Unit 6 The financial sector: Debt, money, and financial markets
6.9 Introducing financial markets
Borrowing and saving (lending) allows households to consume at times when they have little or no income—it can benefit everyone by enabling them to smooth their consumption. Borrowing also allows people to invest and generate future income.
We have learned that the banking system offers one way to borrow and lend, and provides money as a means of exchange and unit of account. But the value of currency is eroded by inflation, and commercial banks pay little or no interest on deposits. People who want to save for their retirement don’t usually hold their wealth in the form of money. In general, money represents a very small fraction of household assets, although this differs across poor and rich households as the data in Section 6.11 will show.
To understand how people borrow, save, and invest in the economy as a whole, we need to go beyond the banking system to the wide range of alternative assets, real and financial, and to the markets where these assets are traded.
People hold wealth in order to save or invest for the future, and they can hold it in the form of real or financial assets. The real assets that constitute aggregate wealth for an economy include a wide range of productive assets, including capital goods, land, and housing. Firms own many of these assets, but since firms are owned by households, households collectively are the ultimate owners of aggregate wealth. There are also many types of financial assets, including bonds and shares as well as bank deposits.
The financial sector, including financial markets, plays an important role in channelling household savings towards investment by companies in the productive assets that will increase the aggregate wealth in the economy. It also enables households to borrow long term to buy a real asset, a house, which is the most important asset for many households, as well as productive capital and real estate for small businesses.
Financial markets and financial assets
In financial markets, savers and borrowers buy and sell financial assets called traded securities; in particular, bonds and shares (introduced in Section 6.3). When companies want to raise funds to invest in productive assets, they can sell shares in the ownership of the company, or reinvest profits already earned, or borrow. Larger companies borrow by issuing bonds (as governments do).
Issuing (that is, selling) a bond is a form of borrowing. But unlike a bank loan, a bond is a tradeable form of debt. The lender who buys the bond initially can sell it to someone else. Both government and corporate bonds (that is, bonds issued by companies) are bought and sold in the bond market. The company makes coupon payments to whoever owns the bond at the time, until it reaches maturity. When companies finance their activities by borrowing (via bank loans or bonds), this is called debt finance.
Likewise, shares (also known as equities, or stocks) are traded in the stock market. Recall that we have used the term ‘equity’ to refer to the net worth of a bank or company. Shares simply refer to a ‘share’ of the equity and the word equities is synonymous with shares or stocks.
When shares are first sold by a company, this is called an initial public offering (or IPO). Since shareholders are co-owners of the company, they legally own any profits the company makes. Shareholders are not entitled to regular payments, but the company may decide to pay out some of its profits to them in the form of dividends. Profits not paid out (retained earnings) are then reinvested into its ongoing operations. If reinvested profitably, the net worth (equity) of the company increases and each share is worth more.
Retained earnings are the main source of funding for investment projects in well-established companies. Small and medium-sized firms use bank loans, and start-ups are often funded by venture capitalists. If a start-up is successful, it ‘goes public’ with an IPO and the initial venture capitalist investors can receive a high payment in exchange for giving up their ownership stake. When companies finance their activities either by selling new shares (via IPOs, venture capital funding) or via retained earnings, this is referred to as equity finance.
Although bond and stock markets enable companies to raise funds by issuing new bonds and shares, most of the trading that takes place is in existing bonds and shares. As a result, the prices of financial assets traded in financial markets can, and do, change on an almost minute-by-minute basis. We shall see that this can result in significant volatility in the rates of return on financial investments.
The role of the financial sector
Figure 6.13 helps to map out the role the financial sector plays in the wider economy. It shows how both banks and financial markets enable households to channel their savings into different forms of productive capital.
Rates of return: Income, capital gains, and inflation
If you are an investor, the choice between different assets—whether financial or real—depends on what you expect to get back in the future: in other words, on the rate of return. This concept applies to any form of asset, whether real or financial.
In the case of a guaranteed bank deposit, the rate of return is simply equal to the rate of interest, but this equivalence between interest rates and rates of return is actually the exception rather than the rule.
For example, in Section 6.4 we defined the rate of return as the amount the borrower actually pays back, compared to the initial loan:
\[\text{rate of return on loans} = \frac{\text{total amount borrower actually pays back – loan}}{\text{loan}}\]
So the rate of return is equal to the interest rate if it is repaid as promised; but if there is a risk of default it is lower.
This formula for the rate of return on a loan can be rearranged as:
This is the same as the definition in Section 6.4, rearranged by adding 1 to both sides of the equation.
\[1 + \text{rate of return}=\frac{\text{total amount borrower pays back}}{\text{loan}}\]
This expression is actually a particular case of a general formula for the rate of return on any asset or investment, which is easy to remember:
By doing some algebra, you can verify that this is the same as the definition in Section 6.4. If you separate out the fraction on the right-hand side into its two components, the second term is equal to –(loan/loan) = –1. If you then add 1 to both sides of the equation you get the new expression.
\[1 + \text{rate of return}=\frac{\text{what you get back}}{\text{what you put in}}\]
For example, suppose you deposit $100 at the bank for a year, and the bank pays interest into the account. If the balance in your account is $110 at the end of the year, the fraction is 110/100 = 1.10, so your rate of return is 10%. The same is true if you buy shares in a business for $1,000, and receive no dividends but sell them a year later for $1,100. And if Julia plants 30 units of grain and harvests 45 units, her rate of return is 50%.
If you are investing in an asset that will be bought and sold in markets (whether financial markets or real estate markets), a crucial determinant of the rate of return is what happens to the market price of the asset.
In Extension 6.9, we show that ‘what you get back’ can be separated into two parts: the value of the asset when you sell it, and any income you receive from it while you own it: for example, interest payments, coupon payments on a bond, or dividends on a share.
This allows us to decompose the percentage rate of return into two components:
\[\text{rate of return } (\%) = \text{capital gain or loss } (\%) + \text{income } (\%)\]
- capital gain
- If the market value of an asset increases, the owner of an asset receives a capital gain equal to the difference between the current and previous market prices.
The capital gain or loss is the change in value (the difference between the future and current prices) and the income is the annual income, both as a percentage of the initial investment (‘what you put in’ = the current price).
Note that any asset that has a future value which depends on future market prices is a risky investment, even if the income component is guaranteed, because market prices are uncertain. We shall see that, both for financial assets like bonds and shares, and for real assets like houses, the volatility of price changes, and hence the capital gain or loss, plays a crucial role in increasing the riskiness of investments.
- real interest rate
- An interest rate corrected for expected inflation (that is, the nominal interest rate minus the expected rate of inflation). It represents how many goods in the future one gets for the goods not consumed now. See also: nominal interest rate, interest rate, rate of interest.
The extension to this section shows how to derive this formula. It is analogous to the calculation of the real interest rate in the Fisher equation (read Section 5.2) except that typically the Fisher equation uses the expected, rather than actual, inflation rate.
Note also that in the case of financial assets, where ‘what you get back’ and ‘what you put in’ is measured in currency terms, the ratio of the two tells us the nominal rate of return. To correct for the effect of inflation, we calculate the real rate of return using a simple approximation.
\[\text{real rate of return} \approx \text{nominal rate of return} − \text{inflation rate}\]
Risky assets, expected rates of return, and risk premia
The previous section showed that for many assets, ‘what you get back’ is uncertain. In that case, you can calculate the expected rate of return—what you expect to obtain, on average over the possible outcomes.
- risk premium
- Risky assets have to offer a higher rate of return than risk-free assets to compensate the buyer for risk. The risk premium is the difference between the return on the risky asset, and the risk-free rate.
But an individual investor’s choice will usually also depend on the degree of risk involved: the variability in the possible outcomes. Some assets are risk-free (such as a savings account offering a guaranteed interest rate). Investing directly in productive assets is typically more risky (for example, buying a van to start a delivery business), and so are shares and many other financial and real assets. If your expected return on two assets is the same, you will probably choose the less risky one. So in general, investors will choose risky assets only if they expect to earn a higher rate of return than safer alternatives. The difference between the expected rates of returns on risky and safe assets is called a risk premium.
Extension 6.9 Decomposing the rates of return on assets
To analyse the rate of return on an investment, it is often helpful to decompose it into the separate effects of different factors. In this extension, we demonstrate two decompositions. First, a rate of return expressed in nominal terms can be decomposed into the real rate and the rate of inflation; we also show that this is the basis of the Fisher equation used in Unit 5. Second, the rate of return on a marketable asset can be decomposed into a ‘capital gain/loss’ from the change in the market price, and an income component.
In the main text, we introduced the general definition of the rate of return, given by
\[1 + \text{rate of return}=\frac{\text{what you get back}}{\text{what you put in}}\]
We can calculate the rate of return over any time period, but we often compare the annual rates of return on different assets, in which case the period is one year. But however long the period, you can apply the formula in the same way: the denominator is the amount you invest now, and the numerator is the amount you will get back at the end of the period.
If you invest some of your wealth in any asset, the rate of return is the proportional rate of change in that part of your wealth at the end of a specified period—the change as a proportion of the initial investment. A key feature of returns in general is that there is no guarantee that the return will be positive: as the ‘small print’ will remind you when you purchase an asset, the value of your wealth can go down as well as up.
To understand why, start with the definition of percentage changes. For example, the percentage change in price levels (inflation) is given by \(\pi = \frac{\Delta P}{P_0}\), where \(\Delta P = P_1 – P_0\). Substitute this definition into the right-hand expression to obtain the left-hand expression: \(\frac{P_1}{P_0} = \frac{P_0 + \Delta P}{P_0} = \frac{P_0}{P_0} + \frac{\Delta P}{P_0} = 1 + \pi\).
We can use similar expressions for other proportional rates of change in economics. For example, the growth rate, \(g\), of GDP, and the rate of inflation, \(\pi\) (the rate of increase of the general price level, \(P\), typically the CPI) satisfy:
\[1 + g = \frac{Y_1}{Y_0} \text{ and } 1 + \pi = \frac{P_1}{P_0}\]
Note that, while we typically think of both \(g\) and \(\pi\) as growth rates, and therefore as positive numbers, these can also both be negative: \(g\) can be negative if output falls in a recession (if \(Y_1\) is less than \(Y_0\)); and \(\pi\) can be negative if prices are falling (deflation).
Nominal and real rates of return
Typically, returns are measured in nominal terms: both the top and bottom of the ratio are measured in units of currency. But what matters to the investor is the real value of what they get back in terms of spending power, and hence the real return.
In the algebra that follows, we shall use the notation RoRn to represent the nominal return and RoRr to represent the real return.
If the value of your wealth invested in the asset increases from \(X_0\) (what you put in) to \(X_1\) (what you get back) in nominal terms, the nominal rate of return RoRn is given by
\[1+ \text{RoR}_\text{n} = \frac{X_1}{X_0}\]
But, with positive inflation, if the price level grows at rate \(\pi\) from \(P_0\) to \(P_1\) over the same period, then the real value of your wealth—in terms of purchasing power—changes from \(X_0/P_0\) to \(X_1/P_1\). The real rate of return on your investment, RoRr, is lower than the nominal return, RoRn.
To calculate RoRr, we need to find the rate of increase of \(X/P\). If \(X\) increases faster than \(P\), \(X/P\) will rise, so RoRr will be positive. But if \(P\) rises faster than \(X\), RoRr will be negative.
Therefore the real rate of return, RoRr, satisfies the expression:
\[1 + \text{RoR}_\text{r} = \frac{X_1 / P_1}{X_0 / P_0} = \frac{X_1 / X_0}{P_1 / P_0}\]
To move from the second expression to the third expression, we used two key rules for handing fractions: 1) dividing by a fraction is the same as multiplying by the same fraction turned upside down , so \(\frac{X_1 / P_1}{X_0 / P_0} = \frac{X_1}{P_1} \times \frac{P_0}{X_0}\), and 2) when we multiply fractions, we can change the order of multiplication of terms in the top or bottom of each fraction, so \(\frac{X_1}{P_1} \times \frac{P_0}{X_0} = \frac{X_1}{P_0} \times \frac{P_0}{X_1}\). Finally, if we then apply rule 1) again, in reverse, we get \(\frac{X_1}{X_0} \times \frac{P_0}{P_1} = \frac{X_1 / X_0}{P_1 / P_0}\), which is the third expression.
By applying our earlier definitions, the top of the fraction in the final expression on the right-hand side is equal to 1 + RoRn, the nominal return, and the bottom is equal to \(1+ \pi\). Therefore:
\[1 + \text{RoR}_\text{r} = \frac{1 + \text{RoR}_\text{n}}{1 + \pi}\]
or equivalently, if we subtract 1 from both sides:
\[\text{RoR}_\text{r} = \frac{1 + \text{RoR}_\text{n}}{1 + \pi} - 1\]
This expression holds precisely, so if you are doing calculations of real returns with actual data, and coding a formula into a spreadsheet or other software, this is the formula you should use. However, to aid our understanding of this expression, we can also use the approximation given in the main text, which we now derive.
A useful approximation
To simplify the expression for RoRr, we can employ a commonly used trick in mathematics: we can always express the number 1 as a fraction, as long as the top and bottom of the fraction are identical. In this case, we can exploit the fact that \(\frac{1 + \pi}{1 + \pi}=1\), and substitute the left-hand expression into the exact expression for RoRr, giving
\[\text{RoR}_\text{r} = \frac{1 + \text{RoR}_\text{n}}{1 + \pi} - \frac{1+\pi}{1+\pi}\]
at which point we can apply another rule of fractions, that is, if two fractions have a common denominator (the same expression on the bottom of each fraction) we can combine them into a single fraction, to give the following:
\[\text{RoR}_\text{r} = \frac{1 + \text{RoR}_\text{n} - (1+\pi)}{1+\pi}\]
We can then simplify by cancelling the 1 and –1 in the numerator:
\[\text{RoR}_\text{r} = \frac{\text{RoR}_\text{n} - \pi}{1 + \pi}\]
This expression is still an exact definition of the real return (so if you were using actual data you could also use this expression, although at the expense of a few more keystrokes). But, at least in countries where the inflation rate is relatively low, the value of \(1+ \pi\) is close to one, so dividing the top of the ratio by \(1+\pi\) makes very little difference to the answer. Therefore, when inflation is low, we get this approximation:
\[\text{RoR}_\text{r} \approx \text{RoR}_\text{n} - \pi\]
Or equivalently, as stated in the main text:
\[\text{real rate of return} \approx \text{nominal rate of return} - \text{inflation rate}\]
How good is this approximation?
If the inflation rate is low, the approximation is quite close to the actual value. For example, if the nominal return RoRn = 8% and the inflation rate is 2%, then the approximation gives:
\[\text{RoR}_\text{r} \approx \text{RoR}_\text{n} - \pi = 0.08 - 0.02 = 8\% - 2\% = 6\%\]
In comparison, the exact expression gives:
\[\text{RoR}_\text{r} = \frac{\text{RoR}_\text{n} - \pi}{1 + \pi} = \frac{0.08 - 0.02}{1.02} = 0.588 = 5.88\%\]
Hence, for any positive rate of inflation, the approximation always overstates the true value; but with low inflation, the approximate return is relatively close to the actual return.
But, as we will discuss in Unit 7, many countries have much higher inflation rates than 2%. The higher the inflation rate, the worse the approximation. For example, with a nominal return RoRn = 56% and an inflation rate of 50%, the approximation would again imply \(\text{RoR}_\text{r} \approx 56\% - 50\% = 6\%\), but the correct answer would be:
\[\text{RoR}_\text{r} = \frac{\text{RoR}_\text{n} - \pi}{1 + \pi} = \frac{0.56 - 0.05}{1.5} = \frac{0.06}{1.5} = 4\%\]
While the approximation still tells you, correctly, that the real return is positive, the approximate return is much less accurate. This is why, when doing calculations with actual data, you should always use the exact expression, rather than the approximation.
Exercise E6.1 Approximating real returns
Go to the International Monetary Fund’s inflation rate webpage and choose four countries, one for each category of inflation shown on their map (0–3%, 3–10%, 10–25%, and 25% or more).
Suppose the real return on an asset, when calculated using the approximate formula and the inflation rates from each of these countries, is always 7%.
- For each of your chosen countries, use the exact formula to calculate the actual rate of return, and calculate how far the actual rate of return is from the approximate rate of return. Comment on your answers.
- Derive a mathematical expression (formula) for the difference between the approximate rate of return and the actual rate of return (approximate minus actual). Use this mathematical expression to explain why (1) the approximation always understates the true value, and (2) the approximation becomes more accurate at smaller (positive) inflation rates.
Nominal and real interest rates: The Fisher equation
In Unit 5, we introduced the Fisher equation that relates nominal and real interest rates:
\[r \approx i - \pi^E\]
In this equation, \(r\) is the real interest rate, \(i\) is the policy rate and \(\pi^E\) is the expected inflation rate. This relationship is a special case of our more general approximation for the real rate of return. The policy rate is risk-free in nominal terms, so in this case the nominal return is equal to the nominal interest rate, \(\text{RoR}_\text{n} = i\). But the real return on an investment paying the policy rate is still risky, due to inflation (as Figure 6.17 showed). So the actual return on such an investment would, using our approximation, be given by:
\[\text{RoR}_\text{r} \approx i - \pi\]
We can then interpret the real interest rate as the expected real return on an investment that pays the policy rate, \(i\). Using our approximation again, we can therefore write the Fisher equation as:
\[\text{RoR}_\text{r}^E \approx i - \pi^E\]
If you are deciding whether to open a risk-free savings account (for example), you know the nominal interest rate. But what you really want to know is what your savings will be worth in a year’s time, in terms of purchasing power. If you have an estimate of the rate of inflation, \(\pi\), over the next year, you could use the equation above to work out your expected real rate of return from the savings account. But since inflation is always uncertain, the real interest rate, \(r\), is always an estimate. In general, this is what we mean when we refer to the ‘real interest rate’ in macroeconomics: it is calculated using expected inflation because that is the information that savers and borrowers have when they make their decisions, and which therefore determines economic outcomes.
Capital gains and income
Suppose now that you invest in an asset that may be bought and sold in a market, and keep it for one year. It could be a real asset, such as a car or a house, or a financial asset such as a share in a firm.
What you put in, \(X_0\), is simply the price, \(p_0\), that you pay for it now—the current market value. But we can separate \(X_1\), or ‘what you get back’, into two parts:
\[X_1 = p_1 + y\]
\(p\) is not to be confused with \(P\), which we are using to represent the CPI, or general price level.
where, \(p_1\), is the price of the asset in one year’s time (the amount you would get if you sold it then) and \(y\) is the income that you receive from the asset during the year: for example, interest payments, coupon payments on a bond, or dividends on a share.
Substituting this expression into the numerator of our expression for the nominal return, we get:
\[1 + \text{RoR}_\text{n} = \frac{X_1}{X_0} = \frac{p_1+y}{p_0}\]
If we add and subtract \(p_0\) from the top of the fraction on the right-hand side, we get \(1+ \text{RoR}_\text{n} = \frac{p_1 - p_0 + y + p_0}{p_0}\).
If we then separate the right-hand expression into two fractions, the second fraction is simply equal to 1: \(1 + \text{RoR}_\text{n} = \frac{p_1 - p_0 + y}{p_0} + \frac{p_0}{p_0} = \frac{\Delta p}{p_0} + \frac{y}{p_0} + 1\)
Subtracting 1 from both sides gives the answer.
Rearranging this expression and writing \(\Delta p = p_1 - p_0\) for the price change over the year, with some further manipulations, we can decompose the nominal rate of return as:
\[\text{RoR}_\text{n} = \frac{\Delta p}{p_0} + \frac{y}{p_0}\]
Interpreted in words, we get the decomposition in the main text:
\[\text{rate of return} (\%) = \text{capital gain or loss} (\%) + \text{income} (\%)\]
where the capital gain, \(\Delta p\), and income, \(y\), are both expressed as percentages of the investment, \(p_0\). As discussed in the main text, for many assets, the riskiness of the return is primarily driven by capital gains or losses.
The nominal return can then be corrected for inflation to give the real return. Alternatively, the same decomposition applies to the real return if all variables in the expression are already measured in real terms.
