Week 6: Risk, Return and the Security Market Line; Lessons from Capital Market History (Ch. 12, 11)
Week 6 explains where the risk premium in the cost of equity comes from. Chapter 12 takes a forward-looking view: expected returns and variances of single assets and portfolios, the split between systematic and unsystematic risk, diversification, beta, the security market line (SML) and the capital asset pricing model (CAPM). Chapter 11 looks backwards at capital market history: how to calculate returns, arithmetic versus geometric averages, variance and standard deviation of historical returns, the (non-)normal distribution of returns, risk premiums and the efficient markets hypothesis.
Expected return of a single asset
Chapter 12 takes a new perspective: the future, not the past. If we know the possible states of the economy, their probabilities and the return in each state, we can compute the expected return as a probability-weighted average.
E(R) = sum over i = 1..N of Prob(State i) x Return i
Example (equity L): recession with probability 0.80 and return -20%; boom with probability 0.20 and return +70%. E(R) = 0.8 x (-0.20) + 0.2 x 0.70 = -0.16 + 0.14 = -2%.
(Book) The projected risk premium is the expected return minus the risk-free rate.
sigma^2 = VAR(R) = sum over i of Prob(State i) x (Return i - E(R))^2; sigma = SD(R) = square root of VAR(R)
Example (equity L): deviations -0.20 - (-0.02) = -0.18 and 0.70 - (-0.02) = 0.72; squared 0.0324 and 0.5184; weighted 0.8 x 0.0324 + 0.2 x 0.5184 = 0.02592 + 0.10368 = 0.1296. SD = square root of 0.1296 = 36%.
Note the difference with the historical variance of Chapter 11 (later this week): with probabilities we weight each squared deviation by its probability; with historical data we divide the sum of squared deviations by n - 1.
Portfolios: expected return, variance and standard deviation
A portfolio is a group of assets (asset classes) such as equities and bonds held by an investor. The portfolio weight is the percentage of the portfolio's total value in a particular asset.
E(RP) = x1 x E(R1) + x2 x E(R2) + ... + xn x E(Rn)
The expected return of a portfolio is the weighted average of the expected returns of its assets (equivalently: compute the portfolio return in each state and take the probability-weighted average).
Example: half of the portfolio in A and the rest equally in B and C. Boom (probability 0.40): returns A 10%, B 15%, C 20%, so the portfolio returns 0.5 x 10 + 0.25 x 15 + 0.25 x 20 = 13.75%. Bust (probability 0.60): A 8%, B 4%, C 0%, portfolio 0.5 x 8 + 0.25 x 4 + 0 = 5%. E(RP) = 0.40 x 13.75% + 0.60 x 5% = 8.5%.
VAR(RP) = sum over states of Prob(state) x (RP,state - E(RP))^2
Example: 0.40 x (0.1375 - 0.085)^2 + 0.60 x (0.05 - 0.085)^2 = 0.00184; SD = 0.00184^(1/2) = 4.3%.
Important: the portfolio variance is generally not the weighted average of the individual variances. Always compute the portfolio return per state first.
Exercise: 25% in A and 75% in B. Boom (probability 70%): A 8%, B 16%; bust (30%): A -2%, B -4%. Portfolio returns: boom 0.25 x 8% + 0.75 x 16% = 14%; bust 0.25 x (-2%) + 0.75 x (-4%) = -3.5%. E(RP) = 0.7 x 14% + 0.3 x (-3.5%) = 8.75%. Deviations 5.25% and -12.25%; variance = 0.7 x 0.002756 + 0.3 x 0.015006 = 0.00643; SD = 8.02%.
Tutorial: boom (0.35): A 7%, B 15%, C 33%; bust (0.65): A 13%, B 3%, C -6%. Portfolio 20% A, 20% B, 60% C: boom 24.2%, bust -0.4%; E(RP) = 8.21%; variance = 0.35 x (0.242 - 0.0821)^2 + 0.65 x (-0.004 - 0.0821)^2 = 0.01377 (1.377%).
Announcements, surprises and expected returns
R = E(R) + U
R = actual total return in the year; E(R) = expected part of the return; U = unexpected part (the surprise). Surprises are, by definition, random events. Only the unexpected part of an announcement moves the price, because the expected part is already in the price. (Book) Announcement = expected part + surprise.
The surprise consists of two parts: R = E(R) + systematic component + unsystematic component. A scatter plot of a stock's return surprise against the market's return surprise shows the systematic component as a clear line (the stock moves with the market) and the unsystematic component as a cloud without relation to the market.
| Systematic risk | Unsystematic risk |
|---|---|
| A risk that influences a large number of assets. Also called market risk. Examples: uncertainty about GDP, interest rates or inflation. | A risk that affects at most a small number of assets. Also called unique, idiosyncratic or asset-specific risk. Because it is unique to the asset, it is unrelated to other assets' unique risk. Examples: a strike, a lawsuit, a new product at one company. |
Diversification
Individual stocks are much more volatile than a broad index: annual standard deviations of Carrefour 41.08%, Fortis 65.55%, Vinci 35.17%, Intesa SanPaolo 31.53%, Saint Gobain 48.80%, Telecom Italia 32.22%, ArcelorMittal 41.07%, Crédit Agricole 57.92% and Adidas 29.47%, each with diversifiable unsystematic risk, against 13.10% for the Euro Stoxx 50 index, which contains mostly non-diversifiable risk.
| Number of equities in portfolio | Average SD of annual portfolio returns (%) |
|---|---|
| 1 | 49.24 |
| 2 | 37.36 |
| 4 | 29.69 |
| 6 | 26.64 |
| 8 | 24.98 |
| 10 | 23.93 |
| 20 | 21.68 |
| 30 | 20.87 |
| 50 | 20.20 |
| 100 | 19.69 |
| 500 | 19.27 |
| 1,000 | 19.21 |
The standard deviation falls quickly with the first few stocks and then levels off at about 19.2%: this floor is the systematic (non-diversifiable, market, portfolio) risk. The part that disappears is the unsystematic (diversifiable, unique, asset-specific, idiosyncratic, firm) risk. Key statement: unsystematic risk is eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk. Total risk = systematic risk + unsystematic risk. (Book) The principle of diversification: spreading an investment across many assets eliminates some, but not all, of the risk.
Beta: measuring systematic risk
Because unsystematic risk can be diversified away at almost no cost, only systematic risk is priced by the market: the expected return on a risky asset depends only on that asset's systematic risk.
beta_i = Cov(Ri, RM) / Var(RM)
Beta measures the covariance of an asset's return with the market return, i.e. the amount of systematic risk in a particular asset relative to that in an average (market) risky asset. The market has beta 1; the risk-free asset has beta 0. Beta is the slope of a regression of the asset's returns on the market returns.
Example: a regression of Apple's returns on S&P 500 returns (2009 to 2011, n = 157) gives y = 0.008223 + 1.0362x with R^2 = 0.56: Apple's beta is 1.036. Tesla's beta on Yahoo Finance (5-year monthly, October 2025) was 2.09. A beta above 1 means the stock tends to amplify market movements; below 1 means it moves less than the market. A stock with a high total risk (standard deviation) can have a low beta if most of its risk is unsystematic, and it will then have a low expected return.
beta_P = x1 x beta1 + x2 x beta2 + ... + xn x betan
Unlike the standard deviation, the beta of a portfolio is simply the weighted average of the betas of its assets.
| Security | Amount invested (€) | Expected return (%) | Beta |
|---|---|---|---|
| Equity A | 1,000 | 8 | 0.80 |
| Equity B | 2,000 | 12 | 0.95 |
| Equity C | 3,000 | 15 | 1.10 |
| Equity D | 4,000 | 18 | 1.40 |
Weights 0.1, 0.2, 0.3, 0.4. Portfolio expected return = 0.1 x 8 + 0.2 x 12 + 0.3 x 15 + 0.4 x 18 = 14.9%. Portfolio beta = 0.1 x 0.80 + 0.2 x 0.95 + 0.3 x 1.10 + 0.4 x 1.40 = 1.16. Tutorial: a portfolio equally invested in a risk-free asset and two mining stocks, Vedanta (beta 2.63) and Eurasian Natural Resources, is as risky as the market (beta 1): 1 = 1/3 x 0 + 1/3 x 2.63 + 1/3 x beta_X, so beta_X = 0.37. Portfolio weights are based on values: 200 shares at £4.50 and 100 shares at £2.70 give a total of £1,170 and weights 0.7692 and 0.2308.
Beta and the risk premium
Consider a portfolio of asset A (expected return 20%, beta 1.6) and a risk-free asset (8%, beta 0). By varying the weight in A we get:
| % of portfolio in asset A | Portfolio expected return (%) | Portfolio beta |
|---|---|---|
| 0 (pure risk-free) | 8 | 0.0 |
| 25 | 11 | 0.4 |
| 50 | 0.5 x 20 + 0.5 x 8 = 14 | 0.5 x 1.6 + 0.5 x 0 = 0.8 |
| 75 | 17 | 1.2 |
| 100 (asset A only) | 20 | 1.6 |
| 125 | 23 | 2.0 |
| 150 | 1.5 x 20 + (1 - 1.5) x 8 = 26 | 1.5 x 1.6 = 2.4 |
Weights above 100% mean borrowing at the risk-free rate to invest more than 100% in A (portfolio risk and return with borrowing). All combinations lie on a straight line from Rf with the same slope.
Reward-to-risk ratio = (E(RA) - Rf) / beta_A
The risk premium per unit of systematic risk: 'rise over run' of the line. Asset A: (20% - 8%)/1.6 = 7.5%. Asset B with beta 0.5 and E(RB) = 11.75%: (11.75% - 8%)/0.5 = 7.5%.
The security market line (SML)
(E(RM) - Rf)/beta_M = E(RM) - Rf, because beta_M = 1
The line through Rf and the market portfolio (beta 1) is the security market line; its slope equals the market risk premium. Example: with E(RM) = 15.5% and Rf = 8%, the slope is (15.5% - 8%)/1 = 7.5%.
(E(RA) - Rf)/beta_A = (E(RB) - Rf)/beta_B = E(RM) - Rf
In equilibrium the reward-to-risk ratio must be the same for all assets in the market, so all correctly priced assets plot on the SML.
What if assets are out of equilibrium? Security C plots above the SML: its expected return is too high for its beta. Using the perpetuity logic P0 = D1/R, a too-high R means the price P0 is too low: C is underpriced, so buy. Buying pushes the price up and the expected return down until C lies on the SML. Security D plots below the SML: its expected return is too low, so its price is too high: D is overpriced, so (short) sell. These trades restore equilibrium.
CML and SML are not the same. The capital market line plots expected return against total risk (standard deviation) and applies to efficient portfolios: combinations of the risk-free asset and the market portfolio (the efficient portfolio); individual stocks lie below it because they carry unsystematic risk. The security market line plots expected return against beta (systematic risk) and applies to all assets and portfolios: in equilibrium every correctly priced asset lies on it.
Quiz: with a market risk premium of 5% and a risk-free rate of 10%, an asset with an expected return of 80% would have beta = (80% - 10%)/5% = 14 (none of the listed options 0.7, 1.4 or 3.6); an expected return of 17% would correspond to beta 1.4.
The capital asset pricing model (CAPM)
E(Ri) = Rf + beta_i x [E(RM) - Rf]
Derived from the equilibrium condition (E(Ri) - Rf)/beta_i = E(RM) - Rf: the return per unit of risk of asset i equals the market risk premium (the slope of the SML).
Three components: (1) the pure time value of money, measured by Rf, the reward for merely waiting without taking risk; (2) the reward for bearing systematic risk, measured by the market risk premium E(RM) - Rf, the reward the market offers for an average amount of systematic risk in addition to waiting; (3) the amount of systematic risk, measured by beta_i, relative to an average asset.
(Book) The SML and the cost of capital: the SML gives the required return (the cost of capital) for an investment with a given beta. A project with an expected return above the SML has a positive NPV. This links directly to the SML approach for the cost of equity (week 5).
Tutorial (Pete plc and Repete plc): Pete has beta 1.35 and expected return 13.2%; Repete beta 0.80 and 10.1%; both are correctly priced, so they have the same reward-to-risk ratio: (0.132 - Rf)/1.35 = (0.101 - Rf)/0.8. Then 0.1056 - 0.8Rf = 0.13635 - 1.35Rf, so 0.55Rf = 0.0308 and Rf = 5.59%. The market return follows from 0.132 = 0.0559 + 1.35(RM - 0.0559): RM = 11.23%. Tutorial (target return): £100,000 in X (15.2%, beta 1.4) and Y (11.6%, beta 0.95) with a target of 14.5%: 0.145 = 0.152wX + 0.116(1 - wX), so wX = 0.81 and you invest 0.19 x 100,000 = £19,000 in Y. Tutorial (systematic risk): Equity I (recession 0.25: 11%, normal 0.50: 29%, irrational exuberance 0.25: 13%) has E(R) = 20.5%; Equity II (-40%, 10%, 56%) has E(R) = 9%. With Rf = 4% and a market risk premium of 8%: beta_I = (0.205 - 0.04)/0.08 = 2.06 and beta_II = (0.09 - 0.04)/0.08 = 0.63. Equity I has more systematic risk, although Equity II has the far larger standard deviation. A portfolio investor is compensated only for systematic risk.
Lessons from capital market history: returns
Dividend yield = D(t+1)/P(t); capital gains yield = (P(t+1) - P(t))/P(t); total return = dividend yield + capital gains yield
Example: price €25 at the start and €35 at the end, dividend €5: capital gain 10/25 = 40%; dividend yield 5/25 = 20%; total return 60% (= (10 + 5)/25).
Tutorial: an equity with year-end prices £60.18, 73.66, 94.18, 89.35, 78.49, 95.05 and dividends 0.60, 0.64, 0.72, 0.80, 1.20 has annual returns (P1 - P0 + D1)/P0 of 23.40%, 28.73%, -4.36%, -11.26% and 22.63%.
Average returns: arithmetic versus geometric
Stock markets of eight European countries (2001 to 2010) show large swings. For the Netherlands the annual returns 2001 to 2009 were -21.75, -41.20, 20.07, 1.98, 25.00, 10.89, -11.62, -43.76 and 31.85%. The arithmetic mean is -28.54/9 = -3.17%, but the index lost almost 50% over those 9 years. Is -3.17% correct? It is the return in an average year, not the average growth rate of your money.
R-bar = (R1 + R2 + ... + RT)/T
The return earned in an average year over a multiyear period.
Geometric average = [(1 + R1) x (1 + R2) x ... x (1 + RT)]^(1/T) - 1
The average compound return earned per year over a multiyear period: the average annual growth rate of your investment.
Question: a share costs €100 (2021), €20 (2022) and €100 (2023). Returns: -80% and +400%. Arithmetic average = (-0.8 + 4)/2 = 160%: would you buy? The geometric average is ((1 - 0.8) x (1 + 4))^(1/2) - 1 = 0%, because 100 x 0.2 x 5 = 100: you ended where you started.
Gains and losses are not symmetric ('falling into a hole is easier than climbing out', Siegel's paradox): the return needed to recover from a loss of L is L/(1 - L): -10% needs +11%, -20% needs +25%, -30% needs +43%, -40% needs +67%, -50% needs +100%, -60% needs +150%, -70% needs +233%, -80% needs +400%, -90% needs +900% and -100% can never be recovered.
Example (Danish returns): 38.80%, -24.21%, -29.28%, 40.32%, 11.49%. Product of (1 + R) = 1.3880 x 0.7579 x 0.7072 x 1.4032 x 1.1149 = 1.1640, which is what an investment of 1 is worth after five years. Geometric return = 1.1640^(1/5) - 1 = 3.08%; arithmetic = 37.12/5 = 7.42%.
Average inflation of beer at the Oktoberfest: one litre cost €1.51 in 1971 and €15.80 in 2025: (15.80/1.51)^(1/54) - 1 = 4.44% per year. Tutorial: the equity above has an arithmetic average of 11.83% and a geometric average of 10.58%. Bitcoin returns 303.16% (2020), 59.67%, -64.27%, 155.42% and 121.05% (2024) give a product of 12.986: €1,000 grows to €12,986 over the five years and the geometric average is 12.986^(1/5) - 1 = 66.99% per year, far below the arithmetic average of 115%.
For US stocks (1927 to 2007) the geometric average is always at most the arithmetic average: large company equities 10.4% versus 12.3%, small companies 12.5% versus 17.1%, long-term corporate bonds 5.9% versus 6.2%, long-term government bonds 5.5% versus 5.8%, T-bills 3.7% versus 3.8%, inflation 3.0% versus 3.1%. The gap is larger the more volatile the returns. We normally only have estimates, and estimates have errors: the arithmetic mean is probably too high for long forecasting horizons and the geometric mean probably too low for short horizons.
R(T) = (T - 1)/(N - 1) x geometric average + (N - T)/(N - 1) x arithmetic average
N = number of years of historical data; T = forecasting horizon in years. For T = 1 it gives the arithmetic average; for T = N the geometric average.
Example: N = 25, arithmetic 12%, geometric 9%. R(1) = 0/24 x 9% + 24/24 x 12% = 12%; R(5) = 4/24 x 9% + 20/24 x 12% = 11.5%; R(10) = 9/24 x 9% + 15/24 x 12% = 10.875%.
The variability of returns
VAR(R) = sum over i of (Ri - R-bar)^2 / (n - 1); SD(R) = sigma = square root of VAR(R)
The variance is the average squared difference between the actual return and the average return. Because historical data are a sample, we divide by n - 1 to get an unbiased estimate of the true population variance.
Example: returns 10%, 12%, 3% and -9%; average 16%/4 = 4%. Deviations 0.06, 0.08, -0.01, -0.13 (summing to 0); squared 0.0036, 0.0064, 0.0001, 0.0169, total 0.0270. VAR = 0.027/(4 - 1) = 0.009; SD = square root of 0.009 = 9.5%. AEX 2018 to 2021: returns -10.4%, 23.9%, 3.0%, 28.0%; average 11.1%; squared deviations 0.04633, 0.01632, 0.00660, 0.02848 (sum 0.09773); VAR = 0.09773/3 = 0.03258 (3.258%); SD = 18.05%. Tutorial: X returns 8, 21, 17, -16, 9% (mean 7.8%) and Y 16, 38, 14, -21, 26% (mean 14.6%): VAR(X) = 0.02067 (2.07%) and VAR(Y) = 0.04868 (4.87%).
Distribution of stock returns
A frequency distribution of annual returns on large US stocks roughly resembles a bell shape. Under a normal distribution, about 68% of outcomes lie within one standard deviation of the mean, 95% within two and 99% within three. For US large-company equities (mean 12.3%, SD 20%): 68% of annual returns between -7.7% and 32.3%, 95% between -27.7% and 52.3%, 99% between -47.7% and 72.3%. Tutorial: Walter White Inc. has normally distributed returns with mean 13% and variance 3.24% (SD 18%); the probability of a return between -5% and +31% (mean plus or minus one SD) is about 68%.
Black swans and fat tails. On Black Monday, 19 October 1987, the Dow Jones fell 23% in six hours, Wall Street's worst day ever. Under a normal distribution this would be a '27 standard deviation event' that should happen once every 10^160 days (20 billion times the age of the universe). The Flash Crash of 6 May 2010 and the collapse of the pound after the UK referendum on leaving the EU show that extreme events are not limited to a single day. In financial markets the probability of outsized events is much higher than a normal distribution predicts (fat tails): the normal distribution is only an approximation, maybe a very bad one. Stocks of small companies are even more volatile, with a wider distribution of returns.
Key insights, also for your own investments: the average real return on stock investments can be expected to be around 6% to 8%; small caps usually yield higher average returns, due to higher risk; risk is usually measured by volatility (standard deviation of returns), about 20% for single stocks; this assumes a normal distribution, but in reality we occasionally see black swans.
Risk premium
Risk premium = return on a risky asset - return on the risk-free asset
Government Treasury bills are used as the risk-free asset. The higher the risk premium, the riskier the investment: it is compensation for risk.
| Investment (US) | Average return (%) | Risk premium (%) |
|---|---|---|
| Large companies | 12.3 | 8.5 |
| Small companies | 17.1 | 13.3 |
| Long-term corporate bonds | 6.2 | 2.4 |
| Long-term government bonds | 5.8 | 2.0 |
| US Treasury bills (risk-free) | 3.8 | 0.0 |
Why don't we all just invest in US small companies? Because their higher average return comes with much higher variability (standard deviation about 31.5% versus 19.8% for large companies over 1926 to 2019): the first lesson of capital market history is that risky assets earn a risk premium; the second is that greater potential reward comes with greater risk.
Market efficiency
An efficient (capital) market is one in which asset prices fully reflect available information. The efficient markets hypothesis (EMH) states that actual capital markets (such as the London Stock Exchange or Euronext) are efficient.
Implications of the EMH: (1) because information is reflected in prices immediately, investors should only expect to obtain an average (normal) rate of return; (2) awareness of information when it is released does an investor no good; (3) the price adjusts before the investor has time to trade on it. (Book) In an efficient market all investments are zero-NPV investments: investors get exactly what they pay for, and firms receive exactly what their securities are worth. Efficiency does not mean prices never move; they move because new information arrives.
| Form | Definition | Example |
|---|---|---|
| Weak form | Prices incorporate all historical information (past prices); future prices cannot be predicted from past prices. | Nothing of value is gained by analysing past stock price changes. Renders technical analysis useless. |
| Semi-strong form | Prices incorporate all publicly available information (historical and current); no delayed response to disclosures. | Information in an SEC filing is incorporated into the price as soon as the filing is made public; fundamental analysis of public data does not beat the market. |
| Strong form | Prices incorporate all information, private as well as public; prices react as soon as new information is generated. | The price reacts to a dividend increase as soon as the board votes, before the public announcement. Even insiders cannot profit. |
Each stronger form includes the weaker ones. Under strong-form efficiency even inside information cannot be used to earn excess returns (Tutorial: the friend with inside information could not profit). No free lunches, because arbitrage is lightning fast: algorithms and high-speed trading exploit mispricings within fractions of a second.
Three pillars (economic forces) of market efficiency:
- Rationality: investors use information rationally and do not systematically over- or undervalue assets; if every investor always acted perfectly rationally, earning an excess return would be very difficult.
- Independent deviations from rationality: even if many investors are irrational, their actions may cancel out if they have different beliefs; irrationality is then just noise that is diversified away.
- Unlimited arbitrage: if collective irrationality does not balance out, well-capitalised, intelligent and rational arbitrageurs who dominate irrational investors will still make the market efficient.
Therefore it is very likely that markets are efficient, and the EMH does not require every investor to be rational (Tutorial: 'wrong'). Behavioural limitations: biases (over-optimism, overconfidence, confirmation bias, representativeness bias, herding), framing effects (loss aversion, the endowment effect, money illusion), heuristics and emotions (Kahneman's 'thinking fast and slow') can cause systematic deviations that arbitrage does not fully remove (behavioural finance).
Can you systematically beat the market? The Wall Street Journal Investment Dartboard contest (1990 to 2002) compared picks of prominent analysts with stocks chosen by throwing darts and with the Dow Jones. The pros won more often (90 to 57 of 147 contests in the book's count), but they probably chose riskier stocks and their public picks boosted prices themselves. The overall message of the lecture: on average it is very hard to beat the market systematically after costs (mandatory podcast: 'Brilliant vs. boring').
Andere hoofdstukken van Corporate Finance
- Week 1: Corporate Finance, Financial Statements and Time Value of Money (Ch. 1, 3, 4)
- Week 2: Corporate Governance and Discounted Cash Flow Valuation (Ch. 2, 5)
- Week 3: Bond and Equity Valuation (Ch. 6, 7)
- Week 4: Investment Criteria, Capital Investment Decisions and Project Analysis (Ch. 8, 9, 10.3 to 10.5)
- Week 5: Cost of Capital and Raising Capital (Ch. 13, 14)
- Week 7: Financial Risk Management and Lessons on Investing (Ch. 20)