a new barn and pay cash for it 7 years from;now. A similar barn to what he wants to build was recently9built by a neighbor for $40,000. (a) Assuming 2% average;inflation over the next 7 years, what will the barn approx.8cost? (b) How much would "Sugar Bear" need to set aside9annually to end up with enough money to pay for the barn,8assuming annual setasides into a "sinking fund" earning&an expected 7% interest (junk bonds?)?+Expected cost of the barn 7 years from now:,(1.149 = FV of one factor for 2% and 7years))(0.1156 = Amt. of an annuity for a future!value at 7% interest for 7 years);(b) Annual sinking fund payment to meet future need::times the FV of an annuity of one factor for 7 years at 7%as shown below:7Note: cross check the aswer to (b) by multipling $5,309,(8.654 = FV of an annuity factor for 7 yearsat 7% interest)Forage$Expected cash on hand 4 years hence:'Expected cost of tractor 4 years hence:'Expected cost of tractor 7 years hence:$Expected cash on hand 7 years hence:6(a) What would be the average annual cost of a $45,0008expected useful life of 25 years at 8.5%? (b) What would5be the amortized cost of the remaining balance if 75%9cost/share is available for this conservation investment?+Amortized or average annual cost w/out C/S:*Amortized or average annual cost with C/S:* =====><(c) If this investment allows for more efficient recycling &7use of manure nutrient content on the farm estimated at5an added $1,250 per year, is the investment worth it?(c) Added nutrient cycling benefits:Added annual cost w/out C/S:Benefit/Cost ratio w/out C/S:Added annual cost with C/S:Benefit/Cost ratio with C/S:Bonus Question:R Calculate the average annual value of benefits and costs using 10yrs. and 6%.S(8) Present Value of a Decreasing Annuity (capitalization of a decreasing annuity):n(i)  1 + (a) Step 1:whereby we derive $10,560)(b) Step 2:r(annual income divided by the expected remaining useful life of the lake = $240/year * PV of a decreasing annuity j(PV of future income * the amortization factor for 6% and 10 years = $1,435/year in average annual income)mThus, Mr. Mabell could pay Ma Kettle, on average, $1,435 per year for 10 years to protect the lake. How muchwLeith Churchbell invested a number of years ago in the development of a small recreational lake in the lovely mountainsyof Virginia (not on a perennial stream of course). He stocked it with sport fish and has very carefully managed the lakeqfor years selling access to outdoor recreationchallenged fisherpersons from the Washington, D.C. area. For manymyears now he has netted on average a handsome $2,400 per year. However, a new neighbor movedin to the upperpwatershed with a boatload of illtempered, freerange, odoriferous goats who have created an erosion problem byrdenuding the entire countryside. Ma & Pa Kettle's overstocked goatrange and the resulting erosion has led to zkindly visit to Dinnerbell's farm and estimated that the lake will be full of sediment within 20 years and that it will bewlargely uninhabitable by sport fish within 10 years. Thus, Doorbell's net annual recreational proceeds will decline torzero within 10 years, or roughly $240 per year. Weighing his options, Churchbell considers use of the AgricultureuStewardship Act, but being no dumbell and in careful consideration of who he was dealing with, he figures negotiationwand persuasion would be preferable to the possibility of a sudden, tragic and mysterious fire sweeping across his farm.pSo Cowbell decides to figure out how much he could afford to pay Klim Cadiddlehopper to help fix the erosion andxsedimentation problem in order to protect his lake. (a) What is the PV of the decreasing stream (annuity) of recreationqincome at 6%? (b) Based only on the projected future income without treating the upper watershed, how much couldGMr. Campbell afford to pay the neighbor every year to protect the lake?psubstantial quantities of sediment being delivered to Mr. Cowbell's lake. A USGS sedimentation geologist paid a$Time Value of Money Example Problems'(1) Present Value of One (discounting):Mathematical FormulaGraphical Depiction?$Example ProblemSolution$3,400 at the end of 10 years.(1 + i)n
=/(0.50835 =< PV of 1 factor for 7% over 10 years)6Planning for tomorrow, Lebano Joe figures he will have5once every 10 years, or three times over the expected3Note: The PV of one factor is the reciprocal of the" compound interest factor.5aside today drawing 7% interest will equal the needed6Parttime farmer, Commissioner Patterson has some cash<($24,000) that he has saved to eventually replace one of his9tractors. He thinks his existing tractor will need to be9replaced about 4 years from now. He also believes he can:earn 8% per year after taxes through a real estate limited9partnership along interstate 81 in Rockingham County (not6counting kickbacks). The tractor he would like to buy:currently costs $33,000 and he expects similar tractors to:appreciate at an average of 3% per year. (a) Will he have9enough cash to buy the tractor 4 years from now? (b) Howabout 7 years from now?(a),(1.360 = FV of 1 factor for 8% over 4 years),(1.126 = FV of 1 factor for 3% over 4 years)Thus the answer to (a) is no.Thus the answer to (b) is yes.(b)0Time Value of Money Example Problems (continued)6(3) Annuity for a Present Value (Amortization, Capital7investment in dairy waste management facilities with an1(0.9771 = Amount of an annuity for a PV = Amort.)Present Value = 1/(1+0.06000)n
+Amortization = 0.06000/(1(1/(1+0.06000)n))(
)Future Value = (1+0.06000)n
mPresent Value of an Increasing Annuity = (((1+0.06000)n+1)(1+0.06000)(n*0.06000))/((1+0.06000)n*(0.06000)2)6
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lVPresent Value of a Decreasing Annuity = (((n*(0.06000)1)+1n/(1+0.06000)n))/(0.06000)2H
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1CAmount of an Annuity for a Future Value = 0.06000/((1+0.06000)n1)?
@:(1,000 lbs. * PV of an increasing annuity for 4 yrs. @ 6%)(a) step 1:(a) step 2:(a) step 3:(a) step 4:@ (conversion of 8,411 lbs. to tons, i.e., 8411/2000 = 4.21):(conversion of 4.21 tons to $'s, i.e., 4.21*$50/tn = $210)> (4,000 lbs. * PV of a constant annuity for 6 yrs. @ 6%)(a) step 5::(conver. of 19,669lbs. to tons, i.e.,19,669/2,000=9.83tns)(a) step 6:@ (conversion of 9.83 tons to $'s, i.e., 9.83*$50/tn = $492)(a) step 7:;(conver.of future $'s to PV, i.e.,$492*PV of 1 factor=$367)(a) step 8:@ (total increase in value, $210 + $367 = $578)/Curly Lewis decides to convert his existing one0rotational grazing system (fencing, gates, water/boundary, conventional grazing operation into a3collection & distribution facilities, establishmentof improved pasture and riparian zone manage/ment). Forage produced is expected to increase0every year for the next 4 years until the entire0expected to continue for years 5  10. (a) What'would be the PV of the increased forage.produced, using $50/ton, 22 total acres and a1discount rate of 6%? (b) Using a projected totalcost for system installation of $4,750, wouldthis investment be worthwhile? $ $ $ $ $ $ $ $ $$? = Unknown to be solved; $ = Known; $ $ 3system is installed at a rate of 1,000lbs./acre/yr.0from 5,000lbs. total dry matter produced/acre up.to 9,000lbs./acre. Maximum production is thenO(b) Yes, the PV of Benefits at $12,710 exceeds the cost of installation $4,750.
(a)step10:q($12,710 * amortization factor for 10yrs.@ 6% = $1,727 = AAB & $4,750 * amort. factor for 10yrs. @ 6% = $645 AAC)Amorofn/aValueItem:PresentFutureAmount tization:Val. of an Val. of aValue ofof an Amt.of an
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Periods:(Mathematical Formulas for each function:No. ofPeriodsTime%Compound Interest & Annuity Table for1the Kettle's would accept are separate questions.uprotection $1,435 per year for years would afford, how long the useful life of the lake would be extended and whether,Prepared by: David L. Faulkner, USDA/NRCS/VA" ``lo69 d ]
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