# Annuities - Financial Management

An annuity is the payment or receipt of equal cash flows per period for a specified amount of time. An ordinary annuity is one in which the payments or receipts occur at the end of each period, as shown. An annuity due is one in which payments or receipts occur at the beginning of each period, as shown.Most lease payments, such as apartment rentals, and life insurance premiums are annuities due. In a 4-year ordinary annuity, the last payment is made at the end of the fourth year. In a 4-year annuity due, the last payment is made at the end of the third year (the beginning of the fourth year).

Future Value of an Ordinary Annuity

A future value of an ordinary annuity (FVANn) problem asks the question: If PMT dollars are deposited in an account at the end of each year for n years and if the deposits earn interest rate i compounded annually, what will be the value of the account at the end of n years? To illustrate, suppose Ms. Jefferson receives a 3-year ordinary annuity of $1,000 per year and deposits the money in a savings account at the end of each year. The account earns interest at a rate of 6 percent compounded annually. How much will her account be worth at the end of the 3-year period? illustrates this concept. Present Value of$100 at Various Discount Rates Timeline of an Ordinary Annuity of $100 per Period for Four Periods Timeline of an Annuity Due of$100 per Period for Four Periods The problem involves the calculation of future values. The last deposit, PMT3, made at the end of year 3, will earn no interest. Thus, its future value is as follows:

FV3rd = PMT3(1 + 0.06)0 = $1,000(1) =$1,000

The second deposit, PMT2, made at the end of year 2, will be in the account for one full year before the end of the 3-year period, and it will earn interest. Thus, its future value is as follows:

FV2nd = PMT2(1 + 0.06)1 =$1,000(1.06) =$1,060

The first deposit, PMT1, made at the end of year 1, will be in the account earning interest for two full years before the end of the 3-year period. Therefore its future value is the following:

FV1st = PMT1(1 + 0.06)2 = $1,000(1.124) =$1,124

The sum of the three figures is the future value of the annuity:

FVAN3 = FV3rd + FV2nd + FV1st = $1,000 +$1,060 + $1,124 =$3,184

Timeline of the Future Value of an Ordinary Annuity The future value of an annuity interest factor (FVIFA) is the sum of the future value interest factors presented in Table I. In this example, the future value of an annuity interest factor is calculated as

FVIFA0.06, 3 = FVIF0.06, 2 + FVIF0.06, 1 + FVIF0.06, 0 = 1.124 + 1.060 + 1.000 = 3.184

Tables of the future value of an ordinary annuity interest factors are available to simplify computations. Table at the end of the book provides a number of future value of an annuity interest factors. A portion of Table III is reproduced here as Table. FVIFAs can also be computed as follows:

FVIFAi, n = (1 + i)n – 1 / i

This formula can be used when you do not have access to interest tables with the appropriate values of i and n or to a financial calculator.

The future value of an ordinary annuity (FVANn) may be calculated by multiplying the annuity payment, PMT, by the appropriate interest factor, FVIFAi, n: FVANn = PMT(FVIFAi, n)

can be used to solve the problem involving Jefferson’s annuity. Because PMT = $1,000 and the interest factor for n = 3 years and i = 6% is 3.184, the future value of an ordinary annuity can be calculated as follows: FVAN3 = PMT(FVIFA0.06, 3) =$1,000(3.184) = $3,184 The spreadsheet solution is shown here. Sinking Fund Problem Future value of an annuity interest factors can also be used to find the annuity amount that must be invested each year to produce a future value. This type of problem is sometimes called a sinking fund problem. Suppose the Omega Graphics Company wishes to set aside an equal, annual, end-of-year amount in a “sinking fund account” earning 9.5 percent per annum over the next five years. The firm wants to have$5 million in the account at the end of five years to retire (pay off) $5 million in outstanding bonds. How much must be deposited in the account at the end of each year? This problem can be solved using either Equation or a financial calculator. Substituting n = 5, FVAN5 =$5,000,000, and i = 0.095 into Equation yields

$5,000,000 = PMT(FVIFA0.095, 5) Since the interest rate of 9.5 percent is not in Table, one must use Equation 5.15 to determine FVIFA0.095, 5. By depositing approximately$827,182 at the end of each of the next five years in the account earning 9.5 percent per annum, Omega will accumulate the $5 million needed to retire the bonds. Future Value of an Ordinary Annuity Interest Factors (FVIFA) for$1 per Future Value of an Annuity Due

Table at the end of the book (future value of an annuity interest factors) assumes ordinary (end-of-period) annuities. For an annuity due, in which payments are made at the beginning of each period, the interest factors in Table III must be modified.

Consider the case of Jefferson cited earlier. If she deposits $1,000 in a savings account at the beginning of each year for the next three years and the account earns 6 percent interest, compounded annually, how much will be in the account at the end of three years? (Recall that when the deposits were made at the end of each year, the account totaled$3,184 at the end of three years.)

We illustrates this problem as an annuity due. PMT1 is compounded for three years, PMT2 for two years, and PMT3 for one year. The correct annuity due interest factor may be obtained from Table by multiplying the FVIFA for three years and 6 percent (3.184) by 1 plus the interest rate (1 + 0.06). This yields a FVIFA for an annuity due of 3.375, and the future value of the annuity due (FVANDn) is calculated as follows:

The spreadsheet solution is shown here. Present Value of an Ordinary Annuity

The present value of an ordinary annuity (PVAN0) is the sum of the present value of a series of equal periodic payments. For example, to find the present value of an ordinary $1,000 annuity received at the end of each year for five years discounted at a 6 percent rate, the sum of the individual present values would be determined as follows: illustrates this concept. Tables of the present value of an ordinary annuity interest factors (PVIFA) are available to simplify computations. Table at the end of the book provides a number of the present value of an annuity interest factors. A portion of Table is reproduced here as Table. PVIFAs can also be computed as follows: Timeline of the Future Value of an Annuity Due (PMT =$1,000; i = 6%, n = 3)

PVIFAi, n = 1- [1/(1+i)n] /i This formula is useful when one does not have access to interest tables with the appropriate values of i and n or a financial calculator. The present value of an annuity can be determined by multiplying the annuity payment, PMT, by the appropriate interest factor, PVIFAi, n: PVAN0 = PMT(PVIFAi, n) Referring to Table to determine the interest factor for i = 6% and n = 5, the present value of an annuity in the previous problem can be calculated as follows: PVAN0 = PMT(PVIFA0.06, 5) =$1,000(4.212) = $4,212 The present value of an ordinary annuity using a spreadsheet is shown next. Timeline of a Present Value of an Ordinary Annuity (PMT =$1,000; i = 6%; n = 5) Present Value of an Ordinary Annuity Interest Factors (PVIFA) for$1 per  Solving for the Interest Rate Present value of an annuity interest factors can also be used to solve for the rate of return expected from an investment.Suppose IBM purchases a machine for $100,000. This machine is expected to generate annual cash flows of$23,742 to the firm over the next five years.What is the expected rate of return from this investment?

Using Equation,we can determine the expected rate of return in this example as follows:

From the 5-year row in Table 5.4 or Table IV, we see that a PVIFA of 4.212 occurs in the 6 percent column.7 Hence, this investment offers a 6 percent expected rate of return. Loan Amortization and Capital Recovery Problems Present value of an annuity interest factors can be used to solve a loan amortization problem, where the objective is to determine the payments necessary to pay off, or amortize, a loan, such as a home mortgage. For example, suppose you borrowed $10,000 from Lexington State Bank. The loan is for a period of four years at an interest rate of 9.0 percent. It requires that you make four equal, annual, end-of-year payments that include both principal and interest on the outstanding balance.This problem can be solved using either Equation or a financial calculator. Substituting n = 4, PVAN0 =$10,000, and i = 0.09 into Equation yields:

By making four annual, end-of-year payments to the bank of $3,086.69 (see more accurate calculator solution), you will completely pay off your loan, plus provide the bank with its 9.0 percent interest return. This can be seen in the loan amortization schedule developed in Table. At the end of each year, you pay the bank$3,087. During the first year, $900 of this payment is interest (0.09 _$10,000 remaining balance), and the rest ($2,187) is applied against the principal balance owed at the beginning of the year. Hence, after the first payment, you owe$7,813 ($10,000 –$2,187). Similar calculations are done for years 2, 3, and 4. Present value of an annuity interest factors can also be used to find the annuity amount necessary to recover a capital investment, given a required rate of return on that investment. This type of problem is called a capital recovery problem.

A spreadsheet solution to this problem is shown next. Loan Amortization Schedule: Lexington State Bank Present Value of an Annuity Due

Annuity due calculations are also important when dealing with the present value of an annuity problem. In these cases, the interest factors in Table must be modified. Consider the case of a 5-year annuity of $1,000 each year, discounted at 6 percent.What is the present value of this annuity if each payment is received at the beginning of each year? (Recall the example presented earlier, illustrating the concept of the present value of an ordinary annuity, in which each payment was received at the end of each year and the present value was$4,212.) Figure illustrates this problem.

The first payment received at the beginning of year 1 (end of year 0) is already in its present value form and therefore requires no discounting.PMT2 is discounted for one period, PMT3 is discounted for two periods, PMT4 is discounted for three periods, and PMT5 is discounted for four periods. The correct annuity due interest factor for this problem may be obtained from Table IV by multiplying the present value of an ordinary annuity interest factor for five years and 6 percent (4.212) by 1 plus the interest rate (1 + 0.06). This yields a PVIFA for an annuity due of 4.465, and the present value of this annuity due (PVAND0) is calculated as follows:

A spreadsheet solution to this problem is shown here. Annuity due calculations are especially important when dealing with rental or lease contracts because it is common for these contracts to require that payments be made at the beginning of each period.

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