To explain the formula for calculating the equivalent capitalisation rate step by step. We'll build it iteratively, starting from the simplest case and adding one layer of complexity at a time. This way, it's like constructing a house: we start with the foundation and add walls, roof, and details gradually. The goal is to find the equivalent capitalization rate (c), which tells us what single rate to use to turn all future rents into one lump sum today, accounting for inflation (RPI) and review timings.

Step 1: The Basic Case – Fixed Rent with No Growth or Reviews

Imagine the rent is fixed at £100 per year forever (or for a long term like 90 years), with no increases due to inflation and no reviews. We just need to discount future payments back to today's value using a fixed capitalisation rate (let's call it y, like 6.5%). This is like valuing a simple savings bond that pays the same amount every year.

The present value factor (f) for an annuity (regular payments) over T years is:

\[ f = a_T = \frac{1 - (1 + y)^{-T}}{y} \]

Here, a_T is the annuity factor for T years. The equivalent capitalisation rate c is just the rent divided by the total present value, but since rent is £100, c = 100 / (100 * f) = 1 / f. (We multiply by 100 to get it as a percentage later.)

Why start here? This ignores RPI changes entirely, so it's too basic for real life where rents grow with inflation. But it shows the core idea: future money is worth less today.

Step 2: Adding Continuous Annual Growth from RPI

Now, let's introduce the changing RPI rate. Assume rents grow every year by a constant RPI growth rate (call it g, like 2.4% or 0.024 in decimal). So, the rent starts at £100 but becomes £100 * (1 + g) next year, £100 * (1 + g)^2 the year after, and so on. This is a "growing annuity."

The present value factor f now accounts for this growth:

\[ f = \sum_{k=1}^{T} \frac{(1 + g)^{k-1}}{(1 + y)^k} = \frac{1 - \left( \frac{1 + g}{1 + y} \right)^T}{y - g} \] (if y ≠ g)

If y = g, it simplifies to f = T / (1 + y).

Then, c = 1 / f.

This step is better because it factors in RPI growth making future rents bigger, so the lump sum today should be higher (meaning c is lower). But in reality, ground rents don't grow every year, they're reviewed in chunks (like every 5 or 25 years), so we need to adjust for that periodicity.

Step 3: Making Growth Periodic (Every n Years) with Full Reviews Only

Next, we account for the frequency of rent reviews (n years, like every 5 or 25 years). Rents stay fixed within each n-year block but jump by the full RPI growth at each review: the growth factor G = (1 + g)^n.

Assume the lease starts right after a review (no initial wait), and T is a multiple of n for simplicity. Number of full blocks: m = T / n.

The present value factor f breaks into blocks:

First block (years 1 to n): fixed rent, so a_n = \frac{1 - (1 + y)^{-n}}{y}

Next block (years n+1 to 2n): rent * G, discounted back: a_n * (1 + y)^{-n} * G

And so on, up to m blocks.

This forms a geometric series: f = a_n * \sum_{k=0}^{m-1} \left( \frac{G}{(1 + y)^n} \right)^k = a_n * \frac{1 - \delta^m}{1 - \delta} where \delta = \frac{G}{(1 + y)^n} (if \delta ≠ 1)

If \delta = 1, the sum is m.

Then, c = 1 / f.

This is more realistic for ground rents, as it captures "stepped" growth every n years instead of smooth annual increases. The frequency n matters: shorter n means more frequent jumps, closer to continuous growth, affecting c.

Step 4: Adding the Years Until the Next Review (r)

Real leases might not start right after a review—there could be r years (1 to n) until the next one. So, add an initial partial period with fixed rent.

The present value factor f now has:

• Initial r years: fixed rent, a_r = \frac{1 - (1 + y)^{-r}}{y}
• Then, the remaining (T - r) years, assuming it's a multiple of n for now: like Step 3, but starting after r years, so discount the blocks by d = (1 + y)^{-r}, and the first block after r has rent jumped by G already.

So, the blocks part becomes: a_n * d * G * \sum_{k=0}^{m-1} \delta^k = a_n * d * S where S = G * \frac{1 - \delta^m}{1 - \delta} (if \delta ≠ 1), or S = G * m if \delta = 1.

Full f = a_r + a_n * d * S

Then, c = 1 / f.

Why add r? It accounts for how soon the first growth hits; if r is small (next review soon), growth starts earlier, lowering c; if r is large, growth is delayed, raising c.

Step 5: Handling Any Total Term T (With Remainder p Years)

Finally, T - r might not be a multiple of n, so there are m full blocks after the initial r, plus a leftover partial block of p = (T - r) mod n years at the end.

The final partial: after m full blocks, rent has grown by G^{m+1} (since the last review jumps it for the remainder), discounted by d * (1 + y)^{-m n} = d * \delta^m / G (but simplified in formula).

In the full equation: add the term if p > 0: a_p * d * G * \delta^m (wait, actually G^{m+1} but adjusted, wait, no: after m blocks, the growth for the partial is G^{m}, but let's use the precise version from earlier).

The complete f = a_r + a_n * d * S + \begin{cases} a_p * d * G * \delta^m & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases}

Where m = \lfloor (T - r) / n \rfloor, and all other terms as before.

Then, the equivalent capitalisation rate c = 1 / f.

This final step ensures the formula works for any lease length, capturing every nuance: RPI growth (g), review frequency (n), time to next review (r), and total term (T).