Alex Braunstein Stat 956 Mini-Project 2: An analysis of TIAA Real Estate Fund The TIAA Real Estate Fund provides a structure unlike any other asset available on the market. Because of its unique structure, it is natural to ask if extra cheese exists for the taking. In this mini-project we first engage in some EDA for the TIAA Real Estate Fund and a quasirelated proxy, the Dow Jones Composite REIT index for the period May 14, 2001 through March 9, 2007. Second, we fit ARIMA models to both of the series. Third, we search for correlation structure between the TIAA asset and the REIT index on a daily, weekly, or monthly levels. Finally, we see if we can find a rule for stepping out of the TIAA asset for a small period based on drops in the REIT index to increase our returns. Our price series could not look more different. The TIAA series is unbelievably smooth… almost too smooth, while the REIT indeed looks like a relatively normal noisy asset. In regards to the self-enforcing nature of its price, we see a repeated trend, where the index goes up for a period, then falls by roughly 1/3 to 1/2 of the value it previously gained. The graphs of the return series look relatively similar, though there are two items of importance for us to note. First, TIAA seems to have more large positive returns and the REIT seems to have more large negative returns, and the scales on these two assets are drastically different. To emphasize this point, consider the histogram of returns for the two assets, with the same scale, and the summary statistics of the return series, which include, the quantiles, mean, and standard deviations of the two series.
Alex Braunstein Stat 956
> summary(TIAAreturns) Min. 1st Qu. -2.911e-03 2.308e-06 > summary(REITreturns) Min. 1st Qu. -0.0529700 -0.0048910 > sd(TIAAreturns) [1] 0.0007569053 > sd(REITreturns) [1] 0.009430154
Median 3.061e-04 Median 0.0008835
Mean 3.660e-04 Mean 0.0005872
3rd Qu. 6.201e-04 3rd Qu. 0.0065700
Max. 9.000e-03 Max. 0.0467900
For each series, we fit with up to 6 AR and MA coefficients, fitting every possible combination. We use AIC to select the “best” model, though its worth noting that the values for most of the models are incredibly close, with only 1 or less separating models with AIC values on the order of –16000. For the TIAA asset, we find an ARMA(1,3) model is the best, which is quite interesting considering that we rarely see significant MA coefficients in this context. Additionally, the significance of both the AR1 and MA1 coefficients are remarkably huge. Again, this is a direct consequence of the unique structure of this asset, which creates an incredibly smooth price path caused by the valuation scheme. Coefficients: ar1 0.9984 s.e. 0.0018 ma1 -0.8713 0.0263 ma2 -0.0512 0.0346 ma3 -0.0650 0.0266 intercept 4e-04 1e-04
sigma^2 estimated as 5.408e-07: log likelihood = 8409.83, aic = -16807.66
Alex Braunstein Stat 956
> aicvals [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] -16733.80 -16764.54 -16776.79 -16775.42 -16773.98 -16777.02 -16776.06 [2,] -16770.58 -16780.03 -16778.36 -16807.66 -16805.77 -16776.15 -16774.10 [3,] -16778.60 -16805.53 -16776.07 -16803.62 -16803.72 -16774.15 -16772.11 [4,] -16776.78 -16774.73 -16802.83 -16804.15 -16802.69 -16772.17 -16770.12 [5,] -16775.82 -16805.63 -16806.67 -16802.76 -16803.97 -16801.08 -16806.63 [6,] -16778.44 -16776.45 -16776.12 -16774.85 -16804.84 -16801.14 -16801.76 [7,] -16776.45 -16774.45 -16772.48 -16772.21 -16804.78 -16799.40 -16802.53 For the REIT index we find the ARMA(1,0) model is the best. Compared to both the magnitude and significance of the coefficients for TIAA asset, the AR1 coefficient for the REIT is both tiny and relatively insignificant (which of course is the norm for these types of assets). Coefficients: ar1 0.0926 s.e. 0.0261 intercept 6e-04 3e-04
sigma^2 estimated as 8.81e-05: log likelihood = 4715.11, aic = -9424.22
Alex Braunstein Stat 956
> aicvals [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] -9413.713 -9423.781 -9422.747 -9422.318 -9420.846 -9419.390 -9418.008 [2,] -9424.216 -9422.300 -9421.420 -9422.558 -9423.266 -9421.548 -9419.847 [3,] -9422.377 -9421.019 -9420.053 -9422.194 -9422.234 -9420.420 -9419.951 [4,] -9422.626 -9422.369 -9424.179 -9422.116 -9420.585 -9417.980 -9418.573 [5,] -9420.968 -9423.477 -9421.630 -9423.387 -9421.124 -9420.815 -9424.465 [6,] -9419.468 -9418.608 -9420.697 -9421.813 -9420.684 -9419.165 -9418.750 [7,] -9418.105 -9416.715 -9414.609 -9415.857 -9418.558 -9416.469 -9414.130 As all of our analysis has shown, these two assets have drastically different return structures and dynamics underlying them. Still, we would expect some correlation between the assets as both are real estate based and a small fraction of the TIAA asset is held in REITS. This expectation is not met. With TIAA returns as our response and the REIT index as our independent variable, we regress the two and find surprisingly little correspondence with a pvalue of .964, the largest I have actually even seen. The results are shown below: Coefficients: (Intercept) REITreturns Estimate 3.660e-04 -9.598e-05 Std. Error 1.992e-05 2.109e-03 t value 18.378 -0.046 Pr(>|t|) <2e-16 *** 0.964
Alex Braunstein Stat 956
We also check to see if any correlation exists on a weekly or monthly basis. We see a significant beta for the weekly data, but not for the monthly data. The results of the regressions for the weekly and monthly returns respectively are: Call: lm(formula = TIAAweekly ~ REITweekly) Coefficients: Estimate (Intercept) 0.0018106 REITweekly 0.0108816 Std. Error 0.0001196 0.0053337 t value 15.14 2.04 Pr(>|t|) <2e-16 *** 0.0422 *
Call: lm(formula = TIAAmonthly ~ REITmonthly)
Alex Braunstein Stat 956 Coefficients: Estimate (Intercept) 0.0072913 REITmonthly 0.0216032 Std. Error 0.0006284 0.0153782 t value 11.603 1.405 Pr(>|t|) <2e-16 *** 0.165
We plot these values with their corresponding regression lines:
Though interesting that some predictive structure exists between weekly returns, the Rsquared value is still very low (les than .02) and seems attributable to the fact that both assets had quite a run during this period. We keep this information in mind as we move on to the final part of our analysis. With our first three sections behind us, we see if there exist ways for us to take advantage of the unique return structure of the TIAA asset to find some extra cheese. More specifically we ask the question: “Given a loss of x% in a REIT index over a period of y days or less can we step out of the TIAA asset for a period of z days and see a larger return?” Though the analysis was done over many different combinations of x, y, and z, we present the results for x= 5, 10, and 15 (no 20% drops were seen and intermediate values such as 7.5% or 12.5% didn’t change the results), y = 20, and z = 126. Changing y did very little to change the results. Changing z to any greater value makes it so you have almost no observations in the 5% case and smaller values did not change the results either. To answer our question, we create a function in R to identify days on which we see a drop of at least x% in y or fewer days. We keep these values and plot them on the prices series for the REIT index price level and the TIAA price level to get an idea of how frequently these drops occur and the way they are spaced. Using the values supplied by our R function, we subset the TIAAreturns by removing the following 126 trading days from each event. Below we show several figures. First, we plot the price level for both the TIAA and REIT asset with lined indicating drops of the specified percent plotted, for 5, 10, and 15% drops. As we can see, the 10% and 15% case are nearly identical and the 5% case leaves very few data points left (under 300 from 1500 originally). As pointed out before looking at more intermediate percentages does not change our results.
Alex Braunstein Stat 956
Alex Braunstein Stat 956 After considering the price series, we turn our attention to the return series, which we subset using our R function. We see very few differences between the shapes of the distributions:
Min 25th percentile Median Mean 75th percentile Max Standard deviation
5% drops -2.268e-03 -1.311e-05 2.899e-04 3.055e-04 6.017e-04 5.258e-03 .0006874679
10% drops -2.268e-03 3.648e-05 3.486e-04 4.227e-04 6.739e-04 6.119e-03 .0007443571
15% drops -2.892e-03 2.828e-06 3.070e-04 3.642e-04 6.224e-04 6.119e-03 .0007217686
Full data set -2.911e-03 2.308e-06 3.061e-04 3.660e-04 6.201e-04 9.000e-03 .0007569053
To emphasize the similarity of these series we consider the CDFs of each. Ideally, we would like to see one uniformly and far to the right of every other CDF. We place the following 4 CDFs of returns on one plot color coded as follows: blue is the full return set, red is 5% rule, green is 10% rule, and orange is the 15% rule. We see that the green line is uniformly to the right up until the .04 or .05 level, and that the red line is almost uniformly to the left of the other CDFs. Of course these results should be taken with a grain of salt, as all four CDFs are quite close to each other so we should hesitate to say that these are “different” from each other in any precise sense of the word.
Alex Braunstein Stat 956
When asking ourselves the question, “Given a loss of x% in a REIT index over a period of y days or less can we step out of the TIAA asset for a period of z days and see a larger return?” our analysis points to an answer of no. Despite the unique structure of the TIAA asset, one would think it would be possible to somehow take advantage of this, however, the TIAA asset is still quite interesting to study and likely a valuable addition to most anyone’s portfolio.