Risk-Return Analysis

Abstract

Risk-return analysis is central to financial decision-making. The basic idea is that risk-averse investors ask compensation for higher risk, in the form of a risk premium on risky assets. The key insight of portfolio theory is that a company’s risk, at least as measured by the distribution of its historical stock returns, can be split into systematic or market-wide risk and idiosyncratic risk. As idiosyncratic risk can be diversified away in a portfolio, investors are only rewarded with a risk premium for the market risk component. This is the beta of the Capital Asset Pricing Model. But historical risk-return analysis has limitations in accurately assessing current and future financial risk. We therefore explore forward-looking measures of financial risk and return. Moreover, we expand the single-factor market model to a multifactor model by adding social and environmental factors.

Yet, another step is to assess social and environmental risk in their own right, as well as their impact on integrated risk. This, in turn, allows us to estimate the cost of integrated capital, which should give corporate managers the tools to make that assessment in their investment decisions. Company examples show that integrated risk-return analysis leads to different, and more sustainable, decisions.

FormalPara Overview

Risk-return analysis is central to financial decision-making. The basic idea is that risk-averse investors ask compensation for higher risk, in the form of a risk premium on risky assets. The chapter starts with an historical overview of risk and realised return over the last century. This overview highlights the risk of the downturn of the business cycle, including major downturns like the Great Depression of the 1930s and the Global Financial Crisis of 2008/2009, putting stock prices downwards.

The key insight of portfolio theory is that a company’s risk, at least as measured by the distribution of its historical stock returns, can be split into systematic or market-wide risk and idiosyncratic risk. As idiosyncratic risk can be diversified away in a portfolio, investors are only rewarded with a risk premium for the market risk component. The Capital Asset Pricing Model, which is commonly used, states that in equilibrium all investors hold a combination of the risk-free asset, such as government bonds, and the market portfolio; and that expected returns only contain a risk premium for market risk as measured by beta.

But historical risk-return analysis has limitations in accurately assessing current and future financial risk. So, this chapter also explores forward-looking measures of financial risk and return. It is important to include the social and environmental risks as well. We expand the single market model to a multifactor model by adding social and environmental factors. This allows us to derive the influence of social risk and environmental risk on financial risk.

Yet, another step is to assess social and environmental risk in their own right, as well as their impact on integrated risk. This, in turn, allows us to estimate the cost of integrated capital, which is the subject of Chap. 13. And that should give corporate managers the tools to make that assessment in their investment decisions. Company examples show that integrated risk-return analysis leads to different, and more sustainable, decisions. See Fig. 12.1 for a chapter overview.

Fig. 12.1

Chapter overview

FormalPara Learning Objectives

After you have studied this chapter, you should be able to:

• analyse risk and return profiles for all types of capital

• differentiate risk-return profiles between various types of financial instruments and various types of capital

• apply rules of thumb in assessing risk-return in corporate investment decisions

• evaluate the pros and cons of various measures of risk and return

• analyse risk in both backward-looking and forward-looking ways

1 Historical Financial Risk and Return

Financial return and financial risk are central to financial decision-making. It all boils down to two key questions for investors:

• 1. Financial return: what can you earn on investing in an asset?

• 2. Financial risk: what can you lose on holding an asset?

Investors are assumed to be risk averse, so they ask compensation for higher risk in the form of a risk premium on risky assets. We start our analysis of return and risk with historical realised annual returns. The average annual return $\overline{r}$ is the average realised return for years n = 1 to N:

$$ \mathrm{Average}\ \mathrm{annual}\ \mathrm{return}:\overline{r}=\frac{1}{N}\bullet \left({r}_1+{r}_2+\dots +{r}_N\right)=\frac{1}{N}\bullet \sum \limits_{n=1}^N{r}_n $$

(12.1)

Table 12.1 documents the global average annual returns from 1870 to 2015 for 16 countries (14 European countries together with the USA and Japan). Analysing the rate of return on a number of asset classes, Jorda et al. (2019) find that the nominal return on equities is about 10.5%, on government bonds 6%, and on Treasury bills 4.5%. Treasury bills are short-term government bonds with a maturity of up to 1 year.

Table 12.1 Global average annual returns from 1870 to 2015 (in 16 countries)

Because of wide differences in inflation across time and countries, it is helpful to compare returns in real terms. Inflation ${i}_{i,t}=\frac{\left(\ {CPI}_{i,t}-{CPI}_{i,t-1}\ \right)}{CPI_{i,t-1}}$ is the realised consumer price index (CPI) inflation rate in a given country i and year t. Inflation-adjusted real returns r^r for an asset class is:

$$ \mathrm{Real}\ \mathrm{return}:{r}^r=\frac{\left(1+r\right)-\left(1+i\right)}{1+i}=\frac{r-i}{1+i}\approx r-i $$

(12.2)

$$ =\mathrm{nominal}\ \mathrm{return}-\mathrm{inflation} $$

Given an average inflation of about 3.5% over the 1870–2015 period, real returns are lower than nominal returns in Table 12.1. The returns are also volatile. While Sect. 12.2 provides more detailed measures of volatility (risk), here we use a broad measure to provide an overview: the decadal moving average is the average return over the last 10 years (decade). It is called a moving average because each year, the current year’s return replaces the latest year’s return. This long-term average provides a straightforward picture of the variation in a time series. The decadal moving average of real returns on equities fluctuates between −4% and + 15% over the 1870–2015 period. The risk premium, an important indicator in asset pricing, is measured as the return on a risky asset class, such as equities, minus the return on a safe asset, such as bills or bonds (see Eq. 12.8 in Sect. 12.2 below). Table 12.1 shows that the historical risk premium relative to bills is about 6% for equities, while the historical risk premium relative to bonds is 4.5%.

The aggregate picture hides significant variation within and between asset classes due to differing risk profiles. Not all equity markets have been equally successful over this long period. Dimson et al. (2021) document real annualised equity returns in local currencies over the 1900–2020 period, ranging from 1% for Austria and 2% for Italy, to over 6% for Australia, South Africa, Sweden, and the USA. They also find significant variation in equity risk premiums (relative to Treasury bills) over that same period, from 3% in Belgium, Spain, and Norway, to 6% in the USA, South Africa, Finland, Australia, Germany, and Japan (see Fig. 12.2).

Fig. 12.2

Equity risk premiums (relative to bills) from 1900 to 2020 (for 21 countries). Source: Data from Dimson et al. (2002, 2021). Note: The risk premium is the return on equities minus the return on bills (the risk-free rate)

While Dimson et al. (2021) report an average world equity risk premium relative to bills of 4.4% and relative to bonds of 3.1%, Jorda et al. (2019) find 6% and 4.5%, respectively, in Table 12.1. Note that the number of countries and time period of analysis are different. We show both sets of figures to highlight differences in reported numbers. When you consult different data sources, it is very common to get slightly different numbers.

We now focus on equity returns. The total annual return on a financial asset r can be divided into two components: the capital gain from the change in the asset price P and a yield component y that reflects the cash-flow return on an investment.

$$ \mathrm{Total}\ \mathrm{return}:{r}_{t+1}=\frac{P_{t+1}-{P}_t}{P_t}+{y}_t $$

(12.3)

$$ =\mathrm{capital}\ \mathrm{gain}+\mathrm{dividend}\ \mathrm{yield} $$

For equities, the yield is the dividend yield (which is calculated as the dividend payment divided by the stock price). Table 12.2 shows that for the total return of equities both the capital gain and dividend income are important. Table 12.2 also shows that more recent returns (from 1950 to 2015) are higher than over the full sample period (1870–2015). A further breakdown of the post-1950 period in Fig. 12.3 shows that real returns on equities and government bonds have been relatively high until the 1990s. Only current returns in the low interest environment from 2015 to 2021 have been very low: 3% for equities and −0.5% for government bonds. The high (until recently) returns on financial capital seem to be at the expense of social and natural capital, which have been reduced (Dasgupta, 2021).

Table 12.2 Equity returns from 1870 to 2015

Fig. 12.3

Real returns over selected time periods (for 21 countries). Source: Data from Dimson et al. (2021)

Example 12.1 shows how the historical return of an individual stock can be calculated using data on stock prices and dividend payments.

Example 12.1 Calculating Historical Stock Returns

Problem

What is the realised annual return for Philips stock in 2020?

Solution

From stock market data, we can take the stock prices and dividends of Philips over 2020. We assume that dividends are reinvested. The table provides these data.

Date	Price	Dividend
31/12/19	43.96
4/6/20	41.97	0.85
31/12/20	43.78

Using Eq. 12.3, we can calculate the historical total return:

$$ {r}_{t+1}=\frac{P_{t+1}-{P}_t}{P_t}+{y}_t=\frac{43.78-43.96}{43.96}+\frac{0.85}{43.96}=1.52\% $$

So, the 2020 return on the Philips stock is 1.52%, which is a combination of a drop in the stock price of −0.41% and a dividend yield of 1.93%.

Historical series show that returns can turn negative during wars and times of crisis. The bill rate, which is a proxy for the risk-free rate, averages about 1% (Table 12.1). The decadal moving average fluctuates from −4% during WWI in the 1910s; to +6% post WWI in the 1920s ahead of the Great Depression of the 1930s; and then again to −5% during WWII in the 1940s (see Fig. 12.4). More recently, the decadal moving average of the bill rate moved up to 3% in the 1980s and subsequently declined to 0% in 2015 (Jorda et al., 2019). This decline was driven by an increase in the premium paid for holding such safe and liquid assets and by lower global economic growth.

Fig. 12.4

The real bill rate from 1870 to 2015. Source: Adapted from Jorda et al. (2019)

2 Traditional Measures of Financial Risk and Return

Asset pricing has several ways to slice and dice financial risks for investors. An investor typically uses standard measures for risk and return.

While assets or projects can be expressed in prices or cash flows, their performance is typically measured as a return r: the percentage increase in the value of an investment per euro invested in the asset. Risky assets have uncertain returns in the future. The probability distribution shows the distribution of returns, by assigning a probability or likelihood p_r to each possible return. Figure 12.5 shows the probability distribution of an asset.

Fig. 12.5

Probability distribution of returns

Given the distribution of returns, we can now calculate the expected or mean return Ε[r] as a weighted average of possible returns, with the probabilities as weights:

$$ \mathrm{Expected}\ \mathrm{return}:E\left[r\right]={\sum}_r\ {p}_r\bullet r $$

(12.4)

$$ =\mathrm{probability}\times \mathrm{each}\ \mathrm{possible}\ \mathrm{return} $$

We illustrate the calculations with a stock that costs €100 today, with possible prices in 1 year of €95 (with 25% probability), €110 (with 50% probability), and €125 (with 25% probability). Table 12.3 shows the set-up. Using Eq. (12.4), the expected return of the stock can be calculated as follows: Ε[r] = 25 %  ∙  − 5 % + 50 %  ∙ 10 %  + 25 %  ∙ 25 %  = 10%.

Table 12.3 Probability distribution of returns for a stock

2.1 Variance and Standard Deviation

The standard statistical measures of the risk probability distribution are variance and standard deviation. ¹ The variance of the return distribution is the expected squared deviation from the mean return in Eq. (12.4):

$$ \mathrm{Variance}: Var\left[r\right]=E\left[{\left(r-E\left[r\right]\right)}^2\right]={\sum}_r\ {p}_r\bullet {\left(r-E\left[r\right]\right)}^2 $$

(12.5)

The standard deviation is simply the square root of the variance:

$$ \mathrm{Standard}\ \mathrm{deviation}: SD\left[r\right]=\sqrt{Var(r)} $$

(12.6)

The variance is typically written as σ² and the standard deviation as σ. The variance shows the spread of the distribution of the returns. On the one extreme, a return is risk-free, when it does not deviate from its mean. By contrast, a distribution of returns with a wide spread has a high variance. The variance of our stock return is: Var[r] = 25 %  ∙ (−5 %  − 10%)² + 50 %  ∙ (10 %  − 10%)² + 25 %  ∙ (25 %  − 10%)² = 0.01125, and the standard deviation is $SD\left[r\right]=\sqrt{Var(r)}=\sqrt{0.01125}=0.106$. As the standard deviation is typically expressed as a percentage, the standard deviation of the stock is 10.6%. The standard deviation can predict future values with a certain confidence level (about 68% for 1 standard deviation). In finance, this standard deviation is also called the volatility of a stock. Table 12.4 reports the standard deviation of the asset classes from Table 12.1. As discussed before, stocks are far more volatile at 22.6% than government bonds and Treasury bills. Treasury bills, which proxy for the risk-free rate, has the lowest volatility at 3.3%.

Table 12.4 Average annual returns and risk from 1870 to 2015 (in 16 countries)

Example 12.2 provides a further illustration of calculating a stock’s expected return and volatility. The advantage of the above risk measures is that they are straightforward to calculate, which explains their popularity in use. However, they do not correspond well with how people experience and interpret risk, and they might not be very representative of future risks, as we will see in Sects. 12.5–12.8 of this chapter.

Example 12.2 Calculating the Expected Return and Volatility

Problem

Suppose stock X is equally likely to have a 20% return or a −10% return. What are the stock’s expected return and volatility?

Solution

We can calculate the expected return by taking the probability-weighted average of possible returns (Eq. 12.4):

$$ E\left[r\right]={\sum}_r{p}_r\bullet r=50\%\bullet 0.20+50\%\bullet -0.10=5.0\% $$

Next, we can calculate the variance (Eq. 12.5):

$$ Var\left[r\right]={\sum}_r{p}_r\bullet {\left(r-E\left[r\right]\right)}^2=0.50\bullet {\left(0.20-0.05\right)}^2+0.50\bullet {\left(-0.10-0.05\right)}^2=0.0225 $$

Finally, the volatility (or standard deviation) is the square root of the variance (Eq. 12.6):

$$ SD\left[r\right]=\sqrt{Var(r)}=\sqrt{0.045}=15\% $$

2.2 Historical Returns and Historical Volatility

As the future probability distribution of returns is not known, investors use historical returns to calculate the expected return and volatility of assets. The historical return is the realised return over a particular time period in the past. The underlying assumption is that the volatility of historical return patterns provides a good indicator or proxy of future risk. However, structural changes at companies and in the wider economy, including sustainability trends, violate this assumption, as we discuss later in this chapter.

In Sect. 12.1, we show how the average return (Eq. 12.1) and total return (Eq. 12.3) of assets for specific time periods can be calculated on the basis of realised annual returns. The variance and volatility of historical returns can also be calculated. Over a timespan of N periods, each return gets an equal weight of $\frac{1}{N}$ which means that ${p}_r=\frac{1}{N}$. But there is one peculiarity. Calculations in Eqs. (12.5) and (12.6) are based on the mean or expected return. As we do not know the mean, we use the best estimate for the mean, which is the average realised return. In doing that, we have to divide by N − 1 rather than N because we lose one piece of independent information (called one degree of freedom) in the estimation. The variance of realised returns then becomes:

$$ \mathrm{Variance}\ \mathrm{of}\ \mathrm{realised}\ \mathrm{returns}: Var\left[r\right]=\frac{1}{N-1}\bullet \sum \limits_{n=1}^N{\left({r}_n-E\left[r\right]\right)}^2 $$

(12.7)

The standard deviation or volatility of realised annual returns is again the square root of the variance of realised returns. We can now calculate the average annual return and volatility of assets classes. Table 12.4 above shows these data on a global level for 16 countries. Table 12.5 shows historical returns and volatility for the USA, the largest market. Berk and DeMarzo (2020) distinguish between the large companies that are part of the S&P 500 index and small stocks. Small stocks or companies are far riskier (with close to 40% volatility) than large companies (with 20% volatility) for several reasons. Small stocks are traded less frequently and are thus less liquid, leading to a larger bid-ask spread. Next, small companies often have less of a track record based on a proven business model. Finally, by their very size, small companies are less able to absorb shocks without defaulting.

Table 12.5 Average annual returns and volatility for the USA, from 1926 to 2017

Example 12.3 calculates the historical return and volatility of an individual stock. The high average annual return at 200% and the very high volatility at 347% in this particular case (Tesla stock) show the importance of diversification. Diversification will moderate the return, but even more so will reduce the volatility, as firm-specific risk is eliminated in a diversified portfolio.

Example 12.3 Calculating the Historical Return and Volatility

Problem

What is the annual historical return on Tesla stock and the volatility from 2017 to 2020? Tesla is a growth company and has not paid any dividends to date. So, all we need to know is the stock price development.

Solution

From stock market data, we can take the stock prices of Tesla over the 2017–2020 period. The table provides these data in the first two columns.

Date	Price	Annual return
31/12/16	42.74
31/12/17	62.27	45.69%
31/12/18	66.77	7.23%
31/12/19	86.08	28.92%
31/12/20	705.67	719.78%

Using Eq. 12.3, we can calculate the historical annual return for 2017:

$$ {r}_{t+1}=\frac{P_{t+1}-{P}_t}{P_t}=\frac{62.27-42.74}{42.74}=45.96\% $$

The annual returns are reported in the third column. The average annual return (Eq. 12.1) is then:

$$ \overline{r}=\frac{1}{N}\bullet \sum \limits_{n=1}^N{r}_n=200.41\% $$

The variance of the Tesla stock is very high. Using Eq. 12.7, the variance is

$$ Var\left[r\right]=\frac{1}{N-1}\bullet \sum \limits_{n=1}^N{\left({r}_n-E\left[r\right]\right)}^2=12.01 $$

Finally, the volatility is the square root of the variance (Eq. 12.6):

$$ SD\left[r\right]=\sqrt{Var(r)}=\sqrt{12.01}=346.61\% $$

So, Tesla is a real growth stock (at least up to 2020) and extremely volatile as well.

Risk-averse investors demand a higher reward—in the form of a higher return—for higher risk. Figure 12.6 suggests that this relationship is more or less linear. As bills are seen as a risk-free asset for investors r_f, the expected return in excess of the bill rate is the risk premium RP that investors receive for holding a risky asset i:

Fig. 12.6

Historical trade-off between risk and return, USA, 1926–2017. Source: Adapted from Berk and DeMarzo (2020)

$$ \mathrm{Risk}\ \mathrm{premium}: RP=E\left[{r}_i\right]-{r}_f $$

(12.8)

$$ =\mathrm{expected}\ \mathrm{asset}\ \mathrm{return}-\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate} $$

Looking at the historical trade-off between risk and return, an interesting question is: over which minimum period would stocks dominate government bonds? In a long time-series of stock and government bond returns, Siegel (2020) finds that US stocks have outperformed US bonds in every 30-year period since 1850, though that neared zero in the Great Depression of the 1930s and again in the Global Financial Crisis of 2008–2009. Over shorter holding periods, the cumulative return on a stock portfolio could occasionally drop below the cumulative return on a bond portfolio. In particular that can happen during financial crises when stock markets collapse (see Fig. 12.7).

Fig. 12.7

Stock market performance and financial crises, 1900–2020

3 Diversification of Financial Risk in Portfolios

The returns of individual companies usually don’t move fully in tandem (or more technically, are not fully correlated). To hedge against risk, an investor may want to diversify its stock holdings in an investment portfolio. The classical example is to hold a portfolio with stock in a company that produces umbrellas and stock in another company that produces ice cream. Whatever the weather in the summer—rain or sunshine—the investor will make a return on one of the stocks. Thus, they limit their potential losses, as well as upside potential. That is the principle behind portfolio diversification.

The risk of fluctuating stock prices can be split into:

• 1. Firm-specific risk that is unique to the company. This risk is called idiosyncratic risk and can be diversified in a portfolio; and

• 2. Market risk that is common to all companies. This risk is called systematic risk and cannot be diversified.

An example of market-wide or common risk is the business cycle. In a downturn, almost all companies will face reduced revenues and thus lower returns. Another source of common risk is changes in the policy rate of central banks. All companies are affected by changes in interest rates.

The question is how many stocks you need for a diversified portfolio. Statman (2004) shows that a well-diversified stock portfolio needs to include just 50–100 stocks to eliminate firm-specific or idiosyncratic variance of stock returns. There are smaller benefits of diversification beyond those 100 stocks, but they are exhausted when the number of stocks surpasses 300 stocks (see Fig. 12.8). Risk management should monitor that the stocks are not overly correlated (reducing their diversification potential) and are spread over sectors and countries. Moreover, diversification gains are mainly driven by a well-balanced allocation over different asset classes, like stocks, bonds, and real estate. Thus, for diversification it is more important to have a portfolio in each asset class (that can be more or less concentrated) than to have a very diversified portfolio (beyond 100–300 securities) in a single asset class (Schoenmaker & Schramade, 2019).

Fig. 12.8

Diminishing benefits of diversification. Source: Adapted from Statman (2004)

3.1 Portfolio Return

We can calculate portfolio return and risk in a more formal way. The expected portfolio return r_p is a weighted combination of individual stock returns r_i, with weights x_i for each stock i:

$$ \mathrm{Expected}\ \mathrm{portfolio}\ \mathrm{return}:E\left[{r}_p\right]={\sum}_i\ {x}_i\bullet E\left[{r}_i\right] $$

(12.9)

3.2 Variance of a Two-Stock Portfolio

Whereas the expected portfolio return requires a simple calculation, the derivation of portfolio risk is more difficult. Only when the stocks are fully correlated (that is, the spread of future outcomes for both stocks move fully in tandem), can we take the average of the individual standard deviations. This case is called perfect correlation: ρ = 1. In practice, most stocks have a correlation of less than 1: 0 ≤ ρ < 1. Stocks can even be negatively correlated: −1 ≤ ρ < 0. That is likely the case for our umbrella and ice cream companies. If it is a rainy summer, the umbrella company has positive returns, while the ice cream company makes a loss. If it is a sunny summer, the profit and loss are reversed between the companies. Box 12.1 explains the role of correlation in stock markets.

Box 12.1 Correlation in Stock Markets

Correlation measures the degree to which two variables move in relationship to each other. In finance, it is often used to measure the co-movement of two stocks or the co-movement of a stock with the market index, such as the S&P 500.

The correlation coefficient ranges between −1 and 1: −1 ≤ ρ ≤ 1. We distinguish three cases:

• 1. Perfect correlation ρ = 1: This is an extreme case. In practice, companies in the same industry with similar characteristics (size, strategy, business model, workforce, etc.) may have a correlation close to 1. For example, the stock price of oil companies may react more or less similarly to news about changes in the oil price.

• 2. Positive correlation 0 ≤ ρ < 1: This is the most common case. Companies’ stock prices may react slightly differently on economy-wide news, but the direction of the movement (up or down) is similar.

• 3. Negative correlation −1 ≤ ρ < 0: Some companies behave counter-cyclically. An outplacement agency has more work in order to help laid-off workers during an economic downturn or recession, when most companies face reduced revenues lowering their stock price.

The portfolio variance is made up of the variance of the individual stocks and the covariance between the individual stocks. The covariance between two stocks σ₁₂ is the product of the correlation coefficient ρ₁₂ and the two standard deviations:

$$ \mathrm{Covariance}:{\sigma}_{12}={\rho}_{12}\bullet {\sigma}_1\bullet {\sigma}_2 $$

(12.10)

The portfolio variance then becomes:

$$ \mathrm{Portfolio}\ \mathrm{variance}: Var\left[{r}_p\right]={x}_1^2\bullet {\sigma}_1^2+{x}_2^2\bullet {\sigma}_2^2+2\bullet {x}_1\bullet {x}_2\bullet {\rho}_{12}\bullet {\sigma}_1\bullet {\sigma}_2-i $$

(12.11)

$$ =\mathrm{variance}\ \mathrm{of}\ \mathrm{two}\ \mathrm{stocks}+\mathrm{covariance}\ \mathrm{between}\ \mathrm{stocks} $$

Again, the standard deviation of a portfolio is the square root of the portfolio’s variance. Example 12.4 illustrates the working of these formulas for a two-stock portfolio.

Example 12.4 Calculating Return and Volatility of Two-Stock Portfolio

Problem

Suppose there is a portfolio with two stocks X and Y, whose returns have the following characteristics:

Stock	Expected return	Standard deviation	Correlation
X	10%	20%	0.6
Y	15%	40%

• (a) What is the expected return and volatility of an equally weighted portfolio?

• (b) What is the portfolio’s volatility, if you demand an expected return of 12%?

• (c) Which portfolio is more efficient?

Solution

• (a) The expected return of the equally weighted portfolio is (Eq. 12.9):

$$ E\left[{r}_p\right]={\sum}_i\ {x}_i\bullet E\left[{r}_i\right]=50\%\bullet 0.10+50\%\bullet 0.15=12.5\% $$

Next, we can calculate the portfolio variance (Eq. 12.11):

$$ Var\left[{r}_p\right]={x}_1^2\bullet {\sigma}_1^2+{x}_2^2\bullet {\sigma}_2^2+2\bullet {x}_1\bullet {x}_2\bullet {\rho}_{12}\bullet {\sigma}_1\bullet {\sigma}_2={0.5}^2\bullet {0.2}^2+{0.5}^2\bullet {0.4}^2+2\bullet 0.5\bullet 0.5\bullet 0.6\bullet 0.2\bullet 0.4=0.074 $$

Finally, the volatility is the square root of the variance (Eq. 12.6):

$$ SD\left[r\right]=\sqrt{Var(r)}=\sqrt{0.074}=27.2\% $$

• (b) The weights of the 12% return portfolio can be calculated using Eq. 12.9. You can solve the following equation: x₁ ∙ 0.10 + (1 − x₁) ∙ 0.15 = 12.0%. The solution is x₁ = 0.6 and x₂ = 1 − 0.6 = 0.4. Alternatively, you can do it by trial and error.

Next, you can calculate the portfolio variance:

$$ Var\left[{r}_p\right]={0.6}^2\bullet {0.2}^2+{0.4}^2\bullet {0.4}^2+2\bullet 0.6\bullet 0.4\bullet 0.6\bullet 0.2\bullet 0.4=0.063 $$

Again, the volatility is the square root of the variance:

$$ SD\left[r\right]=\sqrt{0.063}=25.1\% $$

• (c) The efficiency of the portfolio depends on your criterion. For risk-averse investors, the second portfolio seems to be more efficient. Return is only reduced by 4% (i.e. 0.5% divided by 12.5%), while volatility is reduced by 8% (i.e. 2.1% divided by 27.2%). The first stock has far lower volatility and only slightly lower return; it is thus useful to tilt the two-stock portfolio towards the first stock.

3.3 Variance of Large Portfolios

We can also derive the general formula for the variance of a portfolio with N stocks. The weight of each stock is: ${x}_i=\frac{1}{N}$. Figure 12.9 shows that for such a portfolio we have N cells with the variance of each stock ${x}_i^2\bullet {\sigma}_i^2$ (the shaded diagonal cells), while the remaining N² − N cells contain the covariances between stocks x_i ∙ x_j ∙ ρ_ij ∙ σ_i ∙ σ_j. Equation (12.11) can then be rewritten as follows:

Fig. 12.9

Variance and covariance in a portfolio. Source: Adapted from Brealey et al. (2022). Note: The N-shaded diagonal cells contain the variance of each stock; the N² − N remaining off-diagonal cells represent the covariance between stocks

$$ \mathrm{Portfolio}\ \mathrm{variance}: Var\left[{r}_p\right]=N\bullet {\left(\frac{1}{N}\right)}^2\bullet {\sigma}_i^2+\left({N}^2-N\right)\bullet {\left(\frac{1}{N}\right)}^2\bullet {\rho}_{ij}\bullet {\sigma}_i\bullet {\sigma}_j $$

(12.12)

$$ =\frac{1}{N}\bullet {\sigma}_i^2+\left(1-\frac{1}{N}\right)\bullet {\rho}_{ij}\bullet {\sigma}_i\bullet {\sigma}_j $$

$$ =\frac{1}{N}\times \mathrm{average}\ \mathrm{variance}+\left(1-\frac{1}{N}\right)\times \mathrm{average}\ \mathrm{covariance} $$

As the number of stocks in the portfolio N increases, the portfolio variance becomes the average covariance of the stocks in the portfolio. The correlation of each stock with the portfolio thus determines its contribution to overall portfolio risk, while its own variance no longer matters. This is the core of portfolio diversification. Only the covariance of a stock’s return with portfolio return (the systematic or market-wide risk) counts, while the unsystematic or idiosyncratic risk disappears in the portfolio. Example 12.5 shows how this is calculated.

Example 12.5 Calculating Variance of a Portfolio

Problem

An asset manager holds a very large equally weighted portfolio (that is, all stocks in equal weight). The stocks are from different countries and different industries, with a low correlation of 15% and a volatility of 30%. What is the volatility of the portfolio?

Solution

First, note that the amount of stocks (N) goes to infinity for a very large portfolio. Using Eq. 12.12, $\frac{1}{N}$goes to 0. So, the correct calculation of the portfolio variance is given by:

$$ Var\left[{r}_p\right]= average\ covariance={\rho}_{ij}\bullet {\sigma}_i\bullet {\sigma}_j $$

$$ Var\left[{r}_p\right]=0.15\times 0.30\times 0.30=0.0135 $$

The portfolio volatility equals $\sqrt{0.0135}=11.6\%$, which is lower than the volatility of the individual stocks at 30%. Diversification of stocks (with a correlation of less than one) thus pays off.

4 The Capital Asset Pricing Model

The risk-return measures and the principles of portfolio diversification provide the building blocks for deriving the Capital Asset Pricing Model (CAPM). This is the main theory in finance explaining the relationship between risk and return. The short story is as follows: Investors can construct efficient portfolios of stocks with maximum return given risk. In equilibrium, all investors hold proportions of the market portfolio, which provides the best available risk-return combination. Next, a stock price’s fluctuations can be split into market risk and firm-specific risk. As firm-specific risk can be diversified away in the market portfolio, the investor will only be rewarded for the market risk. The market risk, measured by a stock’s beta, is the co-movement of a stock with the market portfolio. The reward for this market risk is the market risk premium.

The CAPM makes some strong assumptions which are summarised in Box 12.2. The CAPM starts with building the efficient frontier of risky stocks. As discussed in Sect. 12.3, diversification through combining risky stocks in a portfolio eliminates the idiosyncratic or firm-specific risk of individual stocks. An efficient portfolio is a stock portfolio whereby investors cannot increase return given the level of volatility. The efficient frontier is then the combination of highest return portfolios for each volatility level. Figure 12.10 depicts this efficient frontier (blue curve) in the so-called mean-variance framework, whereby the y-axis shows the expected return (mean) and the x-axis the risk measure of volatility (standard deviation as square root of the variance). Individual stocks lie below this efficient frontier (blue curve) because they carry both idiosyncratic (firm-specific) and market risk. Individual stocks are thus ‘less efficient’ in terms of risk-return.

Fig. 12.10

The efficient frontier and capital market line. Note: The efficient frontier is based on monthly returns of the stocks from 2010 to 2021

Box 12.2 Assumptions Behind the CAPM

The CAPM makes several assumptions about individual investor behaviour and about market structure (Bodie et al., 2018).

Investor behaviour

• Risk-averse investors optimise risk-return (maximising return given the level of volatility)

• Investors have homogeneous expectations (they use identical information and draw the same conclusions from this information); this is consistent with the efficient market hypothesis (see Chap. 14)

Market structure

• All assets are publicly held and trade on market exchanges

• Investors can borrow and lend at a common risk-free rate

• There are no taxes or transaction costs

The next step is to choose the best stock portfolio on this efficient frontier for investors. Assuming that investors can freely borrow and lend, they hold a combination of the risk-free asset and an efficient portfolio. The capital market line (straight orange line) in Fig. 12.10 shows this combination. To earn the highest possible return given volatility, investors choose the portfolio with the steepest possible line; that is the line with the highest possible risk-adjusted return. This tangent portfolio provides the highest possible return for a given level of volatility of any (efficient) portfolio available; all other portfolios on the efficient frontier lie below the capital market line. Assuming that investors have homogeneous expectations, all investors want to hold this tangent portfolio. The tangent portfolio then becomes the market portfolio. A key insight of the CAPM is that all investors hold a combination of the risk-free asset and the market portfolio in equilibrium.

As all investors hold the same portfolio, an individual investor basically holds a share of the market portfolio. According to the CAPM, the market portfolio contains all available stocks in the market. In practice, this market portfolio is often represented by a market index of the largest companies in a stock market. Box 12.3 provides an overview of the leading market indices.

Box 12.3 Leading Market Indices

Each stock market has an index representing the largest companies traded in that market, also called the large caps. These stocks are the most liquid. Companies are keen to get in the market index, as that improves the tradability and visibility of their stock. Market indices can be equally weighted (all top 100 or top 500 companies get an equal weight in the index) or value-weighted (the top 100 or top 500 companies are weighted by their market capitalisation).

Some leading market indices are:

• the S&P 500 for the USA

• the STOXX Europe 600 for Europe

• the FTSE 100 for the United Kingdom

• the Shanghai SE Composite Index for China

• the Nikkei 225 for Japan

At the global level, international investors use:

• the MSCI World Index which covers over 1500 large and mid-cap companies across 23 developed markets

• the FTSE All-World index which covers over 3100 companies in 47 countries

Next, we need to derive a stock’s co-movement with, or sensitivity to, the market portfolio. This co-movement is the covariance of the stock with the market portfolio, as explained in Sect. 12.2. A stock’s beta β_i measures the sensitivity of that stock’s return, to the return on the market portfolio. Or more precisely, the co-movement of the stock’s fluctuations σ_i with the fluctuations of the market portfolio σ_mp. We use the correlation coefficient ρ_{i, mp} to measure the co-movement. The beta of stock i is calculated as follows:

$$ \mathrm{Beta}:{\beta}_i=\frac{\sigma_i\bullet {\rho}_{i, mp}}{\sigma_{mp}} $$

(12.13)

Beta thus measures the sensitivity of a stock to market-wide risk factors: to what extent are a company’s revenues and costs related to general economic conditions? Building on Eq. 12.8, the market risk premium RP_MKT is the expected market return minus the risk-free rate:

$$ \mathrm{Market}\ \mathrm{risk}\ \mathrm{premium}:{RP}_{MKT}=E\left[{r}_{MKT}\right]-{r}_f $$

(12.14)

$$ =\mathrm{expected}\ \mathrm{market}\ \mathrm{return}-\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate} $$

We can now estimate the cost of equity capital for a company. Remember that the investor is only rewarded for the systematic or market risk embedded in the company’s stock price. The cost of equity then becomes a combination of the risk-free rate and the market risk:

$$ \mathrm{Cost}\ \mathrm{of}\ \mathrm{equity}:{r}_i={r}_f+{\beta}_i\bullet \left(E\left[{r}_{MKT}\right]-{r}_f\right) $$

(12.15)

$$ =\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate}+{\beta}_i\times \mathrm{the}\ \mathrm{market}\ \mathrm{risk}\ \mathrm{premium} $$

Equation (12.15) is the central risk-return relationship of the CAPM. The calculation of the cost of equity depends on the time period for which the risk-free rate and the market risk premium are estimated. Using historical data from Table 12.1 with a risk-free rate of 4.5% and a risk premium of 6%, the cost of equity capital for a stock with a beta of 1 becomes: r_i = 4.5 %  + 1.0 ∙ [6.0%] = 10.5%. The most recent estimates from Fig. 12.3 are a risk-free rate of 2% (which is a combination of the real bill rate of −0.5% and an inflation of 2.5%) and a market risk premium of 3%. This leads to a far lower cost of equity: r_i = 2.0 %  + 1.0 ∙ [3.0%] = 5.0%, which reflects the low rate situation in the 2010s and early 2020s.

Example 12.6 shows how the beta and the cost of equity of individual companies can be calculated. It appears that with relatively little information—only the stock’s volatility and its correlation with the market portfolio—a company’s cost of equity capital can be calculated in a straightforward way.

Example 12.6 Calculating the Beta and Cost of Equity

Problem

Suppose the S&P 500 has an expected return of 7% and a volatility of 15%. Apple stock has a volatility of 19% and has a correlation of 0.6 with the market. Oracle Corporation stock has a volatility of 35% and a correlation of 0.4. Assume the risk-free rate is 2%. Calculate the cost of equity capital for Apple and Oracle, by first deriving their beta.

Solution

First, calculate the beta of Apple and Oracle using Eq. 12.13:

$$ {\beta}_{Apple}=\frac{\sigma_A\bullet {\rho}_{A,S\&P}}{\sigma_{S\&P}}=\frac{0.60\ast 0.19}{0.15}=0.76 $$

$$ {\beta}_{Oracle}=\frac{\sigma_O\bullet {\rho}_{O,S\&P}}{\sigma_{S\&P}}=\frac{0.4\ast 0.35}{0.15}=0.93 $$

The sensitivity of Apple to the S&P 500 is 0.76. That means if the S&P 500 moves 1%, Apple tends to move 0.76%. Oracle’s market exposure is a bit higher with a beta of 0.93. We use Eq. 12.15 to calculate the firm’s cost of equity capital:

$$ {r}_{Apple}={r}_f+{\beta}_A\bullet \left(E\left[{r}_{S\&P}\right]-{r}_f\right)=0.02+0.76\ast \left(0.07-0.02\right)=5.8\% $$

$$ {r}_{Oracle}={r}_f+{\beta}_O\bullet \left(E\left[{r}_{S\&P}\right]-{r}_f\right)=0.02+0.93\ast \left(0.07-0.02\right)=6.7\% $$

The final step of the CAPM is the security market line (SML), which shows the graphic representation of the risk-return relationship. The security market line in Fig. 12.11 plots the expected return against the beta of each stock or portfolio. The slope is the risk premium of the market portfolio: the market portfolio with β_MKT = 1 on the x-axis delivers the expected return on the market portfolio E[r_MKT] on the y-axis in Fig. 12.11. Following Eq. 12.15, the SML is an easy tool. Given the market-related risk of an investment (measured by its beta), the SML shows the required return necessary to compensate investors for risk as well as the time value of money. ‘Fairly’ priced stocks should be exactly on the SML, whereby only the systematic or market risk of a stock is priced in and compensated.

Fig. 12.11

The security market line. Note: The stocks in the security market line are taken from Fig. 12.10

Limitations to Measures of Historical Risk and Return

The traditional risk-return models discussed so far are essentially based on patterns of historical financial returns and volatility. Using backward-looking statistics, the implicit assumption is that the risk-return relationships remain the same in the future. But that is subject to the Lucas critique. Lucas (1976) basically argues that the structure of the historical relationships will change, when the nature of the assets changes due to policy changes. So historical relationships are not always a good guide for the future. Government policies (or stakeholder pressure) to address sustainability challenges, as discussed in Chaps. 1 and 2, may well turn the tables on the stock market. Profitable companies in the past may become stranded assets in the future (often quoted examples are oil and tobacco companies), while new companies providing solutions to the sustainability challenges rise in value (e.g. Tesla with its electric cars).

Next, there are severe limitations to the benchmarks that are used by investors. Box 12.3 provides an overview of commonly used market indices. However, market indices change a lot over time. Dimson, Marsh, and Staunton (2021) document some major changes in the:

• 1. Weightings of countries:

  • (a) In 1900, the United Kingdom accounted for 24% of global stock market value, followed by the USA (15%), Germany (13%), France (11%), and Russia (6%);

  • (b) In 2021, the USA accounted for 56% of global stock market value, followed by Japan (7%), the United Kingdom (5%), China (4%), and France (3%). This geographic distribution is much more skewed, with a dominant position for the USA.

• 2. Weightings of sectors:

  • (a) In 1900, rail made up over 60% of the US stock market, and almost 50% of the UK stock market. Other large sectors in both markets in 1900 were banking; mining; iron, coal & steel; utilities; textiles; and tobacco.

  • (b) In 2021, large sectors were technology, industrials, health, retail, banking, and oil & gas. Most of these sectors have much fewer physical assets and much more intangible assets.

A case in point is the position of the oil industry. Box 12.4 discusses the rise and decline of the oil industry, supporting the Lucas critique. More broadly, today’s market index represents yesterday’s industry. Old companies and industries are slowly fading out, while new companies and industries are added after much delay. Only when a company becomes large enough, will it be included in the main index. Moreover, some sustainable companies of the future have yet to be established. Historical patterns are therefore an incomplete guide to the future. Section 12.8 discusses forward-looking measures of risk.

Box 12.4 The Oil Industry Now and in the Future

The stock market shows the rise and likely decline of the oil & gas industry. At the turn of the twentieth century, oil companies, such as Standard Oil established in 1870 by Rockefeller, started to emerge. By the late twentieth century, oil companies such as Exxon, Shell, and BP, were among the largest companies in the world. In the USA, Big Tech has already replaced the oil sector, which is now less than 5% of the US stock market. In the United Kingdom, the oil sector still makes up about 10% of the stock market.

Rising carbon taxes shift the focus from fossil fuels to renewable energy. The European Green Deal aims to reduce carbon emissions by 55% in 2030 and to zero by 2050. So, the share of the oil sector is expected to shrink accordingly, as fossil fuels are a major source of carbon emissions.

Another limitation is that stock prices react primarily to news about financial risks, as reported quarterly and annually by companies in their financial reports. Up until now, there has been less attention paid to other risk indicators, such as social and environmental risks. The next sections explore how these limitations can be overcome.

5 Sustainability Adjusted Financial Risk-Return Analysis

As discussed, stock prices react primarily to news about financial risks as reported by companies, as well as broader economic news. There is less attention for other risk indicators, such as social and environmental risks. These risk indicators are not themselves financial in nature, but could have financial implications. Equity analysts are not asking for this type of information in analyst calls with senior management (see Chap. 17). Even if companies report on these other risks, analysts still target their questions towards understanding the quarterly financial results. Following Lukomnik and Hawley (2021), we suggest including social and environmental risks as sources of systematic risk, to get a full picture of a company’s financial performance and risk. Figure 12.12 illustrates the multiple risk sources of systematic risk.

Fig. 12.12

Risk sources of systematic risk

Just like financial risk in the previous section (Sect. 12.4), social and environmental risks can be split into idiosyncratic (firm-specific) risks, which can be diversified, and systemwide risks, which cannot be diversified. An increase in temperature (due to climate change) or a loss of biodiversity now, may lead to lower economic growth (and thus lower financial returns or more financial risk) in the future. These environmental risks are then additional sources of systematic risks, which are priced with a risk premium in the cost of capital or discount rate (see below). By contrast, an instance of water pollution by an individual company is an idiosyncratic risk, which is not priced because it can be diversified.

How to integrate the systematic aspects of social and environmental risks in our cost of equity capital calculations? In the risk-return relationship of the CAPM (Eq. 12.15), a stock’s return depends on its co-movement with the market. The market risk premium captures a wide range of financial and macroeconomic risks in a single factor. We expand this single-factor model to a multifactor model by adding social and environmental risk factors as sources of systematic risk. In this multifactor model, a company’s adjusted cost of equity capital r_i is:

$$ \mathrm{Adjusted}\ \mathrm{cost}\ \mathrm{of}\ \mathrm{equity}\ \mathrm{capital}:{r}_i={r}_f+{\beta}_i^{MKT}\bullet {RP}_{MKT}+{\beta}_i^{SF}\bullet {RP}_{SF}+{\beta}_i^{EF}\bullet {RP}_{EF} $$

(12.16)

$$ =\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate}+{\beta}_i^{MKT}\times \mathrm{market}\ \mathrm{risk}\ \mathrm{premium}+{\beta}_i^{SF}\times \mathrm{social}\ \mathrm{risk}\ \mathrm{premium}+{\beta}_i^{EF}\times \mathrm{environmental}\ \mathrm{risk}\ \mathrm{premium} $$

Similar to the market risk premium MKT in Eq. 12.14, the social and environmental risk premiums RP_SF and RP_EF can be defined as follows:

$$ \mathrm{Social}\ \mathrm{risk}\ \mathrm{premium}:{RP}_{SF}=E\left[{r}_{SF}\right]-{r}_f $$

(12.17)

$$ =\mathrm{expected}\ \mathrm{return}\ \mathrm{on}\ \mathrm{social}\ \mathrm{factor}-\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate} $$

$$ \mathrm{Environmental}\ \mathrm{risk}\ \mathrm{premium}:{RP}_{EF}=E\left[{r}_{EF}\right]-{r}_f $$

(12.18)

$$ =\mathrm{expected}\ \mathrm{return}\ \mathrm{on}\ \mathrm{environmental}\ \mathrm{factor}-\mathrm{risk}\hbox{-} \mathrm{free}\ \mathrm{rate} $$

The market risk premium is the excess return (that is, the return in excess of the risk-free rate) on the market portfolio. Just like the market portfolio has unit exposure β^MKT = 1 to market risk, we can construct factor-mimicking portfolios that have unit exposure to the social factor β^SF = 1 and environmental factor β^EF = 1, respectively, and zero exposures to the other factors. The expected returns on the social and environmental factor portfolios determine the social and environmental risk premiums in Eqs. 12.17 and 12.18.

5.1 Social and Environmental Factor Portfolios

In this setting, we add social and environmental factor portfolios to the market portfolio of the CAPM. How do we derive the other two portfolios? We can devise trading strategies that capture social risk and environmental risk, respectively, which is not captured by the market portfolio.

The construction of the portfolios is based on the S(ocial) and E(nvironmental) pillar of companies’ ESG rating. ESG ratings summarise a company’s performance on environmental, social, and governance issues, as measured by an ESG rating agency (see Chap. 14). For example, the environmental factor portfolio could be constructed by taking the bottom third of the E rating of the STOXX Europe 600 companies in Europe (or the S&P 500 companies in the USA). These companies form a value-weighted portfolio called the brown portfolio. Next, the top third of the E rating of the STOXX Europe 600 companies form a value-weighted portfolio called the green portfolio.

A trading strategy that takes a long position in the brown portfolio, which it finances with a short position in the green portfolio, produces the environmental risk premium. The long position means that the investor owns the brown portfolio, while the short position means that the investor has to deliver the green portfolio. This portfolio which is long in brown stocks and short in green stocks is called the brown-minus-green (BMG) E portfolio. So, this environmental factor portfolio is long on high E risk companies (companies with a low E rating) and short on low E risk companies (companies with a high E rating) and nicely captures the environmental risk premium.

A similar trading strategy can be set up by taking the bottom third of the S rating (bad S companies) and the top third of the S rating (good S companies) of the STOXX Europe 600 companies. This bad-minus-good (BMG) S portfolio captures the social risk premium.

5.2 Challenges of the Multifactor Model

There are several challenges in the application of the multifactor model in practice. The first challenge is to construct the social and environmental factor portfolios based on ESG ratings. As discussed in Chap. 14, the data quality of ESG ratings is currently not very high, but is expected to rise over time. The second challenge is to derive social and environmental risk premiums from financial market data. Remember that the social and environmental risk premium are estimated by trading the social and environmental factor portfolios in the market. As explained earlier, the derivation of the risk-return relationship in the CAPM makes the assumption of efficient financial markets (see Box 12.2). The efficient markets hypothesis states that stock prices incorporate all relevant information (Fama, 1970). By contrast, the adaptive markets hypothesis argues that the degree of market efficiency depends on an evolutionary model of individuals adapting to a changing environment (Lo, 2004, 2017). Chapter 14 discusses the information efficiency of markets in more detail.

Social and environmental risks have only recently been considered as relevant for stock prices, often after major news events. Examples are big litigation cases against tobacco companies based on the health effects of smoking and against oil companies on not adhering to the Paris climate agreement of 2015. Pastor et al. (2022) find an average environmental risk premium of 1.4% (per year) for US companies over the 2012–2020 period. The environmental risk premium increases from 1.2% in 2012 to 1.9% in 2017–2020. Their findings are based on a broad environmental score, across 13 environmental issues related to climate change, natural resources, pollution, and waste.

On a narrow scope, Bolton and Kacperczyk (2023) investigate the relationship between carbon emissions and stock returns. They find that a one-standard deviation increase in emissions yields higher annual stock returns of 3.6% for scope 1 emissions and 4.6% for scope 3 emissions. In their overall sample of 77 countries, Bolton and Kacperczyk (2023) find that there was no significant carbon premium right before the Paris agreement, but a highly significant and large premium in the years after the agreement. This suggests that the Paris agreement has changed investors’ awareness regarding the impending regulatory changes to combat climate change. Interestingly, Bolton and Kacperczyk (2023) interpret the carbon premium (in the form of higher returns) as a reward for carbon transition risk, which reflects the uncertain rate of adjustment towards carbon neutrality.

On the social side, there are no estimates yet based on a broad social score. On a narrow scope, Hong and Kacperczyk (2009) find a risk premium for sin stocks (alcohol, tobacco, and gaming) of 2.5% per year for US companies over the 1965–2006 period. A more recent study finds a risk premium for the same sin stocks of 2.8% per year for US companies over the 1999–2019 period (Zerbib, 2022). Table 12.6 summarises the findings on environmental and social risk premiums. More targeted studies typically find larger premiums than broad-based factor portfolios. Table 12.6 shows an environmental risk premium of 1.9%. The social risk premium is lower and likely to be in the range of 1.0–1.5%.

Table 12.6 Environmental and social risk premiums

But there is also evidence that stock prices do not discount certain social or environmental risks efficiently. Hong et al. (2019) investigate the impact of droughts on food stock prices. They find that food stock prices underreact to increased drought risk. Other social and environmental risks, such as human right violations, underpayment, biodiversity loss, and water scarcity, may also not (yet) be considered as systematic risk sources by analysts and thus not (yet) be incorporated in stock prices.

The market beta ${\beta}_i^{MKT}$ (as in the CAPM) measures the sensitivity of company i to financial and economy-wide risks. The factor betas, ${\beta}_i^{SF}\ \mathsf{and}\ {\beta}_i^{EF}$, measure the company’s sensitivity to social risk (e.g. public health; social inequality; safety and health of workers; low wages in supply chain) and environmental risk (e.g. carbon emissions; biodiversity loss; pollution; waste). The social and environmental beta coefficients can be interpreted as follows:

• ${\beta}_i^{SF},{\beta}_i^{EF}>1$ reflect relatively high exposure indicating that this company is not prepared for the sustainability transition

• $0<{\beta}_i^{SF},{\beta}_i^{EF}<1$ reflect relatively low exposure indicating this company is partly prepared for transition

• ${\beta}_i^{SF},{\beta}_i^{EF}<0$ reflect that the company’s activities will likely benefit financially from transitions.

5.3 Working of the Multifactor Model

The working of the multifactor model can be illustrated with some hypothetical examples. Chapter 13 discusses company cases. Let’s start with the environmental risk premium for carbon emissions. Box 12.5 illustrates that differences in sensitivity to carbon risk lead to different discount factors and project values. A lower carbon exposure leads to a higher value of a project with the same underlying cash flow pattern, due to a lower discount factor.

A similar example is given for the social risk premium. When a company invests in factory safety, it does not only reduce the interruption of the production process (less revenues lost), but it also reduces its social risk exposure as it improves the safety of its employees. An often-used indicator for safety performance is the lost-time injury frequency rate (LTIFR), which refers to the number of lost-time injuries within a given accounting period, relative to the total number of hours worked in that period. Box 12.6 shows how we can calculate the combined production and employee safety benefits of an investment in factor safety.

The factor safety investment already showed how a company can improve its risk profile. This is important for companies that want to put their business model on a more sustainable footing. The choice of the appropriate discount rate for new projects is crucial. Box 6.3 in Chap. 6 illustrated how Shell, a major oil company, turned down an investment opportunity to reduce its carbon emissions, because it used the wrong discount rate (i.e. too high discount rate) leading to a lower valuation.

Box 12.5 Impact of Differing Carbon Intensity on Discounting and Values

We consider three projects with identical cash flows, but differing carbon intensity. The basic set-up is an initial investment of €1000 and four annual net inflows of €300.

The first project is an investment in a forestry project, whereby trees capture carbon. The project has a negative environmental beta of 0.5: ${\beta}_1^{EF}=-0.5$. The second project concerns the investment in a low-carbon technology, with a beta of 0.2: ${\beta}_2^{EF}=0.2$. The third project invests in a high-carbon technology, with a beta of 1.2: ${\beta}_3^{EF}=1.2$. The projects have unit exposure to market risk ${\beta}_i^{MKT}=1.0$ and no exposure to social risk ${\beta}_i^{SF}=0$. We further assume a risk-free rate of 2%, a financial risk premium of 4%, and an environmental risk premium of 2%. We can now calculate the adjusted cost of equity capital with Eq. 12.16, which moves from 5.0 %  = 2 %  + 1 ∗ 4 %  − 0.5 ∗ 2% for project 1, to 6.4 %  = 2 %  + 1 ∗ 4 %  + 0.2 ∗ 2% project 2 and 8.4 %  = 2 %  + 1 ∗ 4 %  + 1.2 ∗ 2% for project 3.

Project	Environmental beta${\beta}_i^{EF}$	Adjusted cost ofequity capital ri	Adjusted Net Present Value NPVi
1	−0.5	5.0%	€63.8
2	0.2	6.4%	€30.1
3	1.2	8.4%	€−15.1

Using Eq. 4.4, we can calculate the adjusted net present value of the project. The first project has the highest net present value of $\text{\EUR}\ 64=-\mathrm{1,000}+\frac{300}{\left(1+0.05\right)}+\frac{300}{{\left(1+0.05\right)}^2}+\frac{300}{{\left(1+0.05\right)}^3}+\frac{300}{{\left(1+0.05\right)}^4}$, as future cash flows are more valuable due to the low discount rate. The low-carbon project has only a small mark-up on the financial discount rate and has an adjusted net present value of €30. The high-carbon project has a high discount factor due to the large exposure to carbon risk, resulting in a negative adjusted net present value of € −15.

Box 12.6 Investment in Safety

A company is contemplating an investment in the safety of one its factories. The factory is currently generating an annual net cash flow of €10,000.

We assume a unit exposure to market risk ${\beta}_i^{MKT}=1.0$, average exposure to social risk ${\beta}_i^{SF}=0.6,$ and no exposure to environmental risk ${\beta}_i^{EF}=0$. We further assume a risk-free rate of 2%, a financial risk premium of 4%, and a social risk premium of 1%. The adjusted cost of equity capital for this company (Eq. 12.16) is r_i = 2 %  + 1 ∗ 4 %  + 0.6 ∗ 1 %  = 6.6%. Using Eq. 4.5, we can now calculate the present value of this factory as $PV=\frac{CF}{r}=\frac{\text{\EUR}\ \mathrm{10,000}}{6.6\%}=\mathit{\text{\EUR}}\ \mathrm{151,515}$.

The safety investment requires an initial investment of €2500 and leads to improved annual cash flows of €100 due to less disruption in the production process. The increased safety of the factory results in a lower beta for social risk: ${\beta}_i^{SF}=0.5$, yielding a lower adjusted cost of equity capital for the company r_i = 2 %  + 1 ∗ 4 %  + 0.5 ∗ 1.0 %  = 6.5%. The present value of this investment is $PV=-\mathrm{2,500}+\frac{\text{\EUR}\ 100}{6.5\%}=\mathit{\text{\EUR}}-962$. On the face of it, the company will decline this safety investment proposal. However, there is also a reduction in the social risk profile of the factory leading to a lower discount factor. The present value of the company now becomes: $PV=\frac{\text{\EUR}\ \mathrm{10,000}}{6.5\%}=\mathit{\text{\EUR}}\ \mathrm{153,846}$, an improvement of € 2331. The safety investment thus turns into an overall profitable project of €1369 and should be done (Table 12.7).

Table 12.7 Impact of differing carbon discount factors

The advantage of the multifactor model is that it accounts for E risks and S risks, but only to some extent. Its effectiveness is limited by the quality of data, and transitions are typically not well accounted for (see Chap. 2). Nor does it give measures of adaptability and robustness to shocks. It is up to further research to explore those issues.

6 Social and Environmental Risk-Return Analysis

The previous Sect. 12.5 derived the adjusted cost of financial capital by including the effects of social and environmental risks on the financial risk-return relationship. The next step for calculating the integrated risk-return is examining the social and environmental risk-return relationship. Chapter 4 already explained that the stakeholders of a company’s social and environmental value are part of wider society. These stakeholders include employees, consumers, suppliers, (local) communities, and future generations. The social discount rate is the appropriate discount rate for these stakeholders. Equation 4.8 provides the basic social discount rate for social and environmental value:

$$ \mathrm{Basic}\ \mathrm{social}\ \mathrm{discount}\ \mathrm{rate}:{r}^s=\delta +\eta \bullet g $$

(12.19)

The first parameter δ reflects the time preference between current and future generations. Equal treatment of current and future generations gives us a time preference of zero: δ = 0. See Sect. 4.3 for a full discussion of the rationale for a zero-time preference in social discounting. The growth rate g is driven by growth in consumption. Given a diminishing marginal utility of consumption, the growth rate is multiplied by the elasticity of marginal utility of consumption η. The elasticity measures how utility changes with consumption.

Next, we introduce risk into the social discount rate. There are several sources of risk related to the growth factor in Eq. 12.19:

• 1. Growth risk: the macroeconomic risk that the growth rate of consumption fluctuates;

• 2. Company risk: the correlation between company risk and growth risk;

• 3. Catastrophe risk: the extreme element of macroeconomic risk of rare disasters (deep recessions) or society collapse.

Fluctuations of the growth rate give rise to uncertainty about future growth of consumption. Gollier (2012) adds a prudence term for uncertainty about consumption growth (based on the variance of consumption growth ${\sigma}_g^2$). Uncertainty about future consumption leads to higher precautionary investing in the future, today. The prudence term is therefore deducted from the social discount rate, as the resulting lower social discount rate increases the present value of future benefits of investing. This in turn leads to higher precautionary investing.

Next, company risk and growth risk can be correlated. As with the CAPM, we can distinguish between a systematic and an unsystematic component. The unsystematic component (unexpected additional social and environmental benefits or costs) can be diversified away. The systematic component presents the relationship between macroeconomy fluctuations (again measured by ${\sigma}_g^2$) and uncertainty about the social and environmental benefits of the company. This systematic component is added to the social discount rate.

The risk premiums for growth and company risk are very small (less than 0.1%) and opposite (Gollier, 2012). The main reason for the small risk premiums is the low variance of consumption growth. We therefore exclude these risk premiums from the social discount rate formula for practical reasons.

The final source of risk is represented by the catastrophic risk parameter L. The rationale for this risk parameter is the likelihood that there will be some catastrophic event so devastating that social and environmental returns from companies are eliminated. Catastrophe risk can be seen as an extreme form of systematic company risk. The futurologist, Toby Ord (2020), distinguishes between man-made catastrophes, such as unaligned artificial intelligence and engineered pandemics, and natural catastrophes, such as supervolcanic eruptions or a comet impact. Ord (2020) argues that the probability of man-made catastrophes is far higher (in the relatively near future) than that of natural ones. His estimate of a catastrophe happening at some point in the next 100 years is 1 in 6. Box 12.7 shows how the 100-year catastrophe risk rate can be translated into an annual risk parameter L of 0.2%.

Box 12.7 Calculating Annual Catastrophe Risk

Ord (2020) estimates the probability of a catastrophe occurring in the next 100 years as 1 in 6, which is 16.7%. The 100-year survival rate of 83.3% (=100–16.7%) can be transformed into an annual survival rate of 99.8% as $\sqrt[100]{0.833}=0.998$. So, the annual catastrophe risk parameter L amounts to 0.2% (=100–99.8%). Of course, this risk parameter is based on Ord’s subjective evaluation of the occurrence of catastrophes. Nevertheless, it provides a ‘ballpark’ parameter for thinking about catastrophe risk.

Building on Eq. 12.19, we expand the social discount rate r^s with a risk parameter L:

$$ \mathrm{Expanded}\ \mathrm{social}\ \mathrm{discount}\ \mathrm{rate}:{r}^s=\delta +\eta \bullet g+L $$

(12.20)

We are now ready to estimate the expanded social discount rate. As explained in Chap. 4, Dasgupta (2021) sets the time preference δ at 0% and growth g at 1.3%. Reviewing several studies, Groom and Maddison (2019) find an elasticity η of 1.5. The final parameter is Ord’s catastrophic risk parameter L of 0.2%. Summing these parameters, Table 12.8 calculates a social discount rate of 2.2%.

Table 12.8 Parameters for the expanded social discount rate

7 Integrated Risk-Return Analysis

We are now ready to wrap the components together in the cost of integrated capital. Eq. 12.16 provides us the adjusted cost of equity capital r_i. For the sake of simplicity, we assume full equity financing—the cost of equity is then the cost of financial capital. Chapter 13 introduces the cost of financial capital as weighted average of the cost of equity and the cost of debt. Eq. 12.20 gives the expanded social discount rate r^s for social and environmental capital. The cost of integrated capital ${r}_i^{IV}$ is then the weighted average of these costs of capital:

$$ \mathrm{Cost}\ \mathrm{of}\ \mathrm{integrated}\ \mathrm{capital}:{r}_i^{IV}=\frac{FV}{IV}\bullet {r}_i+\frac{SV+ EV}{IV}\bullet {r}^s $$

(12.21)

The weights are provided by the company’s respective value components: FV, SV, and EV divided by integrated value: IV = FV + SV + EV. In this way, the weights add up to one. The value components are introduced in Chaps. 5 and 6. Subsequent chapters show how to estimate the value components for companies. Please note that the value components can be negative. The derivation of the cost of integrated capital is relatively easy once one knows the adjusted cost of equity capital r_i and the cost of social and environmental capital r^s.

Given that the adjusted cost of equity capital is higher than the cost of social and environmental capital r_i > r^s, companies with relatively more social and environmental value face a lower cost of integrated capital than companies with lower or negative social and environmental value. Example 12.7 presents differing costs of integrated capital. The medtech company with positive social value faces a cost of integrated capital of 4.7%, while the oil company with negative environmental value faces a cost of integrated capital 20.3%. This difference is very large. The difference in the adjusted cost of financial capital is far smaller: 6% for the medtech company versus 9% for the oil company. The big difference in the cost of integrated capital of the two companies is caused by the difference in the risk profile of the two companies: the medtech’s social assets (lowering its cost of integrated capital) and the oil company’s environmental liabilities (strongly increasing its cost of integrated capital).

Example 12.7 Calculating the Cost of Integrated Capital

Problem

Suppose an oil company’s cost of financial capital is 9% and its cost of social and environmental capital is 2.2%. Next, a medtech company has a cost of financial capital of 6% and a similar cost of social and environmental capital of 2.2%.

The value dimensions are as follows:

Value dimension	Oil company	Medtech company
Financial value	8	4
Social value	−1	3
Environmental value	−4	−1
Integrated value	3	6

What is the cost of integrated capital of the two companies?

Solution

Using Eq. 12.21, we can calculate the cost of integrated capital:

$$ {r}_i^{IV}=\frac{FV}{IV}\bullet {r}_i+\frac{\left( SV+ EV\right)}{IV}\bullet {r}^s $$

For the oil company, the cost of integrated capital is:

$$ {r}_{oil}^{IV}=\frac{8}{3}\bullet 9\%+\frac{\left(-1-4\right)}{3}\bullet 2.2\%=20.3\% $$

And for the medtech company, the cost of integrated capital is:

$$ {r}_{med}^{IV}=\frac{4}{6}\bullet 6\%+\frac{\left(3-1\right)}{6}\bullet 2.2\%=4.7\% $$

The table below gives an overview of both companies’ cost of integrated capital as well as the individual cost of capital components:

Cost of capital	Oil company	Medtech company
Financial	9.0%	6.0%
Social	2.2%	2.2%
Environmental	2.2%	2.2%
Integrated	20.3%	4.7%

8 Forward-Looking Risk

Given the limitations of historical or backward-looking risk measures as discussed in Sect. 12.4, it makes sense to develop forward-looking risk measures that are able to take transitions into account. Figure 12.13 illustrates the time connection between backward- and forward-looking risk. Forward-looking risk measures tend to be of a more qualitative nature. The challenge is to quantify them.

Fig. 12.13

Backward and forward-looking risk

But let’s take a step back and consider alternative definitions of risk. In his book ‘The most important thing’, hedge fund manager Howard Marks says ‘There are many kinds of risk. But volatility may be the least relevant of them all’, with volatility chosen by academics just for its measurement convenience (Marks, 2011, p35). In Marks’ view, risk is the likelihood of losing money, the possibility of permanent financial loss. In a large survey with finance professionals, Holzmeister et al. (2020) also find that finance professionals perceive the probability of losses as the strongest risk indicator, and not volatility (the most common risk measure in finance). ² This financial loss aversion could be extended to the possibility of permanent nonfinancial losses, like permanent negative social and/or environmental impact.

Marks (2011) also identifies other, secondary types of risk that are important because they affect behaviour, such as the risk of falling short of one’s goals, risk of underperformance, career risk, risk of being considered unconventional, and illiquidity risk. And just like the main risk of losing money, these risks are not quantified in a straightforward and standard way. But there are ways to quantify them, such as:

• Scenario analysis;

• Options analysis (see Chap. 19); and

• Replace historical parameters in models with forward-looking estimates.

8.1 Scenario Analysis

Scenario analysis is a process of analysing possible future events by considering alternative possible outcomes (sometimes called ‘alternative worlds’), as discussed in Chap. 2. Scenario analysis can be used to analyse the effects of possible future events on the value of a company. The scenario construction process requires some choices on parameters (TCFD, 2020):

• Time horizon: sufficiently long;

• Number and diversity of scenarios: 3–4 differing scenarios;

• Focal question: critical question that company (or investor) wants to address;

• Drivers of change: main clusters of risk;

• Impact on companies: translate scenarios into impact on companies;

• Probabilities of scenarios: assign probabilities to scenarios.

A first parameter is the time horizon. The time horizon should be short enough to be plausible and long enough for important changes with an impact on future business to take place. For climate transition scenarios, typical horizons are 2030 and 2050. Another parameter is the number and diversity of scenarios. Typically 3–4 scenarios are chosen, which are sufficiently different.

Important parameters are selecting the focal question and the most important drivers of change for that question. Critical questions seek to gain insight into the impact of an overarching trend or phenomenon on the company. Examples of such overarching trends are climate change, water scarcity, digitalisation, demographics, and labour practices in the value chain (e.g. human rights and living wage).

The next step is to identify the underlying drivers of change. In the case of climate change, for example, typical clusters of drivers are policy or technology induced transition risk (risk of transition to low carbon) and environmentally-driven physical risk (risk of flooding or drought). Figure 12.14 provides an example of a scenario matrix for climate risk, which is relevant for carbon-intensive sectors. On the vertical axis, the driver is physical risk: scenarios 1 and 2 have high physical risk with global warming. On the horizontal axis, the driver is transition risk: scenarios 2 and 4 have high transition risk with impact on carbon-intensive companies.

Fig. 12.14

Scenario matrix for climate risk

In the case of other focal questions, a company has to choose the relevant drivers of change, like demographics (early vs late ageing), labour practices (early vs late implementation of living wage), innovation (rapid vs slow), or digitalisation (early vs late digitalisation). The appropriate choice of question and drivers depends on the sector, as each industry is facing its own medium to long-term challenges.

The next step is to assess how a company is affected by the scenarios. Let’s take an oil company with a current stock price of $40. A valuation based on a simple extrapolation of current cash flows (business-as-usual scenario) delivers a stock price of $50. That looks like the company is 20% undervalued in the market. But we can calculate the fair value of the company’s stock based on climate transition risk scenarios. Table 12.9 shows the stock price under the various scenarios: the business-as-usual scenario (company does not change and no transition) with a value of $50, the collapse scenario (company does not change and transition) with a value of $0, the prepared but no transition scenario with a value $40, and finally the prepared and transition scenario with a value of $60.

Table 12.9 Valuations calculated for scenarios for oil company

The final step is to synthesise the scenario results by weighting the probabilities attached to each scenario, which add up to 100%. The probabilities are rough estimates of the likelihood of each scenario. Assigning probabilities to our company in Table 12.9 produces a fair price of $35 (=0.3*$50 + 0.3*$0 + 0.2*$40 + 0.2*$60). So, the stock seems to be overvalued by 14% at the current market price of $40.

8.2 Inditex Case Study

In Chap. 11, we did a similar analysis of Inditex’s transition scenarios. Table 12.10 reproduces the numbers from Tables 11.7 and 11.8. Inditex’s fair price is €24.2 (=0.24*€31.9 + 0.16*€10.4 + 0.36*€22.5 + 0.24*€28.4).

Table 12.10 Transition scenarios weighted valuation for Inditex

8.3 Strategy-Setting

Scenario outcomes can be used as input for the strategy-setting (see Chap. 2). The company can increase its value by better preparing itself for transition, and thus avoid the costly collapse scenario. Good management would take strategic action on its own initiative. Alternatively, investors can demand from the company that it prepares itself for transition in the process of engagement (see Chap. 3). If the company is not prepared to take action (and thus reduce its negative climate impact and increase its value), the investor may divest from the company as it is overvalued on the basis of the climate risk scenario analysis.

8.4 Transition Pathways

Scenario analysis can also be applied to the social and environmental impacts. It is rather time-consuming for investors to do scenario analysis for all its investee companies. There are several investor-led initiatives, in particular in the area of climate transition preparedness, to make these company assessments. An example is the Transition Pathway Initiative (TPI) that provides assessments of companies’ preparedness for the transition to a low-carbon economy (TPI, 2021). TPI uses several climate benchmark scenarios:

• The most stringent is the Paris 1.5 °C global warming scenario;

• The medium benchmark is the 2 °C global warming scenario;

• The least stringent is the Paris pledges (to reduce emissions) by national governments.

Figure 12.15 shows the downward pathway for these benchmarks from 2015 to 2050. The global carbon reduction benchmarks are split into sectoral carbon benchmarks, for example, for electricity and steel. The assigned carbon budget for each sector is divided by activity (e.g. megawatt hours of electricity and tons of crude steel produced) to obtain sectoral carbon intensity benchmarks. Carbon intensity is defined as follows:

Fig. 12.15

Stylised examples of companies aligning with climate benchmarks. Source: Transition Pathway Initiative (2021)

$$ \mathrm{Carbon}\ \mathrm{intensity}: CI=\frac{carbon\ emissions}{activities} $$

(12.22)

The final step is to plot a company carbon reduction pathway against the benchmarks to check whether a company is Paris-aligned, and thus prepared for transition. If a company’s emissions reduction pathway always lies above the Paris Agreement benchmarks, then clearly it cannot be described as Paris-aligned and transition prepared (line A in Fig. 12.15); and vice versa for a company whose pathway always lies below them (line B in Fig. 12.15). The difficult cases are those that lie in between (line C in Fig. 12.15). In this case, company C eventually becomes Paris-aligned, close to the finishing year of 2050. As company C has an overshoot in the early years, which is not compensated in later years, it is not aligned.

There is evidence of backloading in several sectors, like car manufacturing, cement, energy, and oil & gas. This means that TPI expects most reductions to take place towards the end of the Paris agreement (2050). The pathways take into account this backloading. Therefore, there is no ‘excuse’ for companies to be behind on the pathway, since it will stay behind in case of an increase in expected reductions from TPI.

8.5 Uncertainty

Scenario analysis has its limits. Good scenario planning designs diverse scenarios capturing the main sources of risk. And you should be able to assign probabilities to the scenarios. But the challenge is to account for real uncertainty in scenarios. Can you, for example, account for tail risks? Tails risk refers to the chance of a loss occurring due to a rare event (Taleb, 2007). In terms of probability distribution, tail risk involves an abrupt move of more than three standard deviations, while most risks are within one or two standard deviations from the mean (see Sect. 12.2). Real option analysis can be used to deal with uncertainty when the probability distribution is unknown (see Chap. 19).

8.6 Forward-Looking Indicators

There are topic-specific forward-looking indicators that can be applied as well. For example, for climate there is the Implied Temperature Rise (ITR). An asset’s ITR indicates by how many degrees global temperatures would rise if the asset remained on its current pathway and all assets were alike. This may sound very theoretical, but ITR allows for comparison across assets and asset classes and is an indicator of the risk that the asset will go out of business.

9 Conclusions

This chapter reviews historical risk and return patterns for various asset classes. Risk-averse investors demand a higher return for riskier assets. A company’s risk can be split into systematic or market-wide risk; and idiosyncratic risk, which can be diversified away in a portfolio. Investors are only rewarded with a risk premium for the market risk component. The Capital Asset Pricing Model (CAPM) states that in equilibrium, all investors hold a combination of the risk-free asset, such as government bonds, and the market portfolio.

But historical risk-return analysis has a limited capacity for assessing future financial risk. Forward-looking measures of financial risk and return are also needed. Next, responsible investors want to include the social and environmental risks in their financial risk-return analysis. To do so, we expand the single market model of the CAPM into a multifactor model by adding social and environmental factors. This allows us to derive the risk premium for the adjusted cost of financial capital.

The final step is deriving the cost of social and environmental capital (as we did in Chap. 4). Combining the cost of the three capitals yields the cost of integrated capital. This cost of integrated capital gives corporate managers the tool to make an integrated risk-return assessment in their investment decisions. Company examples illustrate that integrated risk-return analysis leads to different, more sustainable, decisions.

Key Concepts Used in This Chapter

• Adaptive markets hypothesis implies that the degree of market efficiency depends on an evolutionary model of individuals adapting to a changing environment

• Beta measures the sensitivity of a company’s stock price to general market movements (with reference to market, social or environmental risk)

• Capital Asset Pricing Model (CAPM) is the main asset pricing model in finance explaining the relationship between risk and return

• Capital market line is the combination of the risk-free asset and the market portfolio

• Correlation measures the degree to which two variables move in relationship to each other

• Cost of capital refers to the required return on an investment

• Discount rate refers to the interest rate used to determine the present value of future cash flows

• Efficient markets hypothesis states that stock prices incorporate all relevant information and thus on average reflect the long-term fundamental value of the firm

• Expected return is a weighted average of possible returns, with the probabilities of these possible returns as weights

• Factor models use risk factors (relating to market, social or environmental risk) to explain a stock’s risk and returns

• Factor portfolio is a portfolio with unit risk exposure to a particular risk factor (market, social or environmental risk) and no risk exposure to other factors

• Financial discount rate or cost of financial capital is the discount rate used to discount financial capital.

• Idiosyncratic risk refers to firm-specific risk that can be diversified in a portfolio

• Loss aversion is the observation that people experience losses asymmetrically more severely than equivalent gains

• Market index represents an entire stock market and thus tracks the market’s changes over time

• Market portfolio refers to the portfolio which contains all available assets in a market

• Portfolio theory shows that a company’s risk can be split in systematic or market-wide risk and idiosyncratic risk, which can be diversified away in a portfolio

• Realised return refers to the return on an asset in the past

• Risk refers to the variation of future returns

• Risk-free asset is a safe asset, such as government bills or bonds

• Risk premium refers the return on a risky asset, such as equities, minus the return on the risk-free asset

• Scenario analysis is a process of analysing possible future events by considering alternative possible outcomes (sometimes called ‘alternative worlds’); it can be used to analyse the effects of possible future events on the value of a company

• Security market line plots the expected return against the risk (measured by the beta) of each stock

• Social discount rate is the discount rate for social projects and can be used to discount social and environmental capital.

• Systematic risk refers to market-wide risk that cannot be diversified in a portfolio