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1 Expected and Required returns: very different concepts Pablo Fernandez. Professor of Finance. IESE Business School, Isabel F. Acín. Independent researcher. University of Navarra. Camino del Cerro del Aguila Madrid, Spain xlippjljamlsj Previous version: October 17, 2017 The expected return is different from the required return. We explain it for an investment project, for the valuation of a share of Coca-Cola and for the market risk premium. However, there are valuations that assume that the expected return is equal to the required return. Similarly, Expected Equity Premium (EEP) and Required Equity Premium (REP) are two very different concepts, although many books and financial literature do not distinguishing them. Both, the REP and the EEP differ for different investors. The topic of this short paper (5 pages) is thinking about valuation : it is important to understand what we are doing. 1. An investment project 2. Shares of Coca-Cola 3. Market risk premium (MRP) 4. Little common sense in many valuations 5. Conclusion Questions Appendix 1. Required return on Coca-Cola's common stock according to a financial analyst Appendix 2. Another valuation of the shares of Coca-Cola. March 2015 Appendix 3. Share price of Coca-Cola ($/share). October 2014-April Appendix 4. Some comments of readers of previous drafts 1. An investment project Table 1 contains investment project A. The equity investment that is required today (year 0) is $100 million. The expected cash flows for the next 3 years are $30, $40 and $50 million. The internal rate of return (IRR) of these expected cash flows is 8.9%. Obviously, 8.9% is the expected return of project A. The company requires a 10% return for project A (and for all projects with risks similar to those of project A). The Net present value (NPV) of the expected cash flows of project A using the 10% (the required return) is - $2.1 million. The NPV is negative because the required return is higher than the expected return. Table 1. Investment project A Year IRR Expected Equity Cash Flows % Expected return Net present = million $ It is obvious that the expected return is different than the required return. And it is also obvious that different persons may have different expected returns (different expected cash flows) and different required returns for investment project A. Ch 40-1

2 2. Shares of Coca-Cola Table 2 contains the valuation of a share of Coca-Cola's common stock according to a financial analyst. The expected Equity Cash Flows of the analyst are in line 1. The required return to equity of the analyst is 7.76%. Lines 2 and 3 show how the analyst values each share of Coca-Cola at $ The current price of the share is $ The internal rate of return (IRR) of the current price ($40.08) and the expected cash flows of line 1 is 9.88%. Obviously, 9.88% is the expected return of buying a share of Coca-Cola according to the analyst (March 2015). The analyst gets a value ($51.56 / share), higher than the current market price ($40.08 / share) because his required return (7.76%) is smaller than his expected return (9.88%) Table 2. Value of Coca-Cola's common stock. Source: A financial analyst. March 2015 line Year (t) Terminal value 5 1 Expected Equity Cash Flow = / (7.76% 0.23%) 2 Present value (t = 0) at 7.76% = $ billion The company has 4,366 million shares 3 Value of each share = $225,130 million / 4,366 million shares = $51.56 / share 4 Current share Price = $40.08 / share It is obvious that the expected return (9.88%) is different than the required return (7.76%). And it is also obvious that different persons may have different expected returns (different expected cash flows) and different required returns for the shares of Coca-Cola. Appendix 2 contains the valuation of another analyst with different expected equity cash flows, different required return (8.5%), different value per share ($40.05) and different expected return. 3. Market risk premium (MRP) The equity premium (also called market risk premium, equity risk premium, market premium and risk premium) is an important, but elusive, parameter in finance. Some confusion arises from the fact that the term equity premium is used to designate different concepts 1. Two of these are: 1. Expected Equity Premium (EEP): expected differential return of the stock market over treasuries. 2. Required Equity Premium (REP): incremental return of the market portfolio over the risk-free rate required by an investor in order to hold the market portfolio 2. Many people use it for calculating the required return to equity (Ke). The two concepts are different. The REP and the EEP are different for each investor and are not observable magnitudes. The Expected Equity Premium (EEP) is the answer to a question we would all like to answer accurately in the short term, namely: what incremental return do I expect from the market portfolio over the risk-free rate over the next years? Numerous papers and books assert that there must be an EEP common to all investors (to the representative investor). Without homogeneous expectations there is not one EEP (but several), and there is not one REP (but several). The CAPM is a useless attempt 3 to calculate the EEP. However, many books, analysts use it to calculate the REP. However, some authors assume that 1) EEP = REP [for example, Brealey and Myers (1996); Copeland et al (1995); Ross et al (2005); Stowe et al (2002); Pratt (2002); Bodie et al (2003); Damodaran (2006); Goyal and Welch (2007); Ibbotson Ass. (2006)]; and that 2) There is a unique EEP [for example, Damodaran (2001a); Arzac (2005); Jagannathan et al (2000); Harris and Marston (2001); Claus and Thomas (2001); Fama and French (2002); Goedhart et al (2002); Harris et al (2003)]. 1 See, for example, Equity premium: historical, expected, required and implied 2 Or the extra return that the overall stock market must provide over the Government Bonds to compensate the investor for the extra risk. 3 It is explained and documented in CAPM: an absurd model Ch 40-2

3 We very much agree with Klarman (1991): It is necessary to understand the rationale behind the formulae in order to appreciate why they work when they do and don t when they don t. 4. Little common sense in many valuations It is obvious that the expected return for a share equals its required return only if the value obtained in the valuation is equal to the current market price. However, we can see in many valuations the following sequence: 1) Expected cash flows of the valuator 2) Calculation of the Expected return of the shares, many times using the CAPM or similar recipes 3) Valuation of the shares using as discount rate the expected return and getting a value different than the current market price. Of course there is a great inconsistency: the use of the expected return would be consistent with the valuation if, and only if, the value obtained was equal to the current market price (because only in that specific situation, the expected return is equal to the required return). There are 2 common schizophrenic approaches to valuation: 1) To admit heterogeneous expectations about expected cash flows, but assuming homogeneous expectations about required rates of return. Also in textbooks: while many authors admit different expectations of equity cash flows, most authors look for a unique discount rate. 4 2) To use an expected return for discounting the expected cash flows, getting a value that is different than the current market price and assume that the expected return used was a required return. 5. Conclusion An anecdote from Merton Miller (2000, page 3) about the expected return in the Nobel context: I still remember the teasing we financial economists, Harry Markowitz, William Sharpe, and I, had to put up with from the physicists and chemists in Stockholm when we conceded that the basic unit of our research, the expected rate of return, was not actually observable. I tried to tease back by reminding them of their neutrino a particle with no mass whose presence was inferred only as a missing residual from the interactions of other particles. But that was eight years ago. In the meantime, the neutrino has been detected. Two questions for the reader. If you find a formula for expected returns that works well in the real markets, 1) would you publish it? 2) Before or after becoming a billionaire? We can find out an investor s REP by asking him, although for many investors the REP is not an explicit parameter but, rather, an implicit one that manifests in the price they are prepared to pay for shares 5. For having an EEP common for all investors we need to assume homogeneous expectations (or a representative investor) and, with our knowledge of financial markets, this assumption is not reasonable. A theory with a representative investor cannot explain either why the annual trading volume of most exchanges more than double the market capitalization. References Arzac, E. R. (2005), Valuation for Mergers, Buyouts, and Restructuring, John Wiley & Sons Inc. Bodie, Z., A. Kane, and A. J. Marcus (2003), Essentials of Investments, 5 th edition. NY: McGraw Hill. Brealey, R.A. and S.C. Myers (2003), Principles of Corporate Finance, 7 th edition, New York: McGraw-Hill. Previous editions: 1981, 1984, 1988, 1991, 1996 and Claus, J.J. and J.K. Thomas (2001), Equity Premia as Low as Three Percent? Evidence from Analysts Earnings Forecasts for Domestic and International Stock Markets, Journal of Finance. 55, (5): Copeland, T. E., T. Koller, and J. Murrin (2000), Valuation: Measuring and Managing the Value of Companies. 3 rd edition. New York: Wiley. Previous editions: 1990 and Damodaran, A. (2001), The Dark Side of Valuation, New York: Prentice-Hall. 4 It seems as if the expectations of equity cash flows are formed in a democratic regime, while the discount rate is determined in a dictatorship. 5 An example: An investor is prepared to pay 80$ for a perpetual annual cash flow of 6$ in year 1 and growing at an annual rate of 3%, which he expects to obtain from a diversified equity portfolio. This means that his required market return is 10.5% ([6/80] ). Ch 40-3

4 Damodaran, A. (2006), Damodaran on Valuation, 2 nd edition. New York: John Wiley and Sons. 1 st edition: (1994). Fama, E.F. and K.R. French (2002), The Equity Risk Premium, Journal of Finance 57 no. 2: Goedhart, M., T. Koller and D. Wessels (2002), The Real Cost of Equity, McKinsey & Company, N.5, Autumn: Goyal, A. and I. Welch (2007), A Comprehensive Look at the Empirical Performance of Equity Premium Prediction, Review of Financial Studies, Harris, R.S. and F.C. Marston (2001), The Market Risk Premium: Expectational Estimates Using Analysts Forecasts, Journal of Applied Finance, Vol. 11. Harris, R.S., F.C. Marston, D.R. Mishra and T.J. O'Brien (2003), Ex Ante Cost of Equity Estimates of S&P 500 Firms: The Choice Between Global and Domestic CAPM, Financial Management, Vol. 32, No. 3, Autumn. Ibbotson Associates (2006), Stocks, Bonds, Bills, and Inflation, Valuation Edition, 2006 Yearbook. Jagannathan, R., E.R. McGrattan, and A.D. Shcherbina (2000), The Declining U.S. Equity Premium, Federal Reserve Bank of Minneapolis Quarterly Review 24: Klarman, Seth A. (1991), Margin of Safety, Harper Business Miller, M.H. (2000), The History of Finance: An Eyewitness Account, Journal of Applied Corporate Finance (13), 2: Pratt, S.P. (2002), Cost of Capital: Estimation and Applications, 2 nd edition, Wiley. Ross, S. A., R. W. Westerfield and J. F. Jaffe (2005), Corporate Finance, 7 th edition, Homewood, IL: McGraw-Hill/Irwin. Previous editions: 1998, 1993, 1996, 1999 and Stowe, J.D., T.R. Robinson, J.E. Pinto and D.W. McLeavey (2002), Analysis of Equity investments: Valuation. AIMR (Association for Investment Management and Research). Questions 1. Is the expected equity premium equal for all investors? 2. The Expected Equity Premium and the Required Equity Premium, should be equal? 3. Which equity premium appears in the CAPM? Which return does the CAPM calculate? 4. How do you calculate the Required Equity Premium? 5. Do you calculate the Expected Equity Premium? How? For what purpose? 6. What are the differences among equity premium, market risk premium, equity risk premium, market premium and risk premium? 7. Please comment: The discount rate is the expected return offered by other equally risky stocks. Source: Brealey & Myers (2nd ed., page 131). What happens when the expected return is negative or very small? Please define and differentiate: Expected Return; Required Return; Historical Equity Premium Expected Equity Premium (EEP); Required Equity Premium (REP) Appendix 1. Required return on Coca-Cola's common stock according to the financial analyst author of table 2 Required return on Coca-Cola's common stock = RKO = RF + βko [E(RM) RF] = = 2.36% [14.00% 2.36%] = 7.76% E(RM) = Expected return on market portfolio = Market portfolio dividend growth rate + Market portfolio dividend yield = 14% Market portfolio dividend growth rate = Retention rate Profit margin Asset turnover Financial leverage = = % = 11.47% Market portfolio dividend yield = Next year expected market portfolio dividends / Current market portfolio price = 2.54% Appendix 2. Another valuation of the shares of Coca-Cola. Source: Another analyst. March 2015 Year (t) value 5 Expected dividend / share 1,29 1,42 1,56 1,69 1,83 51,12 = 1,83 1,0475 / (8.5% 4,75%) Present value (t = 0) at 8.5% = 40,05 $ / share Ch 40-4

5 Appendix 3. Share price of Coca-Cola ($/share). October 2014-April 2015, Source: BigCharts.com Appendix 4. Some comments of readers of previous drafts Q) 1) would you publish it? 2) Before or after becoming a billionaire? Answers. 1) I would publish it 2) Before becoming a billionaire, because a) I cannot hold on to good news for fear of who d be gone before hearing it, and b) My REP (implicit) is different: The value of information horded is inversely related to the passage of time. I do not agree that they cannot be estimated. There is a market oriented expected rate of return in many of the most active securities markets in the developed world. In the US there are many forecasts for the S&P 500: allowing for the law of large numbers the consensus of such forecasts is a realistic figure for what the market participants anticipate. This figure is a reasonable indication of the expected Equity Risk Premiums. Required returns exist for most institutional investors such as pension plans or life insurers as they sell products giving guaranteed future incomes (pensions or annuities). I think that "required return" is an oxymoron. One might reasonably talk about "required expected return" as the minimum expected return that is required before initiating a project or investment. But we cannot require returns in a world where the future is uncertain. In equilibrium they (the two returns) must be the same if expectations are the same. The expected and required returns are identical when determining fair market value (as opposed to investment value to a specific investor). The IRR implied by the expected cash flows is not an 'expected return'. Expected return is the average of future possible returns. A project's IRR is a compound rate of return and is not an arithmetic average of the future possible returns that the project could produce. It was my assumption that if one expected a higher return than required for an investment, that investor would continue to buy more shares until he bid up the price such that his expected return equaled his required return for that particular investment. I didn't understand the motivation for your distinguishing of the two sets of concepts, We have rational expectations models in Macroeconomics (Lucas helicopter drop model or Sargent and Wallace's Natural Rate of unemployment model for e.g.) which are all applications of the Rational Expectations model originally due to, John Muth and then Radner, Lucas and further Grossman & Stiglitz which model rational expectations and show how one can calculate equilibrium prices every time the policy maker wants to do so. If you stick to those approaches you can get a good idea of how to relate expected values with equilibrium values. Analysts chase the numbers to the second-plus decimal point without considering the practical usefulness of what they are doing and whether their conclusions can be used to manage real-world portfolios. The assumptions about long-run Beta, Risk Free rate, PV versus IRR, etc. are so fraught with unknowns that for the analyst s answers to be reasonably accurate it requires both luck and compensating errors. I love this paper! The actual clearing price is based on heterogeneous investors with different expectations of future cash flow, different liquidity preferences, risk aversion etc etc. Consequently, the competitive market clearing price of a small private business (thin) is best described at a price just above the second highest reservation price among the pool of potential buyers. Current BV literature assumes homogeneity. So many things go wrong with this assumption. An elegant set of what Brad Delong calls "finger exercises" that illuminate your point with precise clarity! Common sense is very uncommon in American business schools and it's refreshing to see that at least in Spain, it's making a comeback. Ch 40-5