Using a series of historical market rents and cap rates, Marcato and Key create an income return assumed to be equal to the cap rate. They then estimate the capital growth rate at time t (cgt) of investing in real estate as:
where valuet is the value of a property at time t and is calculated as:
Marcato and Key then use four different approaches to generate an unsmoothed total real estate return series and test whether optimal real estate weights are caused by unsmoothing model selection or by the choice of parameter levels (calibration).
cgt = valuet valuet−1 −1
valuet = rentt capratet
The first unsmoothing procedure is the First Order Autoregressive Reverse Filter (FOARF). Unsmoothed capital rates of growth for real estate investment (ucgt) are estimated as:
where cgt is the capital growth of the valuation-based index at time t, and α1 is the unsmoothing parameter.
Three main assumptions are underlying this model. First, adjusted and unadjusted values of the mean for the series are equal. Second, the model holds over time (stationarity assumption). Third, random errors are left out of the index (the assumption that there is no noise).
The second unsmoothing method is the Second Order Autoregressive Reverse Filter (AR2), that is shown in this equation:
As can be seen in the AR2 equation, this autoregressive process has more than one lag and thus gives a more generalized model. However, Marcato and Key argue that there is no ex-ante motivation to assume the existence of an autoregressive process of an order higher than two when using annual returns. Therefore, they restrict their analysis to an AR2.
The third approach that Marcato and Key use applies a procedure suggested by Fisher, Geltner and Webb (1994) with a First Order Autoregressive specification. Following this procedure, they obtain a Full Information Value Index (FIVI) (also known as FIVI unsmoothing method).
Residuals are computed from (cgt-α1*cgt-1), and their volatility is used to compute the weight (w0):
The weight (w0) is needed to find the unsmoothed rate of capital appreciation from the next equation:
The fourth method – known as STATES – assumes that different phases of the market cycle will tend to produce changes to the unsmoothing parameter. For instance, the unsmoothing parameter will be higher in falling markets versus rising markets because
“valuers” will be inclined to resist downward adjustments more than upward adjustments.
ucgt =cgt−(α1*cgt−1+α2*cgt−2) (1−α1−α2)
w0 = 2 *σresid σequity
ucgt =[cgt−α1*cgt−1]
(1−α1)
ucgt = (cgt −α1*cgt−1) w0
This method also assumes that the stronger the capital appreciation, the higher the unsmoothing parameter, and the stronger the capital depreciation, the lower the unsmoothing parameter.
Different unsmoothing parameters are then applied for different market growth states (hence the name STATES for this method). First, the parameter is fixed for returns ranging between the mean and the mean plus its standard deviation. For returns falling outside this range, new parameters are estimated by adding a varying coefficient to the fixed parameter following the next schedule:
● 0.10 for returns lying between the mean plus 1 standard deviation and the mean plus 2 standard deviations;
● 0.20 for returns falling above the mean plus 2 standard deviations.
● 0.05 for returns lying between the mean and the mean minus 1 standard deviation;
● 0.15 for returns included between the mean minus 1 standard deviation and minus 2 standard deviations; and
● 0.25 for returns falling below the mean minus 2 standard deviations.
The STATES method uses the same equation as in the First Order Autoregressive Reverse Filter to unsmoothed capital growth rates. In this case however, unsmoothing parameters vary, which are then employed for different market growth states.
After the computation of unsmoothed capital growth rates (ucgt) using the four different models just presented (FOARF, AR2, FIVI, and STATES), the next step consists in obtaining an income return (uirt) recalibrated for the unsmoothed capital value index (ucgit) as follows:s
where inct is the income (at time t) and ucgt-1 represents the unsmoothed capital growth index (at time t−1).
Finally, the unsmoothed total return for real estate at time t (utrt) is calculated as the sum of the unsmoothed capital growth and the unsmoothed income return at time t:
This formula reminds us that the unsmoothed total return for real estate has two components. The first is the unsmoothed capital growth (property price appreciation), which is analogous to the capital gains component when investing in stocks. The second is the unsmoothed income return (rents collected from real estate), which is analogous to dividend returns in the case of stocks.
t t
t ucg uir
utr = +
uirt = inct ucgit−1 ucgt =[cgt −α1*cgt−1]
(1−α1)
Example:
We illustrate the use of these equations in a numerical example. Suppose you would like to analyze the time series behavior of the real estate returns for a certain city. After searching for information, you find a real estate time series of capital growth rates that was calculated using the formula [(valuet/valuet-1) – 1], where valuet = rentt / capratet. The corresponding last 16 quarters of real estate capital growth rates were:
Quarter Real Estate Returns
1 2.05%
2 1.75%
3 1.38%
4 1.52%
5 0.84%
6 -1.43%
7 0.05%
8 0.01%
9 -0.33%
10 -5.33%
11 -0.03%
12 -1.03%
13 -0.44%
14 -2.81%
15 0.77%
16 -0.24%
However, you are concerned that this time series may have been the subject of smoothing as it was calculated from an appraisal-based index.
To get a clearer picture from the data, you decide to unsmooth the time series using the First Order Autoregressive Reverse Filter (FOARF). What would the unsmoothed real estate return be for the third quarter using the FOARF? (Note: The value of the unsmoothing parameter α1 was estimated to be equal to 0.5).
Now, suppose that you suspect that the autoregressive process might actually have two lags. What would the unsmoothed real estate return be for the third quarter using the Second Order Autoregressive Reverse Filter (AR2)? (Note: The unsmoothing parameters were estimated to have the following values: α1 = 0.4, and α2 = 0.3).
ucgt = cgt −(α1*cgt−1+α2*cgt−2)
(1−α1−α2) =1.38−(0.4 *1.75+0.3 * 2.05)
(1−0.4−0.3) =0.22%
ucgt = cgt −α1*cgt−1
(1−α1) =1.38−0.5 *1.75
(1−0.5) =1.01%
In this example, what would need to happen for the unsmoothing method known as FIVI to yield the same results as the FOARF? (Note: Assume that the value of the unsmoothing parameter α1 was estimated to be equal to 0.5).
We know the equations for FOARF and FIVI, and how to compute for weight (w0). Then, we can compute the following:
From this equation, we can see that the term (cgt-α1*cgt-1) will simplify since it is in both numerators. Therefore, we have:
Now, since we know that in the case of FIVI w0 is equal to:
We will have, substituting this formula in the previous equation, that:
Now, since a1 = 0.5, then, substituting a1 above by 0.5, we obtain:
We find that:
0 1) 1
( −α =w
resid
equity σ
σ =4 (cgt −α1*cgt−1)
(1−α1) =(cgt−α1*cgt−1) w0
w0 = 2 *σresid
σequity =0.5
w0 = 2 *σresid σequity
(1−α1)= 2 *σresid σequity
That is, the standard deviation of the residuals computed from (cgt-α1*cgt-1) will need to be four times larger than the standard deviation of equity returns for FIVI to yield the same results as FOARF.