Time Value of Money

Time Value of Money - Time Value of Money AAE 320 Paul D....

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Unformatted text preview: Time Value of Money AAE 320 Paul D. Mitchell Goals Learn basic concepts how economists and financial professionals incorporate time into valuing assets and investments Application of these concepts to common (agricultural) decisions Future Value of Investment Suppose you invest $100 at an interest rate of 5% per year, how much would it be worth in 3 years? End Year 1: $100 x (1 + 0.05) = $105 End Year 2: $105 x (1 + 0.05) = $110.25 End Year 3: $110.25 x (1 + 0.05) = $115.7625 = $115.76 This assumes compounding: earn interest on the interest, or equivalently, reinvest earned interest into the principal Future Value of Investment General formula: FV = PV x (1 + r)t PV is the present value, r = interest rate, and t = time period End Year 1: $100 x (1 + 0.05) = $105 End Year 2: $105 x (1 + 0.05) = $110.25 End Year 2: $100 x (1+0.05) x (1+0.05) = $110.25: Start to see the general formula Future Value of Investment General Formula: FV = PV x (1 + r)t How much will $100 be worth in 4 years invested at an annual rate of 11.5%? FV = PV x (1 + r)t = 100 x (1 + 0.115)4 FV = 100 x 1.54561 = $154.56 Future Value Interest Factor: (1 + r)t Tables of these factors exist for different interest rates and time lengths Think Break #18 What's the Future Value Interest Factor for 9.7% for 5 years? How much will $85 be worth in 5 years invested at an annual rate of 9.7%? Present Value of Future Income Instead of calculating the future value of a present value investment, let's reverse it If I know the future value, how much do I have to invest today to have that value? I will have to pay $100 in 3 years. How much do I need to invest today at 8% to have $100 in 3 years? Present Value of Future Income Use the formula, but solve for PV as function of FV: FV = PV x (1 + r)t PV = FV/[(1 + r)t ] How much do I need to invest today at 8% to have $100 in 3 years? FV = 100, r = 0.08, t = 3 PV = FV/[(1 + r)t ] = 100/1.083 = $79.38 Discount Factor Use this formula to convert future income into its present value PV = FV/[(1 + r)t ] = FV x 1/[(1 + r)t ] 1/[(1 + r)t ] = Discount Factor, present value interest factor, present value factor Note, the Discount Rate is r Make tables of discount factors for different interest rates and time lengths Discount Factor If I earn $100 from a project in 3 years, what is this $100 worth to me today? You want to "discount" this $100 back to its present value: What is it worth today? Assume a discount rate of 5% (r = 0.05) PV = FV x 1/[(1 + r)t ] = 100 x 1/[1.053] PV = 100 x 0.8638376 = $86.38 Discount Factor is 0.8638376 What is a Present Value? When we say $100 in 3 years has a present value of $86.38, what's this mean? We are saying that $100 in 3 years is equivalent to $86.38 today, why? Because we could take $86.38 today, invest it at 5% and in 3 years have $100 Discount factors convert future money into its equivalent value today Different discount rates imply different discount factors and so different present values Think Break #19 Suppose you planted ginseng this year, to be harvested in 4 years for $10,000/ac. At a 7.5% discount rate, what is the present value of $10,000 in 4 years? Using Present Values and Discount Factors for Decision Making Evaluate options/opportunities to help choose Determine value of an income stream from an investment: money each year When to harvest trees (How long to wait) Accept a price now or wait for higher price later Convert returns varying over time to an Annuity Planting raspberry bushes to sell berries Compare to other investments Constant payment each year for fixed number years Comparing Options Suppose you could harvest timber from a lot you own today and earn $180/ac. You could wait 1 year and earn $200/ac, or wait 2 years and earn $225/ac. Which plan has the largest present value? Assume a 6% discount rate Option 1: PV = 180 x [1/(1.060)] = $180.00 Option 2: PV = 200 x [1/(1.061)] = $188.68 Option 3: PV = 225 x [1/(1.062)] = $200.25 Effect of Discount Rates Higher discount rate r, future "discounted" more r = 0.03: option 2 = 194.17, option 3 = 212.08 r = 0.06: option 2 = 188.68, option 3 = 200.25 r = 0.09: option 2 = 183.49, option 3 = 189.38 r = 0.12: option 2 = 178.57, option 3 = 179.37 With 3%, 6%, or 9% discount rate, take option 3 With 12% discount rate, take option 1 Put less value on future income and more value on current income. Less "patient"--want the money now Evaluate Opportunities You hear that your farm land is likely to be annexed by the city in the future to be developed into housing or businesses After some research, you find that the land will be worth $12,000/ac if bought by a developer after annexation in 4 years Someone offers $9,000 today to buy your land: Is this a good price? Discount Factors to Evaluate Opportunities Do you take $9,000 today or $12,000 in 4 years? Depends on your discount rate With a 5% discount rate PV = FV/(1 + r)t = 12,000/1.054 PV = $9,872.43: Reject the offer! With a 10% discount rate PV = FV/(1 + r)t = 12,000/1.14 PV = $8,196.16: Take the offer! Think Break #20 At a farm sale, you see a 57 year old John Deere tractor for sale for $6,000. You're an antique tractor aficionado know that in 3 years, when it's 60 years old, it will be worth $7,500. Assuming a 7% discount rate, should you buy it, if you want to make money? If I took $9000 today and invested it, what interest rate would I get? If I used the money to buy more farm land, what is my rate of return on assets? If I took the money from my equity, how much return on equity am I giving up? How much do you need/want cash now? How do you choose a discount rate? The more you need/want money now, the higher your discount rate Solving for Discount Rates Suppose you know the present and future values and want to know the discount rate Use the general formula, but solve for r FV = PV x (1 + r)t (FV/PV) = (1 + r)t (FV/PV)1/t = 1 + r r = (FV/PV)1/t 1 Use this formula to find the discount rate that turns the PV into the FV in t years Solving for Discount Rates Back to the land sale example: Do you take $9,000 today or $12,000 in 4 years? What discount rate r makes the options equal? $9,000 today versus $12,000 in 4 years r = (FV/PV)1/t 1 = (12,000/9,000)0.25 1 r = 0.0746 = 7.46% If you can use the cash to earn more than a 7.46% return, you are better off taking the $9,000 today vs. waiting for $12,000 in 4 years Think Break #21 Back to the antique tractor: suppose you bought it for $6,000. Your spouse is mad, saying it's a waste of money--you could have bought a mutual fund and made 7% annual return. You both agree that you can sell it for $7,500 in 3 years: but what is your rate of return? Suppose you have a project generating an income stream that varies over the years What is the value of this project? Take the income from each year and discount it back to its present value, then add them all up from each year This the project's Net Present Value (NPV) Today's value for the whole income stream Net Present Value of an Income Stream Net Present Value Formula Each year project generates income Yi, where Y is the income and i is the year, and the project lasts t years, then the NPV formula is NPV = t i =1 Yi /(1 + r ) i Net Present Value (NPV) What is a fair price for the right to harvest fruit from an orchard for 3 years if it will produce $3000/year of fruit each year? Assume a 6% discount rate Year 1: PV = 3,000/(1.06)1 = $2,830.19 Year 2: PV = 3,000/(1.06)2 = $2,669.99 Year 3: PV = 3,000/(1.06)3 = $2,518.86 NPV = 2,830.19 + 2,669.99 + 2,518.86 NPV = $8,019.04 Net Present Value (NPV) The income each year does not have to be constant, and can actually be negative (i.e., a cost) in some years Use NPV to compare the value of different projects or investments generating income Enterprise budgets for multiyear crops Plant raspberries to harvest for a few years Raspberry Example Assume 4 year cycle Plant year: cost of $1,200/ac First harvest year: net return of $2,000 Second harvest year: net return of $2,000 Third harvest year: net return of $1,800 Assume a discount rate of 10% Raspberry Example Yea Net Income r 1 $1,200 2 3 4 $2,000 $2,000 $1,800 NPV Present Value* $1,090.91 $1,652.89 $1,502.63 $1,229.42 $3,294.04 formula 1200/(1+0.1)1 2000/(1+0.1)2 2000/(1+0.1)3 1800/(1+0.1)4 *Assuming 10% discount rate Raspberry Example Interpretation: Before you plant, 4 years of raspberries has a NPV of $3,294.04 How does this compare to an alternative, with a constant return each year? Is planting sweet corn that will generate $1000/ac per year better? What constant payment over 4 years is equal in NPV to the variable returns to raspberries? Annuity: A constant payment (C) for a fixed number of years (t) Annuity For a project with varying cash flow over t years generating a net present value of NPV, what is the equivalent annuity? Annuity factor K = (1/r){1 [1/(1 + r)t]} Annuity payment C = NPV/K Can look up K in Annuity Tables Raspberry Annuity Raspberry Example: r = 10%, t = 4 K = (1/r){1 [1/(1 + r)t]} K = (1/0.1){1 [1/(1 + 0.1)4]} K = (1/0.1){ 1 [1/1.14]} K = (1/0.1){1 [1/1.4641]} K = (1/0.1){1 0.683031} = 0.316987/0.1 K = 3.16987 NPV = $3,294.04, C = $3,294.04/3.16987 C = $1,039.17 Raspberry: same as an annuity paying C = $1,039.17/year Think Break # 22 Suppose you calculate the NPV of planting an apple orchard over 15 years is $3,500 using a 5% discount rate. What is the value of the annuity factor K? K = (1/r){1 [1/(1 + r)t]} What is the annuity payment that will generate the same NPV as the apple orchard over 15 years? Summary: Concepts Learned Future value of an investment Present value of future money FV = PV x (1 + r)t PV = FV/(1 + r)t Interest/discount rate These are just the same equation rearranged in different ways r = (FV/PV)1/t 1 Summary: Concepts Learned Net Present Value of an income t stream i NPV = Y /(1 + r ) i =1 i Convert varying income stream into constant Annuity of C over t years K = (1/r){{1 [1/(1 + r)t]} C = NPV/K Extended Case Study Weed Resistance Management Herbicides generally became available for crop production in the late 1940's Use in 2005 in top 19 corn states (93% acres) 24D 1940's, atrazine 1950's, alachlor 1960's, glyphosate 1970's 97% of acres treated with a herbicide Atrazine 66% Glyphosate 31% SMetolachlor/Acetochlor 23% Source: http://usda.mannlib.cornell.edu/usda/nass/AgriChemUsFC//2000s/2006/AgriChemUsFC05172006.pdf Pest Resistance to Control With repeated use of a control method, weed populations can become resistant Has occurred in insects to insecticides and bacteria to antibiotics Process of natural selection (evolution) Growing problem worldwide and in US Number and distribution of resistant weed species globally HERBICIDE RESISTANT WEEDS IN WISCONSIN Weed Situation Herbicide Mode of Action Year Lambsquarters (Chenopodium album) Smooth Pigweed (Amaranthus hybridus) Kochia (Kochia scoparia) Velvetleaf (Abutilon theophrasti) Giant Foxtail (Setaria faberi) Large Crabgrass (Digitaria sanguinalis) Kochia (Kochia scoparia) Common Waterhemp (Amaranthus rudis) Giant Foxtail (Setaria faberi) Green Foxtail (Setaria viridis) Eastern Black Nightshade (Solanum ptycanthum) corn corn corn, railways corn C1/5 - Photosystem II inhibitors (atrazine) C1/5 - Photosystem II inhibitors (atrazine) C1/5 - Photosystem II inhibitors (atrazine) C1/5 - Photosystem II inhibitors (atrazine) A/1 - ACCase inhibitors (fluazifop-p-butyl, sethoxydim) A/1 - ACCase inhibitors (fluazifop-p-butyl, sethoxydim) B/2 - ALS inhibitors (sulfometuron-methyl) B/2 - ALS inhibitors (imazethapyr) B/2 - ALS inhibitors (imazethapyr, nicosulfuron) B/2 - ALS inhibitors (imazamox) B/2 - ALS inhibitors (imazamox, imazethapyr) 1979 1985 1987 1990 1991 1992 1995 1999 1999 1999 1999 carrot, corn, onion, sweet corn carrot, onion roadsides soybean corn, soybean corn, soybean soybean (http://www.weedresistancemanagement.com) Weed Resistance Management Practices Scout fields before and after herbicide application Start with a clean field, using either a burndown herbicide application or tillage Control weeds early when they are relatively small Incorporate other herbicides and cultural practices as part of Roundup Ready cropping systems where appropriate Use the right herbicide at the right rate and the right time Control weed escapes & prevent weeds from setting seeds Clean equipment before moving from field to field to minimize spread of weed seed Use new commercial seed free from weed seed Economics of Weed Resistance Management Weed BMP's slow development of weed resistance to control, but cost money Economic Problem: Do you start spending a little extra money now on weed BMP's so you can keep using an effective herbicide for many years, or do you save the money now and when resistance develops sometime in the future, start paying higher control costs? Economics of Weed Resistance Management Proactive weed resistance management Reactive weed resistance management Start spending time/money now on BMP's to prevent/slow development of resistance Save money now by not using BMPs and pay higher control costs in future when resistance develops Weed Resistance Management Graphics Net Returns ($/ac) R not resistant R with BMP R resistant Reactive Resistance Management Proactive Resistance Management Cost of BMP Use (CBMP) Cost of Resistance (Cr) T resistance Time (years) Intuition: Use BMPs if a) Cost of BMP Use is low and/or b) Cost of Resistance is high Economic Analysis Which strategy do farmer have an economic incentive to use? Which strategy should they use? What are they likely to do? How do you decide? Compare NPV's (or annuity equivalents) of the proactive and reactive strategies What do you recommend to farmers? Economic Model Net Present Value of the 2 income T streams = R NPVProactive /(1 + r )t with BMP NPVReactive = Tresistance t =1 t =1 Rnot resistant /(1 + r )t + t = Tresistance T Rresistant /(1 + r )t Annuity: K = (1/r){1 [1/(1 + r)T]} AProactive = NPVProactive/K (= Rwith BMP) AReactive = NPVReactive/K Economic Analysis Economic values depend on 6 parameters Returns: R not resistant, Rresistant, Rwith BMP Discount rate r Actually only 5 parameters (Costs not Returns) Cost BMP = Rnot resistant Rwith BMP Time: Tresistance and final time period T Costresistance = Rnot resistant Rresistant What the economic model can do Equate the two NPV and determine when it's best to switch from reactive to proactive resistance management Treat any 4 parameters as given and solve for the last parameter Solve for time to resistance (Tresistance) given CBMP, Cresistance, r and T Solve for discount rate r given T, Tresistance, CBMP, and Cresistance Three more possibilities could do Problem with discrete model Discrete time model only allows integer years, when NPV's may actually be equal somewhere in between integers Built spreadsheet to play with and to find the switching points: on class page Better method: convert to a continuous time model (more flexible) Mueller et al. (2005) Weed Technology Continuous Time Model Assume final time period T = infinity i NPVProactive = RBMP e - rt dt = RBMP / r Tresist 0 NPVReactive = RNoResist e - rt dt + 0 Equate NPV's and rearrange Tresist RResist e - rt dt = ( RNoResist / r ) - (CResist / r )e - rTresist i rTresist + ln(CBMP / Cresistance ) = 0 Continuous Time Model Solve this equation for any parameter as function of other 3 to find critical value of the parameter when it's best to switch Tresist = ln(CBMP/CResistance)/r r = ln(CBMP/CResistance)/Tresist CBMP = CResistanceerTresist CResistance = CBMPerTresist Additional tab in the discrete spreadsheet Examples: Discrete Time Rnoresist = 100, RBMP = 95, Rresist = 80, Tresist = 20, t = 30, and r = 10% Use spreadsheet: Reactive better by $25.89 in NPV, or $2.75/yr in annuity What r need so equal? r = 3.66% What RBMP need so equal? RBMP = $97.75 What Rresist need so equal? Rresist = $55.617 Can't get noninteger values for T or Tresist Examples: Continuous Time Rnoresist = 100, RBMP = 95, Rresist = 80, Tresist = 20, and r = 10% Use spreadsheet: Reactive better by $22.93 in NPV, or $2.29/yr in annuity What r need so equal? r = 6.93% What RBMP need so equal? RBMP = $97.29 What Rresist need so equal? Rresist = $63.05 What Tresist need so equal? Tresist = 13.86 yrs Summary of Weed Resistance Extended Case Study Used discrete time NPV analysis to examine the economics of weed resistance management Developed model to determine whether proactive or reactive weed resistance management most economical for farmers Weed Resistance Management Spreadsheet Weed Technology: Mueller et al. 2005 WI Crop Manager: Boerboom and Mitchell 2006 ...
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