Present Value

This topic introduces the concept of present value and discounting. The present value of an investment can be thought of as the initial principal needed to accumulate to a desired amount at the end of a specified period of time. Effective and nominal rates of discount are also introduced, and formulas are derived relating rates of discount to rates of interest.

Definition (Present Value) The present value of can be thought of as the amount to be invested now which will accumulate to after investment periods.
Recall that for compound interest the accumulated value formula is and so the present value
For simple interest, the accumulation formula can be written where the constant interest amount is thought of as a rate applied to the principal. Rewriting, we have and therefore the present value is

Definition (Discount Factor) For a given rate of interest the quantity is the amount of initial investment that will accumulate to 1 at the end of one investment period. The quantity is called the discount factor.
The term discount factor is used exclusively in the context of compound interest. The present value can be written

Example (Present Value) Find the amount of initial investment needed to accumulate to 10,000 in 20 years at 6% compounded quarterly. Using the formula

we have

Note that we can also write where is understood to be the discount factor relative to one investment period; that is,

Definition (Effective Rate of Discount) The effective rate of discount is the ratio of the amount of interest earned during the period to the accumulated amount at the end of the period.
Note the similarity to the definition of Effective Rate of Interest. The key difference is that interest is paid at the end of the investment period on the amount at the beginning of the period, while discount is paid at the beginning of the investment period on the amount at the end of the period.

Example (Effective Rate of Discount) A loan of 1,200 is made for one year at an effective rate of discount of 5%. The discount can be thought of as pre-paid interest; that is, at the beginning of the year, the borrower pays interest in the amount of 0 and at the end of the year repays the 1,200 loan.
The difference between an effective rate of discount and effective rate of interest becomes more clear if another component is added to the problem. Suppose that a loan of 1,200 is made for one year at an effective rate of discount of 5%, for the purpose of depositing in a one-year investment paying 8%. The borrower receives at the beginning of the year. At the end of the year, the borrower's investment has earned The borrower pays back the loan of 1,200 and has earned a profit of 31.20.
If instead the loan was made for 1,200 for one year at an effective rate of interest of 5%, the borrower receives 1,200 at the beginning of the year. At the end of the year, the borrower's investment has earned The borrower repays the loan (plus interest) of and has earned a profit of 36.00. The profit is more in this case because the borrower had the use of the entire 1,200 at the beginning of the year.

Proposition (Relationship Between Interest and Discount) For an effective rate of discount and effective rate of interest , the following relationships hold:

    (i)

    (ii)

    (iii) .

    Proof (Relationship Between Interest and Discount) If 1 is borrowed at an effective rate of discount the interest paid at the beginning of the period is and so the principal at the beginning of the period is By the definition of effective rate of interest, we have i.e. the effective rate of interest is the ratio of the amount of interest earned to the amount of principal at the beginning of the period. Rearranging the terms of this equation, we get

that is, the ratio of the interest earned in the period to the balance at the end of the period.
    From the second equation above, we also see that

Definition (Nominal Rate of Discount) The nominal rate of discount (denoted ) is a rate of discount payable times per investment period. The rate paid is for each of a period.

Proposition (Nominal Rate of Discount)
    (i) For a nominal rate of discount , the corresponding effective rate of discount is
    (ii) For a nominal rate of interest and a nominal rate of discount we have


    Proof (Nominal Rate of Discount) If 1 is borrowed at an effective rate of discount the amount at the beginning of the period is Alternatively, if 1 is borrowed at a nominal rate of discount to be paid times during the period, the amount at the beginning of the period is Therefore, and rearranging we get
    For a nominal rate of interest the accumulated amount after one investment period is From the previously defined relationship we see that From above, we have and so we must have Therefore,

Example (Nominal Rate of Discount)
Find a nominal rate of interest compounded monthly that is equivalent to a nominal rate of discount of compounded quarterly. Using the formula

we have

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