Equations of Value

Two amounts of money payable at different times can be compared by accumulating or discounting the money to a common comparison date. The resulting equation is called an equation of value. This topic contains several examples of problems which can be solved by considering two different streams of money as equivalent. Examples are given to find an initial investment value, an unknown interest rate, and an unknown length of time.

Definition (Equation of Value) An equation which compares two amounts of money payable at different times by accumulating or discounting the payments to a common comparison date is called an equation of value.

Example (Equation of Value) Investor A deposits 1,000 into an account paying interest compounded quarterly. At the end of three years, he deposits an additional 1,000. Investor B deposits into an account with force of interest After five years, investors A and B have the same amount of money. Find
    Consider investor A's account first. The initial 1,000 accumulates at compounded quarterly for five years; the accumulated amount of this piece is The second 1,000 accumulates at compounded quarterly for two years, accumulating to The value in investor A's account after five years is


    The accumulated amount of investor B's account after five years is given by

Setting these two expressions equal, we have the equation

Since we have

Example (Unknown Interest Rate) (i) An initial deposit of 2000 grows to 3432.01 after 6 years when invested at a force of interest of Find

    (ii) A company wishes to repay a debt of 10,000 due now and 12,950 due in three years with a single payment of 20,000 now. If the proposed repayment plan is equivalent to the original plan, what is the implied annual effective rate of interest?
    The present value of the first repayment plan is The present value of the second plan is Therefore, we solve:

Example (Unknown Time) (i) The present value of a payment of 5000 to be made in years is equal to the present value of a payment of  7100 to be made in years. If find .

Substituting we have
    Note that and are equivalent equations (have the same solution). Verbally, this means we can approach the problem by comparing both payments at either "time 0" or "time " and we will get the same solution.
    (ii) A single payment of 3938.31 will pay off a debt whose original repayment plan was 1000 due on January 1 of each of the next four years, beginning January 1, 2006. If the effective annual rate is , on what date must the payment of 3938.31 be paid?
    We will discount all the payments back to January 1, 2006. The present value of the original repayment plan on that date is

The present value of the second repayment plan is just where is the quantity to be determined. Setting the two equations equal, and using we have

Therefore, the single payment of 3938.31 must be paid 1.25 years after January 1, 2006, or on March 31, 2007.

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Equations Of Value
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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