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, re-invest 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 #14
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 #15
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
When to harvest trees (How long to wait) Accept a price now or wait for higher price later
Determine value of an income stream from an investment: money each year
Planting raspberry bushes to sell berries Compare to other investments Constant payment each year for fixed number years
Convert returns varying over time to an Annuity
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
Put less value on future income and more value on current income. Less ―patient‖—want the money now
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
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 #16
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?
How do you choose a discount rate?
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?
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 #17
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?
Net Present Value of an Income Stream
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 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 Yi /(1 r )
i 1
t
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 multi-year 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
Year Net Income Present Value* formula
1
2
–$1,200
$2,000
–$1,090.91
$1,652.89
–1200/(1+0.1)1
2000/(1+0.1)2
3
4
$2,000
$1,800
$1,502.63
$1,229.42
2000/(1+0.1)3
1800/(1+0.1)4
NPV
*Assuming 10% discount rate
$3,294.04
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 #18
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
Interest/discount rate
PV = FV/(1 + r)t
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 stream
NPV Yi /(1 r )
i 1 t 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
2-4D 1940’s, atrazine 1950’s, alachlor 1960’s, glyphosate 1970’s
Use in 2005 in top 19 corn states (93% acres)
97% of acres treated with a herbicide Atrazine 66% Glyphosate 31% S-Metolachlor/Acetochlor 23%
Source: http://usda.mannlib.cornell.edu/usda/nass/AgriChemUsFC//2000s/2006/AgriChemUsFC-05-17-2006.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)
corn
C1/5 - Photosystem II inhibitors (atrazine)
1979
Smooth Pigweed (Amaranthus hybridus) Kochia (Kochia scoparia)
Velvetleaf (Abutilon theophrasti) Giant Foxtail (Setaria faberi) Large Crabgrass (Digitaria sanguinalis) Kochia (Kochia scoparia)
corn corn, railways
corn carrot, corn, onion, sweet corn carrot, onion roadsides
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)
1985 1987
1990 1991 1992 1995
Common Waterhemp (Amaranthus rudis) Giant Foxtail (Setaria faberi)
Green Foxtail (Setaria viridis) Eastern Black Nightshade (Solanum ptycanthum)
soybean corn, soybean
corn, soybean soybean
B/2 - ALS inhibitors (imazethapyr) B/2 - ALS inhibitors (imazethapyr, nicosulfuron)
B/2 - ALS inhibitors (imazamox) B/2 - ALS inhibitors (imazamox, imazethapyr)
1999 1999
1999 1999
Weed Resistance Management Practices
(http://www.weedresistancemanagement.com)
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
Start spending time/money now on BMP’s to prevent/slow development of resistance
Reactive weed resistance management
Save money now by not using BMPs and pay higher control costs in future when resistance develops
Weed Resistance Management Graphics
Net Returns ($/ac) Reactive Resistance Management Proactive Resistance Management R not resistant Cost of BMP Use (CBMP) R with BMP Cost of Resistance (Cr)
R resistant
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?
What are they likely to do? What do you recommend to farmers?
Which strategy should they use?
How do you decide? Compare NPV’s (or annuity equivalents) of the proactive and reactive strategies
Economic Model
Net Present Value of the 2 income streams
NPVProactive Rwith BMP /(1 r )t
t 1 T
NPVReactive
Tresistance
t 1
Rnot resistant /(1 r)t
Annuity: K = (1/r){1 – [1/(1 + r)T]} AProactive = NPVProactive/K (= Rwith BMP) AReactive = NPVReactive/K
t Tresistance
T
Rresistant /(1 r)t
Economic Analysis
Economic values depend on 6 parameters Returns: Rnot resistant, Rresistant, Rwith BMP Time: Tresistance and final time period T Discount rate r Actually only 5 parameters (Costs not Returns) CostBMP = Rnot resistant – Rwith BMP 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
NPVProactive RBMPe rt dt RBMP / r
0
Tresist
NPVReactive
0
RNoResist e rt dt
RResist e rt dt ( RNoResist / r ) (CResist / r )e rTresist
Tresist
Equate NPV’s and rearrange
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 = CResistancee–rTresist 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 non-integer 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