The Time Value of Money The Effects of Compound Interest Concentration of Wealth Present Value

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The Time Value of MoneyThe Effects of Compound
InterestConcentration of WealthPresent Value

• Robert M. Hayes
• 2005

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Overview

• Time Value of Money
• Effects of Compound Interest
• Concentration of Wealth
• Present Value

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Why is there a time value for money?

• A dollar in hand today is worth more than a
  dollar in hand tomorrow. Why is that?
• I could buy something today and thus get the use
  today of what I buy.
• I could invest today and gain the return from
  that investment.
• I could avoid the loss of value due to inflation
  in costs.
• I could lend the money today and gain the
  interest on that loan.
• Why is there interest on a loan?
• There needs to be a return, given the value today
  vs. tomorrow.
• The loss of value from the other potential uses
  must be recognized.
• There are risks that the loan may not be repaid.

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The Relevant Variables

• There are therefore four relevant variables in
  dealing with the time value of money
• The initial amount lent, called the principal
  amount
• The time period of the loan
• The interest rate
• The time period to which the interest rate
  applies
• Note that there are two separate and potentially
  different (in fact, usually different) time
  periods involved (1) the time period of the
  loan, and (2) the time period to which the
  interest rate applies.

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Simple vs. Compound Interest
Methods of Calculating Interest

• Simple interest is applied to the initial amount,
  called the principal, for a given time period for
  interest. If the period of the loan is greater
  than the time period for interest, the simple
  interest will be repeated, at the same amount,
  and accumulate during successive time periods for
  interest until the end of the time period of
  loan.
• Compound interest is applied to the initial sum,
  plus any previous accumulated interest that has
  not been paid, for each successive time period
  for interest.
• The rationale for compound interest is that the
  interest is in fact money that should be in hand
  at the end of the time period for interest, i.e.,
  at the time it is due. Therefore, if that
  interest is not received, it is, in effect, also
  lent and therefore should also bear interest.

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The Relevant Formulas

• Let the four relevant variables be represented as
  follows
• Principal amount, P
• Time period of loan, L
• Interest rate. I
• Time period for interest, T
• Let C be the total amount due at the end of L,
  and let N be the ratio of the two time periods,
  L/T
• For simple interest, the formula for total amount
  due, C, at the end of the time period of loan, L,
  is
• C P (1 (L/T)I) P (1 NI)
• For compound interest, the formula is
• C P(1 I)(L/T) P(1 I)N
• Note that compound interest is exponential.

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The Power of Compound Interest

• Albert Einstein is reputed to have said,
  Compound interest, not E MC2, is the greatest
  mathematical discovery of all time"
• He also is credited with discovering what is
  called the compound interest Rule of 72. The
  Rule of 72 says that the principal amount will
  double in 72/I years, where I is the rate of
  interest. For example, if the interest rate is
  6, the principal amount will double in 12 years.
• To illustrate, suppose that in 1955, a person
  invested 5,000 in a mutual fund and, during the
  ensuing 50 years, all dividends were reinvested.
  Today, that fund is worth 160,000.
• Lets apply the Rule of 72 160,000/5,000 32.
  That is 25, so the original investment doubled
  five times. That means it doubled every 10 years,
  so the average interest rate was 7.2.
• Of course, during that 50 years, the inflation
  rate averaged about 4, so the net gain was
  probably not that much.

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The Effect on Concentration of Wealth

• Interest, especially compound interest, plays a
  significant role in the concentration of wealth.
• In a society, considered from an economic
  standpoint, there are two primary means for
  production Labor and Capital. The former
  represents the results of the contributions of
  individuals as the agents of production the
  latter, the results from investment of
  accumulated past savings in the tools for
  production.
• The relationship between production, on the one
  hand, and labor and capital, as the means for
  production, on the other, is usually represented
  by a production function, a relatively simple
  example of which is the Cobb-Douglas production
  function.
• In a very real sense, interest represents the
  societal income from the investment of the
  capital.
• The question at hand here is the relationship
  between interest and the concentration of wealth.

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• To examine that relationship, lets let the
  Capital owned by person k be C(k), the income
  from Capital be iC(k) (so the interest rate, or
  return on capital, is i), the production
  generated by person k from Labor be L(k), and the
  total expenditures related to person k be E(k).
• It is relevant and even important to note that
  the total expenditures, E(k), related to a person
  consist of three components (1) personal
  expenditures (for self and dependents), (2)
  production expenditures (represented in part by
  overhead, which includes management, space,
  etc., and in part by materials), and (3) societal
  expenditures (represented primarily by government
  and, thus, taxes).
• (The difference between the E(k) and the personal
  expenditures of person k, is what Karl Marx
  refers to as surplus value. That is, it is the
  excess of a persons production over what is
  directly received for it.)
• Normally, one would expect the total
  expenditures, over all persons, to be less than
  or at most equal to the total production over all
  persons (otherwise the accumulated social wealth
  of the past will be dissipated).
• If L(k) E(k) iC(k) gt 0, there will be a net
  addition to societal capital (and, of course, if
  L(k) E(k) iC(k) lt 0, a net reduction to
  societal capital) from person k. Lets suppose
  that person k is permitted to keep the increase
  (or lose the decrease) and add it to (or subtract
  it from) C(k).

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• Now, consider two persons, P1 and P2. Let C(k), k
  1, 2 be their respective ownership of the
  societal wealth. So their income from Capital
  will be iC(k), respectively. Let their
  respective production from their Labor be the
  same, L, and let their respective consumption
  also be the same, E. Thus, in this context, they
  differ only in their relative wealth.
• Then, their respective net savings will be S(k)
  iC(k) L E. The total societal net savings
  will be S(1) S(2) i(C(1) C(2)) 2(L
  E).
• The individual net savings result in a new
  distribution of capital wealth C(k) C(k)
  S(k) C(k) iC(k) L E
• Let C(1) C(2) X, so that, if X gt 0, P1 has
  more wealth than P2.
• Then, C(1) C(2) X i(C(2) X) L E
  (1 i)(C(2) X) L E and C(2) (1
  i)C(2) L E
• C(1)/C(2) 1 (1 i)X/C(2) 1 (1
  i)X/((1 i)C(2) L E)
• If L E lt 0, then (1 i)C(2) gt (1 i)C(2)
  L E and therefore (1 i)/((1 i)C(2) L
  E) gt 1/C(2)
• Therefore, C(1)/C(2) gt 1 X/C(2) C(1)/C(2)
• Thus, if the expenditures related to a person are
  greater than the production related to that
  person, that persons relative share of the
  wealth will be reduced, even though his amount of
  wealth may increase.

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Stages in Wealth Concentration

• I think there is value in understanding the
  stages in wealth concentration, especially as
  represented by the effects of interest.
• At the simplest level, such as a primitive
  agricultural society, every person is effectively
  at the level of subsistence, making just enough
  to meet the needs of themselves and their
  dependents.
• At the next level, there is societal capital, as
  an investment in tools for production, that
  permits a more complex society, with greater
  production than mere subsistence. For a variety
  of reasons, there is almost certain to be some
  degree of concentration of wealth, with some
  persons in the society having more than others.
  And, as just shown, the degree of concentration
  is almost certain to increase over time.

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• At the next level, the degree of concentration
  reaches the point where those with the most
  wealth do not need to subsist on the results of
  their labors, but can do so solely on the income
  from their wealth.
• Lets suppose that subsistence requires an income
  of at least Z. In current economic terms, that
  might be the federal poverty level, which for a
  family of 4 is about 20,000. If the interest
  rate is, just for illustration, at 5, a level of
  wealth of 400,000 would generate that level of
  income, without the need to work. Working would
  then, of course, provide the resources for life
  beyond the poverty level, or for increasing ones
  wealth, or for some mix of the two.

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• At the next level, the degree of concentration
  reaches the point where those with the most
  wealth do not need to subsist on the results of
  their labors, but can do so solely on the income
  from the income from their wealth.
• Continuing with the example of Z 20,000 as the
  subsistence level, that means that the income
  from the wealth generates 400,000 per annum so,
  at 5, the wealth must be 8,000,000.
• The important point here is that the growth in
  wealth no longer depends at all upon labor, but
  can be generated solely from the interest.
  Indeed, if the person could subsist on the
  20,000 per annum, the capital wealth would
  increase by nearly 5 per annum and thus would
  double in 15 years!
• At this level or perhaps at the next one, the
  capital ceases to be money. It becomes power and
  control.

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• I want to examine one final level simply to show
  what happens. Let the wealth accumulated by a
  person be such that subsistence can be obtained
  from the income of the income on the income
  (three levels remove from the need for labor).
• Continuing with Z 20,000 as the subsistence
  level, the wealth would need to generate
  8,000,000 in income so, at 5, that implies
  wealth of 160,000,000. Clearly, this is at the
  level where money represent power.
• And we have not gotten even close to Bill Gates!

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Application to U.S. National Economy

• Simply to illustrate some of the relationships
  among the things I have just discussed, lets
  look at the U.S. national economy.
• From the 1997 Input/Output Tables, we have the
  following data
• Total Intermediate Input
  6.7 Trillion
• Product from Capital, Labor (Value Added)
  8.8 Trillion
• Government taxation
  2.7 Trillion
• Additions to capital
  1.3 Trillion
• Net for Capital and Labor
  4.8 Trillion
• From the Cobb-Douglas production model
• 8.8 a(L)b (C)(1-b)
• Let C KL. Then 8.8 aLK(1-b)
• Net for Capital and Labor L iC 4.8
• If i 5, then L .05C 4.8
• If a K(1-b) 10, then L .88 and C 3.92

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Present Value

• The Role of Present Value of Money
• Calculating Present and Future Value of Money
• Using Net Present Value Analysis
• Selecting a Discount Rate
• Identifying Cash Flows to Consider
• Determining Cash Flow Timing
• Selecting the Best Alternative
• Identifying Issues and Concerns

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The Role of Present Value of Money

• Why is a dollar today worth more than a dollar a
  year from now?
• Investment
• Inflation
• Use and Enjoyment
• The Role of the Discount Rate
• The bases for choice of the discount rate
• The role of risk assessment
• The role of capitalization rates
• The effect of the time period

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Present and Future Value of Money

• Present Value and Future Value
• Effects of inflation
• Present value of a cash stream.
• Present value of a cash stream in perpetuity

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Calculating Present Value

•  
• t
• P ? Fy/(1 i)y
• y 1
•  
• P present dollar value
• Fy future dollar value in year y
• i annual rate of return (e.g., 0.05 is 5 per
  annum)
• y the succession of years
• t number of years in the future
•  

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• If the future dollars are the same for each year,
  say Fy F,
•  
• t 1
  t (1 r) t-1 1
• let S ? so (1 r)S ?
  ?
• y1 (1 r) y
  y1 (1 r) y y0 (1 r) y
•   
• 1 1
  1 (1 r)t - 1
• (1 r)S S 1
• (1 r)0 (1 r)t
  (1 r)t (1 r)t
• Hence 
• (1 r)t - 1
• P FS F
• r(1 r)t
•  

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• There are times when the present value analysis
  needs to consider a cash stream in perpetuityfor
  an infinite period of time. Consider the formula
  shown above
• (1
  r)t - 1
• P FS F
• r(1
  r)t
• but let t be infinity. Note that the second
  term in the expression on the right becomes zero
  and the first term, 1/r. The result is that
• P F/r

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Using Net Present Value Analysis

• Illustrative Contexts for use of Present Value
• Lease-purchase
• Different lease alternatives
• Life-cycle cost
• Trade-off of acquisition costs and costs of
  operation
• Factors Affecting Net Present Value
• The timing of the cash flow
• The discount rate

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Steps in Net Present Value Analysis

• Step 1. Select the discount rate.
• Step 2. Identify the costs/benefits to be
  considered
• Step 3. Establish the timing of the
  costs/benefits.
• Step 4. Calculate net present value of
  alternatives
• Step 5. Select the option with best net present
  value.

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Selecting A Discount Rate

• Nominal Discount Rates
• Real Discount Rates
• Selecting the Rate for Analysis.

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Nominal Discount Rates

• Most benefit-cost analyses should use nominal
  discount rates (i.e., discount rates that include
  the effect of actual or expected inflation or
  deflation).

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Real Discount Rates

• For some projects, it may be more reasonable to
  assess in terms of constant dollars. The real
  discount rate is the nominal discount rate
  adjusted to eliminate the effect of anticipated
  inflation/deflation.

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Determining the Discount Rate

• Once the type of discount rate has been selected
  (whether nominal or real), the values to be used
  are then determined from the appropriate table
  (using linear interpolation to determine values
  for years between those in the table).

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Identifying Cash Flows To Consider

• Cash Flow
• Identify all relevant cash flows, both costs and
  benefits
• Alternatives should clearly identify the cash
  flows that are specifically significant
• Points to Consider in Identifying Costs and
  Benefits
• Include the same cash flows in all alternatives
• Include cash flows in which alternatives will
  differ
• Do not include cash flows that are identical for
  alternatives
• Do not include sunk costs or benefits
• Analysis Period
• For leasing contexts, use the leasing period plus
  renewal
• For acquisition contexts, use life cycle period
• For equipment context, use amortization period

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Representative Costs Benefits

• Net Purchase Price
• Costs for Transportation, Installation, Site
  preparation
• Costs for Design, Training, and Management.
• Repair and improvement costs, including
• Estimated unplanned service calls
• Improvements required to assure continued
  operation.
• Operation and maintenance, including
• Operating labor and supply requirements and
• Routine maintenance.
• Disposal costs and salvage value, including
• Cost of modifications to return equipment to
  original configuration
• Cost or modifications to return facilities to
  original configuration
• Salvage value at the end of the period for
  analysis

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Determining Cash Flow Timing

• Bases for determining cash flow timing
• Offer-Identified Cash Flows.
• Government-Identified Cash Flows.
• End-of-year payment
• When to use End-of-Year Discount Factors
• End-of-Year Discount Factor Calculation.
• Repetitive End-of-Year Cash Flows.
• Mid-Year Payment
• When to Use Mid-Year Discount Factors.
• Mid-Year Discount Factor Calculation.
• Repetitive Mid-Year Cash Flows.

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Calculating Net Present Value to Select The Best
Alternative

• Lease-Purchase Decision, Example 1
• Lease-Purchase Decision, Example 2

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Lease-Purchase Decision, Example 1

• Which of the following will result in the lowest
  total cost of acquisition?
• A Proposal to lease the asset for 3 years. The
  annual lease payments are 10,000 per year, the
  first payment due at the beginning of the lease
  and the remaining two payments due at the
  beginning of Years 2 and 3. 
• B Proposal to purchase the asset for 29,000. It
  has a 3-year useful life. Salvage value at the
  end of 3-year period will be 2,000.

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• Step 1. Select the discount rate. The term of the
  lease analysis is three years, so we will use the
  nominal discount rate for three years, 5.4
  percent.
• Steps 2 and 3. Identify and establish the timing
  of the costs/benefits to be considered in
  analysis. The expenditures and receipts
  associated with the two offers and their timing
  are delineated in the table below (Parentheses
  indicate a cash outflow.)

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• Step 4. Calculate net present value. The table
  below summarizes for each alternative.
• Step 5. Select the offer with the best net
  present value. In this example, it is Offer B,
  the offer with the smallest negative net present
  value.

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Lease-Purchase Decision, Example 2

• Which of the following will result in the lowest
  total cost of acquisition?
• A Proposal to lease the asset for 3 years. The
  monthly lease payments are 1,500 that is, the
  total amount for each year is 18,000. These
  payments are spaced evenly over the year, so the
  use of a MYDF would be appropriate.
• B Proposal to purchase the asset for 56,000.
  It has a 3-year useful life. At the end of the
  3-year period it will have a 3,000 salvage
  value.

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• Step 1. Select the discount rate. The term of the
  lease analysis is three years, so we will use the
  nominal discount rate for three years, 5.4
  percent.
• Steps 2 and 3. Identify and establish the timing
  of the costs/benefits to be considered in
  analysis. The expenditures and receipts
  associated with the two offers and their timing
  are delineated in the table below (Parentheses
  indicate a cash outflow.)

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• Step 4. Calculate net present value. The table
  below summarizes for each alternative.
• Step 5. Select the offer with the best net
  present value. In this example, it is Offer A,
  the offer with the smallest negative net present
  value.

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Identifying Issues and Concerns

• Is net present value analysis used when
  appropriate?
• Are the dollar estimates for expenditures and
  receipts reasonable?
• Are the times projected for expenditures and
  receipts reasonable?
• Are the proper discount rates used in the net
  present value calculations?
• Are the proper discount factors used in analysis?
  
• Are discount factors properly calculated from the
  discount rate?
• Have all cash flows been considered?

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THE END