MATH 1003 Calculus and Linear Algebra (Lecture 3)
Albert Ku
HKUST Mathematics Department
Albert Ku (HKUST) MATH 1003 1 / 16
Outline
1 Future Value of an Annuity
2 Sinking Fund
Albert Ku (HKUST) MATH 1003 2 / 16
Future Value of an Annuity
Future Value of an Annuity
Definition
An annuity is a sequence of equal periodic payments. We call it an
ordinary annuity if the payments are made at the end if each time interval.
The amount, or future value, of an annuity is the sum of all payment plus
all interest earned.
Remark
In this course, all annuities are assumed to be ordinary.
Albert Ku (HKUST) MATH 1003 3 / 16
Future Value of an Annuity
An Example
Example
Suppose you deposit $100 every 6 months into an account that pay 6%
compounded semiannually. If you make six deposits, one at the end of
each interest payment period, over 3 years, how much money will be in the
account after the last deposit is made?
Albert Ku (HKUST) MATH 1003 4 / 16
Future Value of an Annuity
Solution
Idea: We compute the future value of each payment one by one and then
sum them up.
0.06 5
Future value of the 1st payment: 100(1 + )
2
0.06 4
Future value of the 2nd payment: 100(1 + )
2
0.06 3
Future value of the 3rd payment: 100(1 + )
2
······
Future value of the 6th payment: 100
Albert Ku (HKUST) MATH 1003 5 / 16
Future Value of an Annuity
Therefore, the future value of the annuity is
0.06 0.06 2 0.06 5
FV = 100 + 100(1 + ) + 100(1 + ) + · · · + 100(1 + )
2 2 2
This type of sum is called the geometric sum. It can be conveniently
computed by the following formula:
a(r n − 1)
a + ar + ar 2 + · · · + ar n−1 =
r −1
0.06
Now, let a = 100 and r = 1 + 2 . Then by the formula, we have
0.06 6
(1 + 2 ) −1
FV = 100 · 0.06
= $646.84
2
Albert Ku (HKUST) MATH 1003 6 / 16
Future Value of an Annuity
Then how much interest is earned in the annuity?
Interest = FV − Sum of all payments = 646.84 − 600 = $46.84
Remark
If the annuity is not ordinary - For example, payment is made at the
beginning of each period, the method of calculating its future value is
similar but value will be slightly different.
Albert Ku (HKUST) MATH 1003 7 / 16
Future Value of an Annuity
Formula for the Future Value of an Annuity
We now introduce the formula of the future value of an annuity in the
term of the notations used in finance.
Theorem
Let FV = future value, PMT = periodic payment,
i = interest rate per period and n = number of payments. We have
FV = PMT + PMT (1 + i) + PMT (1 + i)2 + · · · + PMT (1 + i)n−1
(1 + i)n − 1
= PMT = PMTsnei
i
where
(1 + i)n − 1
snei = .
i
Albert Ku (HKUST) MATH 1003 8 / 16
Future Value of an Annuity
Another Example
Example
What is the future value of an annuity at the end of 20 years if $2,000 is
deposited each year into an account earning 8% compounded annually?
How much of this value is interest?
Solution
(1 + 0.08)20 − 1
FV = 2000 · = $91523.93
0.08
Interest earned = 91523.93 − 20 × 2000 = $51523.93
Albert Ku (HKUST) MATH 1003 9 / 16
Sinking Fund
Sinking Fund
Any account that is established for accumulating funds to meet future
obligations or debts is called a sinking fund.
Example
Suppose the parents of a newborn child decide that on each of the child’s
birthdays up to the 17th year, they will deposit $PMT in an account that
pays 6% compounded annually. The money is to be used for the college
expenses. What should the annual deposit $PMT be in order for the
amount in the account to be $80,000 after 17th deposit?
Albert Ku (HKUST) MATH 1003 10 / 16
Sinking Fund
Solution
Since the parents hope that the future value of all the periodic deposits
they will made, i.e. the annuity, can cover the child’s college expenses,
then we should set FV = $80000, n = 17 and i = 0.06. Hence we obtain
(1 + 0.06)17 − 1
80000 = PMT
0.06
Solving for PMT , we get PMT = $2835.58.
Albert Ku (HKUST) MATH 1003 11 / 16
Sinking Fund
Formula for Sinking Fund Payment
Suppose the payments of a sinking fund are to be made in the form of an
ordinary annuity, then we obtain the formula for the sinking fund payment:
i FV
PMT = FV = .
(1 + i)n − 1 snei
Albert Ku (HKUST) MATH 1003 12 / 16
Sinking Fund
More Examples
Example
Lion bank offered a money market account with an APY of 5%.
(a) If interest is compounded monthly, what is the annual interest rate?
(b) If a company wishes to have $1,000,000 in this account after 8 years,
what equal deposit should be made each month?
Albert Ku (HKUST) MATH 1003 13 / 16
Sinking Fund
Solution
(a) Given the APY, we need to find the annual interest rate r :
r 12
(1 + ) − 1 = 0.05
12
⇒ r = 4.89%
(b) FV = 1000000 and PMT is the unknown period payment amount.
Then we have
0.0489
12
PMT = 1000000 · 0.0489 96
(1 + 12 ) −1
⇒ PMT = $8532.62
Albert Ku (HKUST) MATH 1003 14 / 16
Sinking Fund
Example
Peter deposits $1000 monthly into MPF (a retirement plan) that earn 3%
compounded monthly. Due to a change in employment, these deposits
stop after 10 years, but the account continues to earn interest until Peter
retires 25 years after the last deposit was made. How much is in the
account when Peter retires?
Albert Ku (HKUST) MATH 1003 15 / 16
Sinking Fund
Solution
First, we need to calculate the future value of Peter’s MPF after 10 years:
0.03 10×12
(1 + 12 ) −1
FV = 1000 · 0.03
12
⇒ FV = $139741.4189
Then, in the next 25 years, no deposit was made. Therefore, the total
amount of money Peter eventually has can be calculated by the
straightforward compound interest formula:
0.03 25×12
Total amount = 139741.4189(1 + ) = $295555.83
12
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