Financial Math FM/Annuities

Learning objectives

The Candidate will be able to calculate present value, current value, and accumulated value for sequences of non-contingent payments.

Learning outcomes

The Candidate will be able to:

Define and recognize the definitions of the following terms: annuity-immediate, annuity due, perpetuity, payable m-thly or payable continuously, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity.
For each of the following types of annuity/cash flows, given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, calculate any remaining item.

Level annuity, finite term.
Level perpetuity.
Non-level annuities/cash flows.

Arithmetic progression, finite term and perpetuity.
Geometric progression, finite term and perpetuity.
Other non-level annuities/cash flows.

Geometric series formulas

Recall the following formulas, which are useful for deriving the formulas for different types of annuities.

$a + a r + a r^{2} + \dots \overset{def}{=} \sum_{k = 0}^{\infty} a r^{k} = \frac{a}{1 - r}, | r | < 1$ ;
$\underset{n + 1 terms}{\underset{⏟}{a + a r + a r^{2} + \dots + a r^{n}}} \overset{def}{=} \sum_{k = 0}^{n} a r^{k} = a (\frac{1 - r^{n + 1}}{1 - r}), r \neq 1$ .

Level annuities

Template:Colored definition Template:Colored remark Template:Colored example

Annuity-immediate

Template:Colored definition Template:Colored remark Time diagram:

     ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...    n      t

Template:Colored proposition

Proof.

PV
v^n                  1
...
v^2         1    
v     1    
*-----*-----*---...--*
0     1     2   ...  n

From the time diagram, we have $a_{\overline{n} | i} = \underset{n terms}{\underset{⏟}{v + v^{2} + \dots + v^{n}}} = \frac{v (1 - v^{n})}{1 - v} = \frac{(1 - v^{n}) / 1 + i}{i / 1 + i} = \frac{1 - v^{n}}{i}$ .

$◻$

Template:Colored remark Template:Colored example Template:Colored example Template:Colored example Template:Colored exercise

Annuity-due

Template:Colored definition Template:Colored remark Time diagram:

↓    ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...   n-1      t

Template:Colored proposition

Proof.

Consider the time diagram for annuity-immediate with payments of 1:

     1     1     1     1
*----*-----*-----*-----*------
0    1     2    ...    n   ... t

The present value of this annuity is $a_{\overline{n} | i}$ .
So, the value of this annuity at $t = 1$ is $a_{\overline{n} | i} (1 + i)$ .
Then, if we regard $t = 1$ as present (i.e. $t = 0$ ) by changing time labels, the time diagram becomes:

     1     1     1     1
*----*-----*-----*-----*------
-1   0     1    ...   n-1  ... t

We can observe that this is the time diagram for annuity-due with payments of 1, and the value at $t = 0$ (i.e. present value) is ${\ddot{a}}_{\overline{n} | i}$ .
It follows that ${\ddot{a}}_{\overline{n} | i} = a_{\overline{n} | i} (1 + i)$ , since these two expressions tell the value at the same time point (with different labels only), with the same series of payment.

$◻$

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Annuities payable Template:Maththly

Sometimes, annuities are not payable annually, and can be payable more or less frequently than annually.

To calculate the present value of these kind of annuities, we can simply change the measurement period, and calculate the interest rate during that period that is equivalent to the given interest (or discount) rate (or force of interest), and the new term of the annuity in the new measurement period.

Using new terms and new interest rates, we can calculate the present value of these kind of annuities by applying previously discussed method.

Template:Colored example Template:Colored example

Annuities payable continuously

Recall from the motivation of force of interest that "payable continuously" is essentially "payable $\infty$ thly (abuse of notation)"
If an annuity pays 1 in each "infinitesimal" time interval, the present value of payments in a measurement period will be infinite, since there are infinitely many such time intervals during arbitrary measurement period.
Because of this, it does not make sense to say an annuity has payments of xxx payable continuously.
Instead, we should use the notion of Template:Colored em to describe the behaviour of continuous payment.

Template:Colored example Template:Colored remark Template:Colored example

Perpetuities

Template:Colored definition Template:Colored remark Time diagram:

     ↓     ↓    ...     
*----*-----*------------------
0    1     2    ...           t

Template:Colored example Template:Colored example

For perpetuities payable $m$ thly, the approach for calculating their present values is the same as that for annuities payable $m$ thly, namely adjusting the measurement period and calculating new interest rate correspondingly.
Also, for perpetuities payable continuously, the approach for calculating their present values is the same, except that we are dealing with improper integrals (upper limits of integrals involved are $\infty$ ).

Non-level annuities

In general, the present value of non-level annuities can be calculated by summing up the present value of each payment.
However, this approach can take a lot of time, and thus may not be the most efficient approach.
We will discuss several special cases of non-level annuities, for which the present value can be calculated in an efficient way.

Arithmetic varying annuities

For some annuities, payments vary (increase or decrease) in arithmetic sequence.
We will develop a formula for calculating their present values in this subsection.

Template:Colored theorem

Proof.

Consider the following time diagram:

                                 Row
                           D     1st
                     D     D     2nd
                     .     .
                     .     .
                     .     .
         D           D     D    n-1 th
   P     P           P     P    nth
---*-----*----...----*-----*
0  1     2    ...   n-1    n    t

For payments in the $n$ th row, the present value is $P a_{\overline{n} | i}$ ;
for payments in the $n - 1$ th row, the present value is $D a_{\overline{n - 1} | i} v = D v \cdot \frac{1 - v^{n - 1}}{i}$ ;
...
for payments in the $n - j$ th row, the present value is $D a_{\overline{n - j} | i} v^{j} = D v^{j} \cdot \frac{1 - v^{n - j}}{i}$ ;
for payments in the 2nd row, the present value is $D a_{\overline{2} | i} v^{n - 2} = D v^{n - 2} \cdot \frac{1 - v^{2}}{i}$ ;
for payments in the 1st row, the present value is $D a_{\overline{1} | i} v^{n - 1} = D v^{n - 1} \cdot \frac{1 - v}{i}$ .
So, the present value of all payments is

$\begin{matrix} P a_{\overline{n} | i} + \sum_{k = 1}^{n - 1} D v^{k} \cdot \frac{1 - v^{n - k}}{i} & = P a_{\overline{n} | i} + \frac{D}{i} \sum_{k = 1}^{n - 1} (v^{k} - v^{n}) \\ = P a_{\overline{n} | i} + \frac{D}{i} (\sum_{k = 1}^{n - 1} (v^{k}) - (n - 1) v^{n}) \\ = P a_{\overline{n} | i} + \frac{D}{i} (\underset{= \sum_{k = 1}^{n} v^{k} \overset{def}{=} a_{\overline{n} | i}}{\underset{⏟}{\sum_{k = 1}^{n - 1} (v^{k}) + v^{n}}} - n v^{n}) \\ = P a_{\overline{n} | i} + \frac{D}{i} (a_{\overline{n} | i} - n v^{n}) \\ = (P + \frac{D}{i}) a_{\overline{n} | i} - \frac{D n}{i} \cdot v^{n} . \end{matrix}$

$◻$

Template:Colored remark Template:Colored example

Geometric varying annuities

Since the expression for present value of annuity is essentially geometric series ^[1], even with payments varying in geometric sequence, the expression is still geometric series, and thus we can use the geometric series formula to calculate the present value.
So, in general, for geometric varying annuities, we use "first principle" to calculate their present value, in the sense that we use geometric series formula to evaluate the expanded form of the present value.

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↑ For example, $a_{\overline{n} |} = v + v^{2} + \dots + v^{n}$ , and ${\ddot{a}}_{\overline{\infty} |} = 1 + v + v^{2} + \dots$ , which are geometric series

[1] For example, $a_{\overline{n} |} = v + v^{2} + \dots + v^{n}$ , and ${\ddot{a}}_{\overline{\infty} |} = 1 + v + v^{2} + \dots$ , which are geometric series

[1]

Financial Math FM/Annuities

Contents

Learning objectives

Learning outcomes

Geometric series formulas

Level annuities

Annuity-immediate

Annuity-due

Annuities payable Template:Maththly

Annuities payable continuously

Perpetuities

Non-level annuities

Arithmetic varying annuities

Geometric varying annuities

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Financial Math FM/Annuities

Learning objectives

Learning outcomes

Geometric series formulas

Level annuities

Annuity-immediate

Annuity-due

Annuities payable Template:Maththly

Annuities payable continuously

Perpetuities

Non-level annuities

Arithmetic varying annuities

Geometric varying annuities

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