The Equity Premium. Bernt Arne Ødegaard. 20 September 2018

Transcription

1 The Equity Premium Bernt Arne Ødegaard 20 September Intro This lecture is concerned with the Equity Premium: How much more return an investor requires to hold a risky security (such as a stock) relative to a risk free investment. We will in particular be concerned with the premium of the whole market, the amount of extra return an investor needs to hold a diversified equity portfolio of all available stocks listed in a given market. 2 Estimating the equity market risk premium When we want to calculate the required return of an investment, typically think in terms of starting with the risk free rate, and adding a premium reflecting the risk of the invesment. For example, in the CAPM we estimate E[r i ] = r f + β i (E[r m ] r f Here the risk premium is proportional to the expectation of the market risk premium: er mt = r mt r ft The risk adjustment is made by scaling up by the beta (β i ) of the investment. We thus need an estimate of the expected market risk premium. There are three different approaches to doing such an estimation. Survey evidence Implied expected market premium in current prices. The historical experience (historical average) 2.1 Surveying investors Possibilities: What are individual investors expectations of returns? What about market professionals? Concentrate on the latter, market professionals Example from the US: Merril Lynch survey of institutional investors 1

2 Year Expectation about market risk premium % 2008 (mar) 4.1% % 2012 (jan) 4.08% 2013 (jan) 4.8% Graham and Harvey (2013) survey of US CFOs numbers for cost of capital for their firm Year Expectation about market risk premium 2012 (jan) 3.95% 2013 (jan) 3.83% Time series of Graham s and Harvey s survey of US CFOs A worldwide survey: Fernandez, Aguirreamalloa, and Linares (2013) 2

3 Surveys problematic: Tend to respond too much to recent stock price movements. Sensitive to how question is asked. Sensitive to who gets surveyed. Do not find much predictive power 2.2 Implied equity premia Simple example: Value = Expected dividends next period Required return on equity Expected growth rate in dividends Possible to estimate three of the variables in the model Value (current level of market) Expected dividends next period Expected growth rate in dividends 3

4 Given these, then one backs out the implicit Required return on equity. Example (from Damodaran (2013), The current level of the S&P 500 is 900. The expected dividend yield on the index for the next period is 3%. The expected growth rate in earnings and dividends in the long term is 6%. Then, it is a matter of solving for r in 900 = 900 3% r 0.06 r = 9%. If the current risk free rate is 5%, we would estimate the risk premium as mrp = 9% r f = 9% 5% = 4%. The next figure is taken from Damodaran (2013), and show such estimates for the US. 4

5 3 Estimating the historical equity market risk premium One way of estimating the expected market risk premium is to base it on the historical equity risk premium. Want to calculate er mt = r mt r ft r mt is typically the return of a broad stock market index, such as S&P 500. r ft is the return on risk free borrowing, typically estimated as the interest rate on long term government bonds. 3.1 Estimating the historical US equity premium Let us use the data provided by Damodaran on his homepage (see Damodaran (2013)) to illustrate the typical calculation. First, look at the time series of the actual annual returns. Return on stocks, bills and bonds Return S&P500 T Bill T bond Annual observations of US markets Calculate the difference S&P 500 T bonds Return

6 Take the historical average. In this picture we illustrate the evolution of the estimate of the average using a longer and longer time period. S&P 500 T bonds Return Stock Bond Arith mean Geom mean Arithm Geom average average S& P 500 T-bonds S& P 500 T-bills The data set we used here, as provided by Damodaran, starts in Researchers have constructed longer time series of stock returns. In the next picture we illustrate calculations of long term means using US data back to 1871, from Goyal and Welch (2008). 6

7 erm rfree logreturn erm Rf Cumulative mean Rolling mean(20y) In addition to the cumulative mean using all the data from 1871, we include a rolling mean, where we use only the last 20 years of data. So basing the estimate on the recent past, which is what this rolling mean does, is going to introduce much variability in the estimate of the expected market risk premium. 3.2 Going beyond the US One argument for the high equity premium in the US is that we are looking at the economy that ex post has turned out to be the most succesful. After the US civil war ( ) there has been no war on US soil. Most other economies in the world has suffered through periods of war and unrest. Using only US data can then introduce a survivorship bias in the estimation (Brown, Goetzmann, and Ross, 1995). We should therefore look beyond the US. The best known world wide evidence on world-wide equity returns is that provided by Elroy Dimson and coauthors (Dimson, Marsh, and Staunton, 2011). 7

8 4 Insight: The link between period length, observation frequency, and estimator accuracy Can we improve on estimates of the market risk premium by being clever with the data? Intuitively, one may think that one may use the fact that returns are observed very often, so frequent as daily (or even more frequent, with todays high frequency datasets), may provide many observations, which can be used to improve estimates. Unfortunately, this is not the case. It turns out that the only thing that improves estimates of mean returns is getting more data in terms of the total period we observe. Slicing the returns into more and more short periods will not improve the accuracy of mean estimates. It will improve accuracy of variance estimates. This important insight is provided in the appendix of Merton (1980). Let us repeat his argument. Assume that the instantaneous rate of return on the market (including dividends), dm/m, can be represented by the Itô type stochastic differential equation dm(t) M(t) = αdt + σdz(t) (1) Let µ be the expected return and σ 2 the variance of the return. These are both constants over a time interval of lenght I t. Suppose that the realized return on the market can be observed over time intervals of length where << I t. Then n = h/ is the number of observations of realized returns over a time interval of length I t. Let X k denote the logarithmic return on the market over the k th observation interval of length during a typical period of length h for k = 1, 2,, n. Using (??), X k kan be written as X k = µ + σ ε k, k = 1, 2,, n (2) 8

9 where the {ε k }, k = 1,, n, are independent and identically distributed standard normal random variables. From (2), the estimator for the expected logarithmic return ˆµ = ( n 1 X k) /h, will have the properties that E[ˆµ] = µ and var(ˆµ) = σ 2 /h Note that the accuracy of the estimator as measured by var(ˆµ) depend only upon the total length of the observation period h and not upon the number of observations n. That is, nothing is gained in terms of accuracy of the expected return estimate by choosing finer observation intervals for the returns and thereby, increasing the number of observations n for a fixed value of h. Consider the following estimator for the variance rate: n ˆσ 2 = 1 h 1 X 2 k From (2), this estimator will have the properties that and E[ˆσ 2 ] = σ 2 + µ 2 = σ 2 + µ 2 h/n var(ˆσ 2 ) = 2σ 4 /n + rµ 2 h/n 2 Because the estimator for σ 2 was not taken around the sample mean ˆµ, ˆσ 2 is biased. However, for large n, the difference between the sample second central and non-central moments is trivial. More important than the issue of bias is the accuracy of the estimator. var(ˆσ 2 ) does depend on the number of observations n for a fixed h, and indeed, to order 1/n, it depends only upon the number of observations. Thus, unlike the accuracy of the expected return estimator, by choosing finer observation intervals, the accuracy of the variance estimator can be improved for a fixed value of h. 9

10 5 The Norwegian Equity Premium Norway is a case study of the problems in predicting the equity premium. Reliable Norwegian Equity data is primarily post If we base the historical risk premium on the difference between the return on a stock market index and the risk free rate, proxied by NIBOR, we get: (Ødegaard, 2016) Index Period Average Annual Excess Return EW ( ) VW ( ) OBX ( ) 6.29 TOT ( ) Here the indices to use are EW (equally weigted) and VW (value weighted), total return indices. The other two are not dividend adjusted. It is hard to argue for an equity market risk premium of this magnitude. If we look at a survey of Norwegian financial analysts and their estimates of the relevant market premium at the end of 2013, we get the responses shown in figure 1. Figure 1 Survey of financial analysts: Market Risk Premium in Norway 2013/2014. Source: NFF (2013) An alternative estimate for Norway is had by imlicit estimates. For the largest stocks in Norway, estimate the stock required return, and then back out the average market risk premium. See figure 2 10

11 Figure 2 Implicit Risk premium Norway Source: NFF (2013) 11

12 6 Predictability of market risk premium Another input to our thinking about the equity risk premium is whether it is predictable. If we are trying to estimate the expected market risk premium, and it is predictable, the predicability should be used in finding the expection. The academic evidence on predictability is however mixed, to say the least. Lettau, Ludvigson, and Wacther (2008) argue that the risk premium is related to the macroeconomy. Goyal and Welch (2008), on the other hand, argue that for long term prediction of the equity risk premium, it is extremely hard to beat the historical average. 7 Literature Damodaran (2013) References S J Brown, W N Goetzmann, and S A Ross. Survival. Journal of Finance, 50:853 73, Aswath Damodaran. Equity risk premiums (erp): Determinants, estimation and implications the 2013 edition. Working Paper, New York University, March Elroy Dimson, Paul Marsh, and Mike Staunton. Equity premia around the world. Working Paper, London Business School, July Pablo Fernandez, Javier Aguirreamalloa, and Pablo Linares. Market risk premium and risk free rate used for 51 countries in Working Paper, IESE Business School, Amit Goyal and Ivo Welch. A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, 21(4): , John R Graham and Campbell R Harvey. The equity risk premium in Working Paper, Duke University, M Lettau, S C Ludvigson, and J A Wacther. The declinging equity risk premium: What role does macroeconomic risk play? Review of Financial Studies, 21: , Robert C Merton. On estimating the expected return on the market. Journal of Financial Economics, pages , NFF. Risikopremien i det norske markedet 2013 og NFF - Norske Finansanalytikeres Forening og PWC, Bernt Arne Ødegaard. Empirics of the Oslo Stock Exchange: Basic, descriptive, results, Working Paper, University of Stavanger, January