**Intro**

Monte Carlo Simulations are a way to model the probability of different scenarios. Why we need Monte Carlo is due to the fact that those scenarios cannot easily be predicted as the input variables which are involved are to some extent random. This thing comes pretty fancy especially when, based on the amount of randomly behaving input parameters, an analytical solution is no more visible.

**Monte Carlo simulations** are used practically everywhere, e.g. in engineering, in biomedicine, with supply chains or in finance to name a few. They are a great help in revealing risk and giving a feeling for the uncertainty in predictive analytics and forecasting.

In this article we have a look at how Monte Carlo simulation works, what the difference is to **Markov Chain Monte Carlo**, how it is to be applied to "real life" situations in real estate and where are the tricky issues waiting in the process.

And we talk about why this is a great tool for predictive analytics applications and a gateway to a world of advanced risk analytics from the perspective of the commercial real estate business.

**Monte Carlo Simulation - Basics**

Suppose, a customer service has two clients. During a month, there is usually a 20 % chance that Client 1 shows up and needs help. In the same time period, the chances that Client 2 shows up for help is around 5 %. In order to make a proper working schedule for the month, the customer service wants to know the probability that one of the customers shows up or that even both customers reach out at the same time.

### The Analytical Approach

We just have 2 parameters (Client 1, Client 2) with an inherent chance of calling the customer service. So, in this case the probability can be derived analytically.

We face the following framework for the whole month:

showing up staying away

-----------------------------------------------------------------------------------------------

Client 1 20 % 80 %

Client 2 5 % 95 %

which gives the following probabilities:

NO client shows up = 0.8 * 0.95 = 76 %

Client 1 OR Client 2 shows up = 0.8 * 0.05 + 0.2 * 0.95 = 23 %

BOTH clients show up = 0.2 * 0.05 = 1 %

-----------------------------------------------------------------------------------------------

sum = 100 %

============

### Monte Carlo Simulation

In a Monte Carlo simulation we would take the same situation, i.e. two clients, a period of one month and the given changes of those clients to show up (20 % for Client 1 and 5 % for Client 2) but we would repeat this situation over and over again (in this example exactly 1 million times) in order to approximate the relevant combined probabilities.

A Monte Carlo simulation for this problem could look like this:

*both_clients <- replicate(1e6, {*

* *

* client_1 <- sample(c(0, 1), n = 30, prob = c(0.8, 0.2), replace = TRUE)*

* client_2 <- sample(c(0, 1), n = 30, prob = c(0.95, 0.05), replace = TRUE)*

* client_1 + client_2*

* }*

Following is an excerpt of possible outcomes with

each column equalling a simulated month

each row equalling a day in that month

0 ... no client showed up

1 ... one of the clients showed up

2 ... both clients showed up

The simulation framework gives the following results for an ordinary month:

no client shows up

one of the clients shows up

both clients show up

The simulation results are approximately the same as with the analytical approach, i.e. 76 % for none of the clients showing up, 23 % for one of the clients showing up and around 1 % for both clients showing up at the same time.

Here is once more the frequency of the various client behaviour during an ordinary month:

Of course, this example is trivial as there is an easy analytical solution. Though, as said above Monte Carlo simulation is a **great help when the amount of input parameters gets larger and an analytical solution is no more possible or is getting too complex to derive**.

**Markov Chain Monte Carlo**

In a Monte Carlo setting, the input parameters are independent of each other. Just look at those two clients in the example before. That Client 1 needs help at one time is completely independent from the needs of Client 2.

This slightly changes for Markov Chain Monte Carlo (MCMC) applications. In this situation the **outcome of the current stage is dependent on the outcome of the previous stage**. It is to be noted that the dependence is just for the immediate previous stage but not for any further stages back!

Here is an easy example to grab this.

Suppose, at stage t_0 a ball is on the position x = 50 and y = 50 on a x/y - grid. Moving from one stage to the next stage, the ball can

stay where it is, i.e. move_x = 0 / move_y = 0

move to the left for 1 tick, i.e. move_x = -1 / move_y = 0

move to the right for 1 tick, i.e. move_x = 1 / move_y = 0

move up 1 tick, i.e. move_x = 0 /move_y = 1

move down 1 tick, i.e. move_x = 0 / move_y = -1

move to the left and up 1 tick each, i.e. move_x = -1 / move_y = 1

move to the right and up 1 tick each, i.e. move_x = 1 / move_y = 1

move to the left and down 1 tick, i.e. move_x = -1 / move_y = -1

finally move to the right and down 1 tick, i.e. move_x = 1 / move_y = -1

Any of those movements have the same probability to be "taken" by the ball.

So, looking at the x/y - grid the ball is in position 50/50 at t_0. At t_1, the position of t_0 is to be taken into account plus one of the possible movements of the ball as described above, e.g. 50 - 1 / 50 resulting in position 49/50 at t_1. Following up, the position of the ball at t_2 is only dependent of its position at t_1 plus any of the possible movements the ball can take. But the position of the ball at t_2 is definitely no more dependent on the situation prevailing at t_0.

To put a long story short, the "random walk" of this ball could look like this:

In principle, that's it. With this knowledge one can build up a proper Monte Carlo/ MCMC application.

**Markov Chain Monte Carlo in Real Estate**

Markov Chain Monte Carlo is an excellent tool to simulate time series. Therefore, it is an optimal pre-requisite for **predictive real estate market models**.

For illustrating purposes, let's have a look at the empiric data of office rent prices in a real estate market.

Based on the empiric time line and a current rent price level of EURO 25.5 we can use MCMC to predict the market rent development for the next, e.g. 5 years:

What this simulation reveals is that by the end of the time period

there is a 30 % chance that the rent price level will be below the current price level

with a 90 % probability the rent price is within an interval of EURO 22 to EURO 32

though, the "hot zone" of a potential rent price development is between EURO 23 and EURO 29

Please keep in mind that the longer the time period for the simulation the broader the range of an expected development gets which reflects the **growing uncertainty about future ongoings**. This is shown in the graph below as the "hot zone", i.e. the deep blue area, is clearly spreading out with time.

**Some "Heavy Work" on Monte Carlo/ MCMC Simulations**

Of course, there are no free lunches!

In order to succeed in a powerful Monte Carlo simulation, this tool has to be prepared and **several serious topics have to be taken into account**.

Just remember the customer service example. Both clients were independent of each other. The derivation of the possibility of both clients showing up at the same time was quite straightforward. Otherwise, the simulation of the ball walking randomly through a x/y-grid had already a first constraint, i.e. the position of the ball in the current stage was dependent on the position of the ball in the immediate previous stage.

Focusing on the real estate market now and hence reflecting once more the rent price scenario, it seems clear that the rent price development at one stage is also dependent on the rent price level of the respective previous stage.

Additionally, the behaviour of the rent price which is basis for the MCMC simulation is to be inferred from the empiric data in a statistical way in order to be usable for the tool.

And so far, we just talked about one market parameter. Though, the rent price is embedded in a market structure with a lot of market features. This other market input has to be **statistically approximated** as well before incorporating it into a simulation. Furthermore, the **association among the market parameters**, expressed as **covariance matrix**, has to be defined before being employed in the MCMC environment. This can be just as tricky.

So, a **lot of technical modelling** and **market expertise** is **necessary** in order to be able **to build up a strong predictive tool** to be **embedded into the risk management systems of real estate investors**.

For different market parameters and their associations, take as an example the empiric data of the yield development of two different real estate markets (but the same asset class). Their development shows a strong association with a correlation amounting to 0.75.

This fact has to be taken into account when setting up a MCMC simulation. And it is not only the relation between those two market features. Any association among market parameters involved has to be addressed in order to be able to reflect the real market in the simulation model.

Doing this, we achieve that the simulated data resemble the behaviour of the observed data as can be seen in the chart below:

The more market parameters the more complex the modelling gets as all those features have to be set in relation to each other. The following chart shows some of the results of a market model based on MCMC (blue points ... empirically observed values, red ... simulated values):

But I guess, you get the point...

**Conclusion**

With a well tuned mixture of statistical modelling and business expertise it is possible to create a **very powerful tool for predictive analytics tasks in the real estate business**. Of course, a sufficient preparation has to be done in order to get the best out of Monte Carlo/ MCMC simulations and to come up to the challenges and pitfalls, more complex models might be prone to.

And, it does not stop there!

Talking real estate, the simulation results can be implemented in the respective risk management environment of real estate companies and are a **very strong foundation for further risk/ performance analytics on project, portfolio or entity level**.

Another gimmick is that Monte Carlo can be handled as a gateway to a more advanced analytical approach implementing the whole field of machine learning/ neural networks and - in a further stage - network analytics and causal inference.

There is a wide variety of toolkits in this sector which just waits to be implemented in a more traditional business as it is the case with commercial real estate.

Implemented with the aim to supporting a real estate company in tackling its risk exposures much more accurately and being able to reduce costs and damages.

**RiskTech at your service!**

For more information, see our page __www.d-darks.com__ or give us a call.

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