Arithmetic progression annuities (finite term and perpetuity)

Exam FM – Financial Mathematics · SOA (Society of Actuaries) · 5 min read

Welcome to the World of Arithmetic Annuities!

Hi there! If you’ve already mastered level annuities (where payments stay the same), you’re ready for the next step. In the real world, payments often change over time. Maybe a company’s profits grow by a set amount each year, or a structured settlement increases annually to keep up with costs. This is where Arithmetic Progression Annuities come in.

In this chapter, we will learn how to value cash flows that increase or decrease by a fixed dollar amount each period. Don’t worry if the formulas look a bit intimidating at first—we’re going to break them down piece by piece until they feel like second nature!

1. What is an Arithmetic Progression?

An arithmetic progression is just a fancy way of saying a sequence of numbers where the difference between consecutive terms is constant.
Example: $100, $110, $120, $130...
Here, the payment starts at $100 and increases by a constant amount (Q) of $10 every period.

In Exam FM, we generally look at two specific types of these annuities:

1. Increasing Annuities: The payments go up (e.g., 1, 2, 3, ..., n).
2. Decreasing Annuities: The payments go down (e.g., n, n-1, ..., 1).

Analogy: Think of an arithmetic annuity like a staircase. Each step you take, you are exactly one height-unit higher (or lower) than the last step.

2. The Basic Increasing Annuity $(Ia)_{\overline{n}|}$

Let's start with the most basic version: an annuity-immediate where the first payment is 1, the second is 2, and it continues until the last payment is $n$ at time $n$.

The Formula

The present value (PV) is denoted by $(Ia)_{\overline{n}|}$:

$ (Ia)_{\overline{n}|i} = \frac{\ddot{a}_{\overline{n}|i} - nv^n}{i} $

Wait, why is there an $\ddot{a}$ in there?

This is a common "stumbling block" for students! Even though we are calculating an annuity-immediate (payments at the end of the period), the formula uses the annuity-due symbol $\ddot{a}_{\overline{n}|}$ in the numerator.
Quick Tip: Just remember that the "I" in the formula "pulls" an extra dot into the "a" in the numerator!

Key Takeaway:

To find the PV of an increasing annuity, you need the interest rate ($i$), the number of periods ($n$), and the level annuity-due value ($\ddot{a}_{\overline{n}|}$).

3. The Basic Decreasing Annuity $(Da)_{\overline{n}|}$

In a decreasing annuity, the first payment is $n$, the second is $n-1$, and the payments decrease until the final payment is 1 at time $n$.

The Formula

The present value is denoted by $(Da)_{\overline{n}|}$:

$ (Da)_{\overline{n}|i} = \frac{n - a_{\overline{n}|i}}{i} $

Did you know? Decreasing annuities are often used to model the interest portion of a loan or certain types of depreciation!

Quick Review Box:

Increasing (Ia): Starts at 1, ends at $n$. Formula: $\frac{\ddot{a}_{\overline{n}|} - nv^n}{i}$
Decreasing (Da): Starts at $n$, ends at 1. Formula: $\frac{n - a_{\overline{n}|}}{i}$

4. The General Case: Payments of $P, P+Q, P+2Q, \dots$

What if the payments don't start at 1? What if they start at $500 and increase by $50? We use a general formula for this. We can split any arithmetic annuity into two parts: a level part and an increasing/decreasing part.

If the first payment is P and the subsequent payments increase by Q each period:

Present Value = $P a_{\overline{n}|} + Q \frac{a_{\overline{n}|} - nv^n}{i}$

However, many students find it easier to think about it this way:
1. A level annuity of amount (P - Q).
2. An increasing annuity $(Ia)$ of amount Q.

Example: Payments are 10, 15, 20.
Here, $P = 10$ and $Q = 5$.
You can view this as:
- A level annuity of 5 (which is $P-Q$)
- An increasing annuity $5, 10, 15$ (which is $Q \times (Ia)_{\overline{3}|}$)

Common Mistake to Avoid: Be very careful with the signs! If the annuity is decreasing, $Q$ will be negative.

5. Arithmetic Perpetuities (Finite isn't always enough!)

What if the payments go on forever? This is an arithmetic perpetuity. Because $n$ goes to infinity, the formulas actually become much simpler because the $v^n$ terms drop out to zero!

Increasing Perpetuity-Immediate

Payments: 1, 2, 3, ... forever.
$ (Ia)_{\infty|i} = \frac{1}{i} + \frac{1}{i^2} $ or more commonly written as:
$ (Ia)_{\infty|i} = \frac{1}{i \cdot d} $

General Perpetuity-Immediate

Payments: $P, P+Q, P+2Q, \dots$ forever.
Present Value = $\frac{P}{i} + \frac{Q}{i^2}$

Analogy: Imagine planting a magical tree that grows 1 extra apple every year forever. To find out what that tree is worth today, you just need this simple formula!

6. Summary and Final Tips

Step-by-Step for Solving Problems:

1. Identify the Pattern: Is it increasing or decreasing? By how much ($Q$)?
2. Find your P and Q: $P$ is the first payment, $Q$ is the change.
3. Check the Timing: Is it an annuity-immediate (end of period) or annuity-due (beginning)?
4. Check the Rate: Ensure your interest rate matches the frequency of the payments.
5. Plug and Chug: Use the formulas above to calculate the value.

Key Takeaways:

Increasing Annuity: $(Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|} - nv^n}{i}$
Decreasing Annuity: $(Da)_{\overline{n}|} = \frac{n - a_{\overline{n}|}}{i}$
Perpetuity: $\frac{P}{i} + \frac{Q}{i^2}$
Relationship: For any $n$, $(Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)a_{\overline{n}|}$. (This is a great way to check your work!)

Don't worry if this seems tricky at first! The "I" and "D" formulas are some of the most algebraic parts of Exam FM. Practice sketching a timeline for every problem; once you see the "stairs" on the paper, the math becomes much clearer.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the defining characteristic of an arithmetic progression annuity?

An arithmetic progression annuity is a sequence of payments where the difference between consecutive terms is a constant dollar amount (Q).

What is the formula for the present value of a basic increasing annuity-immediate $ (Ia)_{\overline{n}|} $?

$ (Ia)_{\overline{n}|i} = \frac{\ddot{a}_{\overline{n}|i} - nv^n}{i} $

Why does the formula for an increasing annuity-immediate contain an annuity-due symbol in the numerator?

It is an algebraic result where the "increasing" nature of the payments shifts the timing of the level components, effectively pulling an extra interest factor into the numerator.

What is the formula for the present value of a basic decreasing annuity-immediate $ (Da)_{\overline{n}|} $?

$ (Da)_{\overline{n}|i} = \frac{n - a_{\overline{n}|i}}{i} $

What is the general formula for the present value of an arithmetic annuity with first payment P and increment Q?

$ PV = P a_{\overline{n}|} + Q \frac{a_{\overline{n}|} - nv^n}{i} $

How can you decompose a general arithmetic annuity into a level part and a basic increasing part?

View it as a level annuity of amount $ (P - Q) $ and a basic increasing annuity scaled by $ Q $, so $ PV = (P-Q)a_{\overline{n}|} + Q(Ia)_{\overline{n}|} $.

What is the present value formula for a basic increasing perpetuity-immediate starting at 1?

$ (Ia)_{\infty|i} = \frac{1}{id} $ or $ \frac{1}{i} + \frac{1}{i^2} $.

How do you calculate the present value of a general arithmetic perpetuity starting at P with increment Q?

$ PV = \frac{P}{i} + \frac{Q}{i^2} $

What is the mathematical identity relating $ (Ia)_{\overline{n}|} $ and $ (Da)_{\overline{n}|} $?

$ (Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)a_{\overline{n}|} $

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Arithmetic progression annuities (finite term and perpetuity)

Welcome to the World of Arithmetic Annuities!

1. What is an Arithmetic Progression?

2. The Basic Increasing Annuity \((Ia)_{\overline{n}|}\)

The Formula

Wait, why is there an \(\ddot{a}\) in there?

Key Takeaway:

3. The Basic Decreasing Annuity \((Da)_{\overline{n}|}\)

The Formula

Quick Review Box:

4. The General Case: Payments of \(P, P+Q, P+2Q, \dots\)

5. Arithmetic Perpetuities (Finite isn't always enough!)

Increasing Perpetuity-Immediate

General Perpetuity-Immediate

6. Summary and Final Tips

Step-by-Step for Solving Problems:

Key Takeaways:

Can you answer these now?

Ready to test yourself?

More Exam FM – Financial Mathematics chapters

Put These Notes into Practice