The Estimation of Expected Stock Returns on the Basis of Analysts' Forecasts

The Estimation of Expected Stock Returns on the Basis of Analysts' Forecasts by Wolfgang Breuer and Marc Gürtler RWTH Aachen TU Braunschweig October 28th, 2009 University of Hannover TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 0

1. Introduction 2. The general bias problem with implied rates of return 3. A bootstrapping approach to quantify the bias problem 4. Conclusion TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 1

1. Introduction ti Starting Point: Long and on-going debate regarding the precise value of the market risk premium, i.e. the difference between the expected one-period return of a broad portfolio of stocks and the corresponding risk-free interest rate equity premium puzzle according to Mehra and Prescott (1985): theoretically justifiable market risk premia are much smaller than those computed on the basis of averages of historical stock returns Several avenues have been taken to address this issue TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 3

1. Introduction ti 1. Improvement of the theoretical analysis in order to explain higher risk premia: Benartzi and Thaler (1995) and Barberis et al. (2001) use behavioral finance arguments like myopia, loss aversion, and ambiguity aversion in order to resolve the equity premium puzzle. Correlation of equity income with consumption changes over the life cycle of an individual (Constantinides et al. (2002)) or Consumption of stockholders differs from the consumption of nonstockholders (Mankiw and Zeldes (1991)) explanation for the empirical failures of the consumption-based CAPM Consideration of taxes, regulatory constraints, t and diversification ifi costs (McGrattan and Prescott (2003)) at least up to now these explanations are not fully convincing (DeLong and Magin (2009)) TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 4

1. Introduction ti 2. Alternative ways to estimate market risk premia: One may refine computations based on historical return data as in Fama and French (2002). One may simply make use of survey data extracted by asking specialists for their opinion regarding market risk premia (Welch (2000) or Graham and Campbell (2007)) only a shifting of the problem: the procedures by which those experts have obtained their market risk premium estimates remain opaque One may make use of credit risk spread data in order to derive equity risk premia on an options-price theoretical basis (Berg and Kaserer, 2008) TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 5

1. Introduction ti 2. Alternative ways to estimate market risk premia (cont.): One may rely on analysts earnings forecasts as the basis of net present value computations. (Claus and Thomas (2001), and Gebhardt et al. (2001)) Equaling these net present values with current stock prices makes it possible to derive an internal or implied rate of return that may be used as an estimator for future expected one-period returns E(d) E (impl) E(d) E(d) E(V r t1) E(d) Vt E(r) (impl) t 1 1r r Vt E(d) (impl) r (impl) TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 6

1. Introduction ti Assertion: (Claus and Thomas (2001)) The last approach might be able to contribute to the resolution of the equity premium puzzle. Reason: The so-called discount-rate effect may be effectively avoided. Meaning of the discount-rate effect: (Fama/French (1988)) If returns are independently and identically distributed over time the sample mean (for a long time period) is a good estimator for the expected return. for long time periods the iid-assumption typically does not hold because of exogenous shocks very slow convergence of the sample mean to the expected return TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 7

1. Introduction ti Example: Assume a decrease in the implied rate of return from 10 % to 5 % for a situation with a constant expected dividend of $1 per period till infinity. (stock) 1 V 0 $10 (stock) 0,1 V0 1 21 r 1 1 110 % (stock) (stock) 1 V0 10 V $20 0 005 0,05 return realizations:, 10 %, 10 %, 110 %, 5 %, 5 %, Large lengths of historical samples are necessary to neutralize outliers like those 110 % TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 8

On the contrary The estimation approach based on implied rates of return will immediately reveal the change in expected rates of return from 10 % to 5 % For dividend forecasts being sufficiently precise, this estimation procedure seems superior to approaches that rely on historical return realizations. Idea of Claus and Thomas (2001): As implied rates of returns are generally smaller than the historical average of realized rates of return, this alternative estimation procedure also may contribute t to resolving (or at least mitigating) the equity premium puzzle. But this raises the question: Are implied rates of returns good estimators for expected rates of return? TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 9

1. Introduction ti Contribution of the paper: p 1) We show analytically and numerically that implied rates of return are only poor predictors of expected one-period stock returns The paper highlights a severe bias problem relating to market risk premia estimation on the basis of implied rates of return. The resolution of the equity premium puzzle by considering implied rates of return is just a mere computational deception. 2) We derive an unbiased forecast equation which is based on (current and historical) implied rates of return and may be utilized as a basis for predicting future one-period returns. It offers the advantage of not depending on dividend volatility. Two advantages of this estimator: - it offers shelter against the discount rate effect and - it is not affected by dividend fluctuations. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 10

1. Introduction ti Presentation is organized as follows: 1) Presentation of our analytical examination of the general bias problem and the derivation of our alternative estimation procedure based on implied rates of return. 2) We undertake a bootstrapping approach based on data from the German stock market in order to - estimate the extent of the bias problem and - determine the welfare gain for investors when referring to our modified approach instead of simple estimates based on realized historical returns. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 11

2. The general bias problem with implied rates of return Assumptions: d t uncertain dividend of a firm at time t with expectation E (d ) at time g "dividend growth-rate", i.e. E(d) (1g)E(d ) E(d) (1g) E(d ). t1 t t1 t 1 t r (impl) t implied rate of return: E(d ) E(d ) V r g. t t t1 (impl) t t1 (impl) t r t g V t Assertion in the literature: The implied return is a reasonable estimator for the expected one-period rate of return from holding the stock from time t to time t+1. V d t1 t1 r t 1 1. Vt TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 13

2. The general bias problem with implied rates of return But a more detailed analysis shows: 1 g 1 (impl) (impl) d t 1 r t1 g r t1 (1r t ) 1 E 1 g (d t t1) 1 (impl) r g 1 g E t 1 (impl) (impl) rt 1 g E(r t t1) (1r t ) 1 1 g 1 (impl) r g t t (*) Consequently: E (r ) r if r r const. (impl) (impl) (impl) t t 1 t t 1 (only) in this situation, estimating expected returns simply by looking at implied returns is superior to any approach that is based on the consideration of historical return realizations as the variance of this unbiased estimator is just zero. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 14

2. The general bias problem with implied rates of return However, in the general case this will not hold true: For the martingale case E(r ) r we directly get (impl) (impl) t t 1 t (impl) (impl) E(r t t1 ) r t if r t1 and d t1 are independent (or negatively correlated) implied rate of return is a biased estimator of the expected rate of return now we analyze the extent of the estimation bias TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 15

Proposition 1. The Estimation of Expected Stock Returns on the Basis of Analysts' Forecasts 2. The general bias problem with implied rates of return Assume the time series of implied rates of return to follow a martingale and all implied rates of return and future dividends to be independent. Then the following statements apply: (impl) (i) Only for Var t(r t 1 ) 0, the implied rate of return at time t is an unbiased estimator of the expected rate of return E(r ). t t1 (ii) The bias in utilizing implied rates of return as an estimator for one-period expected stock returns becomes greater for ceteris paribus greater growth (impl) rates g, i.e. the partial derivative (E (r ) r ) / g is positive. t t1 t (impl,b) (iii) Let the implied rate r (impl,a) be a mean preserving spread of the implied rate r t1 t 1, (impl,b) (impl,a) (impl,a) i.e. r with. In this situation, the estimation t1 r t1 E( t r t1 ) 0 (A) (impl,a) (B) (impl,b) bias for B is higher than the bias for A, i.e. E(r ) r E(r ) r. t t1 t t t1 t Assessment of the assumption: Since changes of implied rates of return should be a consequence of changes in attitudes towards risk, there is no reason why future implied rates of return and expected dividends should be correlated. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 16

2. The general bias problem with implied rates of return However: Things may change if we take into account that the computation of implied rates of return are based on analysts dividend forecasts. if forecasts were independent d of actual market tforecasts (M), then changes in analysts dividend forecasts (A) would not be compensated by corresponding changes in stock prices E (d ) E (d ) E (d ) V,V r (r g) g. r g r g E (d ) (A) (M) (M) t t1 t t1 (impl,m) (impl,a) t t1 t (impl,a) t (impl,m) t t (A) t t t t 1 This leads to TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 17

2. The general bias problem with implied rates of return Proposition 2. Assume analysts dividend forecasts to be more optimistic than market expectations (A) (M) underlying stock price formation, i.e. E t (d t1) E t (d t1), (impl,a) then the corresponding implied rate of return estimator r is greater than the t (impl,b) corresponding estimator r t based on market expectations. The bias is independent of the assumed annual growth rate g of annual cash flows. Empirical evidence: Analysts earnings forecasts are typically too optimistic. (Stickel (1990), Easterwood and Nutt (1999), and Capstaff et al. (2001)) (impl,a) r t (impl,m) r t relying on instead of the corresponding upward bias may partially neutralize the downward bias according to formula (*). it does not seem to be too sensible to fight one estimation error by another. (impl,m) r t r t (impl,a) derive from and then use this estimator directly as an input in (*). Therefore, we need dto know the typical (A) (M) overestimation-factor E (d )/E (d ). t t1 t t1 TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 18

2. The general bias problem with implied rates of return Discussion of the martingale approach : An alternative is to assume that the one-period rates of return r t 1 are independently and identically distributed over time random walk, i.e. we have E(r ) µ for all t. t t 1 (impl) (*) the implied rates of return r t are also independently and identically (impl) (impl) distributed over time i.e. E(r ) µ t t 1 In this case we have an unconditional bias problem: E(r (impl) t1) E(r t ) furthermore, the implied rate of return is a biased estimator of the expected rate of return TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 19

(The Estimation of Expected Stock Returns on the Basis of Analysts' Forecasts Conventional estimator: 2. The general bias problem with implied rates of return Conventional (unbiased) estimator is defined as the arithmetic mean of historical one-period return realizations: 1 g 1 t 1 t 1 (impl) 1 1 (impl) rd(real) 1 g µ t : r Ed(1r ) 1, 1 g t τττ+1t 1 (impl) r g unbiased estimator (independent )An of dividends): An unbiased estimator (independent of dividends): 1 g 1 r g µ : (1r ) 1. 1 g 1 g t1 (impl) (impl) 1 (impl) 1 t t (impl) r (E*) TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 20

2. The general bias problem with implied rates of return Advantage of (E*): This estimator is unbiased for stationary implied rates of return over time. Moreover, its variance is smaller than the variance of the corresponding conventional estimator. Reason: (E*) is not affected by variations of dividends TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 21

2. The general bias problem with implied rates of return Proposition 3. Consider a situation with stochastical historical dividends and implied rates of return being stationary and independent of each other, so that realized rates of return are stationary as well. Then, the conventional estimator exhibits a higher variance then the corresponding estimator (E*). This means that, although both approaches lead to unbiased estimators of E(r ), estimator (E*) will be less volatile. t t1 Proposition 3 is the main finding of our theoretical session as it is describes a new estimator for future expected returns. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 22

2. The general bias problem with implied rates of return Moreover: The lower volatility of estimator (E*) is directly transferred into higher expected utility for investors. Explanation: Investor is maximizing a mean-variance preference function (µ, ) = µ 0.5 2 (constant absolute risk aversion) The investor is combining a portfolio of stocks (with expected return S and return standard deviation S S) with riskless lending or borrowing at a rate of return r f. S is known and ˆµ S is an estimator for S. x ˆµ r * S f 2 S "optimal fraction of stock portfolio" TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 23

2. The general bias problem with implied rates of return From the point of view of a decision-maker who knows the real µ: 2 (µ ˆ r) (µ r) 0,5 (µ ˆ r) (µ(x ), (x )) r. * * S S S f 2 S (1) For two estimation procedures S and S with (1) (2) and Var( ) Var( ) we can show: S (1) (2) (1) (2) E( ) E( ) and Var( ) Var( ) S (2) (1) (2) E( S ) E( S ) S This implies the following Proposition: TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 24

2. The general bias problem with implied rates of return Proposition 4. Consider a mean-variance investor with a preference function (µ, ) = µ 0.5 2 who is searching for the optimal combination of a fixed portfolio of stocks with riskless lending/borrowing. He/she knows the variance of stock returns but not the corresponding expectation value. Then, an unbiased estimation procedure for this expected rate of return will lead to higher preference values the smaller its variance. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 25

2. The general bias problem with implied rates of return To sum up the theoretical discussion: We have seen the following: 1) The current implied rate of return is an unbiased (and perfect) estimator of the future expected rate of return only in situations where there are no changes over time in the implied rate of return. 2) For implied rates of return being stochastic, an unbiased estimator of future expected one-period returns may be based on historical i realizations of implied rates of return according to (E*). This approach is superior to relying on historical realizations of actual one-period returns because this implied-oriented estimator is independent of dividend volatility. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 26

3. A bootstrapping t approach to quantify the bias problem Data: (Datastream) Monthly price data of stocks on the German capital market for the time period from 01/01/1997 to 02/01/2009. For each point in time, we restrict our analysis to those firms in the German stock index DAX for which analysts dividend forecasts are available from IBES. We utilize analysts dividend forecasts for the first following year and then assume a constant growth rate of future dividends. 145 implied rates of return on a monthly basis TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 28

3. A bootstrapping t approach to quantify the bias problem Bootstrapping Approach: We simulate 10,000 time series of implied rates of return with length 100. Each time series starts at t = 0 with an implied rate drawn (by chance) from the 145 implied rates of return, We generate the time series from t = 1 to t = 99 by adding randomly one of the 144 differences (of successive implied rates) to the already drawn implied rate of return. points in time 0 1 2 3 4 99 return series 1 0.2172124% 0.2096499% 0.2250919% 0.2260303% 0.2291524% 0.2277625% return series 2 0.2334629% 0.2239273% 0.1848931% 0.2411112% 0.2337464% 0.2426844% return series 3 0.1014611% 0.1093406% 0.1038452% 0.0991136% 0.0985677% 0.1068582% return series 4 0.1667936% 0.1617047% 0.1748917% 0.1748906% 0.1696767% 0.1555890% return series 5 0.2083799% 0.2038971% 0.2015103% 0.2400118% 0.2107640% 0.2147570% return series 6 0.1014104% 0.1330423% 0.0877093% 0.1077875% 0.0959203% 0.1195374% return series 7 0.1127982% 0.1064252% 0.1247419% 0.1002846% 0.1309252% 0.1275454% return series 8 0.1354291% 0.1280494% 0.1303402% 0.1446505% 0.1433085% 0.1319969% return series 10,000 0.1326933% 0.1409772% 0.1168361% 0.1270246% 0.1253643% 0.1479223% TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 29

3. A bootstrapping t approach to quantify the bias problem Average implied rates of return over all 145 months as well as corresponding return standard deviations: average return ret. std. dev. monthly values g = 0 % 0.1926% 0.0639% g = (1.01) 1/12 1 0.2747% 0.0637% g = (1.02) 1/12 1 0.3561% 0.0635% g = (1.03) 1/12 1 0.4367% 0.0633% g = (1.04) 1/12 1 0.5166% 0.0630% annual values g = 0 % 2.34% 0.79% g = 1 % 3.35% 0.79% g = 2 % 4.36% 0.79% g = 3 % 5.37% 0.80% g = 4 % 6.38% 0.80% Results across all 10,000 runs (g=0): average monthly implied rate: 0.1940 % standard deviation: 0.0653 % Apparently, these values are almost perfectly identical Simulation environment seems to be a good copy of reality To evaluate the estimation procedures we now deal with the case g = 0 TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 30

3. A bootstrapping t approach to quantify the bias problem Bootstrapping Approach (cont.): We draw stochastic dividends from t = 1 to t = 99 assuming dividends to be lognormally distributed with an expectation of time t+1 that is equal to the dividend at time t. Standard deviation of dividends is derived from actual DAX data over the time period from 1997 to 2008. We derive the time series of monthly stock prices for given implied rates of return: V ˆd t t r g (impl) t,month Stock prices enable us to compute actual realized monthly rates of return for each of the 99 periods from t = 1 to t = 99 for all 10,000 runs. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 31

3. A bootstrapping t approach to quantify the bias problem Analysis of the two (unbiased) estimation procedures (g = 0): 1) Conventional moving average of last 36 realized monthly rates of return 2) Moving average according to (E*) TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 32

3. A bootstrapping t approach to quantify the bias problem Results: Both approaches are able to approximate the true expected rate of return (1.118%) across all 10,000 runs almost perfectly. (Conv.: 1.109%, E*: 1.154%) The variance of estimators is much smaller for the approach E* ( = 0.0084%) than for the conventional one based on realized rates of return ( = 0.1274%). Mean-variance preference values of preference values are higher h for the second approach than for the first. This holds true for almost arbitrary values of preference parameter and riskless rate of return r f. one would conjecture that estimator (E*) is superior to an estimator based on historical rates of return TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 33

3. A bootstrapping t approach to quantify the bias problem Average preference values and preference values of preference values: (E*) E( ) (conventional) E( ) r f = 0.1 % r f = 0.3 % r f = 0.5 % r f = 0.1 % r f = 0.3 % r f = 0.5 % = 0.1 0.2339% 0.1472% 0.1239% 9.1904% 9.2771% 9.3004% = 0.5 0.1268% 0.2694% 0.4248% 1.7581% 1.6154% 1.4601% = 1 0.1134% 0.2847% 0.4624% 0.8290% 0 0.6577% 0 0.4800% 0 = 10 0.1013% 0.2985% 0.4962% 0.0071% 0.2042% 0.4020% (E*) (conventional) ( ) ( ) r f = 0.1 % r f = 0.3 % r f = 0.5 % r f = 0.1 % r f = 0.3 % r f = 0.5 % = 0.1 0.2283 % 0.1413 % 0.1176 % 9.3192 % 9.4076 % 9.4338 % = 0.5 0.1257 % 0.2683 % 0.4235 % 1.7838 % 1.6415 % 1.4868 % = 1 0.1128 % 0.2841 % 0.4618 % 0.8419 % 0.6708 % 0.4934 % = 10 0.1013 % 0.2984 % 0.4962 % 0.0058 % 0.2029 % 0.4007 % TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 34

3. A bootstrapping t approach to quantify the bias problem However: It may be objected that (E*) is derived under the explicit assumption of monthly dividend payments. We therefore repeat the whole analysis on the basis of lognormally distributed annual dividend payments. ˆd t (impl) V (1r t (impl) t,month) r g t,ann E* could be used as a heuristic estimation procedure Qualitative findings are essentially the same as for monthly dividend payments. The approach proves superior even under this adverse conditions. Reason: Predominant role of dividend volatility which affects estimators based on realized rates of return quite seriously. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 35

3. A bootstrapping t approach to quantify the bias problem Still to do: The choice of * ˆµ S rf x 2 S may be interpreted as a timing decision, because the fraction that is risky invested will be low in situations with low values for ˆµ S et vice versa. if the estimation (E*) is better than the conventional method it could be possible to achieve better timing results this still has to be tested with real data TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 36

4. Conclusion Implied rates of return are not suited for estimates of future one-period returns. implied rates of return are on average a downward biased estimator for future one-period returns unless implied rates of return are constant over time Estimator (E*) is superior to relying on historical realizations of actual oneperiod returns. We underline our findings by a bootstrapping approach. Next, we want to apply (E*) to get better timing decisions in portfolio management. TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 38

Thank you very much for your attention! Questions? Comments? TU Braunschweig, Institute of Finance Prof. Dr. Marc Gürtler 39