Time Value of Money

1
Time Value of Money

• AAE 320
• Paul D. Mitchell

2
Goals

• Learn basic concepts how economists and financial
  professionals incorporate time into valuing
  assets and investments
• Application of these concepts to common
  (agricultural) decisions

3
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

4
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 (10.05) x (10.05) 110.25
  Start to see the general formula

5
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

6
Think Break 14

• Whats 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?

7
Present Value of Future Income

• Instead of calculating the future value of a
  present value investment, lets 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?

8
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

9
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

10
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

11
What is a Present Value?

• When we say 100 in 3 years has a present value
  of 86.38, whats 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

12
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?

13
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
• Convert returns varying over time to an Annuity
• Constant payment each year for fixed number years

14
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

15
Effect of Discount Rates

• Higher discount rate r, future discounted more
• Put less value on future income and more value on
  current income. Less patientwant 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

16
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?

17
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!

18
Think Break 16

• At a farm sale, you see a 57 year old John Deere
  tractor for sale for 6,000. Youre an antique
  tractor aficionado know that in 3 years, when
  its 60 years old, it will be worth 7,500.
  Assuming a 7 discount rate, should you buy it,
  if you want to make money?

19
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

20
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

21
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

22
Think Break 17

• Back to the antique tractor suppose you bought
  it for 6,000. Your spouse is mad, saying its a
  waste of moneyyou 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?

23
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 projects Net Present Value (NPV)
• Todays value for the whole income stream

24
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

25
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

26
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

27
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

28
Raspberry Example
Assuming 10 discount rate
29
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)

30
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

31
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

32
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?

33
Summary Concepts Learned

• Future value of an investment
• FV PV x (1 r)t
• Present value of future money
• PV FV/(1 r)t
• Interest/discount rate
• r (FV/PV)1/t 1
• These are just the same equation rearranged in
  different ways

34
Summary Concepts Learned

• Net Present Value of an income stream
• Convert varying income stream into constant
  Annuity of C over t years
• K (1/r)1 1/(1 r)t
• C NPV/K

35
Extended Case StudyWeed Resistance Management

• Herbicides generally became available for crop
  production in the late 1940s
• 2-4D 1940s, atrazine 1950s, alachlor 1960s,
  glyphosate 1970s
• 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.p
  df

36
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

37
Number and distribution of resistant weed species
globally
38
 
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Weed Resistance Management Practices(http//www.w
eedresistancemanagement.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

40
Economics of Weed Resistance Management

• Weed BMPs slow development of weed resistance to
  control, but cost money
• Economic Problem
• Do you start spending a little extra money now on
  weed BMPs 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?

41
Economics of Weed Resistance Management

• Proactive weed resistance management
• Start spending time/money now on BMPs 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

42
Weed Resistance Management Graphics
Intuition Use BMPs if a) Cost of BMP Use is low
and/or b) Cost of Resistance is high
43
Economic Analysis

• Which strategy do farmer have an economic
  incentive to use?
• What are they likely to do?
• Which strategy should they use?
• What do you recommend to farmers?
• How do you decide?
• Compare NPVs (or annuity equivalents) of the
  proactive and reactive strategies

44
Economic Model

• Net Present Value of the 2 income streams
• Annuity K (1/r)1 1/(1 r)T
• AProactive NPVProactive/K ( Rwith BMP)
• AReactive NPVReactive/K

45
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

46
What the economic model can do

• Equate the two NPV and determine when its 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

47
Problem with discrete model

• Discrete time model only allows integer years,
  when NPVs 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

48
Continuous Time Model

• Assume final time period T infinity
• Equate NPVs and rearrange

49
Continuous Time Model

• Solve this equation for any parameter as function
  of other 3 to find critical value of the
  parameter when its best to switch
• Tresist ln(CBMP/CResistance)/r
• r ln(CBMP/CResistance)/Tresist
• CBMP CResistanceerTresist
• CResistance CBMPerTresist
• Additional tab in the discrete spreadsheet

50
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
• Cant get non-integer values for T or Tresist

51
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

52
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