Future Value of a Single Amount
Compound interest is the amount of interest earned on a given deposit has become part of the principal at the end of a specified period. The term principal refers to the amount of money on which the interest is paid. Annual compounding is the most common type. The future value of a present amount is found by applying compound interest over a specified period of time.
A general equation for the future value at the end of period n can be formulated:
where future value at the end of period
initial principal (or present value)
annual rate of interest
number of periods the money left on deposit
Example 1: Find the value of $10,000 in 10 years. The investment earns 5% per year.
Example 2: Find the value of $10,000 in 10 years. The investment earns 8% for four years and then earns 4% for the remaining six years.
A financial table also provides values for. The value in each cell of the table is called the future value interest factor. This is the multiplier used to calculate at a specified interest rate the future value of a present amount as of a given time.
The general equation for compounding more frequently than annually is:
where the number of times per year interest is compounded
It is true for any interest rate for any period of time that the more frequently interest is compounded, the greater the amount of money accumulated.
Example 3: What would Fred have at the end of 2 years if he deposited $100 at 8% interest compounded annually, semiannually and quarterly?
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Annually: $116.64
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Semiannually: $116.99
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Quarterly: $117.16
Future Value of an Annuity
An annuity is a stream of equal annual cash flows. These cash flows can be inflows of returns earned on investments or funds invested to earn future returns. Two basic types: ordinary annuity and annuity due. For an ordinary annuity, the cash flow occurs at the end of each period. For an annuity due, the cash flow occurs at the beginning of each period. Because the annuity due’s cash flow occurs at the beginning of the period rather than the end, the future value of an annuity due is always greater than the future value of an otherwise identical ordinary annuity.
Example 4: What is the future value of a 4-year ordinary annuity, if the annual interest is 5%, and the annual payment is $1,000?
Example 5: What is the future value of a 4-year annuity due, if the annual interest is 5%, and the annual payment is $1,000?
Present Value of a Single Amount
Present value is the current dollar value of a future amount. Present value depends largely on the investment opportunities of the recipient and the point in time at which the amount is to be received. The process is referred to as discounting cash flows. It is concerned with the question: “If I can earn k% on my money, what is the most I would be willing to pay now for an opportunity to receive dollars periods from today?”
In other words, the present value, of some future amount,, to be received periods from no, assuming an opportunity cost of , is calculated as:
Example 6: How much do I need to invest at 8% per year, in order to have $10,000 in__.
One year:
Two years:
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Ten years:
Present Value of Cash Flow Streams
Two basic types of cash flow streams are possible: The mixed stream and the annuity. A mixed stream of cash flows reflects no particular pattern; an annuity is a pattern of equal annual cash flows.
Present Value of a Mixed Stream
To find the present value of a mixed stream of cash flows, determine the present value of each future amount, and then add together all the individual present values.
Example 7: A company has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years. If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity?
Present Value of an Annuity
The method for finding the present value of an annuity is similar to that used for a mixed stream, but can be simplified. The present value of an annuity can be found by multiplying the annual cash flows by the sum of appropriate present value interest factors.
Example 8: A company wants to determine the most it should pay to purchase a particular annuity. The firm requires a minimum return of 8% on all investments, and the annuity consists of cash flows of $700 per year for 5 years. Calculate the present value of the annuity.
Present Value of a Perpetuity
A perpetuity is an annuity with an infinite life – in other words, an annuity that never stops providing its holder with a cash flow at the end of each year.
Example 9: Mr. Clark wishes to determine the present value of a $1.000 perpetuity discounted at 10%. Find the present value of the perpetuity.
Means, if he had $10.000 and earned 10% interest on it each year, $1.000 a year could be withdrawn indefinitely without touching the initial$10.000, which would never be drawn upon.
