So when should land development actually occur?

We know that postponing a project has value, just like an option, and this affects the price of land. Using a simplified real options approach, we can calculate the option premium from a one-year delay perspective. We can also make the model more complex, using binomial trees to extend the analysis over several years. However, there’s also a formula that bypasses time limitations and gives us a more direct answer to when it’s optimal to start a project and what the intrinsic value of the land is.

The Samuelson-McKean formula was originally created to value perpetual American warrants. It’s a fairly complex integral formula dealing with stochastic events. Later, this formula was extended by Robert McDonald and Daniel Siegel for capital budgeting purposes; Joseph Williams, Dennis Capozza, and others adapted it to calculate the optimal timing and intensity of urban development.

The version of the formula that applies to land development timing is actually not very complex, because it simplifies the assumption: once the optimal time arrives, the development will indeed occur (i.e., option exercise is not random).

It’s more convenient to break the formula into three parts:

  1. μ – Option Elasticity
  2. V* – Hurdle (or Critical) Value
  3. C₀ – Land Value

Let’s begin:

Option Elasticity (μ)

μ (option elasticity) is itself an important indicator, as it shows how much the land value changes in percentage terms in response to a 1% change in the price of the completed building.

The formula is long, but still simple enough for Excel. It depends on three main components:

  • Y[v] – Cap Rate: capitalization rate of the completed building
  • σ[v] – Volatility of Building Value
  • Y[k] – Construction Cost Yield: the difference between construction cost inflation and the risk-free rate

Y[k]=K[g]−R[f]Y[k] = K[g] – R[f]

(K[g] is effectively inflation)

Let’s use realistic values:

If μ = 4.35, this means that a 1% change in the price of the completed building leads to a 4.35% change in the land value. If we go deeper, this logic also applies to risk premiums:

If the unlevered discount rate for immediate development is 12%, and the risk-free rate is 4.2%, the resulting risk premium is 7.8%. Then, using μ = 4.35, the implied risk premium for land investment becomes 33.9%.

Hurdle Value (V*)

V* is the price level of the completed building at which the project should be initiated—i.e., the critical threshold.

Let K₀ be the construction cost (including developer’s profit).

The formula: V∗K0=μμ−1\frac{V^*}{K_0} = \frac{\mu}{\mu – 1}

This is known as the Hurdle Benefit-to-Cost Ratio.

For example, if the cost to build (excluding land) is $750 per sqm, then: V∗=750×4.353.35≈975V^* = 750 \times \frac{4.35}{3.35} \approx 975

This means the expected sale price per sqm of the completed building must be $975 for the project to become viable.

Land Value (C₀)

The land value is calculated from the formula:

In our case, it comes out to be $200 per sqm of buildable area (not land area). If the current expected sale price is $950 per sqm, this implies that land accounts for 21% of the total revenue.

Important: this $200 figure is the value of land per sqm of building, not per sqm of land.
For example, if the floor area ratio (FAR or K2) is 2.5, and expected build efficiency is 80%, then the actual land value per sqm is $400.

Also, if you exclude the developer’s profit from construction costs, the remaining land value is what needs to be split between the landowner and the developer.

Visual Interpretation

  • The blue line shows land value including the option.
  • The brown line shows the NPV of the development project.

Once the sale price exceeds construction cost, the project starts to generate profit (i.e., positive land value). However, the total NPV may still be negative because the option premium disappears upon execution.

For example, if the sale price is $950, the land value is $200, and the landowner may choose to initiate the project—but they would forgo the option premium, which in this case equals $1/sqm.

Once the sale price reaches the hurdle value (V*), the option component in land value becomes zero. At this point, further delay has no added value, and the project should proceed.

Samuelson-McKean Excel Model

Adapted from:
Commercial Real Estate Analysis and Investment