The Relationship Between Credit Default Swap and Cost of Equity Capital

Transcription

1 Working Paper Series National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No. 668 The Relationship Between Credit Default Swap and Cost of Equity Capital Giovanni Barone-Adesi Moreno Brughelli First version: August 2010 Current version: November 2010 This research has been carried out within the NCCR FINRISK project on Interest Rate and Volatility Risk

2 The relationship between Credit Default Swap and Cost of Equity Capital November, 2010 Abstract We want to assess the relationship between the equity and the debt cost of capital. Using a very simple dividend discount model we compute the implied discount rate and we compare it with the corresponding premium on the corporate credit default swap using a cointegration approach. We demonstrated the existence of a cointegrating relationship between those two variables and we found weak evidence of Granger causality from CDS premium to the discount factor. Our findings are also robust to the choice of different parameter assumptions and model specification. Keywords: Asset Pricing, Cost of Capital, Implied Cost of Capital, Analysts' Forecasts, Discount Rate, Firm Valuation JEL Classification: G12, G31, G32. 2

3 In the traditional models the cost of capital is estimated ex post on the basis of realized returns. The most known and used model is the capital asset pricing model (CAPM) due to Sharpe (1964), Lintner (1965) and Mossin (1966). This model predicts that the expected return on a stock is positively related to its systematic risk, the beta. Many empirical estimates and test of this model based on realized past returns do not support the basic prediction of CAPM see for example Reinganum (1981), Coggin and Hunter (1985), Lakonishok and Shapiro (1986) and Fama and French (1992). Another popular model is the Fama French three factors model (1993). In their specification expected return on any stock depends not only on the market beta, but also on its size and B/M. Due to the poor empirical performance of the models based on past realized returns, many authors have recently proposed alternatives approaches. They use accounting data and stock market prices and by reverse engineering they estimate the implied cost of equity capital. The starting point of all those approaches is the well known proposition that the expected return on any share is the discount factor that equates the share's current price with all the expected stream of future dividends. Since it is not possible to predict dividends up to infinity one must impose some restriction to eliminate the need for an estimate of the terminal value. Gordon (1993) estimates the expected return as the sum of the expected dividend yield and the expected rate of growth in prices. Recognizing the limitation of this measure he finds a significant positive correlation between this variable and the beta. In a following work, Gordon (1997) proposes to estimate the expected return using the finite horizon expected return model (FHERM). In this model the expected return is obtained by finding the discount rate that equates the share's current price to the sum of expected dividends. Where the dividends up to a finite horizon N are obtained from analysts' forecasts and dividends from N+1 to infinity are equal to the forecast for normalized earnings in period N+1. He finds that those estimates can be in agreement with the CAPM predictions. Botosan (1997) uses the accounting based valuation formula developed by Edward and Bell (1961), Ohlson (1995) and Feltham and Ohlson (1995), (i.e. the EBO valuation formula). This model states that current stock prices are a function of today and future book values, future earnings and future stock prices. She analyzes then the association with the implied cost of equity capital with market beta, firm size and a measure of disclosure level. She demonstrates that the expected rate 3

4 of return is negatively associated with disclosure level (in particular for firms with low analysts following); furthermore it is increasing in beta and decreasing in firm size. Ohlson and Juettner (1998, 2000, and 2005), without imposing restriction on dividend policy, develop an alternative parsimonious model relating a firm s price per share to next year expected earnings per share, the short and long term growth in EPS and the cost of equity capital. Gode and Mohanram (2001) use Ohlson and Juettner model to determine the implied cost of equity capital and they find that the expected return is related with conventional risk factors such as earnings volatility, systematic and unsystematic return volatility and leverage. Gebhardt, Lee and Swaminathan (2001) estimate the implied cost of capital using a residual income model for a large sample of US stocks. Examining the firm characteristics that are systematically related to the derived cost of capital, they find B/M, industry membership, forecasted long term growth and the dispersion of analyst earning forecasts explain around 60% of the cross sectional variation in future implied cost of capital. Botosan and Plumlee (2001) want to verify whatever the cost of capital estimated from the unrestricted dividend model is a valid proxy for expected cost of equity capital. They show that the estimates they get are associated with six risk proxies suggested by theory and prior research in a consistent way. In particular, they find a positive and strong relationship between market beta and the derived rate of return. In the second part of their work, they analyze the extend on which restricted form model correlated with the unrestricted form. They find the EBO valuation model correlated the most. However they notice that there is no gain in using such a specification because of the need for forecasts of future stock prices. Among the other models with less need for data the Gordon model shows the highest correlation. Conversely the Ohlson and Juettner model and the Gebhardt, Lee and Swaminathan specification correlate less with the unrestricted dividend model, in addition the association between the derived expected return estimates and the risk factors are less consistent with the theory than the ones obtained using the dividend model or the Gordon model. To summarize, the evidence presented so far seems to indicate that market beta (and other risk factors) computed on realized returns are not able to explain the cross sectional difference in the expected rate of returns. The association with risk factors and cost of equity capital, computed on the basis of accounting variables and current stock prices, seems to be more consistent with the theory. In particular it exists a strong and positive relationship between the implied cost of equity capital and financial leverage of the firm. 4

5 In this paper we want to analyze further the determinant of the equity cost of capital. In particular we want to assess the relationship between the equity and the debt cost of capital. Using a very simple dividend discount model we compute the implied discount rate and we compare it with the corresponding premium on the corporate credit default swap. A CDS is a financial derivative that allows investors to buy protection against the default event of the underlying company. The investor will receive in the case of a default of the underlying company the face value of the defaulted bond, in exchange of this it has to pay periodically a fix amount of money. This premium is approximately equal to the spread between the yield of a bond issued by the firm and the corresponding government yield. For this reason they are a valid and easily available proxy for the cost of debt. An additional important difference compared with previous work is the econometric methodology we use. Since we find evidence that neither the implied cost of equity capital nor the CDS premiums are stationary the usual econometric techniques should not be used. In order to deal with non stationary data we use a cointegration approach. If a long term relationship between the time series exists we should be able to identify a cointegrating vector in such a way that a linear combination of the two variables will be stationary. We find that this is indeed the case. We will provide relatively strong evidence that the cost of equity capital is directly related to the cost of debt and that the CDS premium is able to explain a large amount of the cross sectional difference in the derived implied discount rate. In section one we present the model specification and the source of the data we used. In section two we will summarize the relevant econometric issues and techniques used to deal with cointegration. Section three presents the main results we obtain; in the first part we analyze the aggregate results while in the second part we will present the individual firm numbers. Some robustness check is presented in section four. Finally section five concludes. 5

6 1. Methodology and data: a. Empirical Models Specifications: The implied discount rate or the cost of equity capital is the rate of returns investors need for an equity investment. This rate represents the ex ante expectations about future returns. In most of the practical application however realized (ex post) returns were used. The logic behind this is that in average expectations should be equal to the realized returns. However many studies have shown that estimates based on past realizations are too imprecise to allow reliable conclusions. Despite this fact, the CAPM or the Fama French models remain the most used techniques to estimate the cost of capital. An alternative approach is to estimate the unobservable discount rate from the analysts' consensus forecasts about future cash flows (earning, dividends...) of a firm and its stock's current price. The expected rate of return is thus obtained by equating the current stock price with the intrinsic value of the firm (according to a specific equity valuation model) and solving for the internal rate of return. This methodology, in contrast to the classical ones, requires a model of corporate valuation since the intrinsic value is not an observable variable. In addition it has to rely on the fact that the observed stock price always reflects the true firm value (EMH) and that analysts' forecasts reflect the true market expectations about future cash flows. Although the EMH is generally largely supported by the literature, the last hypothesis is more controversial. Many authors argue that analyst forecasts are in average quite precise and in general they are more accurate than time series model (see for example O'Brien 1987, Brown 1996). Many others instead argue that forecasts errors are too large and they are systematically optimistically biased (see Brown 1995). In the typical neoclassical model the theoretical stock price is defined as the present value of the future cash flows to shareholders. Many different models have been developed, however to keep things as simple as possible we decided to use a simple dividend discount model. We decided not to consider the residual income models because we do not have monthly data on equity book value. We thus proceed as follows; the first step is to obtain the time series of the implied discount rates. To do this we consider the following equity valuation model. 6

7 ,, 1, (1) Where P t,k is the intrinsic price at time t of the security k, E t (D s,k ) denotes the expectation at time t for future dividends payment at time s for security k and finally r t,k is the implied discount rate or cost of equity capital. Since it is not realistically possible to forecast all the stream of dividends up to infinity we must introduce some assumptions about future dividend growth. Specifically we assume a constant growth in dividends after time T (g k ). In this way it is possible to rewrite the above formula as:,,, 1,, 1 (2) Since analysts usually focus on earnings it can be convenient to modify the previous equation in order to deal with EPS. One problem in considering directly earnings is that only a part of them are distributed to shareholders, the rest has to be reinvested in order to allow the firm to grow. The simplest and the most intuitive way to deal with this, it is to assume a constant payout ratio (1 k). The previous model can thus be rewritten as: 1 1 1,, 1, (3) To be sure that the choice of one particular specification does not drive our results, we simply decided to use both the alternatives through all the main part of the paper. At this point we are able to solve for r t,k in such a way that the error between the observed price and the theoretical one is minimized., min,,, (4) 7

8 Where, is the current market price of security k at time t, and, is the cost of equity capital to be estimated. We apply this minimization procedure for every monthly observation. This procedure allows us to take into an account any revisions in market expectations. Finally, given the accuracy of the forecasts and the fact that analysts long horizon figures are usually close to their short term forecasts we have decided to focus only on the three years horizon, since using more estimates do not give us much additional information. We further assume a constant payout ratio of 30% and a long term growth rate of 4%. This rate should mirror the growth rate of the overall economy. Changes in those assumptions over a reasonable range do not alter in a qualitatively way our results. More details are discussed in the robustness check part of this work. The final goal of this work is to try to explain the derived time series of the cost of equity capital as a linear combination with the CDS premiums. The logic behind this is that there exist many common factors that affect simultaneously the cost of the equity capital and the riskiness of the debt of a firm. For this reason we conjecture the existence of a relationship between r t,k and the premium on the corresponding credit default swap (CDS t,k ). To verify this conjecture we investigate for the presence of a cointegrating relationship between those two variables. If this is indeed the case the residuals (, of the following regression should be stationary., CDS,, (5) Since in many cost of capital theories the risk free rate is an important element, we test a second specification in which we include it, specifically we set up the following model:, CDS,, (6) Where RF is the level of the government interest rate. Since it is not clear which maturity is the best we test separately both the three months US T bill rate and the ten years constant maturity US treasury yield. More details on the econometric technique we used are presented in the third part. To reduce the noisiness of the data we repeat all the cointegration analysis aggregating in the following way the data. 8

9 , 1, (7), 1, (8) b. Data Set and descriptive statistics The following analysis is performed on the 30 stocks composing the Dow Jones Industrial Average (at 1st March 2010), covering the period for a total of 75 observations (See Table 1 for details on individual stocks). Data on analysts' forecasts for earning per share (EPS) and dividend per share (DPS) are obtained from I/B/E/S summary statistics database. Given the accuracy and the number of analysts' estimates we use only the mean estimate relative to forecasts up to three years. Summary history consists of chronological snapshots of consensus level data taken on a monthly basis. The snapshots are as of the Thursday before the third Friday of every month. Monthly data on prices and credit default swap premiums are obtained from the DataStream database. The limitation on the time interval we consider is due to the limited data sample of CDSs' premiums, since they are relatively new products we have data only starting from Monthly data on the yield of the three months US Treasury Bill and of the 10 years constant maturity Treasury notes are obtained from the Federal Reserve Statistical Release. Travelers (TRV) and Intel (INTC) have to be discarded from the analysis since we do not have enough CDS data. We also have to discard Cisco (CSCO) from the DPS analysis because all I/B/E/S forecasts for the dividend of this stock are zero. Finally the analysis on General Electric (GE) has to be limited to the period since we do not have CDS data after this date. We decided to focus only on those stocks because are the most liquid and the most followed by analyst. Additionally their cash flows are quite stable and thus forecasts are more accurate. Obviously the sample is quite small and conclusions on the general validity of the findings presented trough this work are difficult do draw. Table 2, Panel A gives descriptive statistics for the selected sample. The mean for the entire data sample of the discount rate estimated on EPS basis is 6.33% while it is 6.88% if computed on DPS. 9

10 Those numbers appear realistic and are consistent with prior researches. Nevertheless the level of those estimates are sensible to the choice of the long term growth rate and the payout ratio, for this reason we decided to repeat the main analysis taking different assumptions about those two parameters. The implied risk premium (computed as the difference between the discount rate and the 10 years treasury constant maturity) are 2.23% and 2.77 % respectively. Those numbers may appear small but they are consistent with the selected sample that is entirely composed by large and stable firms. Given these characteristics they are generally seen as low risk stocks and for this reason the required risk premium is relatively low. During the same period, the mean 3 Months T bill rate was around 2.37% while the average 10 years rate was relatively higher at 4.10%. The average premium on the CDS was 26.32, and basis points for the 1 year, 5 years and 10 years maturity respectively. The year by year statistics indicate that although the overall discount rate did not increase too much during the financial crisis (from around 6% to 6.8%) the risk premium almost doubled from an average of 1.6% to more than 3.5 %. This is consistent by the increase in the risk aversion that is typical during periods of financial instability. The yield on the 3 months T Bill was around 1.36% in 2004, then it rose to an average of 4.73% in 2006 before to drop to 0.14% and 0.10% in 2009 and 2010 respectively. Conversely the yield on the 10 years Notes remained substantially stable around the range % until In 2009 it dropped to 3.18% and finally they recovered to 3.76% in The analysis on the statistics on CDS highlights a dramatic increase in the default risk especially over the short horizon. The average 1 year CDS premium rose in fact from an average of 6 basis points, for the years to more than 70 basis points in 2009 (figure 5). In the first months of 2010 it adjusted to a mean of 27 basis points. Longer horizon CDSs present a similar pattern. During the sample period EPS and DPS forecasts rose to an average rate of around 11 12% per year. This rate of growth however was not uniform, ranging from an average of around 30% in 2004 to a negative average value in 2008 and Interesting the two and three years horizons present the more negative values. This is the results of the financial crisis and the worsening in the future economic perspective. Conversely the ratio between dividends and earnings remained quite stable during the full sample period and through the various horizon averaging around a value of 0.35 (table 3). 10

11 The analysis of the correlation of monthly changes (table 4) shows that CDS premiums are highly correlated across different maturities (ranging from ). With, not surprisingly, the correlation to be the highest between the narrowest maturities. The same appears to be true also for EPS and DPS forecasts over different horizon. As an example the Spearman rank correlation between the one year and the two years horizon EPS forecasts is 0.90, while the correlation between the two years and the three years horizon is 0.8. Instead the correlation between the one year and the three years is only The correlation between monthly changes in EPS and DPS forecasts is quite high (around 0.6 depending on the considered horizon). This may be a consequence of the fact that managers tend to smooth dividends more than earnings. Interestingly, neither the EPS nor DPS monthly changes appear to be significantly correlated with the monthly changes in the level of the risk free rate or in the CDS premium. Even the changes in the short term and in the long term risk free rate seem not to be particularly related (0.33). Finally, we observe a small negative correlation between the changes in the risk free rate and in the CDS premium particularly for the 5 and 10 years maturity. This can be related to a business cycle story: interest rates tend to decrease when economy is slowing, at the same time default risk tends to increase in periods of low economic activity. The correlation between the changes in the discount rates computed according different model specification is quite high (more than 0.7). The value reported are consistent with prior researches (e.g. Botosan and Plumlee (2001)). This is a good result for us since it confirms that the different models give substantially the same output and it is important in the optic of our work because it ensures that the results we are presenting are not driven by the choice of a particular equity valuation model. Monthly changes in CDS premium and in implied discount rates also show a quite remarkable positive correlation. This confirms the existence of a link between the cost of equity capital and the risk premium in the corporate bonds. Although derived from earnings per share and dividends per share forecasts, changes in the implied discount rate present only a small correlation with changes in EPS or DPS. Interesting the relationship with the risk free rates is not strong and it is somehow ambiguous. The correlation with changes in the 10 years yield is around while the correlation with changes in the 3 months rate is negative (around 0.1). This is a little bit surprising since traditionally risk free rates play an important role in the determination of the cost of capital. 11

12 2. Cointegration, unit root and spurious regressions Since neither the premium on the credit default swap nor the implied cost of equity capital are stationary, in order to rule out the possibility of spurious results we have to look for the presence of cointegration. Two or more time series are said to be cointegrated if some linear combination of them has a lower order of integration. If this is the case there exists a statistically significant long run equilibrium relationship between the variables. Several methodologies have been developed to deal with cointegration. In this paper we decided to use two approaches: one based on the VECM ordinary least squares estimates, firstly developed by Engle and Granger (1987) and the other based on the maximum likelihood estimates developed by Johansen (1995). Engle and Granger suggest a two steps procedure. Define a vector y that collect all the observations about the variables of interest (in this case the implied discount rate and the CDS premium). Further assume that the time series are integrated of order one., (9) ~ 1 (10) The first step is to verify whether the series are cointegrated, this is done by estimating by OLS the following cointegrating regression with a constant term: (11) And test whatever the residuals are I(1). One suggested methodology is the augmented Dickey Fuller test on the u t. This test consists in estimating the following autoregression of the residuals u t : (12) And to test whether the is equal to one. The augmented Dickey Fuller t test for the null hypothesis that the two series are not cointegrated is then: 1 (13) 12

13 Assuming the true process for y t to be: Ψ ε (14) It is important to notice that it is not possible to use directly the Dickey Fuller tables to find the critical values, since the residuals u t are generated from a fitting regression. For this reason one needs larger critical values than the standard Dickey Fuller ones. Appropriate values for this statistic are obtained by Monte Carlo simulations and the critical values are tabulated (see e.g. Phillips and Ouillaris, Econometrica 1990 pp ). The second approach was developed by Johansen (1988, 1991 and 1995). His methodology has some advantages over the previous procedure, first he relaxes the assumption that the cointegrating vector is unique, secondly he takes into account the short run dynamics of the system when estimating the cointegrating vectors. His procedure is based on the reduced rank regression method. Suppose that an (nx1) vector y t can be characterized by a VAR(p) in levels of the form: (15) The Johansen algorithm can then be described as follows; the first step consists in estimating a (p 1)th order VAR for y t : Π Π Π (16) Where Π denotes an (n x n) matrix of OLS coefficient estimates and denotes the (n x 1) vector of OLS residuals. And a second set of OLS regression as: (17) Where is the (n x 1) vector of residuals from the second regression. In the second step we have to calculate the canonical correlations from the OLS residual: T Σ 1 T v v (18) 13

14 T Σ 1 T u T Σ 1 T u u v (19) (20) Σ Σ (21) Finally compute the matrix: Σ Σ Σ Σ (22) From this matrix we can easily find the associated eigenvalues. The cointegrating vectors associated with the variables defined in y can be found as the eigenvector of the above matrix associated with the eigenvalues. Finally Johansen proposed two statistics based on the likelihood ratio test: The trace statistics tests the null hypothesis of h=r cointegrating relations against the general alternative of h=n cointegrating relations. The further the eigenvalues are far from zero the larger will be the statistics. It can be calculated as follows: 2 A log 1 (23) The maximum eigenvalue statistics instead tests the null hypothesis of r cointegrating vectors against the alternative of r + 1. This statistic can be computed using: 2 A log 1 (24) Critical values for both tests were tabulated by Osterwals Lenum (1992) using Monte Carlo simulations. Their asymptotic distributions depend on the number of non stationary components under the null hypothesis (n r) and on the form of the vector of deterministic components. 14

15 3. Results Let's start with a brief overlook of the stock market and of the level of the US government bond interest rates during the period taken into consideration by this work. Figure 1 plots the value of the Dow Jones Industrial Average Index (DJIA) for the period The market remained relatively flat during the years While it experienced a quite strong rally starting from years 2006 up to mid The most striking fact however is the dramatic decrease in quotation due to the financial crisis of and the subsequent strong recovery. Figure 2 reports the level of the government bonds interest rates for the USA. The first panel shows the yield on the 10 years constant maturity Treasury note (10YTN). The second panel shows the yield on the three months Treasury bill (3MTB). Finally the third panel reports the difference between the yield on the long term government bond and the short term rate on the Treasury bill (DIFF). During the period ( ) the inflation pressure has pushed up the three months Treasury bill yield, while long term interest rates are remained substantially stable around the 5% level. In the following two years the three months T bill rate has reached the zero level. This is a direct consequence of the extraordinary easy monetary policy that the FED (and many other central banks) had to undertake in order to contrast the deep financial crisis. The long term rates instead showed a slightly different behavior. After a strong decline, that coincide with the more acute phase of the crisis, the yield on the ten years US bonds started in 2009 again to increase as the economy gave some sign of recovery. Consequently the yield differential between long and short term interest rates widened. This measure is often used as an indicator of the investor expectation about the future growth of the economic activity. Although the period under consideration is relatively short, all those different market conditions and dynamics make it very interesting. The presence of those dynamics is an important condition necessary in order to detect any interesting relationship. Figure 3 and figure 4 plot the cross sectional average of earning per share (EPS) and dividend per share (DPS) forecasts made by analysts. As already explained we have considered just forecasts ranging from one year to three years. An important characteristic that emerges looking at those figures is that EPS and DPS measures tend to move together. Also the forecasts for different horizons seem to be highly correlated. 15

16 Interesting it seems that analysts' forecasts tend to lag the DJIA index. We observe an important reduction in forecast some months after the big drawdown of the stock market, and an increase some weeks after the market starts to recover. Figure 5 plots the average credit default swap premium for the 30 firms composing the Dow Jones Industrial Average (as of 1st March 2010). This time series is computed taking the arithmetic mean of the individual CDSs' premiums. 1, (25) As already mentioned the credit spread remained substantially flat during the all pre crisis period. Conversely during the crisis the average premium strongly increases reflecting the overall deterioration of the credit quality of the companies. This will automatically translate into higher financing cost and indirectly it will cause an increase in the cost of equity and a decrease in stock prices. Figure 6 reports the average of the individual stock implied discount rate computed using the different equity valuation models. Although there are some differences in the estimates coming from different models, the overall behavior seem to be the same. The Gordon model seems to be the one that present more variability. From a graphical inspection it appears that there is, especially during the recent financial crisis, a common movement between discount rates and CDS premiums. The two figures appear in fact very similar. Interestingly we can observe on both the graphs a double peak on November the and on March the This seems to indicate a common factor affecting both the cost of equity, approximated by the implied discount rate, as computed before, and the default risk of a company, approximated by the premium on the CDSs. This impression is also confirmed by the correlation between monthly changes in the discount rate and in the premium of CDS. As table 4 reports, the average correlation is in the range depending on the measure we use to compute the implied discount rate and the maturity of the CDS. Over the next sections we want to assess if it exists a cointegration relationship between those two variables. The starting point is to assess if the time series we use have a unit root. To check this we use the common Augmented Dickey Fuller test. That is we estimate the following zero drift (P+1)th order autoregressive [AR(P+1)] model: 16

17 Δ Δ Δ ρy ε (26) And test the null hypothesis that ρ = 1, the t test is then: 1 (27) Table 5 reports the values of the t test for all the time series we use on this paper. As we expect both the implied discount rates computed on EPS and on DPS are not stationary for all the stocks considered. Evidence on the presence of a unit root is weaker on the CDS time series. In fact the ADF t test fails to reject the null of a unit root in four cases at the 5 % probability and on eleven cases at the 10 % level. This however is mainly due to the fact that prior the crisis, premiums where almost flat at a low level. If we consider just the lasts 40 months of the sample we cannot reject the null of a unit root for any series. For this reason we assume that all the time series considered present a unit root. c. Aggregate results Over the next section we will consider the relationship between the cross sectional average of CDS premium and the cross sectional average of the implied discount rates. We first investigate for cointegration between the premium on the average CDS premium and the average implied discount rate obtained using EPS forecasts as in equation 3. We assume the fraction of EPS that is paid out any period to be 0.3 (this value is consistent to the ratio between DPS and EPS that we reported on table 4) and the long term growth rate to be constant at 4%. Different combinations provide qualitatively similar results. Deeper investigation will be presented in the following of the paper. Table 6 panel A reports the results we get using the Johansen procedure. The likelihood ratio test indicates the presence of cointegration between the two variables at the 5 % confidence level. The cointegration vector suggests the existence of a positive relation. The results are similar whatever we consider 1 year, 5 years or 10 years CDS premiums. Panel B instead reports the results we get using the standard OLS regression. The ADF t statistics computed on the residuals also confirms the presence of a cointegration. In addition the beta estimates are very close to the Johansen results. We can notice that the R 2 of the regression is 17

18 quite high (75 80%). The quite low value of the Durbin Watson statistics suggests that the residuals are positively autocorrelated; this may cause to overestimate the level of significance. In panel C and D we perform the same computations as before but on the average implied discount rate computed using dividend per share forecasts as in equation 2. Again we assume a constant long term growth rate of 4%. The results are similar and even stronger than the ones obtained in precedence since the likelihood ratio tests are now all significant at the 1% level and the ADF t statistics are higher than before. Again we observe a positive relationship between the two variables. Furthermore the explanatory power of the model is very high (85 88%), interesting also the DW statistics is a little bit higher than before. All this indicates that at the aggregate level the discount rates obtained using the DPS forecasts provide better results than the ones obtained with EPS. A possible explanation for this may be that with DPS we do not have to make any assumption about the payout rate. On the other side individual DPS forecasts are in general less accurate than EPS forecasts. This trade off will be apparent when we consider the results at individual stock level. There we get better results using EPS than DPS. Finally we want to consider the role of the risk free rate. We individually include in the previous regression the yield on the three months US Treasury bill (3MTB), the 10 years yield on treasury notes (10YTN) and the difference between the two yields (TERM). A priory, it is not clear the impact of those variables, we might expect 3MTB and 10YTN coefficients to be positive since the higher the interest rates are, the higher is the cost of capital and thus the discount factor. On the other hand short term interest rates tends to move with the economical cycle, in period of expansion interest rates tend to be higher and they tend to lower in contraction. At the same time we expect the discount factor to be elevated during period of crisis since stocks tend to be riskier. For this reason we may also have a negative sign for those coefficients. The same story apply to the TERM factor, it is a known fact that the spread between long and short term interest rates can predict the economic cycle. High spreads suggest that investors anticipate an expansion of the economy; a lower spread suggests those investors are not so optimistic about the future growth. For this reason investors may require a lower discount rate when they perceive the economy to recover and a higher premium when they fear a recession. So we could expect a negative sign also on this variable. Table 7 reports the results we obtained. In Panel A and B we consider the discount rate computed from EPS, while in panel C and D we consider the discount rate that comes from DPS forecasts. 18

19 The first point we want to highlight is that the coefficients on CDS still remain significant and indeed does not change too much with respect to the previous case. Also the R 2 and the DW statistics do not change too much. This indicates that the effect due to CDS premiums dominates and it is by far the most important factor that we have considered. Regarding the impact of the risk free variables, as expected, we do not have a clear relationship. Looking first at the 3MTB variable we can observe a positive coefficient. Both the Johansen procedure and OLS give in general the same sign, the only exception is for the DPS, 1 year CDS case. There, using the OLS procedure, the beta on the three months t bill yield presents a negative sign. We have to notice however that this relationship is not very strong, although significant. Things appear messier for the 10YTN. In this case the two methodologies give opposite pictures, OLS suggests a positive coefficient while MLE a negative one. Given those results, a clear relationship between the implied discount factor and the yield on the 10 years bonds cannot be identified. To conclude this section let's give a look on the TERM factor. In this case we observe a negative sign (with again the exception for OLS: 1 year CDS, DPS case). Again we have to notice that this relationship is very weak. An explanation for those weak results can be given by the fact that those "risk free" variables are also cointegrated with the CDS premium as Table 11 shows. Since the explanatory variables tend to be cointegrated we may have some identification issue and thus we are not able to fully capture the effect of the risk free rates. We can notice a negative relationship between the CDS premium and the 3MTB and the 10YTN. The higher is the interest rates the lower is the required compensation for credit risk. An explanation for this is the business cycle story, during expansion interest rates tend to be high and during recession to be low. On the contrary CDS premiums tend to be high during recession when credit risk is more severe and low in booms. This story is also confirmed by the negative sign on the TERM factor. When this spread is high investors perceive a growing economy and thus the perceived credit risk is lower. 19

20 d. Individual results In the previous section all the analysis were conducted using aggregate data. Therefore one might ask if the previous findings apply also at individual level. This section try to answer to this question. As before the individual implied discount rate is computed on the basis of DPS and EPS analysts forecasts according to equation 2 and 3 respectively. We impose for every stock a constant payout ratio of 30 % and a constant long term growth rate of 4%. We are then ready to test for cointegration between the time series of the implied discount rates and of the one year and five years CDS premium. For this purpose, as in the previous part, we use both the standard OLS and the Johansen procedure. Table 9 reports the results obtained applying the Johansen procedure using EPS data. In panel A the one year CDS premium is considered while panel B considers the five years CDS. Focusing first on the one year case, the likelihood ratio test accepts the null of a cointegrating relation, on 25 cases over 28, at the five percent. Two more are significant at the ten percent level. Just for IBM the test fails to find cointegration. For what concerns the sign of the relationship, we find a positive coefficient between the discount rate and the CDS premium for the great majority of stock analyzed. The only exceptions are Caterpillar, Home Depot and General Electric. However concerning General Electric we have to notice that the CDS time series just covers the period January 2004 June In those years the premiums were almost flat at a very low level, this makes very difficult the identification of the relationship between those variable and the discount rate. Results using the five years CDS are very similar (panel B). The numbers presented so far seems to confirm the findings we obtained in the previous section. Even if the implied discount rates are computed on the basis of DPS forecasts the conclusions we made do not change significantly. Table 10 presents the figures for this case. Results are qualitatively similar to the EPS case. We find again a cointegrating relation (at 5% significance level) on 24 over 27 cases. Interesting Caterpillar and Home Depot present the expected cointegrating sign. Only General Electric continues to shows a negative relationship. Again we obtain comparable results also considering the five years CDS case (panel B). The evidence presented so far seems to indicate that the choice between DPS and EPS seems not to alter from a qualitatively point of view too much the results. This is really not surprising given 20

21 the quite high correlation between the two time series (table 4) and since DPS are often estimated as a fraction of EPS. In order to check whether the choice of a different econometric technique leads to different conclusions we perform the same analysis as before applying the standard OLS procedure. Results are presented in table 13 (EPS) and table 14 (DPS). Fortunately the numbers are comparable with those of table 10 and 11 presented before. Almost all the betas are significant and positive confirming the positive relationship between discount rate and CDS premium. Interesting the alphas appears to be quite stable around the 6% level for every stock considered. The level of the alphas depends on the choice of g and k, and it seems to capture the constant common risk for all the 28 stocks analyzed. A less positive point is the ADF test statistic. This test fails to reject the null of a unit root in some cases at 10% probability. This may be due to the small sample we use and to the low power of the test. Figure 10 shows graphically how good this model specification for the discount rate "fits" the observed stock prices. The green line represents the true observed market price of the stock. The blue line is the fitted value of the share. Those values are obtained by firstly estimating the betas and the alphas of the equation 5 according to the methodology described previously. Secondly we get the fitted estimates of the implied discount rate using market CDS premiums. Lastly we combine those numbers with EPS analysts forecasts to get the fitted prices according to equation 3. As we can see even if the model is extremely simple the fitting is quite accurate. More important we are able to maintain a good fitting even during the big crash due to the financial crisis for almost all the stocks we considered. This may indicate that the big drop in stock prices is mainly due to an increase in the discount factor and thus in the risk premium, and that the subsequent recover was mainly due to a normalization of the discount rates. The role of risk free rate is considered in tables 11 (EPS) and 12 (DPS). In Panel A the 3 months Treasury bill rate (3MTB) is considered while Panel B includes the results for the difference between the ten years and the three months yield (TERM). As for the aggregate case, those new variables do not rule out the CDS factor. As for the aggregate case there is not a clear sign for the 3MTB coefficients (Panels A). However the positive one predominates (20/28 cases for EPS, and 14/28 for DPS). This may be explained by the fact that some firms are more sensitive to the business cycle, so the "signaling" aspect of the risk free rate prevails while some others are more concerned with financing cost so the "cost of capital" aspect dominates. The negative sign prevails in front of the TERM coefficients, (20/28 for 21

22 EPS and 17/28 for DPS).The results of the regression is presented in panel B. Those numbers are consistent with the story presented in the previous section; investors require a higher premium when they perceive the economy to do bad in the future, being signaled by the low spread in the TERM factor. As for the aggregate case we may have some identification problem since both TERM and 3MTB appear to be cointegrated with the CDS term (see Table 16). Again the 3MTB (and the 10YTB) are negatively related to the CDS premium (see Panel A and B). While the TERM factor is positively related to the default risk. To summarize, the results obtained suggest the presence of a positive cointegration relationship between the cost of equity capital and the risk premium in corporate bonds. Conversely any of our measure related to the risk free rates seems to be important factors. The picture presented here confirm the findings of the previous section. e. Vector autoregressive specification and Granger Causality As a last check of the relationship we found between the CDS premium and the implied discount rate, we estimate a vector autoregressive model (VAR(p)) on level. (28) Where is the 2 X 1 vector containing the observations And A is a two by two matrix of coefficients The lag length is determined according to the standard likelihood statistics. For the discount rate computed using EPS data, according to the likelihood statistics, the optimal number of lag (p) is three while for the rates computed using DPS it is two. Table 17 reports the coefficients estimates for the VAR(3) model, in Panel A we use the one year CDS premium while in panel B we use the five years maturity. Looking at the first equation, the coefficients for the CDS variable (up to lag two for one year CDS and up to lag one for five years CDS) are significant in explaining the level of the discount rate measure. In addition the value of the estimates of the first lag DR, although significant, it is well below the value of one. This confirms that the level of the DR is related to the level of the CDS premium. In the second equation of the VAR model the DR level seems not to be statistically important in explaining the 22

23 level of the CDS premium, whose value seems to be driven only by past lagged value of the variable itself. Those results are consistent with the random walk hypothesis. We get similar results also using the discount rate that come from the DPS forecasts (table 18). In this case according to the likelihood ratio statistics the suggested specification is a VAR(2). As before the lagged CDS premium is statistically significant in explaining the DR(DPS). Whereas the inverse is not true. The corresponding input response functions (figures 11 and 12) show and confirm the positive relation between changes in the level of one variable and the level of the other variable. Such shocks seem to be quite persistent. The preceding results seem to indicate causality from the CDS premium to the discount factor. As a test of this hypothesis we implement the granger causality test. To illustrate the way we performed the test it may be useful to rewrite the preceding VAR(p) model as: p DR( t) = A DR( t j) + A CDS( t j) + ε ( t) 11 j 12 j 1 j= 1 j= 1 p CDS( t) = A DR( t j) + A CDS( t j) + ε ( t) p p 21 j 22 j 2 j= 1 j= 1 (29) Where p is the maximum number of lagged observation as determined previously, ε1 and ε2 are the residual for each time series and the matrixes A include all the coefficients of the model. CDS are said to Granger causes DR if the inclusion of the CDS terms in the first equation reduces the variance ofε 1. This can be tested by performing an F test of the null hypothesis that all the coefficients in A 12 are jointly significantly different from zero. Obviously the same testing procedure applies also for DR. The results are presented in table 19. It appears that for most of the model specification we used the CDS premiums is not Granger caused by the DR, the Granger causality probabilities for this relation are in fact, except for the [DR(EPS), CDS(1 year)] specification, well above the 10% level. Results seem to be more favorable to causality between the CDS premium and the discount factor. Granger causality probabilities are in fact 0.00 % and 5.66% for the 1 year CDS and 0.13% and 0.03% for the 5 years CDS. 23

24 4. Robustness check: The more sensible part of the paper is the estimate of the implied cost of equity capital. For this reason we want to check whatever the results obtained so far are substantially affected by the choice of a specific equity valuation model and/or a specific parameter setting. We thus repeat our analysis firstly by changing the parameters assumptions, specifically the long term growth rate and payout ratio, and secondly by computing the cost of capital with an alternative model; the finite horizon expected return model (FHERM), for detail on the derivation see Gordon This model is derived from the well known proposition that current stock price equals the discounted sum of all future dividends. In order to derive a treatable formula Gordon imposes that beyond year T, the return on equity (ROE) reverts to the expected cost of equity capital (r). The model can be written as follows: 1 1 (30) Where P t is the price of the stock at date t, r is the expected cost of equity capital, d t is the dividend per share for year t and x t is the earnings per share for year t. Figures 7 and 8 show that different assumptions on the long term growth rate (g) and on the payout ratio (1 k) substantially affect only the level of the implied discount rate. As we can notice we observe a proportional parallel shift of the curve, the higher is the g the higher is the discount rate. Changes in the pay out ratio assumptions lead to similar movements of the curve. However such movements are less pronounced than changes in g. Using the finite horizon model it turns out that the average implied discount rate is more volatile than under our previous specification. In particular one can notice a more pronounced increase in the level of expected return during the turbulent period of the financial crisis. As figures 6 shows the implied cost of equity capital, computed according to the FHERM, rises from an average of 6% in 2004 to a peak of 12% at the end of Nevertheless the overall behavior appears to be very similar under the different specifications. This impression is also confirmed by the correlation analysis between monthly changes in the discount rates derived from different equity valuation models. The correlation matrix is presented in table 3. As already mentioned the correlation between the different expected return models is quite high. The Spearman rank correlations range 24