Valuing ordinary shares and property with growing income

Q: How do you value an asset with variable growth rates (a multi-stage model)?

Calculate the PV of individual payments during the variable period, then add the discounted 'Terminal Value' calculated using the constant growth formula for the remaining perpetuity.

Q: How can the required rate of return (yield) be calculated from the market price and growth rate?

$ i = \frac{D_1}{P} + g $, which means the total yield equals the dividend yield plus the growth rate.

Q: What is the consistency rule regarding nominal and real growth rates in valuations?

Always match nominal interest rates with nominal growth rates, and real interest rates with real growth rates.

Q: What is the 'D0 vs D1' trap in share valuation?

The valuation formula requires the next dividend ($ D_1 $); if given the dividend just paid ($ D_0 $), you must grow it by one period: $ D_1 = D_0(1+g) $.

CM1 – Actuarial Mathematics for Modelling · IFoA (Institute and Faculty of Actuaries) · 5 min read · Updated: 7 Sept 2026

Welcome to Valuing Shares and Property!

In previous chapters, we looked at fixed-income securities like bonds, where the payments are known and stay the same. But what happens when the income grows over time? This is exactly what happens with ordinary shares (where dividends often increase) and investment property (where rents are reviewed and usually raised).

In this chapter, we will apply the Equation of Value to these "growth assets." Don’t worry if the formulas look a bit intimidating at first—once you see the pattern, it’s just like a geometric progression from your school days!

1. The Basics: Shares vs. Property

Before we dive into the math, let's look at what we are actually valuing:

Ordinary Shares: You own a piece of a company. The company pays you dividends. These dividends aren't fixed; they usually grow as the company becomes more profitable.
Property: You own a building and lease it to a tenant. The tenant pays you rent. While rent might be fixed for a few years, it is typically increased at "rent reviews" to keep up with inflation or market demand.

Key Concept: Unlike a bond, these assets theoretically last forever (a perpetuity). Therefore, we are calculating the Present Value (PV) of an infinite stream of growing payments.

2. Constant Growth: The Dividend Discount Model

If we assume the income grows at a constant annual compound rate, we can use a very elegant formula. Let:

$ D $ = The dividend/rent just paid (at time $ t=0 $)
$ g $ = The constant annual growth rate
$ i $ = The required rate of return (interest rate)
$ v $ = The discount factor $ (1+i)^{-1} $

The cashflows would be:

At time 1: $ D(1+g) $
At time 2: $ D(1+g)^2 $
At time 3: $ D(1+g)^3 $
...and so on.

The Equation of Value

The Present Value ($ PV $) is the sum of all these discounted cashflows:

$ PV = D(1+g)v + D(1+g)^2 v^2 + D(1+g)^3 v^3 + \dots $

This is a geometric progression with the first term $ a = D(1+g)v $ and the common ratio $ r = (1+g)v $.

Provided that $ |(1+g)v| < 1 $ (which is the same as saying $ g < i $), the sum to infinity is:

$ PV = \frac{a}{1-r} = \frac{D(1+g)v}{1-(1+g)v} $

A Simpler Version

If you multiply the numerator and denominator by $ (1+i) $, the formula simplifies beautifully to:

$ PV = \frac{D(1+g)}{i - g} $

Note: $ D(1+g) $ is just the dividend expected at the end of the first year. Let's call this $ D_1 $.

$ PV = \frac{D_1}{i - g} $

Quick Review: The "i minus g" rule

This is the most famous formula in this chapter! To find the price of a growing perpetuity, take the next expected payment and divide it by the interest rate minus the growth rate.

3. Variable Growth (Multi-Stage Models)

In the real world, growth isn't always constant. A company might grow very fast for 5 years and then settle down to a "normal" growth rate. To value this, we use a two-step approach.

Step-by-Step Explanation:

Step 1: Calculate the PV of each individual dividend during the "high growth" period.
Step 2: Use the constant growth formula to find the value of all future dividends at the point where growth becomes constant. (This is often called the "Terminal Value").
Step 3: Discount that Terminal Value back to the present day.
Step 4: Add them together!

Example: A share pays a dividend of $ \$2 $ today. It will grow at $ 10\% $ for 2 years, and then $ 3\% $ forever after that. The interest rate is $ 8\% $.

Year 1 dividend: $ 2(1.10) = 2.20 $
Year 2 dividend: $ 2(1.10)^2 = 2.42 $

Value of the "forever" part at the end of Year 2:
Using $ \frac{D_3}{i-g} $, where $ D_3 = 2.42(1.03) = 2.4926 $
$ Value_{t=2} = \frac{2.4926}{0.08 - 0.03} = 49.852 $

Total PV today:
$ PV = 2.20v + 2.42v^2 + 49.852v^2 $
(Where $ v = 1/1.08 $)

4. Calculating the Yield

Sometimes the exam will give you the Price and ask for the Yield (the interest rate $ i $).

Using our favorite formula: $ P = \frac{D_1}{i-g} $

We can rearrange it to solve for $ i $:

$ i = \frac{D_1}{P} + g $

In English: Yield = Dividend Yield + Growth Rate.

Did you know? This explains why "growth stocks" like tech companies can have very low dividend yields. Investors are okay with a tiny $ \frac{D_1}{P} $ because they expect a very high $ g $!

5. Real vs. Nominal Growth

The syllabus mentions calculating "nominal or real" yields. This relates back to Inflation.

Nominal: The actual cash amounts (e.g., your rent goes up by $ 5\% $ in cash terms).
Real: The value adjusted for inflation (e.g., if rent goes up by $ 5\% $ but inflation is $ 3\% $, the "real" growth is roughly $ 2\% $).

Crucial Rule: Always be consistent!
- Use Nominal interest rates with Nominal growth rates.
- Use Real interest rates with Real growth rates.

6. Common Pitfalls to Avoid

Don't worry if this seems tricky; even top students make these mistakes:

The "D0" vs "D1" Trap: The formula $ \frac{D}{i-g} $ requires the dividend at the end of the first year. If the question says "a dividend was just paid," that is $ D_0 $. You must multiply it by $ (1+g) $ before putting it in the formula.
The "g > i" Impossible Scenario: Mathematically, if $ g > i $, the formula gives a negative number, which makes no sense. In the real world, a company cannot grow faster than the economy forever!
Timing of Payments: Usually, dividends are assumed to be paid annually in arrear. If a question mentions "continuous" payments (more common in property), you might need to use the force of interest $ \delta $ or continuous annuity functions, though the "i-g" logic remains the same.

Key Takeaways

1. Constant Growth: Use $ PV = \frac{D_1}{i-g} $.

2. Variable Growth: Calculate the first few payments manually, then use the formula for the "tail" and discount it back.

3. Yield: The total return is the sum of the dividend yield and the growth rate ($ i = \frac{D_1}{P} + g $).

4. Consistency: Match nominal rates with nominal growth, and real rates with real growth.

Quick check

Can you answer these now?

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

What distinguishes the valuation of ordinary shares and property from fixed-income bonds?

Unlike bonds with fixed payments, shares and property involve income (dividends or rent) that typically grows over time and are often treated as infinite streams (perpetuities).

What is the Dividend Discount Model formula for a perpetuity with constant growth?

$ PV = \frac{D_1}{i - g} $, where $ D_1 $ is the dividend at the end of the first year, $ i $ is the required rate of return, and $ g $ is the growth rate.

Under what mathematical condition is the constant growth perpetuity formula valid?

The formula is only valid if the interest rate $ i $ is strictly greater than the growth rate $ g $ (i.e., $ g < i $).

How do you value an asset with variable growth rates (a multi-stage model)?

Calculate the PV of individual payments during the variable period, then add the discounted 'Terminal Value' calculated using the constant growth formula for the remaining perpetuity.

How can the required rate of return (yield) be calculated from the market price and growth rate?

$ i = \frac{D_1}{P} + g $, which means the total yield equals the dividend yield plus the growth rate.

What is the consistency rule regarding nominal and real growth rates in valuations?

Always match nominal interest rates with nominal growth rates, and real interest rates with real growth rates.

What is the 'D0 vs D1' trap in share valuation?

The valuation formula requires the next dividend ($ D_1 $); if given the dividend just paid ($ D_0 $), you must grow it by one period: $ D_1 = D_0(1+g) $.

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