Welcome to the World of Growing Payments!
In our previous lessons, we looked at annuities where the payments were always the same (level payments). But in the real world, things change! Think about a job where you get a 3% raise every year, or a pension that increases to keep up with inflation. These are Geometric Progression Annuities.
In this chapter, we will learn how to value cash flows that grow (or shrink) by a constant percentage each period. Don't worry if this seems a bit more complex than level annuities—once you see the pattern, it’s just like a puzzle that fits together perfectly!
What is a Geometric Progression?
Before we dive into the formulas, let's make sure we understand the concept. A geometric progression is simply a sequence of numbers where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Real-World Analogy: Imagine you have a magic penny that doubles every day. On day one you have 1 cent, day two you have 2 cents, day three you have 4 cents, and so on. Each day is multiplied by 2. That is a geometric progression!
In Exam FM, we usually use the letter \(g\) to represent the growth rate. If your payment grows by 5%, then your common ratio is \((1 + 0.05) = 1.05\).
The Anatomy of a Geometric Annuity
Let's look at a standard annuity-immediate (where the first payment happens at the end of the first period, \(t=1\)):
1. The first payment is \(P\) at time \(t = 1\).
2. The second payment is \(P(1+g)\) at time \(t = 2\).
3. The third payment is \(P(1+g)^2\) at time \(t = 3\).
4. The \(n\)-th payment is \(P(1+g)^{n-1}\) at time \(t = n\).
Quick Review: Notice that the exponent on the growth term is always one less than the time period. For example, at time 10, the growth has happened 9 times.
Calculating the Present Value (PV)
The Present Value is the value of all these future growing payments right now (at time \(t=0\)). To find this, we discount each payment back to the start using the interest rate \(i\).
The formula for the Present Value of a geometric annuity-immediate for \(n\) periods is:
\(PV = P \left[ \frac{1 - (\frac{1+g}{1+i})^n}{i - g} \right]\)
Key Condition: This formula works as long as \(i \neq g\). We will talk about what happens if they are equal in just a moment!
Step-by-Step Example
Scenario: You are promised a series of 10 annual payments. The first payment is \$1,000 one year from today. Each subsequent payment increases by 4%. The annual effective interest rate is 6%.
\n1. Identify the variables: \(P = 1000\), \(g = 0.04\), \(i = 0.06\), \(n = 10\).
\n2. Plug them into the formula:
\n\(PV = 1000 \left[ \frac{1 - (\frac{1.04}{1.06})^{10}}{0.06 - 0.04} \right]\)
\n3. Calculate the ratio: \(\frac{1.04}{1.06} \approx 0.98113\)
\n4. Raise to the power of 10: \(0.98113^{10} \approx 0.82655\)
\n5. Finish the math: \(1000 \left[ \frac{1 - 0.82655}{0.02} \right] = 1000 \left[ \frac{0.17345}{0.02} \right] = 8,672.50\)
Special Case: What if \(i = g\)?
\nIf the interest rate and the growth rate are exactly the same, the formula above would require dividing by zero—which we can't do! But don't panic. If \(i = g\), the math actually becomes much simpler.
\nWhen \(i = g\), every single payment, when discounted back to time 0, is exactly the same value! Specifically, each payment's present value is \(\frac{P}{1+i}\). Since there are \(n\) payments, the formula is:
\n\(PV = n \cdot \frac{P}{1+i}\) (only when \(i = g\))
\nMemory Aid: Think of this as "the simple version." Just multiply the number of payments by the discounted value of the very first payment.
\n\nGeometric Perpetuities
\nA perpetuity is an annuity that goes on forever (\(n = \infty\)). In the exam, you'll see phrases like "payable indefinitely" or "forever."
\nFor a geometric perpetuity to have a finite value, the growth rate \(g\) must be smaller than the interest rate \(i\). If the payments grow faster than the interest can discount them, the value would be infinite!
\nThe formula for the Present Value of a geometric perpetuity-immediate is beautifully simple:
\n\(PV = \frac{P}{i - g}\)
\nDid you know? This is often called the "Gordon Growth Model" in equity valuation, used to determine the value of a stock with constantly growing dividends.
\n\nSummary of Formulas
\nFinite Term (\(i \neq g\)): \(PV = P \left[ \frac{1 - (\frac{1+g}{1+i})^n}{i - g} \right]\)
\nFinite Term (\(i = g\)): \(PV = \frac{n \cdot P}{1+i}\)
\nPerpetuity (\(i > g\)): \(PV = \frac{P}{i - g}\)
Common Pitfalls to Avoid
\nEven the best students can get tripped up by these common mistakes. Keep an eye out for them!
\n1. Mixing up \(i\) and \(g\): Always remember that \(i\) is what the bank gives you (interest), and \(g\) is how much the check grows each year (growth). In the denominator, it is always \(i - g\).
\n2. The First Payment (\(P\)): Make sure \(P\) is the payment at time 1. If the problem says the payment today is \$100 and it grows starting next year, then the payment at time 1 is \(100(1+g)\).
3. Annuity Due vs. Annuity Immediate: If it's an annuity-due (payments at the start of the period), simply calculate the PV as if it were an annuity-immediate and multiply the whole result by \((1+i)\).
4. Negative Growth: Growth doesn't always have to be positive! If payments decrease by 2% each year, then \(g = -0.02\). The formulas still work perfectly—just be careful with your signs (\(i - (-g) = i + g\)).
Quick Review: Key Takeaways
- Concept: Geometric annuities grow by a constant percentage, not a constant dollar amount.
- Variables: \(P\) = first payment, \(g\) = growth rate, \(i\) = interest rate, \(n\) = number of payments.
- The "I Minus G" Rule: Most formulas for these annuities have \(i - g\) in the denominator.
- Perpetuity: If payments go on forever, the PV is just the first payment divided by \((i - g)\).
- Don't Panic: If \(i = g\), the PV is just \(n \times \text{Discounted First Payment}\).
You've got this! Practice a few problems where you identify \(P\), \(i\), and \(g\) first, and you'll see these "scary" formulas are actually very friendly tools.