Confidence Bands for Investment Decisions

CHAPTER 5 Confidence Bands for Investment Decisions 5.1 Introduction A simple buy and hold strategy may not often yield good returns for an investor. Timely booking of profits is essential for making money from the capital market. This need not imply that an investor should do business on a daily basis. An active tracking mechanism can lead to timely booking of profits. Even if investment is made by considering different factors such as interest rate, inflation rate, exchange rate, P/E ratios, share price indices etc, investors have got burnt in many occasions. In this chapter, we propose a model for making investment decisions in stock market that is a hybrid of the fundamentals and technicals. The model will be ideal for taking decisions to buy or sell or hold a particular stock by considering the market conditions prevailing at different points of time. The model will also help in identifying good stocks for retail investors for whom preservation of capital is as important as earning reasonable returns from investment. Generally, investment is distinguished from speculation by the time horizon of the investor and often by the risk-return characteristics of the

97 investment. Any true investor is interested in a good rate of return, earned on a rather consistent basis for a relatively long period of time. Handsome returns can be achieved from the stock market by a variety of techniques. Usually, a prospective investor reaches an investment decision to buy or sell or hold a particular stock by comparing the intrinsic value with the current market price of the stock. The intrinsic value of a stock is calculated by considering factors such as the sales, net profit, book-value, Earning Per Share(EPS), P/E ratios, dividend paid etc. Technical analysts who act solely on the basis of the past behavior patterns in the price and the volume of the particular stock often question this fundamental approach. But the Random Walk Theory formulated by E.F.Fama,(1965) refuted the technical analyst s approach in forecasting the future price of a stock based on past prices. The Random-Walk Theory says that successive price changes are independent. This independence implies that prices at any time will, on the average, reflect the intrinsic value of the security. Furthermore, if a stock s price deviates from the intrinsic value because, among other things, different investors evaluate the available information differently or have different insights into future prospects of the firm. Niederhoffer and Regan (1972) suggested that stock prices were strongly dependent upon earning s changes, both absolute and relative to analysts estimates. They discovered that the common characteristics of

98 the companies registering the best price changes included a forecast of moderately increased earnings and a realized profit gain far in excess of analysts expectations. The worst performing stocks were those characterized by severe earnings declines, combined with unusually optimistic forecasts. The accuracy of earnings forecasts is of enormous value in stock selection. Another important criterion used for stock valuation is the P/E ratio. A high P/E indicates over-valuation of the stock and a low P/E indicates under-valuation of the stock. Therefore, for a proper valuation of a stock, the P/E ratio should be scientifically determined. In a survey, Bing(1971) found that seventy five percent of the analysts used the normal multiplier rules of thumb to determine the P/E ratio. In an attempt to bring some scientific evidence to the problem of normality in P/E s, several studies have been conducted, using statistical techniques to arrive at solutions. Whitbeck and Kisor(1963) studied a number of stocks and speculated that difference in P/E ratios between stocks could be explained by (i) projected earnings growth (ii) expected dividend payout and (iii) the variation in the rate of earning s growth, or growth risk. They also concluded that P/E is an increasing function of growth and payout and inversely related to the variation in the growth rate. In other words, higher P/E ratios were associated with higher growth and payout and less variation in the growth rate. Bower and Bower (1969) showed results

99 similar to Whitbeck and Kisor s(1963) for a cross section of stocks. They saw the same positive effects of earnings growth and payout. They discovered that higher P/E ratios were associated with more rapid earnings growth and higher dividend payout; lower P/E ratios with less marketability, greater conformity to market price movements, and higher price variability. The Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Linter (1965), and Mossin (1966) provide a system whereby investors are able to assess the impact of an investment in a proposed security on the risk and return of their portfolio. But CAPM was called into question by Roll (1977), who argued that the model should be discarded because it was impossible empirically to verify its single economic prediction. This controversial issue is still a subject of heated debate. Fama and French (1992) proposed a model based on size (market capitalization of a firm) and book-to-market ratio as an alternative to the CAPM. But Berk(1995) suggested that size and book-tomarket ratio reflect a combination of different economic mechanisms that are misspecified in the expected return process. The empirically based Fama and French (1996) model is now used by many as a benchmark asset pricing model. According to Pontiff and Schall (1998), the book value of equity proxies for expected cash flows and, therefore, the book-to-market has the power in explaining expected returns. Fama and French (1998)

100 view book-to-market as a measure of value versus growth, with low bookto-market firms being identified as growth firms and high book-to-market firms as value firms. Lakonishok, Shleifer and Vishny(1994) argued that many investors naively extrapolated past growth when evaluating a firm s future performance which frequently resulted in under pricing. But a high book-value combined with a consistent past growth record would gradually eliminate under pricing. Xinting and Ming (2005) viewed size and book-to-market as two significant components of financial distress, growth options, the momentum effect, liquidity, and firm characteristics. Even though different factors such as the EPS, the P/E ratio, the beta factor, book-value, returns from investments etc. are considered in different models, it seems that due importance has not been given to the relationship between the book-value(bv) of the stock, its EPS and the share price index. The movements of the share price index reflects the general economic and market conditions and have some psychological impacts as far as an ordinary investor is concerned. In this chapter an attempt has been made to predict the price of a stock based on its EPS, book-value and the share price index. A confidence band that would enable an investor to take suitable investment decisions such as either to buy or sell or hold a stock has also been proposed. Further, illustration of the performance of two different portfolios, one with high book value to market and low P/E, and another with low book value to market and high P/E is given.

101 This chapter is organized as follows. In Section 5.2, we introduced our model to estimate the price of a stock based on its EPS, book-value and the share price index. In Section 5.3 we suggested the idea of confidence bands, that wouldl be very useful in taking investment decisions. Section 5.4 is devoted to an illustration of our model. In Section 5.5, we considered an empirical analysis of the performance of low P/E stocks versus high P/E stocks listed at NSE. The chapter concluded with section 5.6. 5.2 Stock Valuation Model The price of a stock is a time varying quantity. So, data relating to the price of a stock can be treated as a time series P t where t denotes the period of observation. A general mathematical model of the series can be written as P t = Y t + Є t (5.1) where Y t and Є t represent the systematic part and the random part of the stock price P t. These two components are also known as the signal part and the noise part respectively. The effect of time may be in both the systematic and the random parts. The random part, which is a function of time, is a stochastic process and is the root cause of the variations of the stock price in the short term. The systematic part is a non-random or deterministic function of time. It is the joint effect of the various factors like the EPS, the book value, the general trend of the market measured by

102 the stock price index etc. In fact, the systematic part reflects the intrinsic value of the stock. Block (1995) considered the parameters such as the P/E ratio,eps, ratio of price to book value, return on equity and dividend pay out as the fundamental elements for evaluating the price of a stock. But in our opinion the eps alone is sufficient to represent the parameters P/E ratio, return on equity and dividend pay out for stock valuation. Even though the fundamental elements play a big role in determining the price of a stock, the general mood of the market measured by the share price index also plays a crucial role. Non-consideration of the market movement is one of the main drawbacks of the Block s (1995) theory. We give importance to the book value, eps and the share price index to determine the systematic part of the price of a stock. Let Y t be the intrinsic price of a share, X 1 be its book value, X 2 be the share price index and X 3 be its earnings per share (eps) at time t. Then the plane of regression of Y t on X 1, X 2 and X 3 is Y t = a 0 + a 1 X 1 + a 2 X 2 + a 3 X 3 + u (5.2) where a 0, a 1, a 2, a 3 are constants and u is the error term. The constants can be estimated by the least square method. As a simple thumb rule, sell the stock if P t > Y t and buy it if P t < Y t. An investor desires to get maximum return from his investments. This is possible only if the buy price is minimum and the sell price is maximum as possible. So we must have an idea about the upper limit and

103 the lower limit of the possible price variation of the stock. The variability of the random component Є t provides us a measure to derive the maximum benefit. 5.3 Confidence bands Confidence bands are the limits within which the price of a security is expected to lie with some definite probability. An α% confidence interval for P t is given by Y t ± t α (2, n-3) sy where sy is the standard error of a mean price predicted and is given by s y 2 = s 2 [ 1/n + i j ci j( Xi X i)(xj X j)] (5.3) where n is the sample size, sy 2 is the standard error of the regression and ci j is the (i,j) th element of inverse corrected sum of squares and sum of cross products matrix. Let L 1 and L 2 be the lower and upper confidence limits for P t at a point. Then, P[ P t > L 2 ] = (1- α) /2. (5.4) This means that at a given point the probability that the price of a share exceeds L 2 is (1- α) /2. For example, if α = 0.95, then the chance that the price is more than L 2 is 0.025. Therefore if we are prepared to take a probability risk of 0.025, it is advisable to sell the stock if its price exceeds L 2. In this case, the probability of the opportunity that will be lost for not selling the share for a higher price is only 0.025. Similarly,

104 P [P t < L 1 ] = (1- α) /2. (5.5) This implies that the probability that the price falls below L 1 is (1- α) /2. Therefore, if α = 0.95, a buy can be recommended at this level if its price falls below L 1. In this case also (1- α) /2 can be used as a measure of the risk. If confidence limits were calculated for all points on the regression line, the result would be confidence bands as shown in figure 5.1. The difference L 2 - L 1 can be used as a measure of the range of the possible variation of the price of a stock at a point. If L 2 - L 1 is a small proportion (less than the risk-free return) compared to Y t, then speculation may be avoided. In general, the confidence bands act like the control limits of a control chart. P t * * * * * * * * * * * * * * 0 Y t Figure 5.1 - Confidence bands Note: The S.E. is minimum at the mean values and it increases as estimates are made at values farther from the mean

105 5.4 Illustration Table 5.1 The price(y), book-value (X 1 ), Sensex (X 2 ) and EPS (X 3 ) of Reliance shares. Y X 1 X 2 X 3 Y X 1 X 2 X 3 104.97 89.50 3367.00 14.20 534.90 246.80 5677.00 36.80 154.45 92.00 3361.00 14.00 514.80 246.80 6325.00 36.80 177.20 128.60 3893.00 17.70 423.30 246.80 4756.00 36.80 130.50 132.60 3740.00 18.00 829.30 290.00 8649.50 54.20 318.70 132.80 5001.00 22.40 1253.05 357.40 13382.01 65.10 391.20 140.10 3604.00 25.10 2651.55 440.00 19051.86 82.20 398.40 199.20 3469.00 23.40 1709.90 440.00 14964.12 82.20 276.50 217.20 3156.00 29.30 2600.50 440.00 18280.24 82.20 538.20 246.80 5613.00 36.80 2172.90 440.00 16322.75 82.20 546.20 290.00 6679.00 54.20 1387.35 357.40 14515.90 65.10 293.90 199.20 3469.00 23.40 806.00 357.40 9826.91 65.10 334.30 217.20 3554.00 29.30 810.00 357.40 8799.01 65.10 271.50 217.20 3238.00 29.30 940.00 357.40 10645.99 65.10 410.50 217.20 4311.00 29.30 1075.20 357.40 11444.18 65.10 486.00 217.20 5186.00 29.30 2406.65 560.30 15962.56 133.90 540.30 246.80 5713.00 36.80 2643.60 560.30 16739.33 133.90 576.50 217.20 5786.00 29.30 534.90

106 Table 5.1 gives the price Y (in Rs.) of a share of Reliance Industries Limited (RIL) listed at Mumbai Stock Exchange (BSE), its Book-value (X 1 ), the share price index (X 2 ) and the EPS (X 3 ) observed at different points of time. Tables 5.2, 5.3 and 5.4 give the regression model summary, the ANOVA and the regression coefficients. Stepwise regression is used for determining the model. In the regression, the variable X3 is excluded resulting in the equation Y = 0.114 X 2 + 6.225 X 3 365.744 + u (5.6) The F test indicates that the coefficients of all the independent variables are not zeroes. The Confidence bands for various values of prices, EPS, book values and price indices are given in Table 5.5. Model 1 2 a. b. Model Summary c Adjusted Std. Error of R R Square R Square the Estimate.965 a.931.929 210.8243.971 b.943.939 195.2180 Predictors: (Constant), X2 Predictors: (Constant), X2, X3 c. Dependent Variable: Y Table 5.2 Regression model summary

107 Model 1 2 ANOVA c Sum of Squares df Mean Square F Sig. Regression18586756 1 18586756.07 418.179.000 a Residual 1377854 31 44446.898 Total 19964610 32 Regression18821308 2 9410654.206 246.934.000 b Residual 1143302 30 38110.050 Total 19964610 32 a. Predictors: (Constant), X2 b. Predictors: (Constant), X2, X3 c. Dependent Variable: Y Table 5.3 ANOVA table of the regression Model 1 2 (Constant) X2 (Constant) X2 X3 a. Dependent Variable: Y Coefficients a Unstandardized Coefficients Standardi zed Coefficien ts B Std. Error Beta t Sig. -332.732 69.324-4.800.000.148.007.965 20.449.000-365.744 65.557-5.579.000.114.015.747 7.621.000 6.225 2.509.243 2.481.019 Table 5.4 Regression coefficients

108 P X 1 X 2 X 3 L 1 L 2 514.80 246.80 6325.00 36.80 512.659 662.059 423.30 246.80 4756.00 36.80 318.869 496.646 829.30 290.00 8649.50 54.20 890.789 1032.732 1253.05 357.40 13382.01 65.10 1454.362 1688.318 2651.55 440.00 19051.86 82.20 2117.653 2535.971 1709.90 440.00 14964.12 82.20 1741.363 1976.425 2600.50 440.00 18280.24 82.20 2049.569 2427.402 2172.90 440.00 16322.75 82.20 1872.029 2156.800 1387.35 357.40 14515.90 65.10 1556.508 1845.762 806.00 357.40 9826.91 65.10 1082.596 1246.190 810.00 357.40 8799.01 65.10 951.330 1142.131 940.00 357.40 10645.99 65.10 1179.454 1336.849 1075.20 357.40 11444.18 65.10 1266.314 1432.725 2406.65 560.30 15962.56 133.90 2040.134 2549.927 2643.60 560.30 16739.33 133.90 2142.291 2625.602 Table 5.5 Confidence bands for RIL share prices Let the book value of RIL be X 2 = Rs 357.40(pre-bonus). Then the estimated value of Y when the index X 2 = 14000 is Rs 1266.80. Also

109 L 1 = 1051.24 and L 2 =1482.36 The P-P plot of the regression residuals is given in Figure 5.2 Normal P-P Plot of Regression Standar 1.00 Dependent Variable: Y.75 Expected Cum Prob.50.25 0.00 0.00.25.50.75 1.00 Observed Cum Prob Figure 5. 2 Normal P-P plot of Regression Standardized residual The normal P-P plot of regression-standardized residual given in Figure 5.2 justifies the fit also. In the case of RIL share; the book value and the index determine its price. The EPS has no major role in it. But for some

110 other stocks, the various factors have their own roles in determining the price. 5.5 Importance of Book-Value and P/E The fundamentals of a company play a very crucial and decisive role in determining the price of a stock for a possible buy or sell by a studied investor. Among the fundamentals, the book-to-market (BM) and the EPS are the key players.we have considered an empirical study to compare the performances of two portfolios, one consisting of a sample of ten securities listed at NSE of India with high P/E ratios and the other consisting of ten securities with low P/E ratios but comparatively high BM values. If the P/E ratio of a security is 15 or more, then it is termed as a high P/E stock. If the P/E is less than 8, then it is classified as a low P/E stock. The first category shares were from the Nifty group. We compared the closing prices of these stocks in April 8, 2011 with the corresponding prices in February 12, 2006. Necessary adjustments were made in the prices for the effects of bonus shares, share splits etc. Statistical tests were conducted to test whether there is any significant difference in returns as far as the two portfolios are concerned. The data and test results are given in tables 5.6, 5.7 and 5.8 respectively. The test gives evidence that the returns are significantly different. Further, the mean returns of low P/E shares are about 75% more than that of high P/E shares. Usually, investors rally behind high P/E shares to get

111 Stocks Price in 2006 Price in 2011 Return Percentage ABB 2425.30 4225.00 174.21 Bharathi Airtel 363.80 361.90 99.48 BHEL 1837.70 4402.80 239.58 Grasim Ind 1574.00 2537.70 161.23 HDFC 1363.00 2242.00 164.49 ICICI Bank 603.50 1098.30 181.99 Infosys 2863.00 6455.00 225.46 ITC 163.00 367.60 225.52 Siemens 4419.80 4460.00 100.91 Ranbaxy 432.80 465.60 107.58 Table 5.6 Portfolio of shares with high P/E ratios Stocks Price in 2006 Price in 2011 Return Percentage Canara Bank 263.50 632.50 240.04 LIC Housing 202.60 1145.00 565.15 SRF 276.30 344.80 124.79 Orient Paper 237.60 793.00 333.75 South indian Bank 71.10 280.00 393.81 State Bank of 299.50 756.80 252.69 Travancore Bhushan steel 147.00 478.90 325.78 G.E.Shipping 230.60 284.40 123.33 GSFC 194.90 371.10 190.41 Bank of Baroda 238.90 965.10 403.98 Table 5.7 Portfolio of shares with low P/E ratios

112 Variable 1 Variable 2 Mean 168.0442121 295.373106 Variance 2763.80381 19029.19275 Observations 10 10 Hypothesized Mean Difference 0 df 12 t Stat 2.727523437 P(T<=t) one tail 0.009175221 t Critical one tail 1.782287548 P(T<=t) two tail 0.018350442 t Critical two tail 2.178812827 Table 5.8. t Test: Two Sample Assuming Unequal Variances good returns in a short interval. But, as far as a real investor is concerned, it is advisable to invest in stocks with consistently low P/E ratio and high EPS. 5.6 Conclusion One of the serious drawbacks of the most famous Capital Asset Pricing Model(CAPM) for evaluating stock prices is that it does not take into account the various factors determining the price of a stock. It utilizes mainly the securities β factor that is not enough to determine the price. The importance given to the book-to-market and size by Fama and French(1996), Pontiff and Schall (1998), Xinting and Ming (2005) etc. is also not sufficient to take investment decisions as they are not giving

113 any weightage to stock market index which measures the pulse and direction of the business activity. Our model not only gives importance to the book value and earnings growth but also to the market sentiments. Further, the present model gives a measure of the risk in investment decisions. Investments in low P/E stocks appeared successful in many studies of past performance. Basu (1977) showed that the average annual rate of return is monotonically decreasing from low P/E stocks to high P/E stocks. Our study also confirmed this finding.