Other non-level annuities/cash flows

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

Welcome to the World of Non-Level Cash Flows!

In your studies so far, you’ve likely mastered level annuities—where every payment is the same. But in the real world, things rarely stay the same! Think about your future career: you’ll likely get annual raises (geometric growth), or perhaps you’re paying off a loan where the interest decreases while the principal payments change (arithmetic changes). This chapter is all about handling those cash flows that grow, shrink, or follow a specific pattern. Don't worry if this seems like a lot of formulas at first; we are going to break it down into simple patterns you can recognize on sight.

1. Geometrically Increasing Annuities

A geometrically increasing annuity is a series of payments where each payment is a constant percentage larger than the previous one. This is perfect for modeling inflation or a fixed percentage salary raise.

The Pattern

Imagine the first payment is $1$. The next is $1(1+g)$, the next is $1(1+g)^2$, and so on, for $n$ periods.

How to Calculate the Present Value (PV)

You can memorize a big formula, but here is a secret trick: think of it as a modified version of a standard annuity. If the interest rate is $i$ and the growth rate is $g$, we can use a "net" interest rate.

The formula for the Present Value of an immediate annuity is:
$ PV = \frac{1 - (\frac{1+g}{1+i})^n}{i-g} $

Common Mistake to Avoid:

Watch out for $i = g$! If the interest rate equals the growth rate, the denominator becomes zero. In this special case, the formula simplifies beautifully to:
$ PV = n \times v $ (where $v = \frac{1}{1+i}$)
Actually, a simpler way to think about it is: every term in the PV sum becomes exactly the same! So $PV = \frac{n}{1+i}$.

Quick Summary:

When you see "increases by X percent each year," it’s a geometric annuity. Use the $ \frac{1 - (\dots)}{i-g} $ formula.

2. Arithmetically Increasing Annuities

In an arithmetic progression, the payment increases by a fixed dollar amount rather than a percentage. Think of it like a staircase: each step is exactly the same height.

The Standard Increasing Annuity $ (Ia)_{\bar{n}|} $

This is a specific case where the first payment is 1, the second is 2, the third is 3, and the $n$-th payment is $n$.

The Formula:
$ (Ia)_{\bar{n}|} = \frac{\ddot{a}_{\bar{n}|i} - nv^n}{i} $

Step-by-Step Breakdown:

1. Calculate the annuity-due value $\ddot{a}_{\bar{n}|}$.
2. Subtract the "drop-off" term $nv^n$.
3. Divide the whole thing by the interest rate $i$.

Analogy: The Staircase

Imagine you are building a staircase of blocks. Each year, you add one more block to the pile. The formula above tells you the value of all those blocks today.

3. Arithmetically Decreasing Annuities

This is the reverse! The payments start high and drop by 1 each period. The first payment is $n$, the second is $n-1$, and the last payment is 1.

The Formula:

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

Did you know?

There is a cool relationship between increasing and decreasing annuities. If you add an increasing annuity of $n$ years to a decreasing annuity of $n$ years, you just get a level payment of $(n+1)$ each year! This can sometimes help you check your work.

4. The "P-Q" Formula: For Any Arithmetic Progression

What if the payments don't start at 1? What if they start at \$500 and increase by \$20 each year? This is where the P-Q Formula becomes your best friend.

The Components:

- P = The amount of the first payment.
- Q = The amount of the increase (or decrease, use a negative number).
- n = The number of payments.

The General Formula:

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

Note: The second part of this formula ($ \frac{a_{\bar{n}|} - nv^n}{i} $) is actually equal to $\frac{(Ia)_{\bar{n}|} - a_{\bar{n}|}}{i}$ in some textbooks, but the version above is the most direct for calculation.

Example:

If you receive \$100 in year 1, \$110 in year 2, and \$120 in year 3:
- P = 100
- Q = 10
- n = 3

5. Rainbow or Palindromic Annuities

Sometimes the exam will give you an annuity that increases and then decreases (e.g., 1, 2, 3, 2, 1). This is often called a Rainbow Annuity.

The Trick:

Instead of doing two separate P-Q formulas, remember this beautiful identity:
The Present Value of a payment stream of (1, 2, ..., n-1, n, n-1, ..., 1) is simply:
$ PV = a_{\bar{n}|} \times \ddot{a}_{\bar{n}|} $
Or even simpler:
$ PV = (a_{\bar{n}|})^2 \times (1+i) $

Summary and Tips for Success

Quick Review Box:

- Geometric: Use the growth rate formula. Check if $i=g$.
- Arithmetic Increasing: Starts at 1, goes to $n$. Formula uses $\ddot{a}_{\bar{n}|}$.
- Arithmetic Decreasing: Starts at $n$, goes to 1. Formula uses $a_{\bar{n}|}$.
- P-Q Method: Use for any "non-standard" arithmetic start.

Pro-Tips for the Exam:

1. Draw a Timeline! This is the most important step for non-level cash flows. If the increase starts in year 2, make sure your $n$ and your exponents align.
2. Check the timing: Is it an annuity-immediate (first payment at $t=1$) or an annuity-due (first payment at $t=0$)? The formulas above are for immediate. if it's due, just multiply your final answer by $(1+i)$.
3. Don't Panic: If you forget a formula, you can always discount each payment individually if $n$ is small (like $n=3$ or $4$). It’s better to take an extra minute than to guess a formula you aren't sure of!

You've got this! Mastering these patterns is one of the biggest hurdles in Exam FM. Once you can recognize whether a "step" is a percentage (Geometric) or a dollar amount (Arithmetic), you are halfway to the correct answer.

Quick check

Can you answer these now?

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

What distinguishes a geometrically increasing annuity from other non-level annuities?

Payments in a geometrically increasing annuity grow by a constant percentage (a growth rate, $g$) each period rather than a fixed dollar amount.

What is the formula for the present value of a geometrically increasing annuity-immediate?

$ PV = \frac{1 - (\frac{1+g}{1+i})^n}{i-g} $, where $i$ is the interest rate and $g$ is the growth rate.

How is the present value of a geometric annuity calculated if the growth rate $g$ equals the interest rate $i$?

$ PV = n \times v $ (or $ PV = \frac{n}{1+i} $).

What is a standard arithmetically increasing annuity, and how are its payments structured?

A standard increasing annuity, denoted $(Ia)_{\bar{n}|}$, features payments that increase by 1 unit each period, starting with a payment of 1.

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

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

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

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

What is the mathematical relationship between standard increasing and decreasing annuities of the same duration?

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

What is the P-Q formula for a general arithmetic progression?

$ PV = P a_{\bar{n}|} + Q \frac{a_{\bar{n}|} - nv^n}{i} $, where $P$ is the first payment and $Q$ is the constant increase.

What is the shortcut for calculating the present value of a palindromic (Rainbow) annuity?

$ PV = a_{\bar{n}|} \times \ddot{a}_{\bar{n}|} $ or $ (a_{\bar{n}|})^2 \times (1+i) $.

How should you adjust non-level annuity formulas if the payments are made at the start of the period (annuity-due)?

Calculate the value as if it were an annuity-immediate and then multiply the entire result by $(1+i)$.

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