Business Mathematics: Annuities and Their Applications - Annuities

An annuity is a sequence of payments made at regular intervals. Payments may be made to you by an insurance company or a retirement fund, for example, or by you to an account which you may use later to make a purchase or pay for your children’s education.

The time between payments is called the payment interval or conversion period, and the time over which the money is to be paid is called the term of the annuity.

When the term of the annuity is fixed, the annuity is said to be certain', otherwise it is called a contingent annuity. A 30-year mortgage is an example of a certain annuity because there are precisely 360 monthly payments made during the duration of the loan.

An example of a contingent annuity is a monthly retirement payment paid during the life of a retiree. The number of payments to be made is unknown and is estimated by actuaries using mortality tables. In this book we consider only certain annuities.

Interest is paid at regular intervals called payment intervals. When the payment period coincides with the conversion period, the annuity is said to be simple; otherwise it is called a general annuity. We shall considei only simple annuities in this book.

When payments are made at the end of each payment interval the annuity is called an ordinary annuity.

An annuity due is an annuity whose payments are made at the beginning of the payment. Unless otherwise stated, all annuities in this guide, are assumed to be ordinary annuities.

In dealing with annuity formulas we shall use the following symbols:

R. The periodic payment of the annuity.
n: The number of payments made.
i:  The interest rate per payment interval.
S: The accumulated value of the annuity; the amount of money you will have at the end of the annuity’s term.
A: The present value of the annuity, or discounted value', the amount of money which must be set aside today to allow a specified payment for a predetermined period of time. The present value is the amount of money the annuity is worth today.

Remember that i = annual interest rate (expressed as a decimal) divided by the number of payments per year, so for a monthly annuity at 6% annual interest, i = 0.06/12 = 0.005.

The accumulated value of an ordinary annuity is computed by the formula 1
S = R × accumulation factor

The accumulation factor is computed by a rather complex mathematical formula. Table 6.1 shows accumulation factors for a variety of periods and interest rates.


Example 1
To determine the accumulation factor for an ordinary annuity of four payments whose periodic interest rate is 2%, we go to Table 6.1 and look in the row corresponding to n = 4 and the column corresponding to i = 0.02. The accumulation factor is 4.12161.


Example 2
Compute the accumulated value of an ordinary annuity of $500 which is compounded quarterly for one year at an annual rate of 8%.
In this example R — 500, n = 4, and i = 0.08/4 = 0.02. From Example 1 the accumulation factor is 4.12161: S = R × accumulation factor = $500 × 4.12161 = $2,060.80.


Example 3
To understand how money accumulates in an ordinary annuity, we can compute the value of the annuity after each payment. Since each payment of $500 is made at the end of the quarter, interest is compounded at most three times. The following table illustrates the growth of the money.

Next we consider an annuity due, where each payment is made at the beginning of the payment interval.

Table: Accumulation Factors for Simple Annuities



Example 4
Compute the accumulated value of an annuity due of $500 which is compounded quarterly for one year at an annual rate of 8%.

As in Example 3, R = 500, n = 4, and i = 0.08/4 = 0.02. However, since each $500 payment is made at the beginning of the payment interval rather than the end, the value of each payment is multiplied by a factor of 1 + i. The total value of the annuity at the end of its term is also increased by a factor of 1 + i. Therefore
S = R × accumulation factor × (1 + i) = $500 × 4.12161 × 1.02 = 2,102.02

The growth of the money is illustrated below. Note that the end-of-year value of each $500 payment is 1.02 times what it was in Example 3 (above):

The accumulated value of an annuity due is computed by the formula
S = R × accumulation factor × (1 + i)


Example 5
Compute the accumulated value of an annuity of $100 invested at the end of each month for one year at an annual rate of 6%.

Since the payment interval is one month, the annual rate of 0.06 must be divided by 12 to give i = 0.005. R = $100 and n = 12. The accumulation factor is 12.33556 from the Table above.
S = R × accumulation factor = $100 × 12.33556 = $1,233.56


Example 6
Suppose that Ryan deposits $1,000 every 3 months in a savings account which pays 8% annual interest compounded quarterly. How much will Ryan have in the bank at the end of 5 years?
 

R = $1,000, n = 5 × 4 = 20 payments, and i = 0.08/4 = 0.02
Assuming that Ryan deposits his money at the beginning of each 3-month period,
S = R × accumulation factor × (1 + i) = $1,000 × 24.29737 × 1.02 = $24,783.32

The next example illustrates how much money to put aside each period in order to accumulate a specified amount.


Example 7
How much must Sylvia save each month if she would like to accumulate $5,000 in 3 years? Assume interest is paid at the rate of 6% compounded monthly and that Sylvia makes her deposit at the end of each month.

In this problem we wish to compute R. S = $5,000, n = 12 × 3 = 36, i = 0.06/12 = 0.005. The accumulation factor from Table 6.1 is 39.33610. Since S = R × accumulation factor,
$5,000 = R × 39.33610

We divide both sides of the equation by 39.3361 and obtain R = $127.11. Sylvia must save $127.11 each month.
If the deposits are made at the beginning of each payment interval, the calculation is just a bit different.


Example 8
How much must Sylvia save each month (see Example 7) if deposits are made at the beginning of each month?
This is an annuity due. As in the previous example, the accumulation factor is 39.33610, so
$5,000 = R × 39.33610 × 1.005
$5,000 = R × 39.53278

Dividing $5,000 by 39.53278, we obtain R = $126.48. Note that Sylvia’s deposits are a bit less if she makes them at the beginning of the month. (Does this make sense to you?)
The present value of the annuity, or discounted value, is the amount of money which must be set aside to allow a specified payment for a specified period of time. The present value indicates, for example, how much money should be invested in order to provide a fixed monthly stipend for retirement for the next 20 years.
 

The present value of an ordinary annuity is computed by the formula
A = R × discount factor

The table below shows discount factors for a variety of periods and interest rates.


Example 9
Bill would like to invest some money so that he can receive a monthly retirement supplement. How much money must he invest at 9% compounded monthly in order to receive $500 at the end of each month for the next 5 years?
In this example R = $500, n = 5 × 12 = 60, and i = 0.09/12 = 0.0075. The discount factor from Table 6.2 is 48.17337.
A = R × discount factor = $500 × 48.17337 = $24,086.69
 

The present value of an annuity due is computed by the formula
A = R × discount factor × (1 + i)


Example 10
How much would Bill (see Example 9) have to invest if payments are to be made at the beginning of the month?
A = R × discount factor × (1 + i) = $500 × 48.17337 × 1.0075 = $24,267.34
Note that this amount is larger than in Example 9. (Does this seem reasonable?)

Table: Discount Factors for Simple Annuities



Example 11
Gene’s grandmother left him $10,000 when she died. So that he would not squander the money, she stipulated in her will that the money be invested in a bank account paying 8% interest compounded quarterly and paid out in equal installments every 3 months for a period of 10 years. How much will Gene receive each quarter?

In this problem A = $10,000, n = 10 × 4 = 40, and i = 0.08/4 = 0.02. The discount factor from Table 6.2 is 27.35548.
A = R × discount factor
$10,000 = R × 27.35548
Dividing $10,000 by 27.35548, we obtain R = $365.56 per month.

Solved Problems:

6.1 Find the accumulated value of an annuity of $750 invested at the end of each quarter for 5 years at an annual rate of 8% compounded quarterly.
Solution
R = $750, n = 5 × 4 = 20, i = 0.08/4 = 0.02. The accumulation factor (from Table 6.1) is 24.29737:
S = R × accumulation factor = $750 × 24.29737 = $18,223.03

6.2 Find the accumulated value of an annuity of $50 invested at the end of each month for 2 years at an annual rate of 9% compounded monthly.
Solution
R = $50, n = 12 × 2 = 24 payments, and i = 0.09/12 = 0.0075. The accumulation factor (from Table 6.1) is 26.18847.
A = R × accumulation factor = $50 × 26.18847 = $1,309.42.
The value of the annuity is $1,309.42.

6.3 XYZ Savings Bank pays interest at the rate of 4% annually compounded quarterly. How much money will Roger have in the bank at the end of 5 years if he deposits $250 at the end of each quarter?
Solution
R = $250. There are 20 quarters in a 5-year period, so n = 20, and i — 0.04/4 = 0.01. The accumulation factor is 22.01900 (from Table 6.1):
S = RX accumulation factor = $250 × 22.01900 = $5,504.75
Roger will have saved $5,504.75.

6.4 How much money will Roger (see Example 6.3) have in the bank at the end of 5 years if he deposits $250 at the beginning of each quarter?
Solution
As before, R = $250, n = 20, and i = 0.01:
S = R × accumulation factor × (1 + i) = $250 × 22.01900 × 1.01 = $5,559.80

6.5How much interest is earned in 10 years if $100 is deposited at the end of each month in an account that pays 15% compounded monthly?
Solution
R = $100, n = 10 × 12 = 120, i = 0.15/12 = 0.0125. The accumulation factor (Table 6.1) is 275.21706:
S = R × accumulation factor = $100ȕ275.21706 = $27,521.71
Since the total amount deposited is $100 × 120 = $12,000, the interest earned is $27,521.71 — $12,000.00 = $15,521.71.

6.6 Barney makes a New Year’s resolution to put $100 into the bank at the beginning of each month, beginning January 2000. If the bank pays 6% interest compounded monthly on the last day of each month, how much will Barney have one year later?
Solution
R = $100, n = 12, i = 0.06/12 = 0.005. The accumulation factor (Table 6.1) is 12.33556. Since this is an annuity due, it follows that
S = R × accumulation factor × (1 + i) = $100 × 12.33556 × 1.005 = $1,239.72

6.7 Find the present value of an annuity of $350 at the end of each month for 5 years at 9% compounded monthly.
Solution
R = $350, n = 12 × 5 = 60, i = 0.09/12 = 0.0075. The discount factor obtained from Table 6.2 is 48.17337:
A = R× discount factor = $350 × 48.17337 = $16,860.68

6.8 A refrigerator can be purchased for $150 down and $50 a month for 18 months. What is the equivalent price if the refrigerator is purchased for cash? Assume that the interest rate on credit is 15% compounded monthly.
Solution
R = $50, n = 18, i = 0.15/12 = 0.0125. The present value of an annuity of $50 per month for 18 months is computed using a discount factor of 16.02955, obtained from Table 6.2:
A = R× discount factor = $50 × 16.02955 = $801.48
The equivalent cash price for the refrigerator is $150.00 + $801.48 = $951.48.

6.9 Mr. Smith would like to receive $4,000 each quarter for 10 years after he retires. How much money (to the nearest dollar) does he have to save in a money market fund which pays at the rate of 8% compounded quarterly?
Solution
R = $4,000, n = 10 × 4 = 40, and i = 0.08/4 = 0.02. The discount factor (Table 6.2) is 27.35548:
A = R × discount factor = $4,000 × 27.35548 = $109,422

6.10   Bob TUrner has saved a total of $170,000 for retirement. He has put the money in a mutual fund which pays 9% annual interest compounded monthly. How much should he withdraw each month in order to have enough money to last for 15 years?
Solution
A = $170,000, n = 12 × 15 = 180, and i = 0.09/12 = 0.0075. The discount factor (Table 6.2) is 98.59341:
A = R × discount factor
$170,000 = R × 98.59341
Dividing $170,000 by 98.59341, we get $1,724.25. This is the amount Bob should withdraw each month.

6.11   When Sarah James’ husband died, she became the beneficiary of a $100,000 life insurance policy. Instead of taking the money in a lump sum, she elects to receive a monthly stipend over a period of 20 years. If the insurance company pays interest at the rate of 6% compounded monthly, what will her monthly income be?
Solution
A = $100,000, n = 20 × 12 = 240, i = 0.06/12 = 0.005. The discount factor is 139.58077:
A = R × discount factor
$100,000 = R × 139.58077
Dividing by 139.58077, we obtain R = $716.43 as her monthly income.

6.12   At age 30, Mr. Bixby begins to save for his retirement by depositing $200 every 3 months into a savings account that pays 5% interest compounded quarterly. At age 60 he decides to retire, using his savings account as the basis of an annuity. How much will he get every quarter if he wants to get equal payments for the next 20 years? Assume the interest rate to be fixed at 5% over the full 50-year period.
Solution
First we must compute how much Mr. Bixby will have saved up in 30 years: R — $200, n = 30 × 4 = 120, and i = 0.05/4 = 0.0125. The accumulation factor from Table 6.1 is 275.21706.
S = R × accumulation factor = $200 × 275.21706 = $55,043.41
Now we compute the quarterly payment. A = $55,043.41, n = 20 × 4 = 80, i = 0.0125. The discount factor from Table 6.2 is 50.38666.
A = R × accumulation factor
55,043.41 = R × 50.38666
R = $1,092.42
Mr. Bixby will receive $1,092.42 every 3 months for the next 20 years.

6.13   A sofa sells for $600. As an incentive, a furniture store offers to accept $50 per month for one year with no finance charge. If the actual rate of interest charged by the bank is 12% compounded monthly, what is the actual cost of the sofa?
Solution
The present value of the annuity is the actual cost of the sofa. (The difference between this number and the $600 paid goes to the bank as interest.) R = $50, n = 12, and i = 0.12/12 = 0.01. The discount factor (Table 6.2) is 11.25508:
A = R × discount factor = $50 × 11.25508 = $562.75

6.14 Tom Evans purchases a life insurance policy which has an annual premium of $1,500 due at the beginning of the year. If he elects to pay his premium in quarterly installments, how much should he pay at the beginning of each quarter if the interest rate is 10% compounded quarterly?
Solution
The present value of the premium, A, is $1500, n = 4, and i = 0.10/4 = 0.025. The discount factor is 3.76197 (from Table 6.2):
A = R × discount factor × (1 + i)
1.500 = R × 3.76197 × 1.025
1.500 = R × 3.85602 R = 389.00
Tom should pay $389 at the beginning of each 3-month period.

6.15   The monthly rent for a one-bedroom apartment in Manhattan is $1,500, payable at the beginning of the month. If the current interest rate is 9%, what would be a fair amount to charge someone if they wish to pay their yearly rental in advance?
Solution
A = 1,500, n = 12, i = 0.09/12 = 0.0075. The discount factor from Table 6.2 is 11.43491:
A = R × discount factor × (1 + i)
A = $1,500 × 11.43491 × 1.0075
A = $17,281 (to the nearest dollar)