Time Value of Money
Annuity
Derivation of the Annuity Calculation Formula
Introduction
The purpose of this document is to explain the mathematical foundation for annuity based loans, to briefly discuss some of the advantages and disadvantages of this type of loans, and to provide the reader with sufficient knowledge to make informed decisions regarding annuity loans.
Definition of Annuity Payment
The re-payment of an annuity loan consists of a series of fixed payments (annuities) over a number of periods, usually expressed in years or months, that constitute the duration of the loan. Each annuity payment consists of principal (original loan amount) re-payment as well as interest payment. The sum of these components is constant. The proportions principal re-payment and interest payment vary during the course of the loan, because the loan re-payment each period is a fixed amount. As a result, the re-payment of the principal gradually increases from a small to a large proportion of the annuity, while the interest amount gradually changes in the opposite direction. Mortgage loans are frequently designed as annuity loans.
Advantages and Disadvantages of an Annuity Loan
As a result of paying a fixed amount every period, the re-payment of the principal is initially very slow. This is why building equity in property, such as a home, is very slow for a good part of the duration of an annuity loan. This is because the interest payment is high since the loan amount is still high, and both the interest payment and the principal re-payment must fit within the annuity. For those who prefer predicable and unchanging loan payments, an annuity loan is perhaps the preferred type of loan. However, the total cost of the loan in the form of interest payments is higher than for other types of loans with the same interest rate and duration. However, the high cost of the loan, relative to the original loan amount, can be significantly reduced by paying more than the agreed upon annuity (the fixed amount) every period. This is because the additional loan re-payment is applied directly to the re-payment of the principal, thereby reducing the amount of interest paid each period.
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Time Value of Money
Annuity
Derivation of the Annuity Calculation Formula
Variables
PV = The original loan amount, a k a known as the Present Value, denoted PV (all
amounts in this document are expressed in the duration of the loan. US$).
n = The number of periods, usually expressed in years or months, that constitute a = Annuity payment; a fixed amount paid at the end of each period of the duration
of the loan.
i=
The annual interest rate, percent.
R = A factor based on the annual interest rate, and defined as follows:
i R = 1 + 100
The derivation of the annuity calculation formula is performed in two steps:
(1)
Step 1: Calculation of the future value of the original loan amount PV ; that is, the value at the end of the loan duration at i percent annual interest rate. Step 2: Calculation of the sum of the future values of the individual loan re-payments (annuities) made at the end of each period for the duration of the loan. Step 1: The future value of
PV
after one period is
PV ⋅ R = PV ⋅ R1
After two periods, the future value of
(2)
PV
is
PV ⋅ R1 ⋅ R = PV ⋅ R 2
Consequently, after
(3)
n
periods, the future value of
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Time Value of Money
Annuity
PV
is
PV ⋅ R n −1 ⋅ R = PV ⋅ R n
(4)
Thus, the equation 4 represents the future value of the original loan amount (principal) at the end of the loan duration. This is the amount of money a borrower must repay in the form of fixed periodical (usually monthly) installments, known as annuities, during the course of the loan. Step 2: Since each loan re-payment (annuity) is made at the end of a particular period, one must calculate the future value of each annuity, and then sum them up in order to determine the collective future value of the annuities. The future value of each annuity is its value at the end of the last period of the loan duration. The duration of the loan is
n
periods. Since each annuity is made at the end of a period, the 1st annuity is paid at the end of the 1st period, denoted
n1
Since the interest rate
i
is constant for the duration of the loan, the future value
Ta1
of the 1st annuity
a1
after
n −1
periods is
i Ta1 = a1 ⋅ 1 + 100
By substituting
n −1
(5)
R
for
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i 1 + 100
the equation 5 can be written as
Ta1 = a1 ⋅ R n −1
Consequently, since the 2nd annuity
(6)
a2
is paid at the end of the 2nd period, its future value after
(n − 2)
periods is
Ta2 = a2 ⋅ R n −2
Finally, the future value of the last annuity
(7)
an
is
Tan = an ⋅ R n −n
The sum of the future values of the annuities
(8)
Tatot
is
Tatot = Ta1 + Ta2 + Ta3 + ...... + Tan
(9)
Note that the future value of each individual annuity includes compounded interest from the period following the period in which the annuity was paid, until the last period for the duration of the loan. Since the sum of the future values of the annuities must equal the future value of the original loan, it follows that
Tatot = PV ⋅ R n
The equation 9 can be written as
(10)
Tatot = a1 ⋅ R n −1 + a2 ⋅ R n− 2 + a3 ⋅ R n −3 + ...... + an ⋅ R n −n
But
(11)
a1 = a2 = a3 = ...... = an = a
Therefore, the equation 11 can be written as
(12)
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Tatot = a ⋅ R n−1 + a ⋅ R n − 2 + a ⋅ R n −3 + ...... + a ⋅ R n −n Tatot = a ⋅ R n −1 + a ⋅ R n− 2 + a ⋅ R n −3 + ...... + a ⋅1
Tatot = a ⋅ R n −1 + a ⋅ R n− 2 + a ⋅ R n −3 + ...... + a ⋅ R n− n = a ⋅ R n −1 + R n − 2 + R n −3 + ...... + 1
Based on the equation 10, the equation 15 can be expressed as
(13) (14)
(
)
(15)
PV ⋅ R n = a ⋅ R n −1 + R n − 2 + R n −3 + ...... + 1 −R
results in
(
)
(16)
Multiplying both sides in the equation 16 by the factor
(− R ) ⋅ PV ⋅ R n = (− R ) ⋅ a ⋅ R n −1 + R n − 2 + R n −3 + ...... + 1 − R ⋅ PV ⋅ R n − R ⋅ PV ⋅ R n
n −1 n−2 n −3
( = (−1) ⋅ R ⋅ a ⋅ ( R + R = a ⋅ (−R − R − R
n n −1
n−2
) + R + ...... + 1) − ...... − R ) ) )
(17) (18) (19)
Adding the equations 16 and 19 results in
PV ⋅ R n = a ⋅ R n −1 + R n − 2 + R n −3 + ...... + 1
(
(16) (19) (20)
− R ⋅ PV ⋅ R n = a ⋅ − R n − R n −1 − R n− 2 − ...... − R PV ⋅ R n ⋅ (1 − R ) = a ⋅ 1 − R n
(
(
)
Verification of the Equation 20
Adding the left hand sides of the equations 16 and 19 results in
PV ⋅ R n + − R ⋅ PV ⋅ R n = PV ⋅ R n − R ⋅ PV ⋅ R n = PV ⋅ R n ⋅ (1 − R )
Adding the right hand sides of the equations 16 and 19 results in
(
)
a ⋅ R n −1 + R n − 2 + R n−3 + ...... + 1 + a ⋅ − R n − R n −1 − R n − 2 − ...... − R = a + a ⋅ − R n
(
)
(
)
(
)
since all but one term on the right hand side in each of the equations 16 and 19 cancel each other out. Further modification results in
a + a ⋅ −Rn = a ⋅ 1 − Rn
(
)
(
)
Further, multiplying both
Consequently, the verification confirms the equation 20. sides of the equation 20 by the factor
( −1)
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results in
( −1) ⋅ PV ⋅ R n ⋅ (1 − R ) = ( −1) ⋅ a ⋅ (1 − R n )
PV ⋅ R n ⋅ ( R − 1) = a ⋅ R n − 1
(21) (22)
(
)
Rn − 1 PV ⋅ R n = a ⋅ R −1
The equation 23 is the annuity formula, which can also be expressed as
(23)
a=
PV ⋅ R n ⋅ ( R − 1)
(R
n
−1
)
(24)
The equation 23
Rn − 1 PV ⋅ R n = a ⋅ R −1
is sometimes expressed as
(23)
1 − R − n PV = a ⋅ R −1
The equations are identical, as shown below:
1 1 − R n PV = a ⋅ R −1
Rn −1 PV = a ⋅ n R ( R − 1) Rn −1 PV ⋅ R = a ⋅ ( R − 1)
n
Conversion of Time Related Variables in the Equations 23 and 24
The interest rate
i
is usually expressed as the annual interest rate, known as the APR, or Annual Percentage Rate, and the duration
n
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Annuity
of the loan as a number of years. However, since annuities almost always are paid on a monthly basis, the variables
i
and
n
must be converted accordingly in the equations 23 and 24, since the value of the annuity
a
depends on the chosen period. Assuming that the equations 23 and 24 are based on year as the chosen period, then, if month is the preferred period, the equation 23 must be written as
PV ⋅ R
n⋅12
(R = a⋅
n⋅12
−1
)
( R − 1)
(25)
where
i 12 R = 1 + 100
Similarly, the equation 24 must be expressed as
(26)
a=
PV ⋅ R n⋅12 ⋅ ( R − 1)
(R
n⋅12
−1
)
(27)
where
i 12 R = 1 + 100
(28)
Example
A home buyer secures a 30-year mortgage loan of $150000 at a 6 percent APR. What is the annuity if the loan is repaid in the form of monthly installments? Applying the equation 27 and 28 results in
a=
150000 ⋅ R 30⋅12 ⋅ ( R − 1)
(R
30⋅12
−1
)
(E1)
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and
6 12 R = 1 + 100 R = 1.005
Inserting the equation E3 in the equation E1 results in
(E2)
(E3)
a=
150000 ⋅1.00530⋅12 ⋅ (1.005 − 1)
(1.005
30⋅12
−1
)
(E4)
a=
150000 ⋅ 6.022575212 ⋅ 0.005 5.022575212
(E5) (E6)
a = 899.33
The sum of the annuities paid over 360 months is
n =360 n =1
∑ a = 360 ⋅ 899.33 ∑ a = 323758.80
n =1
(E7)
n =360
(E8)
The total cost
Ctot
of the loan in the form of interest charge is
Ctot = 323758.80 − 150000 Ctot = 173758.80
The average interest cost per month is
(E9) (E10)
Cavg =
173758.80 360
(E11) (E12)
Cavg = 482.66
Thus, the average equity (principal re-payment) per month is
Eavg = 899.33 − 482.66 Eavg = 416.67
Thus, the sum of the equity over 360 months is
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Time Value of Money
n =360 n =1
Annuity
∑E ∑E
n =1
avg
= 360 ⋅ 416.67 = 150000
(E15)
n =360
avg
(E16)
which is equivalent to the original loan amount.
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