Annuities
3.13 Appendix
Deriving Annuity Formulas
FV Formula – Simple Annuity
Before we delve into deriving the future value formula for annuities, it’s essential to understand geometric sequences. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio and is represented by [asciimath]r[/asciimath]. The first term of a geometric sequence is denoted by [asciimath]a[/asciimath]. An example of a geometric sequence is [asciimath]{3, 6, 12, 24, 48,96}[/asciimath] where each term is obtained by multiplying the previous term by 2. Here, the first term is [asciimath]a=3[/asciimath] and the common ratio is [asciimath]r=2[/asciimath].
In general, a geometric sequence with n terms is written as [asciimath]{a,ar,ar^2,ar^3,...,ar^(n-1)}[/asciimath]. A partial sum of a geometric sequence, denoted as [asciimath]S_n[/asciimath], is the sum of the first n terms of the sequence. This sum can be calculated using the formula:
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
where [asciimath]n[/asciimath] is the number of terms to be summed.
Returning to our main topic, we aim to find a simple formula for calculating the future value of annuities. We begin with an ordinary simple annuity, where payments are made at the end of each payment period. The total value accumulated at the end of the term is the sum of the future values of each individual payment using the formula for the future value of compound interest.
[asciimath]"FV"=PMT(1+i)^0+PMT(1+i)^1[/asciimath] [asciimath]+...+PMT(1+i)^(N-2) +PMT(1+i)^(N-1)[/asciimath]
[asciimath]=PMTubrace( [(1+i)^0+(1+i)^1 +...+(1+i)^(N-2) +(1+i)^(N-1) ])_(S_n)[/asciimath]
After factoring out [asciimath]PMT[/asciimath] from this summation, we observe that the expression inside the square brackets is the partial sum of terms of a geometric sequence. This sequence starts with a first term of [asciimath](1+i)^0=1[/asciimath] and has a common ratio of , meaning each term is derived by multiplying the previous term by .
Therefore, the partial sum of this geometric sequence with [asciimath]N[/asciimath] terms, the first term [asciimath]a=1[/asciimath], and the common ratio [asciimath]r=1+i[/asciimath] can be determined by
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
[asciimath]S_n=1[(1-(1+i)^N)/(1-(1+i))][/asciimath]
[asciimath]=[(1-(1+i)^N)/-i][/asciimath]
[asciimath]=[((1+i)^N-1)/i][/asciimath]
Thus, the future value of all the payments in an ordinary simple annuity is calculated as follows
[asciimath]FV=PMT(S_n)[/asciimath]
[asciimath]FV=PMT[((1+i)^N-1)/i][/asciimath]
FV Formula – Simple Annuity Due
Recall that a geometric sequence with [asciimath]n[/asciimath] terms is written as [asciimath]{a,ar,ar^2,ar^3,...,ar^(n-1)}[/asciimath]. A partial sum of a geometric sequence, denoted as [asciimath]S_n[/asciimath], is the sum of the first [asciimath]n[/asciimath] terms of the sequence. This sum can be calculated using the formula
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
where [asciimath]n[/asciimath] is the number of terms to be summed.
We aim to find a simple formula for calculating the future value of a simple annuity due, where payments are made at the beginning of each payment period. The total value accumulated at the end of the term is the sum of the future values of each individual payment using the formula for the future value of compound interest.
[asciimath]"FV"=PMT(1+i)^1+PMT(1+i)^2[/asciimath] [asciimath]+...+PMT(1+i)^(N-1) +PMT(1+i)^(N)[/asciimath]
[asciimath]=PMTubrace( [(1+i)^1+(1+i)^2 +...+(1+i)^(N-1) +(1+i)^(N) ])_(S_n)[/asciimath]
After factoring out [asciimath]PMT[/asciimath] from this summation, we observe that the expression inside the square brackets is the partial sum of terms of a geometric sequence. This sequence starts with a first term of [asciimath](1+i)^1[/asciimath] and has a common ratio of, meaning each term is derived by multiplying the previous term by.
Therefore, the partial sum of this geometric sequence with [asciimath]N[/asciimath] terms, the first term [asciimath]a=1+i[/asciimath], and the common ratio [asciimath]r=1+i[/asciimath] can be determined by
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
[asciimath]S_n=(1+i)[(1-(1+i)^N)/(1-(1+i))][/asciimath]
[asciimath]=(1+i) [(1-(1+i)^N)/-i][/asciimath]
[asciimath]=(1+i) [((1+i)^N-1)/i][/asciimath]
Thus, the future value of all the payments in simple annuity due is calculated as follows
[asciimath]FV=PMT(S_n)[/asciimath]
[asciimath]FV=PMT[((1+i)^N-1)/i](1+i)[/asciimath]
PV Formula – Ordinary Simple annuity
Recall that a geometric sequence with [asciimath]n[/asciimath] terms is written as [asciimath]{a,ar,ar^2,ar^3,...,ar^(n-1)}[/asciimath]. A partial sum of a geometric sequence, denoted as [asciimath]S_n[/asciimath], is the sum of the first [asciimath]n[/asciimath] terms of the sequence. This sum can be calculated using the formula
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
where [asciimath]n[/asciimath] is the number of terms to be summed.
We aim to find a simple formula for calculating the present value of an ordinary simple annuity, where payments are made at the end of each payment period. The total present value is the sum of the present values of each individual payment using the formula for the present value of compound interest.
[asciimath]"PV"=PMT(1+i)^-1+PMT(1+i)^-2[/asciimath] [asciimath]+...+PMT(1+i)^-(N-1) +PMT(1+i)^(-N)[/asciimath]
[asciimath]=PMTubrace( [(1+i)^-1+(1+i)^-2 +...+(1+i)^-(N-1) +(1+i)^(-N) ])_(S_n)[/asciimath]
After factoring out [asciimath]PMT[/asciimath] from this summation, we observe that the expression inside the square brackets is the partial sum of terms of a geometric sequence. This sequence starts with a first term of [asciimath](1+i)^-1[/asciimath] and has a common ratio of , meaning each term is derived by multiplying the previous term by.
Therefore, the partial sum of this geometric sequence with [asciimath]N[/asciimath] terms, the first term [asciimath]a=(1+i)^-1[/asciimath], and the common ratio [asciimath]r=(1+i)^-1[/asciimath] can be determined by
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
[asciimath]S_n=(1+i)^-1 [(1-((1+i)^-1)^N) /(1-(1+i)^-1)][/asciimath]
Applying the product of powers property of exponents, which states that [asciimath](b^x)^y=b^(x*y)[/asciimath], we can express [asciimath]((1+i)^-1)^N[/asciimath] as [asciimath](1+i)^-N[/asciimath], thereby simplifying the numerator. Additionally, considering that [asciimath](1+i)^-1[/asciimath] equals [asciimath]1/(1+i)[/asciimath], we can multiply the term [asciimath](1+i)[/asciimath] by the denominator.
[asciimath]S_n=1/(1+i) [(1-(1+i)^-N)/(1-(1+i)^-1) ][/asciimath]
When you distribute [asciimath](1+i)[/asciimath] across the terms in the denominator and recognize that multiplying a term by its reciprocal results in 1, it follows that [asciimath](1+i)*(1+i)^-1=1[/asciimath].
[asciimath]S_n= [(1-(1+i)^-N)/((1+i)-1) ][/asciimath]
[asciimath]= [(1-(1+i)^-N)/i ][/asciimath]
Therefore, the present value of all payments in an ordinary simple annuity can be calculated using the following formula:
[asciimath]PV=PMT(S_n)[/asciimath]
[asciimath]PV=PMT [(1-(1+i)^-N)/i ][/asciimath] Formula 3.9
PV Formula – Simple Annuity Due
Recall that a geometric sequence with [asciimath]n[/asciimath] terms is written as [asciimath]{a,ar,ar^2,ar^3,...,ar^(n-1)}[/asciimath]. A partial sum of a geometric sequence, denoted as [asciimath]S_n[/asciimath], is the sum of the first [asciimath]n[/asciimath] terms of the sequence. This sum can be calculated using the formula
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
where [asciimath]n[/asciimath] is the number of terms to be summed.
We want to find a simple formula for calculating the present value of a simple annuity due, where payments are made at the beginning of each payment period. The total present value is the sum of the present values of each payment using the formula for the present value of compound interest.
[asciimath]"PV"=PMT(1+i)^0+PMT(1+i)^-1[/asciimath] [asciimath]+...+PMT(1+i)^-(N-2) +PMT(1+i)^-(N-1)[/asciimath]
[asciimath]=PMTubrace( [(1+i)^0+(1+i)^-1 +...+(1+i)^-(N-2) +(1+i)^-(N-1) ])_(S_n)[/asciimath]
After factoring out [asciimath]PMT[/asciimath] from this summation, we notice that the expression inside the square brackets is the partial sum of terms of a geometric sequence. This sequence starts with a first term of [asciimath](1+i)^0[/asciimath] and has a common ratio of , meaning each term is derived by multiplying the previous term by.
Therefore, the partial sum of this geometric sequence with [asciimath]N[/asciimath] terms, the first term [asciimath]a=(1+i)^0=1[/asciimath], and the common ratio [asciimath]r=(1+i)^-1[/asciimath] can be determined by
[asciimath]S_n=a((1-r^n)/(1-r))[/asciimath]
[asciimath]S_n= [(1-((1+i)^-1)^N) /(1-(1+i)^-1)][/asciimath]
Applying the product of powers property of exponents, which states that [asciimath](b^x)^y=b^(x*y)[/asciimath], we can express [asciimath]((1+i)^-1)^N[/asciimath] as [asciimath](1+i)^-N[/asciimath], thus simplifying the numerator. To simplify the denominator, we multiply both the numerator and the denominator of the expression on the right-hand side by the factor [asciimath](1+i)[/asciimath]:
[asciimath]S_n= [(1-(1+i)^-N)/(1-(1+i)^-1) ] xx (1+i)/(1+i)[/asciimath]
Next, we distribute the factor [asciimath](1+i)[/asciimath] across the terms in the denominator. recognizing that multiplying a term by its reciprocal equals 1, we see that [asciimath](1+i)*(1+i)^-1=1[/asciimath].
[asciimath]S_n= [(1-(1+i)^-N)/((1+i)-1) ] (1+i)[/asciimath]
[asciimath]= [(1-(1+i)^-N)/i ](1+i)[/asciimath]
Therefore, the present value of all payments in a simple annuity due can be calculated using the following formula:
[asciimath]PV=PMT(S_n)[/asciimath]
[asciimath]PV=PMT [(1-(1+i)^-N)/i ] (1+i)[/asciimath] Formula 3.11
