**Kenneth M. Lusht, Ph.D., MAI**

# Income-based models for estimating hotel values

## T*he conceptual relationship between the direct capitalization and discounted cash flow models*

*W**hen considering a hotel property for potential sale or purchase, business owners and investors base their decisions to proceed on valuation models. They want to know how much investment is required and what the return on that investment is likely to be. To make these estimates, income-based valuation models are typically used. Although a number of such models exist, the two most widely used are the direct capitalization model and the discounted cash flow model. These two models differ in well-known ways with respect to the processes used to arrive at value estimates. What is more important, however, is to recognize that despite their procedural differences, the models share fundamental consistencies.*

In this article, we begin by reviewing the conceptual relationship between the direct capitalization and discounted cash flow models. Then, we use some of the conclusions from that review as the basis for considering some key relationships embedded in the discounted cash flow model. They include:

- the relationship between the assumed holding period (the time period the property will be held), and the resulting value estimate.
- the relationship between the going-in capitalization, or cap, rate (the current market rate) and the going-out capitalization rate (the expected capitalization rate at the end of the holding period).
- the relationship between the holding period assumption and the going-out cap rate.

The intent is to reinforce our understanding of these relationships and their implications for investment decision making and value estimating. In doing this, we emphasize not the numbers themselves, but why the numbers relate as they do.

## The starting point: what drives value?

Considering what drives values in real estate markets will quickly produce a long list of variables. On inspection, however, all the variables can be fit into one of three overall categories:

- how much income is expected
- when the income is expected
- how certain the income is

These three categories of variables drive the value of all investments—not just real estate investments. They are reflected in the present value calculation, PV = FV (1/ (1 + y)n, where “FV” is the estimate of the how much, the “n” reflects the when, and “y” reflects the certainty, or risk, in the form of a required yield or rate of return.

When considering these drivers of value, it is important to recognize that the how much, when, and how certain estimates refer to all future income. This includes annual operating income, as well as the expected proceeds from the sale at the end of the holding period (commonly called the reversion).

In other words, the prices that investors are paying and the capitalization rates that appraisers and consultants observe are the result of a process that either implicitly or explicitly considers all of the how much, the when, and the uncertainty.

## Justifying the direct capitalization model

When it is recognized that all of the “how much” drives value, we are faced with the potentially uncomfortable observation that the direct capitalization model explicitly uses only a fraction of the how much. That is, direct capitalization relies explicitly on only the first year of expected income to develop the value estimate.

How then can its use be justified when values are driven by potentially many years of operating income plus the reversion? The answer, of course, is that the rest of the income and the reversion are implicit in the capitalization rate.

This is a key insight, and it is reflected in the familiar relationship between the discount rate and the capitalization rate: R = Y – g, where “g” is the annual growth rate for income and value.

For example, if investors are discounting cash flows at 9.0 percent, and if the forecasts of income and reversion assume a 3.0 percent annual rate of growth, the resulting cap rate (the one we observe) will be 6.0 percent. Applying this cap rate to the first year of expected income produces the same value estimate as discounting all of the how much at 9.0 percent.

Implications

- For investment decisions, the difference between the cap rate at purchase and the required yield is the annual change in income and value necessary to achieve that yield. For example, buying at a cap rate of 6.0 percent with a required yield of 10.0 percent means income and value must grow by 4.0 percent per year over the assumed holding period to achieve the 10.0 percent yield. And using the same approach, we can also solve for the markets’ required yield by adding the forecast of annual income and value change to the observed cap rate.

There is a caveat. The R = Y – g model works perfectly only if both income and value grow at the same rate, and that rate is constant. Will this ever happen? Of course not—like all models, it is a simplification of reality that includes some built-in error. However, it is close enough often enough to be useful.

- The mathematical simplicity of direct capitalization should not be allowed to obscure an important fact: it does not relieve the analyst of the necessity to make at least informal forecasts of income and value change for both the comparable properties and the subject. Clearly, a 6.0 percent cap rate extracted from properties expected to have 3.0 percent growth rates should not be used to estimate the value of a property expected to have a 5.0 percent annual rate of growth (holding other factors equal). To do so would produce a value estimate that is wrong by a substantial amount.

# “ R = Y – g ”

where “g” is the annual growth rate for income and value

## Discounted cash flows: does the holding period assumption matter?

The understanding that capitalization rates reflect all expected future income and value changes is the logical underpinning for arriving at the correct answer to the question of whether the holding period assumption is relevant to the resulting value estimate.

Recall that the estimate of the reversion is the estimate of what the property’s market value will be at the end of the holding period. It then follows that capitalizing year n + 1 income at the appropriate going-out rate will produce an estimate of market value at that time. This is simply applying direct capitalization to estimate the reversion.

We also know the value at reversion will reflect all income beyond year n+1. This can be written as follows: V = VNOI + VREV, with VREV = value of net operating income (NOI) beyond the holding period.

Thus, if a five-year holding period is assumed, V would equal the value of NOI for years 1–5 (VNOI), plus the value of the reversion (VREV), which would be the value of NOI for years 6 and beyond. If the holding period assumption is changed to, say, 10 years, then V would equal the value of NOI for years 1–10, plus the value of the reversion, which will be the value of NOI for years 11 and beyond.

The value estimates are allocated differently between the holding period and the reversion, but they sum to the same value. It is like cutting a string—wherever it is cut, the two pieces together will produce the same length.

Implications

- Because capitalization rates implicitly reflect all future income, the holding period assumption should not change the value estimate. In the discussion below, we will see that this conclusion also has important implications with respect to the choice of the going-out cap rate.

There is a caveat here, as well: we are discussing the association of the holding period assumption and the value estimate. There may be other good reasons, for example a client’s request, to choose a certain holding period.

- The direct capitalization model is the extreme case of the V = VNOI + VREV model, as all anticipated growth is reflected in the going-in cap rate. Thus, it can be thought of as the discounted cash flow model with all of the value captured in the reversion. It is in fact, the same as the prior holding period’s reversion.

# “ V = VNOI + VREV ”

with VREV = value of net operating income (NOI) beyond the holding period

## The relationship between going-in and going-out cap rates

The going-out cap rates that are used to estimate the value at the end of the assumed holding period (the reversion) are more often than not higher than the corresponding going-in cap rates.

The main reason for that relationship is best seen by example. Imagine two buildings side by side. They appear almost identical physically, and they have the same single tenant paying the same rent for each building. This in turn produces the same income for each. Without any more information about the properties, it is reasonable to believe they would have the same value, meaning they would have the same capitalization rates. However, suppose you find out that one building is almost new, whereas the other is 10 years old. Which building is more valuable? The newer building, because it will likely be more productive for more years into the future, as the result of its longer remaining economic life. In terms of their capitalization rates, the newer building’s rate would be lower.

Now think about the relationship between the going-in and going-out rates for a specific property. This also involves both a newer and an older building—except, here, it is the same building, older by the length of the assumed holding period.

All other things being equal, it follows that the older version of the building should have a higher expected capitalization rate (its going-out rate) than the newer version (its going-in rate).

This, in fact, is the relationship we almost always see reflected in investor surveys that report on current going-in and going-out rates. It is nothing more than recognition of the fact that capitalization rates reflect future income—that is, the use of a higher going-out rate reflects the assumption that a building has fewer remaining years of income production as it gets older.

There will be exceptions to this relationship, but the underlying principle should be the starting point for valuation analysis.

*The newer building, because it will likely be more productive for more years into the future, as the result of its longer remaining economic life. In terms of their capitalization rates, the newer building’s rate would be lower. *

## The relationship between the holding period assumption and the going-out cap rate

Earlier, we concluded that the holding period assumption should not impact the market value estimate. Now we see that for this to be properly reflected in practice, it requires the use of a going-out cap rate that is correlated with the assumed holding period.

Cap rates exist on a continuum, from today’s going-in rate (typically lowest) through the going-out rate (typically highest) at reversion. All other factors being equal, each year that passes puts upward pressure on the capitalization rate. Thus, if the assumed holding period is changed, a change in the going-out cap rate must also be considered.

This point is especially relevant if the selected relationship between the rates is based on results of investor surveys. A careful reading of those surveys will often reveal an assumed holding period—often 10 years. Assumed holding periods that differ will likely require a different rate relationship.

## In summary

The two most widely used income-based value models are direct capitalization and discounted cash flows. These models are conceptually equivalent, as both produce value estimates that reflect all the how much, the when, and the certainty. This means both should in theory produce the same estimates of value. Procedurally, direct capitalization implicitly captures expected future income and value changes in the capitalization rate, while those future numbers are considered explicitly in discounted cash flow models.

The recognition that capitalization rates reflect all future productivity has some important implications for the proper use of discounted cash flow models. First, it means the holding period assumption should not affect the value estimate. Second, it explains why going-out cap rates are typically assumed to be higher than going-in cap rates. Third, and finally, it links the holding period assumption and the choice of the going-out cap rate.

Copyright © 2017-2018 by Hotel Asset Management Magazine. All rights reserved.

## About the Author

Dr. Kenneth M. Lusht, Ph.D., MAI, is Distinguished Professor of Real Estate at Florida Gulf Coast University, and Professor Emeritus of Business Administration and Zimmerman Endowed University Fellow at Penn State University. There, he held various positions, including Chairman of the Department of Insurance and Real Estate, Associate Dean for Research and Director of the Doctoral and MS Programs, Director of the Institute for Real Estate Studies, Academic Program Director for the Continuing Education Program in Real Estate, and Interim Dean of the Smeal College of Business.

He is a Fellow of the Homer Hoyt Institute, has served as a Faculty Associate of the Lincoln Institute of Land Policy, and as a Trustee of the Appraisal Foundation. He has served on the Board of Trustees of the Center for Economic Studies, and has been a visiting scholar at universities in Australia, New Zealand, Hong Kong, Singapore, Turkey and Finland. He has been a member of the Education Advisory Committee to the Pennsylvania Real Estate Commission. Dr. Lusht is a past president of the American Real Estate and Urban Economics Association, and he served two terms on that Association’s Board of Directors. He is widely published in academic and professional journals, and has authored or coauthored three texts as well as numerous seminars and courses for the Appraisal Institute.