Behavioral finance may play an increasingly important role for astute hotel investment decision-makers in the future. Using insights from psychology to understand how human behavior influences investment decisions, behavioral finance may help us understand the puzzling decisions made by some hotel investors in recent times, e.g. why were some hotel investors willing to pay more for a dollar of real net operating income (NOI) when that income was rationally expected to erode in the future. Investors normally pay more when real NOI is expected to increase?

In other words, why do hotel cap rates decline near the peak of a market and increase near the trough of a market? If investors understood that real NOI is mean-reverting, they would expect high growth when real NOI is low and low growth when real NOI is high! Relatively high cap rates should be applied to high real NOI and relatively low cap rates should be used for low real NOI. Without a clear understanding of mean-reversion many investors are likely to repeat the mistakes of the past by undervaluing hotels near the bottom and overvaluing hotels near the peak of a market cycle. Supply, demand and mean-reversion are fundamental concepts for hotel investors.

#### Mean-Reversion: The Search for Equilibrium

A time series, such as Real RevPAR, shows mean-reversion¹ if it tends to fall when its level is above its mean and rise when its level is below its mean. The concept is illustrated in Exhibit 1 which shows the mean-reverting nature of Real RevPAR for the U.S. Lodging Industry since 1958. Nominal or current RevPAR has grown from $8.45 in 1958 to $83.57 in 2017, a CAGR of 3.1% which compares with inflation with a CAGR of 3.7% over the same period. Adjusting nominal RevPAR for inflation it is easy to see that Real RevPAR has cycled around a long-term average of $70.60 ($2017). Real RevPAR has grown from $71.69 in 1958 to $83.57 in 2017, a CAGR of 0.3%.

Exhibit 2 compares the frequency distribution of annual Real RevPAR for the period 1958-2017 with a normal distribution and shows that ninety-five percent of annual Real RevPAR fell between $60.63 and $80.57. Apart from its mean-reverting characteristic, annual Real RevPAR is auto correlated which means that it is correlated with itself.

Annual Real RevPAR has a large positive autocorrelation of 0.7769 at lag 1 year which means that Real RevPAR in successive years is correlated – if Real RevPAR in any year is high (or low), the following year’s Real RevPAR will tend to be high (or low). It also has a statistically significant negative autocorrelation at lags 4, 5 and 6 years of -0.3819, -0.5004 and -0.3952 respectively. This means that if Real RevPAR in any year is high (or low), Real RevPAR in subsequent years 4, 5 & 6 will tend to be low (or high). Based on our proprietary mean reversion model which incorporates these autocorrelations we determine the mean reversion for total U.S. Real RevPAR to be $70.80 with a standard deviation of $4.78.

We are therefore able to provide standard deviation bands around our mean-reversion forecast of $70.80. Based on the mean-reverting nature of the U.S. Lodging Industry there is a 68% probability that future Real RevPAR will fall between $66.02 and $75.58 (one standard deviation) and a 95% probability that it will fall between $61.25 and $80.35 (two standard deviations). There is a 99% probability that future Real RevPAR will fall between $56.47 and $85.13.

The data can be used to determine how far the industry has diverged from its equilibrium state and the probability of further upside or downside. Providing a similar analysis at the metro, tract and competitive set level has proved to be a significant investment and asset management tool for investors.

¹ Mean-Reversion- A time series shows mean-reversion if it tends to fall when its level is above its mean and rise when its level is below its mean. Therefore, a mean-reverting time series tends to return to its long-term mean, although its long-term average value is not the same as its mean-reverting level but in many instances they may be similar. If a time series is currently at its mean-reverting level, then the model predicts that the value of the time series will be the same in the next period. At its mean-reverting level, we have the relationship xt+1=xt. For an autoregressive AR (1) model (xt+1=b0 +b1xt), the equality xt+1=xt implies the level xt=b0 +b1xt, or that the mean-reverting level xt, is given by xt=b0÷(1b1). So the AR(1) model predicts that the time series will stay the same if its current value is b0÷(1-b1), increase if its current value is below b0÷(1-b1), and decrease if its current value is above b0÷(1-b1). We now have sufficiently long time series to effectively determine the mean reverting value of occupancy, real ADR, real RevPAR, real expenses and income and hotel values.

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