This chapter shows how we can apply simulation to obtain useful insights and guidance on an important practical problem. It demonstrates the “how” and “why” of flexibility analysis for real estate. It shows how we can do the analysis in a realistic setting, and the results demonstrate by example the value of flexibility for real estate management.

Along the way, we present two methodological additions to the traditional, single‐stream pro forma DCF analysis. These are:

• Extension of the time horizon in the DCF model; and
• Use of IF statements.

We demonstrate these additional tools in a very simple, concrete example. But they will be even more useful down the road in more complex development project and management decision analysis.

9.1 The Resale Timing Problem

The question of when and how to decide to sell an investment property is a basic and universal consideration in real estate, as it is in other arenas of investment. Yet, the standard pro forma DCF analysis and valuation of an income‐generating property does not address this question. When we limit the analysis to a fixed period, such as 10 years, as is the traditional practice, we act as if this period represents the length of our interest in the property. Indeed, the traditional model explicitly presumes a resale at the end of that period, the liquidation of the investment. But is this realistic? And if not, does this traditional analysis practice matter for valuing the property or understanding investment strategy?

Let’s now recognize that prices do go up and down, and that investors generally have the flexibility to decide when to sell. How do we play the game of “buy low, sell high” (or, at least, “sell high”)? Flexibility analysis using the simulation modeling we’ve described in previous chapters is well suited to address this question.

In the investment world, people often think about timing the resale of an asset in terms of a “stop‐gain” decision rule. The rule is to sell the asset as soon as its price rises above a pre‐specified (trigger) level, no matter what, and not to sell the asset before then.

The idea behind the stop‐gain rule is to not be too greedy. In practical terms, the idea is to cash in on your “super‐normal” gains and reinvest those profits elsewhere, even if the subject investment might still have some more gain to run up—in other words, the property could still face seemingly bright future prospects.

The stop‐gain rule can be rational if you believe that the price dynamics in the asset market are mean‐reverting—that once the asset value has pierced some pre‐specified high level, then it must surely soon “fall back to earth.” To the extent that real estate prices have a strong tendency to go through cycles, that they tend to revert to a mean value, then there is a good chance that a stop‐gain rule might be a good investment strategy, a good way to profit from the flexibility that property owners realistically have about when to sell.

Contrarily, if prices do not revert to a mean, if they are truly memoryless, as analysts often assume they are in the stock market, then a stop‐gain rule might not be productive, implying that resale timing flexibility might not be as valuable.

Let’s explore how the stop‐gain rule might work for investing in real estate. We now show how to set up the DCF model to accommodate a stop‐gain rule for flexible resale timing for a typical property investment.

9.2 Extending the Time Horizon of the Discounted Cash Flow Model

We have already demonstrated the value of flexibility in the timing of the resale of an asset. In Section 4.3, we considered the case of a rental property whose value depended on whether an attractive complementary development did or did not materialize nearby. In that case, we were able to show that the flexibility to time the resale had significant value. But that example was very simplistic, because we had not yet introduced the use of probability distributions and simulation analysis, which were described in Chapters 5 and 6. We can now make the analysis of resale timing much more realistic.

We need to expand the possible time horizon considered in the DCF model in order to explore fully the general case of resale timing. Indeed, we should not limit the analysis to the 10 years of the conventional DCF analysis. This is a particular instance of a generic issue in handling the analysis of complex choices for managing uncertainty. Different issues require different extensions to the pro forma DCF analysis.

For the analysis of resale timing, an obvious modification to the traditional DCF model is to expand the number of future years in the analysis. In this case, we expand the model from the traditional 10‐year horizon to a much larger number, 24 years. (In fact, investors rarely hold investment properties for longer than 24 years, at least in the United States.)

The other crucial addition will be to allow the model to represent a resale before the 24‐year horizon, indeed, even before the 10‐year horizon in the original traditional pro forma, should the stop‐gain rule trigger this decision.

9.3 IF Statements

The key to modeling flexibility to make choices is to embed IF statements in the spreadsheets for the DCF analysis. IF statements are commands that trigger a decision when the spreadsheet encounters pre‐specified conditions. They provide the means to automate the process of mimicking the decisions that investors or managers would take. (This is the analog nature of our simulation modeling approach.)

In a sense, IF statements take simulation analysis “to the next level.” A simulation without IF statements is passive; it simply describes the outcomes that would occur without any managerial intervention. The IF statements enable the simulation to represent the potentially active managerial or investor interventions. IF statements allow us to capture the outcomes that occur when managers react purposely, meaningfully to events that may occur. And this is how flexibility obtains its value.

An IF statement is a standard command in spreadsheet programs such as Microsoft Excel^®. IF statements instruct the analysis to implement management decisions at the proper time. They are remarkable commands that do a lot. They:

• Monitor the development of each scenario over time, period by period;
• Check to see if it is time to implement a predetermined decision; and then
• According to predetermined instructions, alter the inputs to the DCF for all subsequent periods, in effect implementing management instructions to react to specified circumstances.

These abilities allow IF statements to effectively mimic the actions of a human manager or decision‐maker following the progress of a project in any future scenario. They thus enable us to create a realistic “what if” analysis, to reflect contingencies: if the scenario develops in this way, then the managers would do such and such, and the outcome would then be so and so.

As an example, suppose we wish to estimate the value of a stop‐gain rule to sell on a 20% gain above what had been predicted in the original pro forma. Based on our previous definition of pricing factors, this would be indicated by a pricing factor realization of 1.20 in some year of a given scenario. We would refer to 20% as the stop‐gain resale decision “trigger.” Suppose that the simulation develops a scenario in which the sequence of pricing factors period by period are as in Table 9.1. The appropriate IF statement checks each pricing factor, doing nothing until the simulated pricing factor gets to or exceeds the trigger level (1.20 in this case). It then gives the command to “sell,” and this reorganizes the inputs to the DCF to create a sale and value the reversion of the asset at that point.

Importantly, IF statements enable us to consider forms of flexibility far more complex and realistic than those accessible to standard economic equilibrium models of financial options. As we demonstrate in later chapters, this is because they can:

• Respond to complex combinations of past events; and
• Coexist with many different IF statements, so we can easily value simultaneous forms of flexibility. For example, we can “nest” IF statements within other IF statements, such as: IF(current price is above trigger, and IF(we have held the property past some minimum holding requirement)), THEN(sell the property now).

9.4 Trigger Value for Stop‐Gain Rule

The stop‐gain rule requires the setting of a “trigger” value, the gain at which (and not before which) the resale will occur. For this illustration, we have set this trigger to +20% (that is, a pricing factor of 1.20). In any randomly generated scenario (a Monte Carlo trial), the property will be sold once and only once, in the year when the realized property cash flow outcome first exceeds the original pro forma projected outcome for that year by more than 20%, or in Year 24. We simulate 2000 independent random scenarios for the analysis, each one containing up to 24 future years. Box 9.1 gives details on how to implement this in a spreadsheet.

9.5 Value of Example Stop‐Gain Rule

Let’s now examine the value of resale timing flexibility under the stop‐gain rule. To illustrate, we continue the example of the DCF valuation of the simple rental property we introduced in Chapter 1, and for which we considered the binary scenario analysis in Chapter 4. The DCF model estimates the present value (PV) of the property at a 7% discount rate, or the IRR of the investment assuming a $1000 price.

As we indicated in Section 4.3, the value of the flexibility is the difference it makes to a project:

(4.2)

To obtain the value of flexibility, we must therefore compare the project with the flexibility against a base case or alternative that lacks the particular flexibility or rule we are evaluating.

In this example, the base case alternative is the situation that assumes that resale always, automatically, and inflexibly occurs in Year 10. This case corresponds to the assumption in the traditional pro forma DCF valuation.

However, the traditional pro forma valuation is unrealistic. It does not take into account the price dynamics in the market place. The traditional pro forma considers only the single‐stream future cash flow projection, thereby producing a single PV number (or IRR number at a given price), and a single estimate: “the” PV, or “the” IRR of the investment (ex‐ante). In the simulation, we will sample the entire future outcome (ex‐post) PV and IRR possibility distributions.

To assess the value of flexibility properly, we must expose the inflexible case to the same independent, random future scenarios as the flexible case. We must compute the outcomes for the two cases, inflexible and flexible, under exactly the same scenarios of pricing factor realizations. We can then compare the results of both the inflexible and flexible cases, not only against the (single‐number) traditional pro forma metrics, but also side by side against each other for the entire distribution of possible (ex‐post) outcomes, recognizing the uncertainty and price dynamics that realistically exist.

The target curves in Figures 9.2 and 9.3 show the difference in the distributions of the investment performance outcomes between the flexible and inflexible approaches to resale timing. (These images are identical to Figures 8.4 and 8.5, which we used to explain target curves.) They now demonstrate the value of the flexible stop‐gain rule in the case of our rental property investment. Figure 9.2 shows the cumulative sample distributions of the ex‐post present value based on the 7% discount rate. Figure 9.3 shows the cumulative sample distributions of the ex‐post IRR based on the $1000 price. The inflexible case assumes automatic resale at Year 10. The flexible case implements the stop‐gain rule with the 20% trigger, selling the property either before, after, or right at the 10‐year horizon of the traditional pro forma. The vertical dashed lines represent the means of the ex‐post sample distributions. The vertical black dotted line is the corresponding single number from the traditional pro forma DCF valuation.

Consider first the target curves for the inflexible case. These show the effect of explicitly recognizing uncertainty, without recognizing any management flexibility to time the resale. The simulation reveals the entire range and distribution of possible ex‐post outcomes for the property asset investment, as well as the central tendency or expectation of those outcomes. In terms of PV, the mean of the simulated distribution is virtually identical to the traditional pro forma–based PV, $1000. Without recognizing any resale timing flexibility, the traditional approach gives an essentially correct PV, in terms of a single number representing the central tendency of the likely ex‐post result. This is because we have assumed that the traditional cash flow pro forma is a “good” pro forma; that is, it presents unbiased expectations of the future cash flows for the property.

But the traditional approach does not give the correct IRR results, as we can see in Figure 9.3, even in spite of the fact that the pro forma contains unbiased cash flow projections. The mean of the simulated ex‐post IRR distribution at the $1000 price, assuming the 10‐year resale, is not the 7% of the single‐stream pro forma, but rather less than 6.75%. This reflects the concave nature of the IRR metric as a function of the cash flows, and reflects the uncertainty that we have modeled of typical real estate pricing factor probabilities and dynamics, as described in Chapter 7. Moreover, the simulation reveals not just a single estimate for the IRR, but the entire range and distribution of possible ex‐post IRR realizations. You can think of this as a visual representation of the risk in the investment return. We see that there is about a 90% chance that the actually realized ex‐post 10‐year IRR could lie anywhere between 2.5% and 11.5% per annum, an approximately symmetrical distribution around a mean of about 6.75%. Thus, the simulation not only explicitly quantifies the expected return but also indicates the investment risk in the return (in the sense of the range in the likely outcome).

Now consider the target curves for the flexible case. These represent the results allowing for flexible resale timing, as implemented by our stop‐gain rule with the 20% trigger. This flexibility clearly improves expected value, as the target curves for the flexible case are clearly to the right of those for the inflexible case. The flexible resale rule is much more likely to lead to a more favorable investment result measured by the PV at 7% than the inflexible resale rule. In fact, the stop‐gain flexible resale timing provides an average ex‐post present value some 25% higher than the traditional pro forma, around $1250 instead of $1000. The result for the IRR is perhaps even more impressive. At the $1000 price, the mean ex‐post IRR is around 14%, with a much wider range that is strongly positively skewed, ranging in terms of 90% confidence between roughly 5.5% and well over 30%. (The long, upper‐right‐hand tail in the cumulative target curve indicates the positive skew.) These findings are interesting in themselves, because, while the simple example rental property from our Chapter 1 pro forma is only an illustration, its numbers are in fact typical of stabilized income‐generating investment properties in the United States.

Using a scatterplot like that described in Figure 8.6, we can see that the flexible resale timing (with the 20% stop‐gain trigger) beats the inflexible 10‐year resale in about three‐quarters of the outcomes, while the inflexible beats the flexible in about one‐fifth of the outcomes. The few remaining outcomes, where the results are identical, correspond to scenarios in which the stop‐gain rule happened to indicate a sale in Year 10, the same as the inflexible rule. In fact, Figure 8.6 is based on the simulation analysis reported in this chapter. If you look back at that figure, you can see that there are some dots below the zero‐difference horizontal axis. We therefore do not have first‐order stochastic dominance, as per our definition of that term in Box 8.1. However, if you look at the scatterplot of Figure 8.6, you will note that the only outcomes in which the inflexible rule beats the flexible rule are in scenarios in which the investment has performed rather well, at least nearly average and with positive IRRs, mostly upside outcomes and no downside outcomes (no dots below zero in the left‐hand tail). Thus, it seems unlikely that any investor would prefer the inflexible rule, even a very risk‐averse investor who is mostly very worried about the possibility of extreme downside outcomes.

The scatterplot allows us to see the clear investor preference for the flexible approach, even though that approach does not strictly dominate the inflexible approach (it has greater IRR variance). The much greater variance in the flexible decision rule’s IRR distribution as compared to that of the inflexible rule’s is due to the positive skew in the IRR distribution (which appears prominently in Figure 9.3), and is therefore not an apples‐to‐apples measure of risk for comparison with the variance in the inflexible rule’s IRR distribution.

9.6 Conclusion

This chapter applies simulation to a specific, concrete example that is realistic and contains useful and practical implications for decision‐makers. The question of timing the resale of investment property is universal and fundamental.

This chapter showed how we can explicitly recognize and model uncertainty and flexibility. Procedurally, this requires us to extend the conventional DCF analysis and to implement IF statements in the spreadsheet to enable us to model a possible type of active management of the property.

The results of the analysis demonstrate that the simulation tool we are presenting does indeed enhance the traditional DCF valuation model. In this analysis, flexibility increased the expected present value by 25% (from $1000 to $1250). In the following chapter, we will discuss what this means.