Annuities Certain
1Introduction
2Annuities-immediate
3Annuities-due
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Annuities: Certain vs. Contingent and Immediate vs. Due, Lecture notes of Game Theory
Definitions and examples of annuities, distinguishing between annuities certain and contingent, and annuities-immediate and annuities-due. It explains the concept of present value and the equations for calculating the present value of annuities, using examples of loan repayments.
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2021/2022
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Annuities Certain
1 Introduction
2 Annuities-immediate
3 Annuities-due
Annuities Certain
1 Introduction
2 Annuities-immediate
3 Annuities-due
General terminology
- An annuity is a series of payments made
at specified intervals (e.g., yearly - whence the name) called the payment periods
for a certain (defined in advance) length of time
- (^) If the length of time is fixed (deterministic), then the annuity is called annuity certain
- (^) If not, then it is called a contingent annuity; an important example is the life annuity
- (^) If the length of time is deterministic and infinite, the annuity is called a perpetuity
General terminology
- An annuity is a series of payments made
at specified intervals (e.g., yearly - whence the name) called the payment periods
for a certain (defined in advance) length of time
- (^) If the length of time is fixed (deterministic), then the annuity is called annuity certain
- (^) If not, then it is called a contingent annuity; an important example is the life annuity
- (^) If the length of time is deterministic and infinite, the annuity is called a perpetuity
General terminology
- An annuity is a series of payments made
at specified intervals (e.g., yearly - whence the name) called the payment periods
for a certain (defined in advance) length of time
- (^) If the length of time is fixed (deterministic), then the annuity is called annuity certain
- (^) If not, then it is called a contingent annuity; an important example is the life annuity
- (^) If the length of time is deterministic and infinite, the annuity is called a perpetuity
General terminology
- An annuity is a series of payments made
at specified intervals (e.g., yearly - whence the name) called the payment periods
for a certain (defined in advance) length of time
- (^) If the length of time is fixed (deterministic), then the annuity is called annuity certain
- (^) If not, then it is called a contingent annuity; an important example is the life annuity
- (^) If the length of time is deterministic and infinite, the annuity is called a perpetuity
The definition
- If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate
- Caveat: The terminology may seem counter-intuitive...
- If the amount paid at the end of each period is equal to $1, we call this annuity the basic annuity-immediate
- In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity
The definition
- If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate
- Caveat: The terminology may seem counter-intuitive...
- If the amount paid at the end of each period is equal to $1, we call this annuity the basic annuity-immediate
- In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity
The definition
- If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate
- Caveat: The terminology may seem counter-intuitive...
- If the amount paid at the end of each period is equal to $1, we call this annuity the basic annuity-immediate
- In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity
Notation: Present value
- (^) an... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods
- Assume that the rate of interest is constant and equal to i in each payment period
- Writing the time 0 equation of value (with the help of a time-line), we get
an = v + v 2 + · · · + v n^ = v ·
1 − v n 1 − v
1 − v n i
- If we want to emphasize the exact value of the interest rate used, we write an i
Notation: Present value
- (^) an... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods
- Assume that the rate of interest is constant and equal to i in each payment period
- Writing the time 0 equation of value (with the help of a time-line), we get
an = v + v 2 + · · · + v n^ = v ·
1 − v n 1 − v
1 − v n i
- If we want to emphasize the exact value of the interest rate used, we write an i
Notation: Present value
- (^) an... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods
- Assume that the rate of interest is constant and equal to i in each payment period
- Writing the time 0 equation of value (with the help of a time-line), we get
an = v + v 2 + · · · + v n^ = v ·
1 − v n 1 − v
1 − v n i
- If we want to emphasize the exact value of the interest rate used, we write an i
Notation: Accumulated value
- sn... denotes the accumulated value at the time of the last payment of all the payments made by a basic annuity-immediate over the length of time equal to n periods, i.e.,
sn = a(n)an
or, equivalently,
an = v (n)sn
- In the case of compound interest with the fixed interest rate of i per period, we have that
sn i = (1 + i)nan i =
(1 + i)n^ − 1 i or, equivalently,
an i = v nsn i
Examples
I. Consider an annuity which pays $500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. ⇒
500 · a40 0. 045 = 500 · 18 .4016 = 9200. 80
II. Roger invests $1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? ⇒ We look at Roger’s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the $1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals $1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation
R · a40 0. 02 = 1000
We get that R = 36. 56