An annuity is a series of payments made at fixed intervals of time. Examples of annuities
are regular deposits to a savings account, monthly home mortgage payments, monthly insurance
payments and pension payments. Annuities are classified by the frequency of payment dates.
The payments may be made weekly, monthly, quarterly, yearly, or at any other interval
of time.
Valuation The valuation of an annuity entails concepts
such as time value of money, interest rate, and future value.
Annuity-immediate If the number of payments is known in advance,
the annuity is an annuity-certain. If the payments are made at the end of the time periods,
so that interest is accumulated before the payment, the annuity is called an annuity-immediate,
or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before
being paid.
The present value of an annuity is the value of a stream of payments, discounted by the
interest rate to account for the fact that payments are being made at various moments
in the future. The present value is given in actuarial notation by:
where is the number of terms and is the per period interest rate. Present value is linear
in the amount of payments, therefore the present value for payments, or rent is:
In practice, often loans are stated per annum while interest is compounded and payments
are made monthly. In this case, the interest is stated as a nominal interest rate, and
. The future value of an annuity is the accumulated
amount, including payments and interest, of a stream of payments made to an interest-bearing
account. For an annuity-immediate, it is the value immediately after the n-th payment.
The future value is given by:
where is the number of terms and is the per period interest rate. Future value is linear
in the amount of payments, therefore the future value for payments, or rent is:
Example: The present value of a 5 year annuity with nominal annual interest rate 12% and
monthly payments of $100 is:
The rent is understood as either the amount paid at the end of each period in return for
an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an
interest-bearing account at the end of each period when the amount PV is invested at time
zero, and the account becomes zero with the n-th withdrawal.
Future and present values are related as:
and
Annuity-due An annuity-due is an annuity whose payments
are made at the beginning of each period. Deposits in savings, rent or lease payments,
and insurance premiums are examples of annuities due.
Because each annuity payment is allowed to compound for one extra period. Thus, the present
and future values of an annuity-due can be calculated through the formula:
and
where are the number of terms, is the per term interest rate, and is the effective rate
of discount given by . Future and present values for annuities due
are related as:
and
Example: The final value of a 7 year annuity-due with nominal annual interest rate 9% and monthly
payments of $100:
Note that in Excel, the PV and FV functions take on optional fifth argument which selects
from annuity-immediate or annuity-due. An annuity-due with n payments is the sum
of one annuity payment now and an ordinary annuity with one payment less, and also equal,
with a time shift, to an ordinary annuity. Thus we have:
Perpetuity A perpetuity is an annuity for which the payments
continue forever. Since:
even a perpetuity has a finite present value when there is a non-zero discount rate. The
formula for a perpetuity are:
where is the interest rate and is the effective discount rate.
Proof of annuity formula To calculate present value, the k-th payment
must be discounted to the present by dividing by the interest, compounded by k terms. Hence
the contribution of the k-th payment R would be R/(1+i)^k. Just considering R to be one,
then:
We notice that the second factor is an infinite geometric progression of the form,
therefore,
Similarly, we can prove the formula for the future value. The payment made at the end
of the last year would accumulate no interest and the payment made at the end of the first
year would accumulate interest for a total of years. Therefore,
Amortization calculations If an annuity is for repaying a debt P with
interest, the amount owed after n payments is:
because the scheme is equivalent with borrowing the amount to create a perpetuity with coupon
, and putting of that borrowed amount in the bank to grow with interest .
Also, this can be thought of as the present value of the remaining payments:
See also fixed rate mortgage. Example calculations
Formula for Finding the Periodic payment(R), Given A:
R = Am) )〗^(-(n-1))m)) Examples:
Find the periodic payment of an annuity due of $70000, payable annually for 3 years at
15% compounded annually. R = 700001) )〗^(-(3-1))1))
R = 70000/2.625708885 R = $26659.46724
Find the periodic payment of an annuity due of $250700, payable quarterly for 8 years
at 5% compounded quarterly. R= 2507004) )〗^(-(32-1))4))
R = 250700/26.5692901 R = $9435.71
Finding the Periodic Payment(R), Given S: R = S\,m) )〗^(n+1)-1)m)-1)
Examples: Find the periodic payment of an accumulated
value of $55000, payable monthly for 3 years at 15% compounded monthly.
R=5500012) )〗^(36+1)-1)12)-1) R = 55000/45.67944932
R = $1204.04
Find the periodic payment of an accumulated value of $1600000, payable annually for 3
years at 9% compounded annually. R=16000001) )〗^(3+1)-1)1)-1)
R = 1600000/3.573129 R = $447786.80
Other types Fixed annuities – These are annuities with
fixed payments. The insurance company guarantees a fixed return on the initial investment.
Fixed annuities are not regulated by the Securities and Exchange Commission.
Variable annuities – Registered products that are regulated by the SEC in the United
States of America. They allow direct investment into various funds that are specially created
for Variable annuities. Typically the insurance company guarantees a certain death benefit
or lifetime withdrawal benefits. Equity-indexed annuities – Annuities with
payments linked to an index. Typically the minimum payment will be 0% and the maximum
will be predetermined. The performance of an index determines whether the minimum, the
maximum or something in between is credited to the customer.
See also Amortization calculator
Fixed rate mortgage Life annuity
Perpetuity Time value of money
References
Samuel A. Broverman. Mathematics of Investment and Credit, 5th Edition. ACTEX Academic Series.
ACTEX Publications. ISBN 978-1-56698-767-7. Stephen Kellison. Theory of Interest, 3rd
Edition. McGraw-Hill/Irwin. ISBN 978-0-07-338244-9.