4E Savings Annuities

A savings annuity is a savings plan wherein equal regular deposits, or payments, are made into an account, and these deposits earn compound interest. The time between payments is called the payment period of the annuity; typical payment periods are one month, two weeks, and one week. Often the payment period is set up to coincide with the time between the depositor's paychecks, so that a fixed amount can be automatically set aside from each check and applied to the annuity. As we will see, long-term annuities can be surprisingly rewarding to an investor, in that the amount taken out of the annuity can far exceed the amount put into it.

Some annuities are open-ended, allowing the investor to continue deposits indefinitely, while other annuities have a fixed expiration date, when the investor is to cease all deposits. For the latter type of annuity, the term of the annuity is the time from the beginning of the first payment period to the end of the last payment period. When this term is finished the annuity is said to have expired, at which time the investor either withdraws the money or reinvests it in some other agreed-upon plan.

We will consider ordinary annuities, in which the payment is deposited at the end of each time period. (The other kind is called an annuity due, wherein the deposit is made at the beginning of the time period; there is not much difference between the two kinds, but to avoid confusion we will discuss only the ordinary kind.) We will consider also only simple annuities, in which the compounding period is the same as the period between payments.

Before discussing the general formulas for annuities, we look at an illustrative example. Suppose that Elaine, planning a mainland vacation next year, decides to open a one-year simple ordinary annuity at American Savings to help with the financing. She will make regular monthly payments of $100, at an annual interest rate of 6%, and the annuity will begin on January 1 and expire on December 31 of the same year. We will calculate the amount in the annuity at the end of the year.

Since the annuity is simple and payments are made monthly, interest likewise will be compounded monthly. Since the annuity is ordinary, Elaine will make the payments at the end of each month. Thus the first payment will be made January 31, the second February 28, the third on March 31, and so on, until the last payment is made on December 31, the same day that the annuity expires. The annual interest rate is r = 6%, and so the monthly rate is R = r/12 = 6%/12 = .5% = .005; as we have seen, this means that all deposits are multiplied by the factor 1.005 for each month they remain in the bank.

In calculating the value of the annuity on its expiration day, it is perhaps easier to begin with the last payment and work backwards. Since the last payment is made on the expiration day, it earns no interest at all - thus the December 31 payment contributes only $100 to the expiration value of the annuity. The next to last payment however, made at the end of November, has been in the bank for one month at the expiration date and thus is worth $100 · 1.005. The payment made at the end of October has been in the bank for two months at the expiration date, and it is worth $100 · 1.005². We proceed with this logic until we reach the first payment, made on January 31; it has been in the bank for 11 months on the expiration date and is worth $100 · 1.005¹¹. Finally, the total value of the annuity on expiration is the sum of the contributions from all the monthly payments - namely, the amount

A = $100 + ($100 · 1.005) + ($100 · 1.005²) + … + ($100 · 1.005¹¹) .

The above calculation is rather tedious to perform on a hand calculator, as it involves many multiplications and then a big addition - however, there is a handy shortcut mathematical formula for the final sum that avoids most of this effort. If we specify the notation

p = payment amount		r = annual interest rate
n = # of payments per year		R = r/n = periodic interest rate
t = # of years		A = expiration amount
N = n · t = total # of payments

then the formula for the amount in the annuity at expiration is

In our example with Elaine, the payment amount is p = $100, the periodic interest rate is R = .06/12 = .005, and the total number of payments is N = 12; thus the amount in Elaine's annuity at expiration will be

Notice that the total amount that Elaine pays into the annuity is 12 · $100 = $1200. Hence the amount of interest that her money earns over the life of the annuity is the difference

I = $1233.56 − $1200.00 = $33.56 .

example 1

On his 30^th birthday, Carlos began depositing $50 twice monthly into a 40 year simple annuity, paying 4.5% annual interest. We will calculate

(a) the value of the annuity at expiration,

(b) the total amount Carlos will pay into the annuity,

Since Carlos makes 2 payments each month, the number of payments per year is n = 2 · 12 = 24. The number of years is t = 40, and so the total number of payments over the life of the annuity is N = 24 · 40 = 960. The annual interest rate is r = 4.5% = .045, and the periodic interest rate is

R = r/n = .045/24 = .001875 .

(This figure represents the percentage interest earned - as a decimal - over each half-month period.) The amount of each payment is p = $50. Substituting into the formula for the amount A at expiration, we find that after the 40 year term the annuity will be worth

Over the life of the annuity Carlos makes 960 payments of $50 each; thus the amount he will pay into the annuity is

960 · $50 = $48,000 .

The amount of interest Carlos earns is the difference between what he gets back and what he pays in - namely,

I = $134,385.60 − $48,000.00 = $86,385.60 .

As you can see, the amount of interest Carlos earns is quite striking, even at the relatively modest annual interest rate of 4.5%. His $48,000 investment earns him more than $86,000 in interest.

example 2

Suppose that, on their wedding day, Jose and Josephine open a 30 year simple ordinary annuity. We calculate the worth of the annuity on expiration when they are old and gray, under the following conditions:

payments are monthly at $300 each, and the annual interest rate is 7%,
payments are twice a month at $100 each, and the annual interest rate is 9%,
payments are every two months at $500 each, and the annual interest rate is 6%,
payments are every six months at $1000 each, and the annual interest rate is 8%.

In each case the number of years is t = 30. The other relevant figures for the four cases are listed below:

a)	n = 12	b)	n = 24	c)	n = 6	d)	n = 2
	r = .07		r = .09		r = .06		r = .08
	p = $300		p = $100		p = $500		p = $1000

For each of the four cases we substitute into the formula

and for the four amounts at expiration we get

You can check that the amounts put into the annuity, in the four cases, are

a)		$300 · 360 = $108,000 ,
b)		$100 · 720 = $72,000 ,
c)		$500 · 180 = $90,000 ,
d)		$1000 · 60 = $60,000 ,

while the corresponding interest totals earned in the four cases are

a)		I = $365,991.30 − $108,000.00 = $257,991.30 ,
b)		I = $368,127.47 − $72,000.00 = $296,127.47 ,
c)		I = $249,790.10 − $90,000.00 = $159,790.10 ,
d)		I = $237,990.69 − $60,000.00 = $177,990.69 .

example 3

Emily, having just embarked on her teaching career at age 25, intends to open a simple annuity at a 6% annual interest rate, taking automatic monthly payments from her salary. If she wants the annuity to be worth one million dollars when she retires at age 65, what should be her monthly payments into the annuity?

We have n = 12 payments per year, r = 6% = .06, and the term of the annuity will be t = 65 − 25 = 40 years. She wants the final amount to be A = $1,000,000. We substitute into

and get

Solving this equation for p, we find that Emily's monthly payment must be

p = $1,000,000 ÷ 1991.4907 = $502.14 .

EXERCISES 4E

Louise, wanting to save money to begin her own business, decides to open a three-year ordinary simple annuity, making monthly payments of $500 at a 7.2% annual interest rate. Determine
1. the value of the annuity in three years at expiration,
2. the total amount she will pay into the annuity,
3. the total interest her money will earn.
4. Suppose that Louise discovers an additional expense, and now estimates that it will cost her $25,000 cash to begin her business in 3 years. What should be her revised payment amount?

The Nemoto family, stung by Christmas bills the previous year, have elected to join a Christmas club. Beginning in March they will deposit $50 twice monthly into a simple ordinary annuity at First Hawaiian Bank. Their money earns 4.8% annual interest, and the annuity expires at the end of November of the same year, just in time to start their Christmas shopping. Calculate
1. the amount in the annuity at the end of November,
2. the total of all their deposits,
3. the amount of interest they will earn on their deposits.

Mr. and Mrs. Sharpe are elated, having just received a check from the Bank of Hawaii for the value of their recently expired ordinary simple annuity. They began the 50-year annuity promptly after their graduation from U.H. Calculate the size of the check, under the assumption that the annual interest rate was
1. 6%, with payments monthly at $100 each,
2. 5.2%, with payments weekly at $30 each,
3. 4%, with payments quarterly at $200 each,
4. 4.8%, with payments twice a month at $75 each.
Also, in each of the four scenarios determine the amount of interest their money earned.
Suppose you open an ordinary simple annuity today at 6.5% annual interest, making payments at the end of every week. If you want the annuity to be worth $500,000 when you retire in 45 years, what should be the size of your payments?