Continuous Money Flow 8'3

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Continuous Money Flow8.3

• JMerrill, 2007

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Total Money Flow

• Total money flow does not take into account the
  interest the money could earn after it is
  received. It is simply the total income.
• If f(x) is the rate of money flow, then the total
  money flow over the time interval from x 0 to x
  y is given by

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Total Money Flow

• So, if a uniform rate of money flow (lets say
  2,000/yr) happens over a specified time period
  (lets say 5 years), its a simple multiplication
  problem 10,000.
• For a variable function, we need the definite
  integral. If the money flow increased by 8/yr

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Present Value of Money Flow

• A specific amount of money that can be deposited
  today at a specified interest rate that yields a
  given sum in the future is called the present
  value of this future sum.
• The future sum is the future value (or final
  amount).

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Present Value of Money Flow

• To find the present value of a continuous money
  flow (with interest charged continuously), we
  use
• Where f(x) is the rate of continuous money flow
  at an interest rate, r, for t years.

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Example

• A company expect its rate of annual income during
  the next 3 years to be given by f(x) 75,000x, 0
  lt x lt 3
• What is the present value of this income over the
  3-year period, assuming an annual interest rate
  of 8 compounded continuously?

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Example
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Example

• The actual income over the 3-year period is
• This means that it would take a lump-sum deposit
  of 288,064 today paying a continuously
  compounded interest rate of 8 over 3 years to
  equal the cash flow of 337,500 with interest.

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Accumulated Amount of Money Flow at Time t

• To find the amount of money flow at an interest
  rate, r, at time x, the amount of flow at time t
  is
• This represents the final amount of money
  flowincluding interest

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Example 1

• If money is flowing at a constant rate of
  2000/yr over 5 years at 12 interest compounded
  continuously, find
• A) The total amount of the flow over the 5-year
  period

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Example 1

• B) The accumulated amount, compounded
  continuously, at t 5

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Example 1

• C) The total interest earned
• The amount of money flow over the 5-year period
  was 10,000
• The total amount, with interest, was 13,701.98
• 13,701.98 - 10,000 3701.98

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Example 1

• D) The present value of the amount with interest

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However

• If the rate of money flow is increasing/decreasing
  exponentially, then f(x) Cekx, where C is a
  constant that represents the initial amount of k
  is the (nominal) continuous rate of changewhich
  may be positive or negative.

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However

• So the Accumulated money flow is
• And the Present Value of Money Flow is

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Example 2

• A continuous money flow starts at a rate of
  1000/yr and increases exponentially at 2/yr
• a) Find the accumulated amount at the end of 5
  years at 10 interest compounded continuously
• b) Find the present value at 5 compounded
  continuously

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Example 2

• A) with the wacky formula, we have

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Example 2

• B)