The Rank-Size Rule and Challenges in Diversifying Commercial Real Estate Portfolios

Abstract

The strategy of geographically diversifying a portfolio of commercial real estate assets is an intuitive approach for risk management. However, due to high concentrations of these assets in major metropolitan areas, investors may face additional constraints in the portfolio optimization process. The rank-size rule, a log-linear relationship between city rank and size, provides one of the greatest empirical regularities in regional science. As such, it serves as a possible theoretical guide to the weights given to properties by location in a commercial real estate portfolio. This paper sets forth some ideas relating to the concentration side of portfolio variance and the limiting effect that large concentrations may have on the ability to diversify risk. Two variants of the rank-size relationship – the Zipf distribution and the parabolic fractal distribution – are fitted to a variety of datasets to provide a sense of the degree of concentration in the commercial real estate industry. These empirical findings suggest the presence of limitations to geographical diversification that have varying degrees of severity across different property types or sectors of the commercial real estate market.

Appendices

Appendix : Appendix A: Technical Details and Estimation Issues

A.1 Expressions for Median Ranks

As alluded to in Section “The Zipf Distribution”, the median rank of the Zipf distribution decreases as the underlying concentration gets larger. This median rank, which we label p₅₀, indicates the number of sites needed to attain a 50% share of the implied distribution. In the context of modeling the concentration of CRE assets, this p₅₀ rank represents how many of the largest cities are required to account for a 50% share of a given market.

In the equal-weighted case of α = 0, the median rank is given in Eq. 33 given the usual conventions for medians with even and odd n.

$$ \text{Med}(x)=\frac{n+1}{2} $$

(33)

For α = 1, we can use the approximation in Eq. 10 to yield an expression for the median rank site by using Eq. 6 and setting the sum of the probabilities of sites with below median rank to 1/2, as done in Eq. 34.

$$ \begin{array}{@{}rcl@{}} \frac{H_{m,1}}{H_{n,1}}=0.5 \text{ where }m\text{ represents the median} \end{array} $$

(34)

Going into more detail, Eq. 35 follows from Eq. 34. Substituting the approximation Eq. 10 in Eq. 35 leads to Eq. 36, and solving for m yields Eq. 38. The 3/4 in Eq. 38 arises because $\exp (-0.5\gamma )$ yields 0.7493, a close approximation to 3/4.

$$ \begin{array}{@{}rcl@{}} H_{m,1}&=&0.5H_{n,1} \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} \ln(m)&\approx& 0.5\ln(n)-0.5\gamma \end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} m&\approx&\sqrt n\cdot \exp(-0.5\gamma) \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} \text{Med}(x)&\approx&\frac{3\sqrt n}{4} \end{array} $$

(38)

A.2 Zipf Distribution Estimation Issues

For an empirical cross-section of data, the estimation of Zipf’s α produces a measure of the concentration of that variable. The log-linear regression from Eq. 7 can be fitted using ordinary least squares and the $\hat {\beta }_{1}$ parameter produces an estimate of − α. The variable of interest, x, is regressed on a shifted rank, r = rank − 1/2. A rank shift of 1/2 follows from Gabaix and Ibragimov (2011) to reduce small sample bias. Thus, the top rank is 0.5, followed by 1.5, and so on.

In regards to the standard errors, Gabaix (2009) note that positive autocorrelation in the residuals from the ranking procedure makes the typical OLS standard errors incorrect. To address this, we estimate the coefficients and standard errors across 10,000 bootstrap iterations.

With the estimated parameters, the fitted values for the n observations can be converted into implied weights of the rank-size distribution. To revert back to the original units of the respective variable, we take the exponential of both sides of Eq. 7, yielding Eq. 39 and Eq. 40.

$$ \begin{array}{@{}rcl@{}} x &=& \exp(\beta_{0} + \beta_{1}\ln(r) + \varepsilon) \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} &=& \exp(\beta_{0}) \cdot r^{\beta_{1}} \cdot \exp(\varepsilon) \end{array} $$

(40)

Since β₀ and β₁ are fixed parameters, Eq. 40 reduces to Eq. 41. If ε is normal, then $\mathbb {E}[\exp (\varepsilon )]=\exp (\sigma ^{2}/2)$. Since both $\exp (\beta _{0})$ and $\mathbb {E}[\exp (\varepsilon )]$ are constant for each site, these do not affect the relative weights given to each site.

$$ \mathbb{E}[x|\beta] = \exp(\beta_{0}) \cdot r^{\beta_{1}} \cdot \mathbb{E}[\exp(\varepsilon)] $$

(41)

From these predicted values, a probability mass function is created by dividing the vector of estimates by its sum as in Eq. 4. Following the cumulative density function from Eq. 6, we examine how many of the largest sites are needed to make up specific shares of the population. For example, in Eq. 34, the median rank is obtained by setting Eq. 6 equal to 1/2 and solving for m. This solves for the rank where 50% of the variable concentration is located in and above, which we denote p₅₀. Alternatively, one can interpret this median rank as where a random selection from this distribution would have a 50% probability of being located in one of the top p₅₀ ranked sites. We present estimates for p₂₅, p₅₀, and p₇₅ along with the estimated model parameters in Tables 2–6.

A.3 Parabolic Fractal Distribution Estimation Issues

The expansion of the functional form in the rank-size relationship extends the constant elasticity from the linear Zipf model to a variable elasticity as in Eq. 42.

$$ d(r) = \frac{d\ln(x)}{d\ln(r)} = \beta_{1}+2\beta_{2}\ln(r) $$

(42)

Since the construction of a rank-size relation means that the slope cannot be positive, we need to ensure the fitted curve is non-increasing of the relevant domain, r ∈ [0.5,n − 0.5]. Accordingly, we estimate the model under the following first-order constraints for both the top and bottom ranked sites.

$$ \begin{array}{@{}rcl@{}} d(0.5) &\leq &0 \implies \beta_{1} \leq -2\beta_{2}\ln(0.5) \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} d(n-0.5) &\leq& 0 \implies \beta_{2} \leq \frac{-\beta_{1}}{2\ln(n-0.5)} \end{array} $$

(44)

Since the derivative of a parabola is a line, a negative slope at both bounds of an interval ensures that the slope is negative over the entire domain. This constraint is imposed in the estimation.

Appendix B: Economic Census Variable Definitions

See Table 8 for variable definitions from the 2012 Economic Census for the variables used in Section “Economic Census Data”.

Table 8 Variable descriptions from the 2012 Economic Census

Appendix: Appendix C: Demonstration of Portfolio Variance with Correlated Asset Returns

As demonstrated in Section “Portfolio Variance”, the correlation among asset returns dampens the limiting effect of high concentrations. This was shown by examining a scenario where asset variances were normalized to equal one and all combinations had an equal correlation of ρ. This appendix devises a Monte Carlo experiment to examine whether some simple theoretical approximations work well for typical n and generate positive definite covariance matrices.

We generated a Toeplitz matrix based on a vector equal to ρ plus or minus small equal increments to yield an vector with a mean of ρ and a very small standard deviation. This creates a positive definite covariance matrix, which we convert to a correlation matrix that is very close to the one assumed in theory. This effectively introduces variation to the individual asset correlations rather than simply assuming a constant, equal correlation across all asset pairs, as in the example from Section “Portfolio Variance”.

A common way to generate variation within a correlation matrix is through the use of a Wishart distribution (Bru 1991; Gouriéroux et al. 2009). Using the Toeplitz matrix as an underlying matrix with 20 degrees-of-freedom, we generated other instances of the covariance matrix, which are standardized to correlation matrices. This leads to more variation in the levels of correlation as shown in Table 9. For example, in Case 9 with n = 450 and ρ = 0.8, the minimum correlation entries equaled 0.43 and the maximum entries equaled 0.97.

Table 9 Minimum, mean, and maximum of pairwise correlations from simulated correlation matrices using Wishart distribution with 20 degrees-of-freedom

For each case, we only used one randomly generated matrix but then examined 10,000,000 symmetric random permutations of the rows and columns of this matrix. That is, we reordered the rows and columns using the same permutation vector. This preserves the diagonal and symmetry. We then computed the portfolio variance $w^{\prime }Vw$ for the different levels of α = 0, 1, 2 for each of 9 cases consisting of the combinations of n = 50, 150, 400 and ρ = 0, 0.4, 0.8. Table 10 shows the measured portfolio variance in each case along with the error of the theoretical prediction below the measured variance for α = 1, 2. The errors of the measured and theoretical variance are quite small with the largest error (0.00927 for Case 1) as expected for small n (n = 50) where the asymptotic approximations do not work as well.

Table 10 Portfolio variance (with difference from predicted variance below) across different sample sizes (n), levels of dependence (ρ), and rank-size regimes (α)