The mathematical analysis of annuities is very important to make the financial projections that the company needs in the study of new projects.
Deferred annuities
A deferred annuity is one in which the first payment is made after a certain number of periods have elapsed.
Example 1
A debt of $ 800,000 is to be paid off with 20 quarterly payments of $ R each. If the first payment is made exactly one year after the money was lent, calculate R with a rate of 36% CT.
Solution
It is observed that the first payment is in period 4 that corresponds to the end of the first year. The annuity must start at point 3 and end at point 23, in addition, its present value must be transferred to point 0 where the focal date has been set. The value equation will be:
800,000 = R (1 - (1 + 0.9) -20 / 0.09) (1.09) -3
R = $ 113,492.69
Annuities
- Ordinary Overdue Deferred Perpetual General
Perpetual annuities
An annuity that has an infinite number of payments is called an infinite or perpetual annuity, in reality, infinite annuities do not exist, because in this world everything has an end, but it is supposed to be infinite when the number of payments is very large.
This type of annuity occurs when a capital is placed and only interest is withdrawn.
The perpetual annuity is represented:
Obviously, there is only present value that comes to be finite, because the final value will be infinite
VP = Lim n – µ R (1- (1 + i) -n) / i)
VP = R Lim n – µ 1-0 / i
VP = R / i
Example 1
Find the present value of a perpetual income of $ 10,000 per month, assuming an interest of 33% CM.
Solution
i = 33% / 12
i = 2.75%
VP = R / i
PV = 10,000 / 0.0275
PV = 363,636.36
General annuities
Ordinary and advance annuities are those in which the interest period coincides with the payment period. In the case of general annuities, the payment periods do not coincide with the interest periods, such as a series of quarterly payments with an effective semi-annual rate.
To carry out a reliable financial analysis, it is necessary to apply all the necessary and correct tools in each case
A general annuity can be reduced to a simple annuity, if we match the time periods and the interest periods, there are two ways you can do it:
1. The first way is to calculate equivalent payments, which must be made in accordance with the interest periods. It consists of finding the value of payments that, made at the end of each interest period, are equivalent to the single payment made at the end of a payment period.
2. The second way is to modify the rate, making use of the concept of equivalent rates, to match the interest and payment periods.
Example 1
Find the amount of S of 30 quarterly payments of $ 25,000 each assuming a rate of 24% CM. doing it by both methods.
Solution
1. A. The payment of $ 25,000 at the end of a quarter is replaced by payments at the end of each month like this:
B. Then there is a simple annuity, because the payments are monthly of $ R each and the rate of
i = 24% / 12
i = 2%
C. It then follows that:
25,000 = R (1 + 0.02) 3) -1 / 0.02
R = 8,168.87
D. The number of monthly payments will be 30 x 3 = 90, so S will be:
S = 8,168.87 (1 + 0.02) 90 -1 / 0.02
S = 2,018,990
2. A. We are looking for a quarterly effective rate equivalent to 24% CM
(1 + 0.02) 12 = (1 + i) 4
i = 6.1208% Quarterly cash
B. We then have:
S = 25,000 (1 + 0.061208) 30 -1 / 0.061208
S = 2,018,990
