**Introduction**

In several previous posts on bankruptcy, we have described the evolution of share prices for banks just before their bankruptcy or on their approach to negative prices: Colonial Bank (CNB), the CIT Group (CIT), American International Group (AIG), and Lehman Brothers (LEH). It was shown that the CNB sank below the edge of bankruptcy and CIT is on the brink with unclear future. AIG has been fluctuating around the waterline for a while, and LEH was definitely ready for a bankruptcy.

On the other hand, the CIT Group and AIG are bailed out, and thus, our model predicting share prices should fail to predict the artificially supported prices. Therefore, it would be instructive to estimate the fictitious evolution of share prices after the bankruptcies or alternative evolution after the bailouts. In this post we compare Lehman Brothers, Fannie Mae (FNM) and Freddie Mac (FRE). All three had a sharp fall in the second half of 2008.

**Model and data**

To predict share prices we use the 3-C model [1-3]: a share price, sp(t), is described with the following empirical function:

sp(t) = A*CPI1(t+t1) + B*CPI2(t+t2) + C*(t-2000) + D (1)

, where CPI1 and CPI2 are empirically determined CPI components; t is the calendar time; A, B, C, and D are empirical coefficients. Because the prices for goods and services (in the headline CPI) can lag or lead relevant share prices the time lags t1 and t2 are introduced. Both time lags can be positive or negative. In our model, we test all possible lags between 0 and +13 months. So, we chose the best from 34*33*14*14 ~ 200,000 possible models, with two extra degrees of freedom introduced by linear and free terms.

The full set includes 34 CPI components, which are tested as predictors for each share price. Among these components are major expenditure categories: the headline CPI (C), the core CPI, i.e. the headline CPI less food and energy (CC), food and beverages (F), housing (H), apparel (A), transportation (T), medical care (M), recreation (R), education (ED), communication (CO), and other goods and services (O). All components used in the model are not seasonally adjusted ones and retrieved from the Bureau of Labor Statistics.

**Results**

When applied to the LEH time series between July 2003 and September 2008, the 3-C model (1) gives the following empirical relationship:LEH(t) = -10.40*FB(t-4) + 5.84*DUR (t+2) + 67.96*(t-2000) + 952; STDEV=$4.73 (2)

, where t is the calendar time, FB(t) is the (not seasonally adjusted consumer price) index for food leading the LEH by four months, and DUR(t+2) in the index of durable goods lagging two months behind the LEH. Figure 1 compares the stock price predicted according (2) with the observed one. This model differs from that in the previous post in two aspects. First, it uses DUR instead of the personal care index (PC). This is not a fundamental change, however. Each empirical model based on two components of the CPI includes one index, which is sensitive to the change in a given share price, and one index, which expresses the overall reaction of prices for goods and services induced by the change in the share price.

Here I would like to illustrate the process on the example of isostatic compensation of ice floating in a water-filled tank. If you put some load on a piece of ice it will sink a bit deeper, but the level of water in the tank will also increase. Similarly, if some stock price increases it will affect prices of many goods and services, not only those related to the stock. There should be a component of the CPI which expresses the overall reaction the best. One can consider it as a reference component.

Then the difference between these two components maps the change in the price. Since there are many components of the CPI, which might represent the overall reaction of the CPI on the given price change, their exchange does not influence the model much. In the case of LEH, the defining component is the index for food (or food and beverages), and the reference component is the PC or DUR.

The empirical models for FNM and FER are as follows:

FNM(t) = -7.38*FB(t-8) – 8.47*R (t+3) + 42.57*(t-2000) + 2119; STDEV=$4.42 (3)

FRE(t) = -6.07*FB(t-4) - 3.42*O (t-9) + 57.96*(t-2000) + 1917.8; STDEV=$4.16 (4)

, where R is the index of recreation and O is the index for other goods and services in the CPI. Figures 2 and 3 compare the observed and predicted prices for FNM and FER, respectively.

All three empirical models just describe the evolution of corresponding prices using available CPI estimates. The models do not foresee stock the price evolution beyond current knowledge. However, since relevant CPI estimates are available for the last year, i.e. 11 months after the bankruptcy of Lehman Brothers, one can formally calculate the fictitious evolution of these share prices, as shown in Figures 1through 3. The LEH price demonstrates sustainable decrease between October 2008 and March 2009. Then the price regains some strength and approaches the zero line in August 2009. If bailed out in September 2008, the Lehman Brothers would start to grow in September 2009. One cannot say the same thing about AIG – it is still unstable.

The price of Fannie Mae lags behind both defining CPI components. Therefore, the prediction is possible only till April 2009. Unfortunately, the price continues its fall. There is no sign a turn. On the contrary, the price of Freddie Mac is well predictable into 2010. In March 2009, the price reached the bottom and then started to grow. It is still negative and will stay below the zero line in 2009.

We have predicted fictitious share prices of three companies. This prediction might be helpful for the assessment of future policy on Freddie Mac and Fannie Mae.

**Conclusion**We have predicted fictitious share prices of three companies. This prediction might be helpful for the assessment of future policy on Freddie Mac and Fannie Mae.

Figure 1. Comparison of the observed and predicted LHE’s share price. The latter is obtained using (2).

Figure 2. Comparison of the observed and predicted FNM’s share price. The latter is obtained using (3).

Figure 3. Comparison of the observed and predicted FRE’s share price. The latter is obtained using (4).

**References**

[1] Kitov, I., Kitov, O., (2009). Predicting share price of energy companies: June-September 2009, MPRA Paper 15863, University Library of Munich, Germany, http://mpra.ub.uni-muenchen.de/15863/01/MPRA_paper_15863.pdf

[2] Kitov, I., Kitov, O., (2009). Modelling selected S&P 500 share prices, MPRA Paper 15862, University Library of Munich, Germany, http://mpra.ub.uni-muenchen.de/15862/01/MPRA_paper_15862.pdf

[3] Kitov, I., Kitov, O., (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.

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