Introduction: Moving Beyond Level Payments
Welcome to one of the most practical chapters in the CM1 syllabus! So far, you have likely mastered level annuities—where the payment is the same every single time. But in the real world, things change. Inflation makes prices rise, people get annual salary raises, and sometimes we buy a product today but don't want the payments to start until several years later.
In this chapter, we will learn how to value cashflows that wait to start (deferred), grow over time (increasing), or change in other ways (varying). Don't worry if the formulas look a bit intimidating at first; we will break them down into simple logic that you can apply to any problem.
1. Deferred Annuities: The "Wait for it" Payments
A deferred annuity is simply a standard annuity that starts at some point in the future rather than right now.
The Concept: Imagine you are 20 years old and you buy a plan that will pay you £1,000 every year for 10 years, but the payments only start when you turn 30. You have "deferred" the start of the annuity for 10 years.
Notation and Calculation
The standard notation for an annuity for a term of \(n\) years, deferred for \(m\) years, is \(_{m|}a_n\). There are two easy ways to calculate the Present Value (PV) of these payments:
- The Discounting Method: Calculate the value of the annuity at the time it actually starts, and then discount that single value back to today.
\(_{m|}a_n = v^m \cdot a_n\) - The Subtraction Method: Imagine an annuity for the whole period (\(m + n\)) and subtract the part you didn't receive (the first \(m\) years).
\(_{m|}a_n = a_{m+n} - a_m\)
Quick Tip: Always draw a timeline! It is the best way to ensure you haven't missed a year or used the wrong power of \(v\). If the first payment is at time \(t = m+1\), it is an annuity in arrear deferred for \(m\) years.
2. Increasing Annuities: Watching Your Money Grow
In this section, the payments are no longer level. For a standard increasing annuity, the first payment is 1, the second is 2, and so on, up to the \(n\)-th payment which is \(n\).
The "Ia" Formula
The Present Value of an increasing annuity in arrear is denoted by \((Ia)_n\). The formula is:
\((Ia)_n = \frac{\ddot{a}_n - nv^n}{i}\)
Wait! Why is there an \(\ddot{a}_n\) (annuity-due) in a formula for an annuity in arrear?
This is a common point of confusion. The mathematical derivation leads us there, but you can think of it as a "balancing act" between the increasing payments and the interest rate. Just remember: if the annuity is in arrear, you use \(i\) in the denominator. If it is in advance (an increasing annuity-due), you use \(d\):
\((I\ddot{a})_n = \frac{\ddot{a}_n - nv^n}{d}\)
Continuous Increasing Annuities
If the payments are increasing continuously (like a stream of water getting stronger and stronger), we use the force of interest \(\delta\):
\((\bar{I}\bar{a})_n = \frac{\bar{a}_n - nv^n}{\delta}\)
Did you know? These formulas are very sensitive to the interest rate. Even a small change in \(i\) can significantly change the value of an increasing annuity because the largest payments happen furthest in the future!
3. Varying Annuities: The General Case
Sometimes payments don't increase by exactly 1 unit each time. They might increase by £500 every year, or they might decrease.
Arithmetic Progressions
If you have an annuity where the first payment is \(P\) and each subsequent payment increases by \(Q\), you can think of it as two separate parts:
1. A level annuity of amount \((P - Q)\).
2. An increasing annuity of amount \(Q \cdot (Ia)_n\).
Example: Payments of £100, £120, £140... for 10 years.
Here, \(P = 100\) and \(Q = 20\).
Value = \((100 - 20)a_{10} + 20(Ia)_{10} = 80a_{10} + 20(Ia)_{10}\).
Decreasing Annuities
While the syllabus highlights increasing annuities, you might encounter payments that go \(n, n-1, ..., 1\).
A simple trick: \((Da)_n = (n+1)a_n - (Ia)_n\).
4. P-thly Varying Annuities
The syllabus mentions pthly variants. This is where payments are made \(p\) times a year (e.g., monthly where \(p=12\)).
The most common version in exams is where the payment amount stays constant throughout each year but increases once at the start of every year. For example, £100 per month in Year 1, £200 per month in Year 2, etc.
The formula for this is: \((Ia)_n^{(p)} = \frac{\ddot{a}_n - nv^n}{i^{(p)}}\)
Key Difference to Watch Out For:
- If the payment amount increases every year, use \(i^{(p)}\) or \(d^{(p)}\) in the denominator.
- If the payment amount increases every pthly period (e.g., every month), the formula becomes a bit more complex, but the logic remains the same: \(\frac{1}{p} \cdot \text{standard formula using } i^{(p)}/p\).
5. Summary and Key Takeaways
Quick Review Box:
- Deferred (\(_{m|}a_n\)): Just a level annuity "pushed" into the future. Multiply by \(v^m\).
- Increasing (\((Ia)_n\)): Payments are \(1, 2, 3... n\). Formula: \(\frac{\ddot{a}_n - nv^n}{i}\).
- Continuous Increasing (\((\bar{I}\bar{a})_n\)): Use the same logic but divide by \(\delta\).
- Varying (\(P, P+Q, ...\)): Split into a level part and an increasing part.
Common Mistakes to Avoid:
- Mixing up \(i\) and \(d\): Remember, "Arrear uses \(i\), Advance uses \(d\)".
- The \(n\) in \(nv^n\): Students often forget that \(n\) is the total number of payments, not the deferred period.
- Timing: Always check if the first payment of an increasing annuity is at time 1 (arrear) or time 0 (advance). This changes everything!
Don't worry if these formulas feel heavy. The more you practice splitting complex cashflows into level and increasing parts, the more intuitive it becomes. You're building the toolkit to value almost any financial contract!