Yield is one of the key terms for share analysis, i.e. degree of profit from an investment into the share concerned. This in most cases is formed by the capital yield and dividends paid during the period in question. If we disregard the staggered payment of dividends, the corresponding return for the period of concerned is
ri = ( (pi - pi-1) + di) / pi-1,
where pi-1 is the quotation at the beginning and pi is the quotation by the end of the period concerned and di is the paid dividend.
In order to determine return from the development of the share market price at a certain time, it is necessary to adopt a certain model. The value of shares is most often expected to increase by a interest and market price for the shares Pt at time t than can be expressed as
Pt = P0 eµt,
where P0 is the market price of the share at time t = 0 and µ is the rate of return.
The parameters of the given market price model can be estimated from the previous share quotations and the dividends paid, using the least squares method, from condition of the minimum
∑i=1...n [(pi + di) - P0 eµti]2 → min,
for the given time period. The P0 parameter and the µ rate of return are then the solution of non-linear equations obtained by partial derivations
P0 ∑i=1...n (eµti)2 - ∑i=1...n (pi + di) eµti = 0,
P0 ∑i=1...n ti (eµti)2 - ∑i=1...n ti (pi + di) eµti = 0.
The comparison focused on the annual rate of return in 2001-2006 for the following shares traded on the Prague Stock Exchange in the SPAD system
For the calculation of the parameters of the market price model, we opted for a sliding period of the previous year and used daily closing quotations on the Prague Stock Exchange. The P0 parameter was eliminated by substitution from the above-mentioned system of non-linear equations
P0 = ∑i=1...n (pi + di) eµti / ∑i=1...n (eµti)2.
and the following non-linear equation
∑i=1...n (pi + di) eµti ∑i=1...nti (eµti)2 / ∑i=1...n (eµti)2 - ∑i=1...n (pi + di) eµti = 0
was solved with the use of the numeric method. For a summary of the reflected dividends paid in 2000 - 2006, see Table 1.
The accuracy of the market price model is shown in an example comparing the daily closing quotations for the shares of TELEFÓNICA O2 C.R. on the Prague Stock Exchange, as of 1 September 2004.
Until January 2002, the annual rate of return practically kept on decreasing as a consequence of the quotation decreasing from CZK 125.09 in February 2000 to CZK 57.82 in October 2001, reaching its minimum value during the period concerned, i.e. ‑48.8%. Subsequently, the annual rate of return increased along with the increasing quotation, to reach the maximum value of 107.1% in late March 2005. At the end of period concerned, as a consequence of the quotation decreasing from CZK 862.20 to CZK 565.50, the annual rate of return decreased from the value of 82.5% to the current level of 28.8%.
The maximum annual rate of return during the period concerned, i.e. 63.8%, was achieved by the end of April 2004, reflecting the growing quotations practically during the entire period from the commencement of trading this issue at the Prague Stock Exchange in October 2002 until the splitting of the issue in July 2004, from CZK 1,016 to CZK 4,229. Subsequently, the annual rate of return gradually decreased, reaching the current value of 2%. The splitting of the issue did not influence the development of the annual rate of return to any significant extent.
The minimum annual rate of return during the period concerned was at the beginning of the period concerned, i.e. ‑7,7% in January 2001. This value reflects the sharp increase in the quotation in early 2000, i.e. to CZK 1,152, followed by a decrease and fluctuation around the mean value of CZK 843.96. Subsequently, the annual rate of return grew - as a consequence of the growing quotation to CZK 1,184 and its subsequent decrease - up to 33.5% by the June 2001 but decreased again down to ‑5.3% in January 2002. The maximum value during the period concerned, i.e. 88.9% was achieved in June 2002, after a relatively dramatic quotation growth from September 2001 (CZK 806.70) to May 2002 (CZK 1,920). Since June 2002, the annual rate of return has been gradually decreasing with some fluctuations, reaching the current medium value of 5.6%.
The maximum value during the period concerned, i.e. 59.6% was achieved in early October 2002, after a gradual quotation growth from to CZK 12,766. Since October 2002, the annual rate of return has been gradually decreasing with some fluctuations. The minimum annual rate of return during the period concerned was ‑40.3% at the end of the period concerned, which reflects the decrease of the quotation from CZK 20,926 to CZK 10,276 in the last six months.
The minimum annual rate of return during the period concerned, in October 2001, was ‑77.6%. This value reflects the decrease in the quotation from March 2000 to September 2001, from CZK 965.90 to CZK 184,01. Subsequently, the annual rate of return grew and reached 15.3% by the end of June 2002, but subsequently decreased again down to ‑46% in December 2002. This trend is caused by the quotation growing to CZK 383.50 and its subsequent decrease to CZK 210.80. With some further fluctuations of smaller intensity, the annual rate of return began increasing by the end of 2002 to reach the last maximum of 46.2% in early July 2005.
The annual rate of return reached its minimum value in May 2002, i.e. ‑61.1%. This value reflects a long-term decrease in the quotation, culminating in early March 2002 with the minimum value of CZK 28.03. Since March 2002, the annual rate of return has reported practically continuously increasing trend and the annual rate of return increased with certain fluctuations. The first maximum (78.7%) was achieved in October 2003, the second (95.9%, i.e. global during the period concerned) in early May 2005. At the end of period concerned, as a consequence of the quotation decreasing from CZK 300.00 to CZK 156.80, the annual rate of return decreased from the value of 87.0% to the current level of 23.2%.
For the purposes of this study, the issue has been traded on the Prague Stock Exchange for a relatively short period only, i.e. since June 2004. Since June 2005, the annual rate of return dropped from 59.2% to the current value of 20.2%.
Roman Pospíšil, Karel Pliska, Michal Patka, 09-Jul-06