Semi-annual report on SP 500: from April to September 2009

This is a semi-annual report on the evolution of S&P 500 index. It compares the actually observed monthly closing levels to those predicted by our model, as originally presented in [1]. The major conclusion is that the prediction of a monotonic growth in S&P level made in March 2009 is correct. There is no indication that the growth will end before May 2010, where we foreseen the turn to a negative slope. Accordingly, the S&P 500 annual return will soon reach positive figures and will be growing at an accelerated rate till May 2010.

The model links S&P 500 annual returns to the number of 9-year-olds and has passed econometric tests for cointegration, with goodness-of-fit reaching 0.9. Since the number of 9-year-olds can be accurately predicted using younger cohorts, one can predict S&P returns at a several-year horizon. (Standard demographic projections may be used to predict over a decade ahead.)

We revealed the link between S&P 500 and the number of 9-year-olds in December 2007 using historical data since 1985. Since the very beginning of 2008, we have been carefully tracking the evolution of S&P 500. The original model actually predicted a sharp fall in 2008 [1]. (As in many scientific studies, the attempt to improve the original model, in order to fit data between 2003 and 2007, had failed, as we reported in this blog at several occasions.) The re-calibrated version of the original model developed in March 2009 is as follows:

Rp(t) = 165dln[N3(t+6)] - 0.17 (1),

In (1), Rp is the 12-month cumulative return; N3 is the number of three-year-olds; t+6 – time shifted by six years ahead to extrapolate the number of 3-year-olds into the number of 9-year-olds; dln is the rate of growth, i.e. the monthly increment in N3 normalized to the contemporary level. The time step is one month or 1/12 of a year.

Figure 1 compares the initial prediction of the S&P 500 evolution, the observed trajectory for the period since February 2009, and a new prediction till 2011. Relationship (1) is used with the number of 9-year-olds extrapolated from the number of 3-year-olds. In March 2009, we used a preliminary calibration of the original model and assumed the monthly increment in S&P 500 would be 80 points. This assumption was too optimistic and the growth was weaker. The original prediction is shown by blue lin with solid circles in Figure 1, and the observed values are shown by red diamonds. The black line represents an updated prediction since September 2009. It reproduces the old prediction but with a 46-point monthly increment since October 2009, as discussed below.

All in all, Figure 1 presents strong evidence in favour of our original model. Six months of almost monotonic growth were not expected by many market players in March 2009. The next nine months should bring additional validation to the model. The most important event will be the turn in May 2010. But the growth before this date is of crucial importance as well.

Figure 1. The evolution of S&P 500. Blue line - the original prediction using (1) with 80 unit per month increment. Red line – observations. Black line – the updated prediction since October 2009 with a 46-point monthly increment.

In any case, the model predicts a sudden drop in 2008 and 2009 to the level of 700, which has been followed by constant growth to the level 1050 in September 20009. Initially, we estimated the peak value in 2010 as 1800. But the last six months demonstrated the necessity to re-calibrate the model using new data. (In physics, even fundamental constants are under permanent re-estimation, and empirical and even fundamental models are constantly recalibrated.) Therefore, we also re-estimated coefficients in (1) to fit the last six S&P 500 (monthly) readings. The new model is as follows:

Rp(t) = 135dln[N3(t+6)] - 0.17 (2),

Actually, we needed to reduce the coefficient of linear term from 165 to 135 with free term unchanged. Figure 2 depicts the updated prediction of the S&P 500 annual returns. The peak S&P 500 value in May 2010 should be 1425 (not 1800) if the future increment will be 46 points per month, as observed between March and September 2009.

Figure 2. Observed and predicted S&P 500 returns. The September level of S&P 500 index is 1057. The past six months are relatively well predicted.

Conclusion
Between 1985 and 2009, the S&P 500 returns can be accurately described by population estimates. The model based on the number of 9-year-olds produces a time series which is cointegrated with the S&P 500 returns, i.e. reveals a weak causality, as proved by the cointegration tests.

The re-calibrated model predicts the continuation of S&P 500 growth into 2010 with the peak level of ~1400 in May. The annual returns will reach positive zone soon and also peak in May 2010 at the level of 50% to 70%.


References
[1] Kitov, I., Kitov, O., (2007). Exact prediction of S&P 500 returns, MPRA Paper 6056, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/6056.html

Time to buy Treasuries

A deflationary period is coming. As we wrote in 2006, deflation should start in 2012. Some funds (Pimco, for example, as Bloomberg reports) are getting prepared to these poor times. Do not miss the quick start and re-visit our old paper for the prediction of the deflationary period lenght:

Inflation, unemployment, labor force change in the USA
Ivan O. Kitov
Abstract

A model is developed linking the measured inflation (consumer price index or GDP deflator), unemployment and change in labor force. During the last twenty-five years, unemployment in the USA has been a lagged linear function of inflation. In turn, inflation has also been a lagged linear function of relative change in labor force with time. The lag is currently three years. Only a small decrease in labor force participation rate is currently observed in contrast to a strong increase between 1965 and 1990. According to the indicated relationship, the well-known stagflation period clearly resulted from the lag: the sharp increase in inflation coincided in time with the high unemployment induced by the high inflation period two years before. One can predict the unemployment rate in the USA in the following two years within the accuracy of inflation measurements. For example, the end of 2005 is a pivot point from a period of decreasing unemployment to one of moderate growth from 5% in 2005 to 6% in the middle of 2008. Starting in 1960, cumulative values of the observed and the model predicted unemployment are in agreement with the lag between inflation and unemployment. Inflation is defined by a lagged linear function of rate of change in labor force. The observed and predicted inflation almost coincide for the last forty years of annual measurement values, smoothed by a five-year wide moving window curves and as cumulative curves as well. Deviation of the curves before 1960 can be explained by a degraded accuracy of the measurements. A severe decrease in the rate of change of labor force is expected after 2010. This drop can potentially induce a long-term deflationary period. The same effect has been observed for Japan starting in 1990. There are numerous implications of the results for monetary and social policy-makers. The most important is an absence of any means to control inflation and economic growth except though a reasonable labor policy. In addition, some urgent measures are necessary to prevent the start of a deflationary period in 2010-2012.

CPI components in Japan: Medical care and education

As we discussed many times in this blog, Japan has an eternal problem with deflation. The headline CPI and its components are on a long-term decline. It will continue another 40 years because the level of labor force in Japan has been and will be decreasing in line with the ageing and overall depopulation. Today, Japan Statistics released an estimate of the consumer price index for August 2009 - http://www.stat.go.jp/english/data/cpi/1581.htm.

It is instructive, that the only two components (from ten), which did not suffer annual decrease, are the medical care and education. This is in agreement with the extraordinary high rate of growth in both these components in the USA .

Despite the surging prices for medcare and education, the US is awaiting the fate of Japan - a longer deflationary period since 2012.

Goldman Sachs on oil in 2009 and 2010

Reuters cites an updated forecast on oil demand and price just issued by GS: $85 per barrel by the end of 2009. Currently, oil price is down to $66-67, but it should reach $100 in 2009.

The Feds must bail out Lehman Brothers

There is a fundamental question on relative merits of the previous bank bailouts - which bank did deserve a bailout and how much would it really cost? Our model of stock price, developed in articles and posts, provides an additional measure of the size of bank problems - total debt. One can estimate the debt as a product of the number of shares and their market price, which was negative for the bailed out and not bailed out banks. We base our estimate on relevant stock prices described in this blog. Table 1 lists the number of stocks, their peak negative price, and the debt:

Table 1.
....................vol............ price,$.......debt,$
LEH...... 689,000,000.......... -40...... -2.76E+10
C ..........11,000,000,000..... -10.......-1.10E+11
AIG....... 134,000,000........ -870 ......-1.17E+11
FRE...... 648,000,000......... -40 ........-2.59E+10
FNM...... 1,110,000,000...... -45 .......-5.00E+10
(see fictitious stock price estimates for Citigroup and AIG in Figures below)

The table shows that Lehman Brothers had smaller problems than Citigroup and AIG. So, it was was easier to bail out LEH from purely mathematical and financial point of view.

Also, the joint negative price of AIG, FRE and FNM is less than 200bn. It could be some (deliberate?) miscalculation of the >1000bn help package. The toxic debt might be bigger than the total debt associated with stock prices, but there should be unaccounted positive assets.

Beyond the bankruptcy – Fannie Mae, Freddie Mac, and Lehman Brothers

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.

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.

Predicting bankruptcy : the case of Lehman Brothers

We continue testing our model againts bancruptcy cases. Lehman Brothers presents an old but crucial case. Before we start our quantitative analysis, we have to clarify the the term "predicting" has a narrow sense of quantitative description. In other words, we do not forecast the bankruptsy, but decompose relevant stock price, sp(t), into a sum of two CPI components, a linear and a free term:

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

where CPI1 and CPI2 are defining CPI components; t is the calendar time; A, B, C, and D are empirically determined coefficients. Because prices for good and services 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 do not pretend that one could see the fall of LEH months before it had happened.

So, the empirical model describing the time history of LEH share price is as follows:

LEH(t) = -9.75*F(t-4) – 8.13*PC (t-2) + 94.9*(t-2000) + 2802 (2)

, where t is the calendar time, F(t) is the (not seasonally adjusted consumer price) index for food and beverages, leading the LEH(t) by four months, and PC(t-13) in the index of personal care leading the LEH(t) by 2 months. Or this model we used LEH prices between July 2003 and September 2008; the CPI components were taken till December 2008, i.e. three months ahead of the last LEH reading. Figure 1 demonstrates that the model predicted the fall below zero but three months later. Standard deviation of the residual is $4.7.

Once again, this model could not be obtained in August 2008.

Figure 1. Observed and predicted LEH share price.