2. Annuities
1. Basic Annuities
1.1 Introduction
Annuity: A series of payments made at equal intervals of time.
Examples: House rents, mortgage payments, installment payments on
automobiles, and interest payments on money invested.
1
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10(155.82) − 1000 = $558.20. Annuity-Due The payments are made at the beginning of the period. 1) Present values more than one period before the first payment.
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2. Annuities
1. Basic Annuities
1.1 Introduction
Annuity: A series of payments made at equal intervals of time.
Examples: House rents, mortgage payments, installment payments on automobiles, and interest payments on money invested.
Annuity-certain: An annuity such that payments are certain to be made for a fixed period of time.
Term: The fixed period of time for which payments are made
Contingent annuity: An annuity under which the payments are not certain to be made. A common type of contingent annuity is one in which payments are made only if a person is alive (Life Annuity).
Payment period: Interval between annuity payments.
A verbal interpretation to the formula for an|
1 = ian| + vn.
Relationship between sn| and an|.
sn| = an|(1 + i)n
and 1 an|
sn|
- i
This relationship can be derived as follows:
1 sn|
- i =
i (1 + i)n^ − 1
- i
i + i(1 + i)n^ − i (1 + i)n^ − 1
i 1 − vn
=
an|
Example 1 Find the present value of an annuity which pays $500 at end of each half-year for 20 years if the rate of interest is 9% convertible semiannually.
The answer is
500 a 40 |. 045 = 500(18.4016) = $9200. 80.
Example 2 If a person invests $1000 at 8% per annum convertible quarterly, how much can be withdraw at end of every quarter to use up the fund exactly at the end of 10 years.
Let R be the amount of each withdrawal. The equation of value at the date of investment is Ra 40 | 0. 02 = 1000.
Thus, we have
R =
a 40 | 0. 02
(3) Let the level payment be R. An equation of value for R at the inception of the loan is Ra 10 | = 1000
which gives
R =
a 10 |
Thus the total amount of interest paid is equal to
1.3. Annuity-Due
Annuity-Due The payments are made at the beginning of the period.
The present value of the annuity a¨n|:
a ¨n| = 1 + v + v^2 + · · · + vn−^1 =
1 − vn 1 − v
1 − vn iv
1 − vn d
The accumulated value of the annuity s¨n|:
s ¨n| = 1 + i + (1 + i)^2 + · · · + (1 + i)n−^1 + (1 + i)n
= (1 + i)
(1 + i)n^ − 1 1 + i − 1
(1 + i)n^ − 1 iv
=
(1 + i)n^ − 1 d
Example 4 An investor wishes to accumulate $1000 in a fund at end of 12 years. To accomplish this the investor plans to make deposits at the end of each year, the final payment to be made one year prior to the end of the investment period. How large should each deposit be if the fund earns 7% effective. Since we are interested in the accumulated value one year after the last payment, the equation of value is Rs¨ 11 | = 1000
where R is the annual deposit. Solving for R we have
R =
s¨ 11 |
- 07 s 11 |
1.4. Annuity Value on Any Date
Present values more than one period before the first payment.
Accumulated values more than one period after the last payment date.
Current values between the first and last payment dates.
Present values more than one period before the first payment date
This type of annuity is often called a deferred annuity, since the payments commence only after a deferred period. In general, the present value annuity- immediate deferred for m periods with a term n periods after the deferred period is vman| = an+m| − am|.
1.5. Perpetuities
A perpetuity is an annuity whose payments continue forever, i.e. the term of the annuity is not finite.
a∞| The present value of a perpetuity-immediate
a∞| = v + v^2 + v^3 + · · ·
=
v 1 − v
v iv
=
i
provide v < 1 , which will be the case if i > 0. Alternatively we have
a∞| = (^) nlim→∞ an| = (^) nlim→∞
1 − vn i
i
since (^) nlim→∞ vn^ = 0.
By an analogous,for a perpetuity-due, we have
a ¨∞| =
d
It should be noted that accumulated values for perpetuities do not exist, since the payments continue forever.
Example 5 A leaves an estate of $100,000. Interest on the estate is paid to beneficiary B for the first 10 years, to beneficiary C for the second 10 years, and to charity D thereafter. Find the relative shares of B, C, and D in the estate, if it assumed the estate will earn a 7% annual effective rate of interest.
The value of B’s share is
7000 a 10 | = 7000(7.0236) = $49, 165
to the nearest dollar.
1.6. Nonstandard term and Interest rates
Consider first what the symbol an+k|, where n is a positive integer and 0 < k < 1.
an+k| =
1 − vn+k i
1 − vn^ + vn^ − vn+k i
= an| + vn+k
[
(1 + i)k^ − 1 i
]
It is the present of an n-period annuity-immediate of 1 per period plus a final payment at time n + k of (1+i)
k− 1 i.
(1+i)k− 1 i is an irregular payment, seems rather unusual.^ A payment^ k^ is a good approximation to it.
In practice many courts use an annuity-certain for a person’s life expectancy in measurement of economic damages in personal injury and wrongful death lawsuits.
1.7. Unknown Time
In general, problems involving unknown time will not produce exact integral answer for n
These problems could be handle along the liens of the section above in which a small payment is made during the period following the last regular payment. However it is seldom done in practice. The final smaller payment date is not convenient for either party to the transaction.
What is usually done in practice is either to make a smaller additional payment at the same time as the last regular payment, in effect making a payment larger than the regular payment ( ballon payment), or to make a smaller payment one period after the last regular payment (drop payment).
- The equation of value at end of 15th year is
100¨s 14 | + X 2 = 1000(1.05)^15
Thus, X 2 = 1000(1.05)^15 − 100(s 15 | − 1) = 2078. 93 − 2057 .86 = $21. 07. It should be noted that 20 .07(1.05) = 21. 07 , or that in general X 1 (1 + i) = X 2.
- In this case the equation of value becomes
100 a14+k| = 1000
where 0 < k < 1. This can be written as
1 − vn+k i
= 10 or v14+k^ = 1 − 10 i =. 5.
Thus (1.05)14+k^ = 2. Hence k =. 2067. Thus, the exact final irregular payment is X 3 = 100
(1.05).^2067 − 1
paid at time 14.2067. The exact answer obtained between the answer to 1 and 2, as we expect.
1.8. Unknown rate of Interest
There are three methods to use in determining an unknown rate of interest.
- Solve for i by algebraic techniques.
an| = v + v^2 + · · · + vn
is an nth degree polynomial in v. If the roots of this polynomial can be determined algebraically, then i is immediately determined. We can express an| and (^) a^1 n| in terms of i
an| = n −
n(n + 1) 2!
i +
n(n + 1)(n + 2) 3!
i^2 + · · · ,
an|
n
[
n + 1 2
i +
n^2 − 1 12
i^2 + · · ·