4. Discount Rates and Scarcity of Capital

Discount rates of financial capital are determined in a setting of markets with supply and demand of financial funds. Figure 4.2 illustrates the flow of financial funds from investors to users through financial markets and intermediaries (such as banks and institutional investors). A large supply of funds relative to demand lowers the price or discount rate of financial capital, ceteris paribus. Next, financial markets are influenced by government policies. While central banks can supply or withdraw (short-term) funds, governments make regulations to ensure a proper functioning of financial markets. Examples of such government regulations are protection of property rights, enforcement of contracts, transparency of price discovery, and supervision of financial markets and intermediaries (De Haan et al., 2020).

A flow diagram. The flow is as follows. Investor-savers, financial markets and intermediaries, and user spenders. Investor savers include households, companies, and governments. User spenders include households, companies, and governments.

Most of the time, people prefer money today over money tomorrow, because of inflation and opportunity costs. This is called the time value of money: the difference in value between money now and money in the future. So, how to compare €100 today with €100 in 1 year? The difference is calculated by means of the discount rate, which is the interest rate r used to determine the present value PV of future cash flows. The discount factor then is the factor by which a future cash flow CF over n periods must be multiplied in order to obtain the present value:

Another way to express this: the €103 in 1 year is the future value of €100 today. Future value FV is the value of a cash flow now, expressed in euros in the future.

One can also calculate the present value of a stream of cash flows. This often includes negative cash flows as well, in which case one refers to the net present value NPV, i.e. net of cash outflows. Let’s take the example of a 6-year bond (a debt security issued by governments or companies to investors). Figure 4.3 visualises the annual cash flows of the bond, from the perspective of the bondholder (the investor).

A timeline represents the cash flows for 6 years from 2022 to 2028. In 2022, the cash flow is minus 1000 Euros. In 2028 the cash flow is 1030 Euros. The remaining years have a cash flow of 30 Euros each.

The negative signs are for cash outflows, in this case the payment of the bond’s principal (of €1000) in 2022 to the company or country issuing the bond. The principal or face value of a bond is the amount the investor pays to the issuer of the bond. In the years 2023–2028, a €30 coupon (reflecting an annual interest rate payment of 3% of €1000) is received, and in 2028 the principal is paid back by the issuer. Let’s suppose the discount rate of the bond is 3%, equal to the coupon. The NPV of the bond is then calculated as presented in Table 4.1. It starts by multiplying each annual cash flow with its associated discount factor. Remember that the discount factor is 1/(1 + r)ⁿ. If 2022 is right now, then n = 0 and its discount factor is 1/(1 + r)ⁿ = 1/(1.03)⁰ = 1. For 2023, the discount factor is 1/(1 + r)ⁿ = 1/(1.03)¹ = 0.971. Multiplying the annual cashflows with their discount factors gives the annual present values. The sum of those PVs is the NPV. The NPV of a cash flow stream from n = 0 to N then becomes:

It might seem a coincidence that the NPV is exactly zero. However, in competitive markets, NPVs will often be zero or close to zero: competition between investors (arbitrage) will quickly result in the adjustment of prices of overpriced or under-priced securities (see Box 4.1 on arbitrage). In the above case, the NPV should be exactly zero, since the discount rate equals the coupon rate.

When we looked up the value of the 10-year Brazilian government bond in June 2021, almost 6 months had passed since its issuance. Over this period, the price of the bond had fallen from $1218.7 to $1104.3 and its yield had risen from 6.9 to 9.05% (Table 4.6). Both are reflected in the calculations: the lower price means a less negative initial cash flow; and the yield raises the discount factor. Moreover, the discount factor changes, since we don’t use full year but partial year discounting. For example, the 2022 discount factor now becomes 1/(1.0905)^(194/365) = 0.955. Since 1-1-2022 is 194 days away from 21-6-2021, the 9.05% yield only applies to 194/365 = 0.53 of a year.

The difference between Tables 4.5 and 4.6 illustrates the inverse relation between bond prices and bond yields. But it does not tell us why the price went down and the yield went up. Prices and yields can change for various reasons. We will discuss this in Sect. 4.2.

The above example showed how the present value of a stream of cash flows can be calculated, and how that present value may change in relation to the discount rate used. Another matter is the development of value over time as a return is earned on it.

Let’s suppose that you have €30,000 in a savings account and earn a 2% annual interest on it. What will be the value of your savings account in 50 years’ time? The answer depends on what you do with the intermediate returns. 2% on €30,000 is €600. So, if you cash in on your interest each year, then you will earn €600 each year and the value of your account stays at €30,000. The €600 is the simple interest: interest without the effect of compounding. However, if you keep the interest in your savings account, then your annual interest receipts will rise over time since the interest is calculated over a growing amount of capital. You then receive interest not just on the original amount of €30,000 but also on the interest previously earned. This is called compounded interest. The difference in value grows exponentially over time, as Table 4.7 illustrates.

With compounded interest, the value after 50 years is €80,748 instead of €30,000. Of course, a fairer comparison is to add the received interest (50 × €600 = €30,000), in which case the value is €60,000. This is still €20,748 short, i.e. earned from interest on interest. People tend to underestimate this power of compounding. The effects are especially striking at higher returns and over longer periods of time. Figure 4.4 shows that interest on interest is insignificant at the beginning, but grows to 25% of total value (€20,748 out of €80,748).

A stacked bar graph plots the values of interest on 30000, interest on interest, and original 30000 in E U R from 1 to 49 years. The interest on 30000 and interest on interest follow an increasing trend. The value for the original 30000 is approximately equal in all years.

The compounding effect is also visible when comparing rates of compounding. As seen in Fig. 4.4, 50 years of compounding at 2% results in a final value of €80,748 and a total return of 169%. But compounding at 4% gives a final value of €213,201 (2.6 times higher) and a total return of 611% (3.6 times higher). At 8% compounding, the final value is €1.4 million, over 17 times higher than at 2% (Table 4.8).

A perpetuity is a stream of regular and equal cash flows into infinity. For example, a 3% perpetual bond may pay a €30 coupon on an annual basis forever, as Fig. 4.5 illustrates.

A timeline represents the cash flows for 5 years from 2022 to 2027. In 2022, the cash flow is minus 1000 Euros. The remaining years have cash flows of 30 Euros each.

The principal of €1000 is never repaid as such, but its value is earned back by means of the perpetual stream of €30 per year. The value of a perpetuity can be calculated as follows:

which can be reduced¹ to:

In practice, perpetual bonds do end at some point as their issuer ceases to exist. Nevertheless, some survive for centuries. For example, Yale owns a Dutch water bond from 1648 that still pays interest.²

Like a perpetuity, an annuity is a stream of equal cash flows paid at regular intervals. Unlike a perpetuity, an annuity ends after a predetermined number of payments. So, an annuity is effectively a perpetuity with an end date (n = N) and its calculation reflects that: it is a perpetuity minus that same perpetuity with a later start date (and hence discounted). The value of an annuity can be calculated as follows:

Annuities are often used in mortgages to smooth out the interest payments that are higher at the beginning than later on (when the residual amount has fallen). The borrower pays then equal amounts (divided over interest payment and principal repayment) over the lifetime of the mortgage.

Market discount rates refer to the interest rates on government securities in specific markets. The highest quality government securities are considered ‘risk-free’. Market discount rates are the benchmarks against which discount rates are determined. If they move, then all discount rates move. Such moves are effectively shifts in the scarcity of capital. When supply of capital is tight and demand is high, market discount rates (the price of capital) will be high. Conversely, when supply is ample, such as in the recent period of quantitative easing, market discount rates will be low and sometimes even negative. The scarcity is partly set by the central bank’s supply of money. But the scarcity is also a function of the risk appetite in the market and the need for capital.

The investment horizon is another consideration. Short-term interest rates are strongly influenced by the monetary policy of central banks. Box 4.4 explains the different money market segments for funds up to 1 year. Yields of government bonds are influenced by expected short-term interest rates and the term premium. Risk-averse investors demand a term premium (or risk premium) for investments in long-term bonds to compensate them for the risk of losses due to (unexpected) interest rate hikes; those losses increase with a bond’s duration, which is the sensitivity of a bond’s price to changes in interest rates (see Chap. 8).

Corporate bonds tend to carry more serious default risk and lower liquidity than government bonds. These are the main drivers of the corporate yield spread, which is the difference between yields on corporate bonds and government bonds with the same maturity and rating. For example, yields of AAA-rated corporate bonds are generally higher than AAA-rated government bond yields. The corporate yield spread can be calculated per rating class and per maturity (Table 4.9).

Default risk is the risk that a bond will not make its promised payments. This is higher for corporate bonds since, unlike governments, they do not have the option of raising taxes to meet their payment obligations. They have to meet their obligations from their own cash flow. Defaults of corporate bonds are related to the business cycle and clustered in recession times. Bonds with higher operational and sustainability risk, and higher sensitivity to the business cycle, tend to have lower ratings and higher yields. Operational and sustainability risks can stem from social and environmental issues such as labour unrest or environmental costs that put a burden on financial performance (see Chap. 8 for more detail on default risk).

The lower liquidity of corporate bonds stems from lower trading frequencies and higher transaction costs, which are due to the smaller sizes of individual corporate bond markets (i.e. per security) and less competition among bond traders.

Shareholders are the so-called residual claimants, in that they are paid out of the profits that remain after the other stakeholders and the providers of credit have been paid. As a result, equity typically carries a higher risk than corporate bonds. This higher risk is reflected in the equity risk premium, which is the expected excess return of equities over the risk-free rate. The equity risk premium is mostly related to market risk, but can also be related to social and environmental risk. The equity risk premium tends to be higher for smaller companies, more cyclical companies, and companies with weaker corporate governance. This is discussed more thoroughly in Chap. 12 on risk-return analysis.

Stock markets trade public equity, but equity can be private as well. In fact, most equity in most companies starts as private and remains private. That means there are high transaction costs involved in buying and selling it. As a result, private companies typically have a liquidity discount over public companies (i.e., the equity value of private companies is lower than that of similar public companies).

The above sections described the determinants of discount rates from the perspective of the securities and markets under consideration. But discount rates are also affected by the ones using them, i.e. the investors. Investors may have behavioural biases that induce them to use other, lower or higher, discount rates than expected. For example, people have the tendency to use the same discount rate for all investment projects within a company, even if they differ significantly in their risk profile. This is called the ‘one discount rate fits all heuristic’.

Many moral philosophers (e.g., Krznaric, 2020; Rawls, 1971) and economists (e.g. Ramsey, 1928; Stern, 2006) argue for an equal treatment of current and future generations. In economic terms, this means that the well-being of future generations gets the same weight as the well-being of the present generation (Dasgupta, 2021). So, when evaluating investment proposals for combatting climate change or preserving biodiversity, the well-being or ‘interest’ of future generations is taken fully into account alongside the interest of the present generation. This implies a zero-time preference between current and future generations. A positive time preference would advantage the present generation at the expense of future generations (facing global warming or biodiversity loss). Krznaric (2020, p.142) describes lucidly the practice of discounting the future as follows ‘Discounting is an iconic expression of the colonisation of the future, treating it as virtually empty of inhabitants’.

But there is more to social discounting than the time preference δ between current and future generations. Ramsey (1928) derives the social discount rate r^s for the appraisal of social projects:

The growth rate g is driven by growth in consumption as well as by total factor productivity growth (i.e., innovations increase efficiency of production). The latter means that future generations have cheaper and more innovative solutions than those in the present. 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. So, if future generations have better technology or if there is a greater elasticity, you should discount more. That is because the discount rate is set in such a way that consumers derive the same utility from current and future consumption. Higher future consumption thus increases the discount rate to equalise it with current consumption.

The social discount rate in Eq. (4.8) is a combination of time preference and (adjusted) consumption growth. The Ramsey rule measures the willingness to pay for a sure transfer of consumption through time. The risk premium for the social discount rate is introduced in Chap. 12.

Economists writing on global climate change (Cline, 1992; Nordhaus, 1994; Stern, 2006) have used Eq. (4.8) to identify optimum policies for correcting climate externalities (external impacts). Table 4.10 shows their different choice of parameters, assuming a growth rate of consumption of 1.3% (Stern, 2006). William Nordhaus is the outlier with a relatively large time preference of 3%. By contrast, William Cline and Nicholas Stern have a time preference close to zero, leading to discount rates of 1–2%. Dasgupta (2021) finds that the vast majority of economists find a social discount rate of 1–3% appropriate for long-run public projects. Low discount rates lead to high investment in combatting climate change, as the future benefits of restricting global warming are almost fully included because of limited discounting.

More broadly, low social discount rates imply that future environmental and social capital is almost as scarce as current environmental and social capital. Companies that add to these scarce capitals create integrated value and will be rewarded, while companies that draw these capitals down will face an increasing cost.

The financial balance sheets for all the above companies look the same now, but will be different after internalisation of social and environmental externalities (e.g. through regulation) or the anticipation of internalisation (e.g. taking provisions for future litigation), as discussed in Chap. 2. The social discount rate is developed to analyse public investment projects. We apply it in this section to show scenarios in which companies want to (or must) meet societal expectations. The empirical prediction is that companies with large social and environmental liabilities will have a higher cost of integrated capital, ceteris paribus. This risk premium will rise when the risk of internalisation—that is the likelihood of (future) regulation or change in technology or consumer preferences—rises. By contrast, companies with social and environmental assets will enjoy a lower cost of integrated capital. Chapters 12 and 13 provide an in-depth analysis of the cost of integrated capital and Chap. 15 introduces integrated balance sheets. The cost of integrated capital can be used to discount projects on an integrated basis, including the financial, social, and environmental value dimensions.

As suggested in Chap. 2, scenario analysis is a useful tool to get insight into the possible internalisation of external impacts (externalities). In the scenario analysis, a company can identify the most important societal trends for the industry in which it operates (including the likely internalisation of the main impacts) and assess its relative position (for example, degree of pollution or payment of living wages) within the industry.