Future value of an ordinary annuity:
. .
. .
. , .
A 500
0 01 1 0 01 1
500 0 01 0 488864 500 48 8864 24 443 19
40
= + -
=
= =
^ h
R T
SSS ;
6
V X
WWW E
@
The amounts are off by a few cents, which is due to rounding, of course. (If you need a refresher on computing compound interest, refer to Chapter 9.)
Computing the Payout from an Annuity
As I mention earlier in this chapter, an annuity is set up to collect money and allow that money to earn interest over some period of time. Annuities allow people to make donations to organizations; regular payouts are made as donations over many years — or forever. Scholarships and business-starting grants are often funded with annuities.
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Chapter 10: Investing in the Future
When you have an annuity, you can start the payout program as soon as the money is deposited, or you can defer the payments for a number of years, allowing the initial deposit to grow even more and the account to increase in size. You can even set up an annuity with payments in perpetuity,meaning that they pay out forever; as long as only the interest is paid out, the annuity can keep providing money.
The payouts of an annuity are just the reversal of the payment into an annu- ity. In the earlier section, “Finding the present value of an annuity,” you see how a lump sum can be equal to making regular payments. In fact, depositing a certain amount of money can accumulate as much in an account as paying in over a number of years. The payout of an annuity is like taking that lump sum and parceling it out in regular, periodic payments, until it’s gone. The diminishing amount in the account still earns interest, but the amount of growth of the account gets smaller and smaller over time.
Receiving money from day one
If you make a lump sum contribution into an annuity, and if you want to with- draw regular amounts of money from the annuity, how much can you take out, and how long will the money last? Both of these questions are dependent on each other, and both are dependent on how much money is in the account.
But one thing is obvious: The less you take out each time, the longer the money will last. Be frugal, my friend!
Figuring out the amount of upfront money that’s needed
When an endowment (a type of annuity) is made, the benefactor often has a goal in mind — an amount of money that he or she would like to see given each year for a certain number of years. With that goal in mind, the benefac- tor then determines how much to invest or deposit so that the desired alloca- tions can be made.
Consider this example: Imagine that an entrepreneur wants to give an endow- ment to a small business catalyst to help local fledgling businesses get a good start. The arrangement says that the entrepreneur will give $5,000 quarterly to young businesses over the next 5 years. The endowment is invested in an account earning 5% compounded quarterly. Upfront, how much did the entre- preneur have to put into the account?
The present value of 20 payments of $5,000 earning interest at 1.25% per quarter (5% ÷ 4 = 1.25%) is found with the formula for the present value of an ordinary annuity (see the earlier section, “Finding the present value of an annuity,” for more on this formula):
, . .
, . $ , .
P R i
i
1 1
5 000
0 0125 1 1 0 0125 5 000 17 599316 87 996 58
n 20
= - +
= - +
= =
- -
^ h ^ h
R T SSS
R T SSS 6
V X WWW
V X WWW
@
It looks like the benefactor gave approximately $88,000 to help the new businesses.
Discovering how much time the annuity will last
If you deposit a lump sum into an annuity, regular amounts can be withdrawn from the account over a period of time until the money is gone. So you’ll likely want to figure out how long that annuity will last based on your withdrawals.
Say, for example, that Hank gets a large insurance settlement and deposits it in an account earning 6% compounded monthly. He arranges for monthly payments of $5,000 to be made from that annuity to help offset his business expenses. If the insurance settlement was for $400,000, how long will he be able to get the monthly payments?
Using the formula for the present value of an ordinary annuity, you can solve for the number of payments, n. The interest rate per month is 6% ÷ 12 = 0.5%, which is 0.005. The regular payments, R, are the amounts being paid to Hank each month. Here’s what your equation should look like:
, ,
. 400 000 5 000 .
0 005 1 1 0 005 n
= R -^ + h- T
SSS
V X WWW
Simplify the equation by dividing each side by 5,000 and then by multiplying each side by 0.005, like so:
, ,
,
, .
.
. .
. .
. .
. .
5 000 400 000
5 000 5 000
0 005 1 1 0 005
0 005 80
0 005 1 1 0 005
0 005 0 4 1 1 0 005
1 0 005 0 6
n
n
n
n
# #
=
- +
= - +
= - +
+ =
-
-
-
-
^
^
^
^
h
h h h
R T SSS
R T SSS
V X WWW
V X WWW
Now take the natural log of each side of the equation, which allows you to bring the exponent, –n, down as a multiplier. Divide each side by ln(1.005), and use a scientific calculator to do the computation:
145
Chapter 10: Investing in the Future
. .
. .
.
. .
.
ln ln
ln ln
ln ln n
n n
1 005 0 6
1 005 0 6
1 005
0 6 102 42 102 42
n
. .
=
- =
- = -
^ - ^
^ ^
^
^
h h
h h
h h
Are some of these steps a bit unfamiliar to you? If it has been a while since you’ve seen an algebraic solution like this, refer to “Doubling your money, doubling your fun,” earlier in this chapter, where you can see a similar process used to solve an equation.
The value of n, the number of monthly payments, is about 102.42. Divide that amount by 12 months, and you see that Hank will receive about 8.5 years of monthly payments of $5,000 from his insurance settlement. $5,000 ×102.42 =
$512,100, which is the total amount of money that Hank will receive during those 8.5 years. The settlement was for $400,000; the difference between
$512,100 and $400,000 is $112,100. So about $112,100 of that total is interest.
Deferring the annuity payment
You can deposit an amount of money into an account with the understanding that regular payments will be made, but only after several years. You may have deposited the money in the account all at once, or you may have accu- mulated it over a period of years as an annuity. In any case, the amount in the account grows with compound interest before being disbursed. To figure the present value of a deferred annuity, you need to determine the number of payment periods in the payout and the number of payment periods the payout will be deferred before starting the disbursement.
The present value of a deferred annuity is found with this equation:
P R
i i
i 1
1 1
n n
1 2
= +
- + -
^
^ h R h
T SSS
V X WWW
where Ris the regular payout amount, iis the interest per period, n1is the number of periods the annuity is deferred, and n2is the number of periods that payments are to be made from the annuity.
The formula for deferred payments combines two other formulas: the present value of an ordinary annuity (to determine the amount of money needed at the beginning of the disbursement period) and the present value of that pre- sent value (to determine how much you need to deposit right now to have the required amount some time in the future).
Check out this example: A trust fund is set up to pay for the college educa- tion of a 2-year-old boy. With the arrangement, $60,000 will be paid to the stu- dent per year for 5 years, starting when he’s 18 years old. The fund earns 5.5% compounded quarterly. The payout will also be quarterly. What’s the present value of this fund?
The quarterly payout amounts are $60,000 ÷ 4 = $15,000. The interest rate per quarter is 5.5% ÷ 4 = 1.375%, which is 0.01375 in decimal form. The number of periods that the annuity is deferred is 16 ×4 = 64, and the number of periods over which the payment will be made is 5 ×4 = 20. Putting all the numbers in their proper places, you get this equation:
, . .
.
, .
.
, . , .
P 15 000
0 01375 1 0 01375 1 1 0 01375
15 000
0 0329517 0 2390035
15 000 7 253146 108 797 19
64 20
=
+ - +
=
= =
-
^
^
h R h
T SS S
; 6
V
X WW W E
@ So the present value is $108,797.