Post 6 - A Solution

So, what is the point of the lengthy previous five posts? It’s this: Your standard DCF valuation technique, using dedicated commercial real estate software or excel, is severely inadequate. To summarize, DCF (or more generally, your typical model using point-estimates) does not account for or value the variability (and other higher moments) of assumptions, which in extreme domains such as the financial realm, can be more important than the average. It also does not account for optionality, as one has to price in upsides and downsides that may not exist. So, what to do about deterministic models?

Funny enough, the answer, I propose, is hidden in the very last line of Portfolio Selection by Harry Markowitz. He says this:

Portfolio Selection - Harry Markowitz, the father of Modern Portfolio Theory, 1952

I agree. We need to throw out deterministic modeling, which only allows for simple point estimates that take into account Gaussian assumptions, and use a more probabilistic model of valuation via Monte Carlo simulation. The jump from a deterministic model to a Monte Carlo model is a jump in dimensionality; instead of a point estimate, we get a range of estimates. Instead of assuming one rate of rental growth, we “stochasticize” (randomize, or perturb) the input. Instead of getting one NPV, we get a range of NPVs whose probability to be above, below, or between certain amounts can be calculated. For those who desire a precise number, one can take the average of all scenarios to arrive at a more mathematically sound point estimate: one that values all possible scenarios.

Monte Carlo models for real estate aren’t brand new; similar points have been brought up in prior papers and by the team over at A.CRE (adventuresincre.com, a great resource). However, there are caveats to Monte Carlo modeling. Garbage in, garbage out is still a motto that very well holds true. The current Monte Carlo models available for commercial real estate users (none had existed for retail real estate, until now) all make a mistake: They assume that their inputs, or assumptions, vary via a normal, uniform, or another thin-tailed distribution. In essence, their outputs (values) are determined using similar assumptions that got LTCM in trouble. If our inputs/assumptions are fatter-tailed than the Gaussian in reality, but we model with thinner tails, then we are ripe for a Turkey Problem.

So, let us build the Monte Carlo model for retail real estate and think deeper about how the data and assumptions truly behave. We will start with how they have behaved in the past and see if we can better extrapolate into the future than simply blindly assuming the Gaussian.

Two Monte Carlo models were built: one for acquisitions and one for developments. We will pretend to acquire/develop the vanilla shopping center example from the previous post (Post 4). Let’s address the problems with the legacy, deterministic models and the current Monte Carlo models one by one.

Static Assumptions

A Shot In the Dark

I am under the belief that we should want to make our models as realistic as possible. Assuming static rent growth, vacancy, etc. is too simplistic and not real. So let’s start with a basic description of how assumptions have to be created in these new models:

Step 1: Access as much past data as one can on each assumption needing to be stochasticized. Do not exclude any “outliers” from any data analysis. Past behavior, while not necessarily indicative of future behavior, at least proves to us what is possible.

Note: If the past data proves to be fat-tailed, then we can eliminate any usage of thin-tailed distributions. However, the opposite is not true. If the past data shows to be thin-tailed, unfortunately, we cannot rule out fat-tailedness. A million observations of white swans (or Gaussian data) do not confirm the statement “all swans are white” (or the data is Gaussian). One observation of a black swan (or that the data is fat-tailed), immediately proves “all swans are white” incorrect (Taleb, 2007). I’ll remind you of the instances of pseudo-convergence to the average with the Pareto 80/20 mentioned in a previous post. Only when we are certain the Gaussian is true (a human cannot physically weigh 50 billion pounds, so we can be confident that it will not appear in future samples), can we be comfortable using thin-tailed distributions.

Step 2: Fit the past data to either a discrete or continuous distribution, depending on the type of data. The @Risk software by Palisade was used for these analyses. We fit the data using Akaike's Information Criteria (AIC) by testing how well each distribution fits the relevant data set. Then, parametric bootstrapping (at a 95% confidence interval) is used to arrive at a better estimate of the confidence intervals for the parameters of each fitted distribution.

Step 3: Once we have a list of the best-fitting distributions, we choose one. This is where the art of modeling comes into play. Ask yourself, does the fitted distribution make sense? Am I adequately accounting for what you believe could be worst- and best-case scenarios? If two models were close in ranking, it is recommended that the fatter-tailed model (the one with the higher kurtosis) is chosen. This is the more-conservative route – higher kurtosis allows for wilder swings. Again, prior knowledge of the assumption’s behavior, experience in the field, etc. are all important here; we do not blindly accept the #1 best-fitting distribution (especially if it is thin-tailed). The sample distribution is just that, a sample. It does not necessarily represent the population, especially if little data exists or the data in the sample is actually fat-tailed but the information-rich outlier has yet to show itself.

Step 4: Implement the chosen distribution into the model. Theoretically, that data point should now behave as it can in real life.

Simple! Repeat these steps for each data point you want to stochasticize. Once complete, you have a model that models potential future realities of the assumption by modeling how that data can actually behave.

Take annual market rent growth, for example. Steps 1 through 4 were followed for neighborhood shopping centers in each market (N=143) CoStar keeps track of. Best-fitted distributions ranged from thin-tailed triangle distributions (the most common) to thicker-tailed Laplace distributions (second most common). Our bias is for thicker tails (and we use our intuition of market rent growth; we know a bad recession can crater asking rents when there is a quick and large jump in the supply of space), so we choose the Laplace.

There is an argument to make even further adjustments to this Laplace distribution (rent growth is one of the most important assumptions to get as correct as possible, remember). The Laplace distribution requires a location and a shape parameter input, which is analogous to the mean and standard deviation of a normal distribution. Since we could potentially be a turkey and would like to avoid Thanksgiving, we do not assume that any one-fitted Laplace distribution’s location and shape parameter is the correct one. Rather, we assume probability distributions for the location and shape parameters themselves. Current real estate Monte Carlo models miss this crucial point: there is a risk that a data point carrying most of the information within the true distribution is outside of the tested sample. By stochasticizing the location and shape parameters themselves, we move, expand, and contract the probability distribution of rent growth itself. This makes sense; we are uncertain of how the true probability distribution of rent growth will look in the future. The “empirical” (historical) distribution usually isn’t the empirical distribution. The future is usually crazier than we think.

Finally, this model does not assume any correlations between assumptions given it is impossible to gauge how future correlations will look or will behave in the future. There is a big argument to be made to stochasticize the correlation coefficients between the variables, but that complexity goes beyond the depths of this current iteration. This is 100% part of the future development of our model, however.

The biggest advantage of a Monte Carlo model is that it is the ultimate sensitivity analysis; one can still model their mean rent assumption, but how that assumption can vary, skew, and spike (analogous to a variable’s variance, skewness, and kurtosis) is now taken into account, something that is impossible for the legacy software to do. Now, an assumption’s ability to vary, skew, and spike all has an effect the value, as it should.

Blending

Blending Reality vs Reality Blended

Static modeling forces us to take shortcuts. Modeling a blend of rent or releasing CapEx, numbers that won’t occur in reality doesn’t make a ton of sense if you don’t have to do so. The beauty of Monte Carlo is the fact that each simulation should be able to model a reality that can actually occur. Let’s illustrate with an example:

In our Vanilla Shopping Center development, say our shop space has a 70% of accepting their option or signing a renewal (based on the “empirical” or historical average). In the legacy models, one has to blend static renewal and new leasing CapEx via a static probability, the simple math of which is shown below (Alarm bells should be ringing: “But wait, how can the releasing CapEx vary, skew, and spike? How can the option/renewal probabilities change during bear and bull markets?” etc.).

Unfortunately, we are quite certain that ~$11 per square foot of releasing CapEx for shops will never be spent, yet we model it anyway. No longer does this have to be the case.

Now, instead of a blended number, we can model the binary outcomes: the tenant either renews or vacates. If the model has them renewing, then the renewal-releasing CapEx is pulled into the model. If the model has them vacating, then the new releasing CapEx number is pulled. Each tenant has a different renewal probability and different required CapEx numbers for new and renewal (technical: renewal releasing CapEx is modeled as a conditional probability (X | x > 0) after a certain probability, which is why those numbers may seem high) just like reality. Of course, once more certainty arises, then one can override formulas for actuals just like any other model. However, this development model shown above is accounting for the full uncertainty of future CapEx spend; nothing is known today. Now each simulation will be modeled with a reality, not a blend.

Now we must address a pragmatic problem: Presenting a probability distribution as your assumption to an investments committee (or to whoever greenlights your projects) is likely ahead of its time. If you answer the question “What is your assumption for rent growth?” with “A Laplace with a location parameter of X and a shape parameter of Y” (not to mention the parameters’ distributions), you may get some blank stares. If one needs a singular number to attach to, such as an average, then that could be argued to be somewhat of a blend. But there is a significant distinction here: Monte Carlo can take what can actually happen with assumptions and then blends. Deterministic DCF models blend assumptions and then model something that can’t actually happen. Said another way, Monte Carlo models are blending potential realities while deterministic DCF models are reality blended

Yes, theoretically, the blended releasing CapEx number would occur in the “average scenario” if stochasticizing around the assumed renewal probabilities and releasing CapEx spend via a normal distribution. However, as soon we start assuming that crazier, non-Gaussian things can happen (which is reality), then that is not at all the case. In the acquisition model of a simple purchase of the Vanilla Shopping Center, we assume our average scenario of releasing CapEx spend as a percent of the new lease’s rent, whether they are a shop or anchor tenant, and whether it will be a new lease or a renewal. We also assume our average anchor and shop renewal probabilities (anchors and shops are distinguished because they often have much different dynamics, lease terms, CapEx spend trends, etc.). The first graph shows the total projected releasing CapEx spend when we operate as we do today: blend the scenarios and model like normal. We receive one outcome (also known as a Dirac stick) of ~-$268,000 (negative because it is a cost).

Remember, by blending, we are assuming a moderate level of releasing CapEx spend on each and every space. However, in reality, releasing CapEx spend can vary wildly (I have proven this using proprietary data which, unfortunately, cannot be publicized). However, one does not need data to understand that this makes intuitive sense:

· Perhaps we get unlucky and at each rent expiration, for whatever reason, each tenant decides to vacate. Lots of turnover will mean lots of new releasing CapEx spend.

· Depending on the rent being charged on each lease (assuming the real estate owner has a basic level of competence), the releasing CapEx amount can be larger or smaller.

· For renewals, a lot of the time, a tenant doesn’t need any capital infusion. Sometimes, depending on the market, they have the power to demand a bit from the landlord.

· And, of course, different tenants’ leases expire at different times, so no one renewal probability should be the same given the market is constantly shifting.

Simply, on one extreme, we could have a lot of turnover and releasing CapEx could be expensive. On the other, every tenant renews to market and it costs us virtually nothing (and everything in between!). Releasing CapEx spend belongs to a much more dynamic system that can vary wildly. Both the historical data and this simple logic agree.

So, let us model potential releasing CapEx spend how it can actually play out (using past data, which is quite extensive and very much showed we are in a much fatter-tailed environment, thus, no adjustments were made to the probability distributions - AKA how the assumptions can behave going forward. We mistrust past data more if it “proves” to be Gaussian with relatively limited data; a lot of data telling us that we are well within the borders of Extremestan is safer). We use the same exact “average” assumptions as the normal case: The same average new and renewal CapEx spend as a percent of new rent for both anchors and shops, the same average renewal probabilities for both shops and anchors, etc. However, the difference here is that how these “average” values can fluctuate, skew, and spike are now modeled and simulated.

As it turns out, we get a wildly different average (bad news: much more average CapEx spend than previously thought) of ~-$423,000. But is this more accurate? Well, I’ll let you in on a little secret. This vanilla shopping center is modeled close to a property in real life. This property has spent ~$740K in releasing CapEx over the last 10 years (I’ll note that is within one standard deviation of my mean). My model, which I remind you, uses the same exact “average” assumptions as the deterministic model I built to compare the two. The only difference is that my Monte Carlo model allows for fluctuations. It takes into account variability, skew, kurtosis, etc. of this spend based on real historical data. By nature of this alone, the Monte Carlo model came in closer to reality (of course, this is N=1 so more testing is needed, but we’re off to a good start!)

This illustrates such a key point repeated in previous posts: In the Gaussian domain, fluctuations (or errors) in “x” (our assumptions) result in much smaller fluctuations (or errors) in F(x) relative to an extreme, non-Gaussian domain. Since the behavior of releasing CapEx data can be extreme, it is no surprise the blended assumption of releasing CapEx was significantly further away from reality: we are attempting to forecast (and blend) single-point estimates in a fat-tailed domain. As I’ve said, in this domain, getting the dynamics of the system correct matters much more than getting the initial “average” assumption correct.

Optionality

Your Values Are Wrong

Now twist your brain away from acquisition and pretend we are developing Vanilla Shopping Center instead. In our pretend scenario, the market is showing enough demand for our Vanilla Shopping Center development without the pads. Let’s pretend it’s the year 2010, so we aren’t sure if or when the pads can be built. However, as soon as we see sufficient demand for the pads come to fruition, we will build them immediately.

Let’s say that sufficient demand for extra space would manifest itself as a lower vacancy rate than a market or stabilized/normal rate. For this development Monte Carlo model, we will define our stabilized market vacancy level as 92.5%. If demand for space rises above that level, then we will build extra space. If it does not, then we won’t. But as of now, we are confident we will at least reach market vacancy rates with our Vanilla Shopping Center scenario without the pads.

Here are two sample pro formas for our development Monte Carlo model. The first one shows a scenario where we realized demand for space over the market vacancy rate of 92.5% (See the “optional construction” line item). The second pro forma shows a scenario where demand for space never exceeded 92.5%.

Remember from Post 5: our problem with deterministic DCF models was that they cannot account for the inequality of the upside and downside scenarios. Here is that point using the Vanilla Shopping Center example.

The first chart shows the distribution of levered NPVs of the Vanilla Shopping Center development without any chance the pads were to be built. This is analogous to just underwriting the plain shopping center scenario without consideration for the potential pads. Our expected NPV from the Vanilla Shopping Center is ~$700,000. Our maximum loss is ~$6.4M and our maximum gain is ~$11.3M.

The next distribution of levered NPVs shows the same analysis with the same assumptions, just this time the development of the pads is guaranteed to happen. This is analogous to traditional DCF underwriting the Vanilla Shopping Center plus the pads.

As expected with a grander, riskier project, both our expected minimum and maximum NPVs are significantly more extreme, with the maximum loss increasing by about 50%. However, partially due to the non-linear effects of a potential reduction in terminal cap rate, potential rent growth, etc. we have a larger expected NPV. This makes sense too – larger projects should add more value to the firm. More risk, more reward. Now we will see how optionality makes things even better.

This final distribution is our real option properly modeled: If the demand is there we build. If it is not, we do not.

Just as predicted, when optionality is introduced, the maximum downside is close to the downside of the smaller-scale project (~-$6M for both) while the maximum upside is closer to the upside of the larger-scale project ($13M and $14M, difference due to chance). Even better, our expected return is higher than both of the other scenarios ($1.8M in our real option model vs $700K in our Vanilla Shopping Center model and $1.6M in our full development model). How does that make sense?

This ties back to the problem of blending. When you are traditionally underwriting these two scenarios, you would think you could average (blend) the NPVs of each scenario and that would be your expected value of the project (($700M + $1.6M)/2 = $1.15M). That would be incorrect. The combination of the downside of the smaller scenario and the upside of the larger scenario shifts the entire distribution to the right. This gives you an elevated expected value, just due to the chance that one day you could take advantage of a bull market (and there almost always is – real estate cycles). The value of the real option is the magnitude of the distribution’s expected NPV movement relative to the static scenarios. This value of optionality is almost never taken into account when a development is first underwritten.

Here is this same real option distribution but for the levered IRR in scatter plot form. We can see right at the trigger point of 92.5% that our distribution changes. The relative frequency of data points falling below our cost of capital (~8.8% in this case) is visibly different on either side of the trigger point.

The lesson: Heightened optionality allows your developments to be more concave to uncertainty, and the future is certainly quite uncertain.

Point Estimates

The Difference Between Trees and Forests

We’ve now stochasticized our assumptions, intelligently accounting for how they can actually behave in reality, fixed the blending problem, and properly modeled optionality. So, what to do with the result of a Monte Carlo simulation?

If you go down this rabbit hole and start living by Monte Carlo, you will likely still need to report a singular projected NPV or IRR of a project to whoever is making decisions. In this case, the average and/or median of the distribution are your likely candidates. However, most humans take that average number and translate it to mean “the most likely”, which is incorrect. It may be wiser to just report the median - 50% of the NPVs fall above and below the median NPV. It also may be wise to report the likelihood of success. In this case, ~30% of the simulations fall below an NPV of $0, implying a ~70% chance of success. One can take these probabilities of success and failure and compare them across other investments. Ceteris paribus, one could choose to accept the project with the highest likelihood of success or the highest expected value. Or, if simply deciding between one project or doing nothing, then one would accept the project if the odds of success meet or exceed your subjective risk-tolerance threshold.

Compare the sophistication of the Monte Carlo to the deterministic DCF models employed by the legacy software companies today. DCF models take in point-estimate assumptions and spit out point-estimate NPVs or IRRs. There is no information regarding how that NPV can vary, skew, or spike. The point estimate is like studying a single tree to understand the complete ecology of the forest; you need much more information beyond that.

Even traditional sensitivity analyses are inadequate. Usually, they are performed by changing one variable while keeping the other static. This is useful to understand how one assumption might affect the overall NPV, but if you are still keeping every other assumption static, it isn’t very informative of how the project can perform relative to Monte Carlo. Even if you manually change a few or several variables to model a downside and upside scenario, there are only so many scenarios you can underwrite manually. A low sample size (perhaps one would traditionally underwrite 3-5 scenarios) does not allow for confidence in statistical information, even in a Gaussian context.

However, the real value of this information is that now we can focus on what could ruin us – something LTCM could have benefitted from. A Monte Carlo model can help us define how much we can lose; we have a map of Thanksgiving and how we could potentially get there. If time plays out and we perform at or exceed expectations, great – we are lucky. However, we could get unlucky - the downside of the distribution could play out. If the downside of the distribution is unacceptable, i.e. would, in any way, lead to ruin (bankruptcy, bad reputation, etc.) then it is necessary to focus on and hedge the downside (left-tail risk) the best we can.

One must ask themselves; how do I limit the downside scenario? How can I structure deals to increase the odds of success (shifting the distribution of NPVs to the right)? How do I maximize the potential geometric average return of this investment (if you’re a Kelly believer)? The hedging of tail risk and maximization of the geometric average return (preferably, that is one-in-the-same) in a real estate context is the next piece of the puzzle. This will be our next topic.