Unit 4: The Financial Sector
4.1: Money
Money is a strange object. We all associate money with wealth, but money itself doesn’t make us wealthy. We’re not really out to get money per se. Rather, what we’re interested in are the things
that you can use money to buy. Money is just an easy way to facilitate the exchange.
History of Money and the Banking System
Money has been around in human society for a long time, mainly because barter trading is not always practical. If you raise cows but you want to buy a car, how can you guarantee that the person selling cars is interested in getting cows in exchange? And how many cows or oranges trade for a car? Money avoids this problem by translating the value of everything to the same scale.
Historically, most moneys were gold or precious metals with some inherent value. But nowadays we use paper money that has no real value in and of itself. The history of paper money is interesting. Back when gold was the primary unit of exchange, people would deposit their gold with banks, which would in turn issue a paper certificate entitling the depositor to return to the bank to claim the gold. Eventually, traders realized that rather than carrying around the gold itself, they could just trade the deposit certificates. So that’s how paper money was born – basically receipts that private banks issued for gold that was on deposit at the bank.
Eventually, the bank realized that most people didn’t return to the bank very often to collect their gold. Rather, they were just trading their certificates. So the bank didn’t actually have to keep all of the gold on hand that people had on deposit. It started lending out the gold for interest, which is how the modern banking system was born.
Functions of Money
Lots of things have functioned as money in different economies. Most modern economies use paper money, but stones, cows and seashells have all arisen as money in different societies. In order to be a money, an item needs to satisfy three properties.
Unit of account: Money is a common measure for the value of all goods and services.
Store of value: Money retains its value over time.
Desirable Properties of Money
Some things work better than other things as money. Here are some desirable properties of money.
Scarce (but not too scarce): Dirt is not a good money because there is too much of it around for it to have much value. At the same time, original Rembrandt paintings would not make a good currency because there are not enough of them for a monetary system to function. Something like our paper money is ideal, because the supply can be controlled.
Portable: Two-ton stones are not a good money because they are difficult to carry around to transact business.
Divisible: A good money can be practically used for both large and small transactions. Live cows are not a good money because they are not divisible into smaller amounts.
Fiat Money
For most of history, we used commodity money, which is money that has some value other than
its use as a currency. Gold and cows were commodity moneys when they were used for transactions because they have inherent value and a use other than their use as money.
By contrast, what we use now is fiat money, which is money that has basically no intrinsic value other than its use as currency. The paper currency issued by the government has value only because the government has passed laws making it acceptable in exchange for goods and services. The currency itself has no inherent value.
It is important to note currencies today are not “backed up” by gold or anything else. At one time, dollars were redeemable and equivalent to gold. But that’s not true anymore. The only value that your dollar has is whatever someone else is willing to give you for it.
will accept them because they can’t use them on day 29! And so on and so on, if you keep going with this argument, then US dollars lose all their value immediately. The point is that the only reason a store owner accepts your money today is because he expects to be able to use it in the future. But the only reason the next person accepts it is because she has faith that she’ll be able to keep using it, etc… What gives money its value now is confidence in its continued future value.
Central Banking
Each country’s money supply is controlled by its central bank. In the United States, the central bank is called the Federal Reserve Board. In the UK, it is the Bank of England and in the European Union, there is the European Central Bank. Central banks are responsible for the following.
Supervise and regulate private banks
Act as a lender to private banks – if a private bank does not have enough cash in reserve to pay claims to depositors, the central bank will lend money to the private bank
Issue paper currency
Check-clearing service – arranging for the transfer of funds between banks when a check written on one bank is deposited to another bank
Maintain the government’s accounts
Monetary policy – managing the supply of money
Measuring the Money Supply
In general, money is any asset that is widely acceptable as payment and is highly liquid, where liquidity refers to how easily something can be converted to cash. Cash is the most liquid asset, and the money in your checking account is also highly liquid since it is easy to convert it to cash. On the other hand, a piece of property or artwork – although they are assets with value – are not considered to be money since they are not liquid.
The money supply is the amount of money in the hands of the public at any point in time. There are three definitions.
M1 is the narrowest definition of money. It includes currency and also checking account balances, which are highly liquid.
M2 includes M1, and also savings accounts, money market accounts and certificates of deposit less than $100,000. These are not quite as liquid as cash or checking account balances, but still fairly liquid.
M2 = M1 + savings account balances + money market accounts + CDs less than $100,000
M3 is the broadest definition of the money supply. It includes everything in M2, in addition to certificates of deposit greater than $100,000.
M3 = M2 + CDs greater than $100,000.
Notice that M3 > M2 > M1 by definition, since M3 includes M1 and M2, plus other things. To give you an idea of the numbers, M1 in the US is currently around $2.5 trillion and M2 is around $10.7 trillion. Of this, hard currency is about $1.1 trillion.
Fractional Reserve Banking
In addition to holding on to deposits, the main function of banks is to broker transactions between borrowers and lenders. When you deposit money in a bank, the bank doesn’t just hold on to the money – it lends it out. The bank makes profits by charging a higher interest rate on loans than what it pays to customers on their deposits.
Why do we use banks instead of just lending the money out ourselves? First, the bank brings together depositors and borrowers, which makes the transactions much easier to arrange. Second, the bank absorbs the risk of defaults. If a bank lends money out and the borrower doesn’t repay it, the bank is on the hook – the depositors are still entitled to their money.
Banks are required to hold on to a certain percentage of their deposits in reserve. They can’t lend out all their money since some depositors might want to show up and withdraw money. The
required reserve ratio (RRR) is the percentage of total deposits that banks are required to hold in reserve. The RRR in the United States is 10%. What this means is that, if a bank has $50 million total deposited by its customers, it must keep $5 million in reserve. It can either keep reserves as cash or it can keep reserves as deposits at the Fed.
The Money Multiplier
An individual deposits $1000 cash in Bank A. This immediately adds $1000 in deposits to the banking system.
Bank A is required to keep $100 of the $1000 in reserve but lends out the other $900. This $900 will, at some point, end up deposited in another bank. Say it is deposited at Bank B. This adds another $900 in deposits to the banking system.
Bank B is required to keep $90 out of the $900 in reserve but lends out the other $810. This $810 will, at some point, end up deposited in another bank. Say it is deposited at Bank C. This adds another $810 in deposits to the banking system.
Bank C is required to keep $81 out of the $810 in reserve but lends out the other $729. This $729 will, at some point, end up deposited in another bank. Say it is deposited at Bank D. This adds another $729 in deposits to the banking system, etc…
Overall, the total increase in bank deposits resulting from this initial $1000 deposit is:
Δ deposits = $1000 + $900 + 810 + $729 + ⋯
We can derive this total precisely by using the money multiplier, which is defined as:
money multiplier = 1
𝑅𝑅𝑅
To use the money multiplier, the total change in deposits resulting from an initial injection into the system is:
Δ deposits = (money multiplier) × (initial injection)
For the example above, the money multiplier is 1
𝑅𝑅𝑅 = 1
0.1= 10, and so the total increase in
deposits that results from the initial $1000 deposit is:
Δ deposits = (10) × ($1000) = $10,000
The initial deposit of $1000 actually creates $10,000 of new deposits in the banking system after we take account of the successive loans and deposits that result from the initial injection.
is known as a bank run, and if too many depositors try to withdraw their money at once, it can lead to a collapse of the banking system.
The money multiplier actually gives the maximum change in deposits that can result from an initial injection into the banking system. The actual increase in deposits may be less than what the money multiplier gives for two reasons.
Excess reserves: In the example above, each bank held only the minimum 10% in reserve and lent out the absolute maximum that it could. But banks sometimes hold more in reserve than they are required to. This reduces the amount of lending and in turn reduces the total increase in deposits.
Currency drain: We assumed in the example above that, when the first bank lent out $900, this $900 ended up eventually in a bank. But that’s not true if the person who borrows the $900 holds on to some of it as cash. This money that is held as cash and not re-deposited into the banking system thus will also reduce the total increase in bank deposits.
Demand for Money
When economists talk about demand for money, we mean the demand for holding wealth as cash rather than in other forms. If you have $1 million of wealth, your demand for money is the amount of your wealth that you want to keep as money instead of in some other form.
Why hold any money at all? Financial assets like stocks and bonds pay interest. Even if you don’t like risk, you could buy a bond from the US Treasury Department that is risk-free. So, why would you hold on to any of your wealth as money – which pays no interest – instead of holding it as some investment that pays interest? There are three sources of demand for money.
Transactions demand: People need a certain amount of cash to pay for everyday expenditures.
Precautionary demand: People sometimes hold extra cash that they don’t necessarily intend to spend right away, but as a reserve for emergencies.
Speculative demand: If financial markets are unstable, people sometimes hold on to cash while they are waiting for other financial investments to look more attractive.
your bed, you give up the interest that you could have earned by investing the money in something like a bond that pays interest. The cost of choosing to hold cash is exactly the interest that you could have earned if you had put the money into a bond instead of holding on to it in cash.
This concept leads to a demand curve for money. As the interest rate rises, the opportunity cost of choosing to hold on to cash rises, and so people want less money. At higher interest rates, people with a lot of wealth would rather have bonds or something that pays a return, rather than holding on to their wealth as money. In short, at higher interest rates, the amount of money that people want to hold falls. This relationship is illustrated below.
As usual, changes in the interest rate 𝑟 cause a movement along the demand curve. But there are a couple of things that shift the demand curve for money. In other words, these are factors other than the interest rate that affect the amount of money that people want to hold.
Changes in income: People with higher income spend more. As income rises, demand for money rises (shifts right). As income falls, demand for money falls (shifts left).
Changes in the price level: If the things that people buy increase in price, they will need to hold more money. As price level rises, demand for money rises (shifts right). As price level falls, demand for money falls (shifts left).
The Supply of Money
The supply of money is fixed by the Federal Reserve Board at any particular time. It can be raised or lowered by policy, but at any particular point in time it is fixed.
Bond Prices and Interest Rates
Consider a consumer who has two choices for holding her wealth. She can hold cash, which pays no interest but which is liquid. Alternatively, she can hold a bond, which pays interest. A bond is a promise to pay some fixed amount in the future.
Bonds sell for a price lower than what they will pay in the future. For example, a bond that will pay you $1000 in one year sells for less than $1000 today. The difference is the return that the holder of the bond earns.
The interest rate on a bond is simply the annual percentage return that a holder earns over what he paid for the bond. Suppose that a bond will pay $1000 in one year and sells for $800 today. The interest earned on the bond is:
𝑟 =1000 − 800
800 = 0.25 = 25%
On the other hand, suppose you have to buy the bond for $900 today rather than $800. In that case, the profit that you earn is lower. The interest earned on the bond is now:
𝑟 = 1000 − 900
From these examples we see that bond prices and interest rates are inversely related. The higher the price of the bond today, the less of a return you will earn on the bond when you cash it in for $1000 in one year. If the bonds sell for a low price today, then the interest you earn is higher. By definition, bond prices falling is equivalent to interest rates rising.
Money Market Equilibrium
The equilibrium interest rate in the economy is the interest rate where the supply of money and the demand for money are equal. In the diagram below, the equilibrium interest rate in the economy is 7%.
It is important to understand the dynamics of the economy that cause the interest rate to adjust to the equilibrium level.
What happens when the interest rate in the economy is higher than 7%?
From the diagram, you can see that supply of money will be higher than demand for money. In other words, there is an excess supply of money. When the interest rate is high, people want more bonds and less money.
The excess demand for bonds will cause the price of bonds to rise. This higher price of bonds is equivalent to saying that the interest earned on the bonds falls. The interest rate will fall.
What happens when the interest rate in the economy is lower than 7%?
The excess supply of bonds will cause the price of bonds to fall. This lower price of bonds is equivalent to saying that the interest earned on the bonds rises. The interest rate will rise.
4.2: Monetary Policy
The Federal Reserve Board can control the supply of money in the economy. In this section, we will discuss how they control the money supply and the effects on the economy of money supply changes.
The Tools of Monetary Policy
The Fed uses three policy tools to manage the supply of money.
Changing the required reserve ratio (RRR): By increasing the RRR, banks are forced
to hold on to their deposits instead of lending them out. This slows down the deposit multiplier process and reduces the supply of money in the hands of the public. By lowering the RRR, banks are free to hold on to less of their deposits and make more loans to customers. This results in more money circulating in the hands of the public.
Open market operations: The Fed can sell bonds to banks or private citizens. By doing so, it reduces the amount of money in the hands of banks and private citizens, since they pay the Fed for the bonds using money. The Fed can also buy bonds from banks or private citizens. By doing so, it increases the amount of money in the hands of the public since the Fed is buying the bonds by paying the bondholders with new money.
Changing the discount rate: The discount rate is the rate at which the Fed lends money to private banks. If the discount rate is high, banks are likely to hold excess reserves (making fewer loans) in order to ensure that they have enough cash on reserve when depositors want to withdraw their money – because it would be expensive for the bank to borrow it from the Fed if it runs out. These excess reserves slow the deposit multiplier process and reduce the amount of money in the hands of the public. Conversely, if the discount rate is low, banks feel comfortable making lots of loans because they’re not worried about running out of reserves – if they run out of reserves, they can just borrow from the Fed at a low rate. This increase in lending from private banks, because of the low discount rate maintained by the Fed, therefore increases the supply of money circulating in the hands of the public.
Tool Increase Money Supply
Decrease Money Supply
RRR Decrease Increase
Open Market
Operations Buy bonds Sell bonds
Discount Rate Decrease Increase
Practically speaking, the Fed hardly ever changes the RRR because it disrupts the banking system. Open market operations are the day-to-day tool that the Fed uses to manage the money supply. There is a Federal Open Market Committee (FOMC) that regularly manages bond purchases and sales. The Fed occasionally changes the discount rate as a signal about its desired change in the money supply.
Expansionary Monetary Policy
Suppose that the Fed raises the supply of money by using one of the tools shown in the table above. The diagram below shows the effect on the money market.
The intuition for this change is straightforward. Expansionary monetary policy increases the liquidity in the banking system by increasing the money that banks have on hand. By doing so, loans are relatively easy to get because banks have plenty of funds. Since loans are in abundant supply, the interest rate charged on loans falls.
Contractionary Monetary Policy
Suppose instead that the Fed lowers the supply of money. The diagram below shows the effect on the money market.
Notice that the equilibrium interest rate rises. Decreases in the money supply raise the interest rate.
Again, the intuition makes sense. Contractionary monetary policy reduces the liquidity of banks and the funds that banks have available. Thus, funds are scarce and so banks can charge high interest rates on loans.
Effects of Interest Rate Changes
Consumption: When interest rates rise, consumers spend less. One reason is that consumers will choose to save more of their money instead of spending it if they can get high interest rates by saving and investing. Another reason is that high interest rates make it more expensive for consumers to borrow and spend money.
Investment: When interest rates rise, firms spend less on investment spending like new factories and new machines. This is true whether the firm borrows the money for its investment spending or whether it uses its own money. If the firm is borrowing the money for its investment projects, then increases in the interest rate make it more expensive for the firm to borrow money, so it won’t pursue as many investment projects. If the firm uses its own money, then at higher interest rates it is more worth it for the firm to use its funds to buy financial assets and earn high interest rates rather than using its money to build factories and machines. Either way, higher interest rates mean less business spending on investments.
Net exports: When interest rates rise, foreigners buy fewer American goods and services, and Americans buy more foreign goods and services. Thus, higher interest rates reduce net exports. We will cover the reason for this in detail in the unit on international economics and exchange rates. Basically, the reason is that higher US interest rates make American assets more attractive to foreigners, which causes the US dollar to rise in value. This rise in the value of the US dollar makes US exports expensive for foreigners to buy and makes foreign products cheaper for Americans who hold dollars.
Summarizing, we can say the following.
Higher interest rates reduce consumption spending, investment spending and net exports.
Overall, higher interest rates reduce aggregate expenditures.
Lower interest rates increase consumption spending, investment spending and net exports.
Overall, lower interest rates increase aggregate expenditures.
Monetary Policy – Putting it Together
Monetary policy raises or lowers the supply of money. The immediate effect is to change the interest rate, and we just saw in the previous section that changes in the interest rate affect aggregate expenditures. Let us review both expansionary and contractionary monetary policy.
Expansionary monetary policy involves increasing the supply of money (lower RRR, buying bonds or lower discount rate)
→ Supply of money rises
→ Interest rate falls
→ Aggregate expenditures rise
→ GDP rises (with a multiplier)
Contractionary monetary policy involves reducing the supply of money (higher RRR,
selling bonds or higher discount rate)
→ Supply of money falls
→ Interest rate rises
→ Aggregate expenditures fall
→ GDP falls (with a multiplier)
Both fiscal and monetary policy attempt to manage the economy by managing the level of aggregate expenditures. In turn, changing the level of aggregate expenditures impacts GDP through the multiplier process.
So now we can see the dual policy tools of fiscal and monetary policy. Fiscal policy changes aggregate expenditures directly by changing government spending or taxes. Monetary policy changes aggregate expenditures indirectly by changing the interest rate, which in turn changes aggregate expenditures.
4.3: Accounting for time – present value
Many choices in economics involve decision making with a time element. That is, costs are paid and revenues are received at different points in time. One example is firms making decisions about capital investments. Ford has to shell out a large amount of money to build a factory today, with the profits generated by the factory not coming until later. This is also important for any financial investments. You buy stocks and bonds today, but the returns on the stocks and bonds come in the future.
Present and Future Value
The core principle is that $1 paid in the future is worth less than $1 paid today. The reason is simple. If I pay you $1 today, you could invest it and earn interest, which means in the future you would have more than $1. So it’s better to have the $1 today rather than to have it in the future. This is the idea behind present value.
Suppose a bank pays a 5% interest rate and you put $100 into the bank.
After 1 year, your account is worth 100(1 + .05) = $105. Think about what you’re doing by distributing the $100 across the parentheses. 100 × 1 is your original $100 principal. The other piece 100 × .05 is the $5 interest that you earned. Combined, your account is worth $105.
In the second year, you earn 5% interest not just on the $100 but on the whole $105 balance. This is an important concept known as compounding. You earn interest not only on your principal, but also on your previously accumulated interest. Since you start the second year with 100(1 + .05) = $105, you earn 5% return on this whole sum, so the value of your account at the end of the second year is:
[100(1 + .05)](1 + .05) = 100(1 + .05)2 = 110.25
Notice that this is slightly higher than $110, to reflect not only another year of interest on your original $100 balance, but also on the interest from the first year.
In the third year, you start out with 100(1 + .05)2 = 110.25 in your account. You earn 5% interest
on this entire amount, so the value of your account at the end of the third year is:
Notice that the compounding is increasing at an increasing rate. Financial advisors often do these kinds of calculations for people to show them that, even with a little bit of money starting out, if you leave it for long enough, it can turn into a large sum by the end because of compound interest. It’s good to start saving when you’re young. Even if you only save a little bit, you can get a lot by just leaving it alone and letting the interest compound.
Continuing on like this, after 𝑛 years, the amount of money in your account is:
100(1 + .05)𝑛
If we generalize this, suppose you put $𝑃𝑉 into an account today. Here, 𝑃𝑉 stands for “present value”. The interest rate on the account is 𝑟. You leave the money in your account for 𝑛 years. Then the value of your account in the future $𝐹𝑉 can be calculated as:
𝐹𝑉 = 𝑃𝑉(1 + 𝑟)𝑛
In words, this formula gives the future value 𝐹𝑉 of some sum of money today 𝑃𝑉. It takes present
values and translates them into future values.
If we turn this formula around, we can solve it for 𝑃𝑉:
𝑃𝑉 = 𝐹𝑉
(1 + 𝑟)𝑛
This formula gives you the value today of some amount of money that is to be paid in the future. It takes future values and translates them into present values.
For example, suppose the interest rate is 3% and I offer to give you $100 five years from today. The value today of this promise can be calculated using the formula:
𝑃𝑉 = 100
(1 + .03)5 = 86.26
Let’s do the same calculation again, but with a different interest rate. Suppose now that the interest rate is 10% and I offer to give you $100 five years from today. The value today of this promise is:
𝑃𝑉 = 100
(1 + .10)5 = 62.09
When the interest rate is low, present and future values are close together – not much interest is lost by getting the money in the future instead of getting it today, so the future value is almost the same as the present value. But as the interest rate rises, the present value gets lower and lower. There is a lot of lost interest, so getting the $100 in the future is worth a lot less today because of the opportunity cost of the foregone interest.
Example: Investment in a New Drug
A pharmaceutical company is considering whether to invest in developing a new drug. Development of the drug requires a $10 million investment today. If they develop the drug, the company will earn a profit of $4 million in one year, another $4 million in two years and then another $4 million in three years. The patent expires after three years, so there’s no more profit.
A very incorrect answer is to say that the investment should be made because the profits add up to $12 million and the development only costs $10 million. This is the wrong answer because the profits are coming in the future, whereas the cost of the development has to be paid today. And we that future values are worth less than present values. The $12 million and the $10 million are not comparable to each other.
In order to solve the problem, we need to take the future profits and convert them into present values. This will allow us to take the $10 million payment today and compare it to the future profits, once they have been translated back to their equivalent present values.
Suppose the interest rate is 8%. Since $4 million is received in one year, $4 million is received in 2 years and $4 million is received in 3 years, we can use the present value formula to calculate the present value of the profits:
𝑃𝑉 = $4 million
(1 + .08)1+
$4 million (1 + .08)2+
$4 million
The project costs $10 million profit and generates profits that, translated back to their present equivalent today, are worth $10.31 million. So, when the interest rate is 8%, the project is worth doing because the profits are greater than the development costs.
Suppose instead that the interest rate is 12%. In this case, the present value of the profits from the projects is:
𝑃𝑉 =$4 million
(1 + .12)1+
$4 million (1 + .12)2+
$4 million
(1 + .12)3 = $9.61 million
In this case, the project is not worth doing. The company should not pay $10 million to develop a drug that will only yield profits of $9.61 million when translated back to today’s present values.
What’s going on here? When the interest rate is 12%, the company is better just taking the $10 million and buying financial investments to collect the 12% return. This is more profitable than investing in the drug. But when the interest rate is 8%, the company won’t make so much profit by earning interest on the money, so in that case it’s better to develop the new drug. This is a simple way of explaining why firm investment falls when the interest rate rises.
Infinite-Horizon Payments
Suppose you win a prize that will pay you $100 a year forever, starting in one year. If the interest rate is 4%, the present value of the prize is:
𝑃𝑉 = $100
(1 + .04)1+
$100 (1 + .04)2+
$100 (1 + .04)3+
$100
(1 + .04)4+ ⋯
This would be hard to add up by hand because there are an infinite number of terms! Luckily, there is a simple formula for solving problems of this variety. For a payment of $𝑋 per year starting one year from today and continuing forever into the future, the present value is:
𝑃𝑉 =𝑋
𝑟
So, for our example above, we can calculate that the present value is exactly:
𝑃𝑉 =100
One way to think about this is that, if you had $2500 in a bank that paid 4% interest. You could just leave the $2500 there and collect 4% interest every year, which is exactly $100. In other words, a stream of $100 a year forever is equivalent to getting a $2500 lump sum today.
4.4: Financial Assets and Returns
Stocks and Bonds
A bond is a promise to pay some fixed amount at a certain time in the future. The maturity date
is the date at which you can cash in the bond and the face value is the amount that you cash the bond in for on the maturity date. A bond with a maturity of 20 years will pay the face value of the bond 20 years after the date of the original sale. Some bonds also pay coupon payments, which are small payments made along the way until the maturity date.
Bonds are issued by firms or governments to raise money. A bond is a legal contract. Unless the organization that issues the bond goes bankrupt, the holder of a bond is guaranteed to receive the face value of the bond when the bond matures.
A stock is a share of ownership in a corporation. Unlike a bond, a stock does not involve any promise by the company to pay anything in the future. Some companies pay out part of their annual profits as a dividend. Also, if the stock goes up in value, you can sell it for a higher price than what you bought it for. But the stock itself is only worth whatever someone will pay you for it; there is no guarantee from the company. While bonds are legal debt instruments, you are not guaranteed anything by owning a stock. Thus, stocks are much riskier than bonds.
Like bonds, stocks are issued by corporations to raise money. However, notice that the corporation only receives money from the initial sale of the stock. Google raises money through initial stock issuance. When investors trade the stocks back and forth after this point, it is the investors themselves who make money, not Google.
Bonds represent a higher proportion of financial assets than stocks. For the US economy, the ratio of bond assets to stock assets is about 3 to 1. However, the stock market receives more attention in the news because it is more volatile. Bond values are relatively stable because bonds involve promises to pay. But people’s perceptions of what a company is worth can vary greatly, so stock prices fluctuate a lot.
Bond Prices
𝑃𝑉𝑏𝑜𝑛𝑑=
𝐶 (1 + 𝑟)1+
𝐶 (1 + 𝑟)2+
𝐶
(1 + 𝑟)3+ ⋯ +
𝐶 + 𝐹𝑉 (1 + 𝑟)𝑛
Basically, you receive 𝐶 every year until the bond matures, and then in 𝑛 years when the bond matures you also receive the face value of the bond 𝐹𝑉. The value of the bond is the present value of the stream of payments.
Incidentally, from this equation you can see that increases in the interest rate reduce the present value of the bond. This confirms what we said earlier that interest rates and bond prices are inversely related.
For example, consider a bond with a face value of $100 that matures in 5 years and pays a coupon payment of $10 each year. The market interest rate is 5%. The present value of the bond is:
𝑃𝑉𝑏𝑜𝑛𝑑 = 10
(1 + .05)1+
10 (1 + .05)2 +
10 (1 + .05)3+
10 (1 + .05)4+
10 + 100
(1 + .05)5 = $121.65
In words, the value today of the stream of future payments that the bond will generate is $121.65. Since the market for bonds is competitive, the price for which the bond sells will be driven up very close to $121.65. After all, if it only sold for $110 – someone else would be willing to step in and pay more since it is worth $121.65. For this reason, the formula above is often called the bond pricing formula.
The only uncertainty in the bond pricing formula is that the company might go bankrupt, in which case the bond cannot be redeemed. Also, the market interest rate in the future might be uncertain. In particular, the real rate of interest is uncertain because the future inflation rate is unknown.
Term Structure of Interest Rates
Suppose an asset is worth $200 today and is worth $210 in one year. Then we can just use the formula for a percentage change to calculate the return as:
𝑟 =210 − 200
200 = .05 = 5%
Suppose you buy a CD from a bank for $100 and it will be worth $125 in 5 years. It’s misleading to say that the return is 25%, because your money is tied up for five years. Instead, we usually talk about the annualized return on assets – what would have been the equivalent return each year. The formula for calculating annualized return is:
𝑟𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑= (𝑉𝑓𝑢𝑡𝑢𝑟𝑒 𝑉𝑡𝑜𝑑𝑎𝑦)
1/𝑛
− 1
So for the CD that you buy from the bank, the annualized return that you are earning each year on this CD is:
𝑟𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑= (125 100)
1/5
− 1 = .04564
So an asset worth $100 today that is worth $125 in 5 years is equivalent to getting a 4.564% return
each year. To confirm this answer, suppose that you put $100 in a bank and you leave it there for 5 years, earning a 4.564% return each year. After 5 years, you would have:
100(1 + .04564)5 = $125
Any time the term structure on an asset is not 1 year, the returns are usually stated in annualized terms so that the returns on various assets can be compared irrespective of the term structure. All returns are then stated in terms of what would have been the equivalent one-year return.
Stock Prices
A stock pays a dividend 𝐷 each year, and in 𝑛 years the holder of the stock expects to be able to sell the stock for 𝐸𝑃. The present value of this stream of payments is:
𝑃𝑉𝑏𝑜𝑛𝑑= 𝐷
(1 + 𝑟)1+
𝐷 (1 + 𝑟)2+
𝐷
(1 + 𝑟)3+ ⋯ +
𝐷 + 𝐸𝑃 (1 + 𝑟)𝑛
Risk and Return
Assets that involve a higher level of risk generally pay their investors a higher return in order to compensate for the risk associated with the investment. If two assets paid the same return, but one was safe and the other was risky, all investors would obviously go for the safe one. Thus, riskier assets have to pay higher returns to attract investors. But how do we think about risk?
For a simple example, suppose that there are two stocks. Stock in an ice cream company pays you a 5% return if it turns out to be a cold year but a 7% return if it turns out to be a hot year. Stock in a hot chocolate company, on the other hand, pays you a 12% return if it turns out to be a cold year but a 0% return if it turns out to be a hot year. There’s an equal 50/50 chance of a cold year or a hot year.
You might be tempted to say that the hot chocolate stock is riskier. After all, both stocks on average give you a 6% return. But the ice cream stock either pays 5% or 7% – which is not particularly volatile. But the hot chocolate stock either pays nothing or pays a huge 12% return.
But this is not necessarily the right answer. Suppose all the rest of your money is invested in a water park that you own – which is obviously more profitable during hot years. In that case, it is safer to invest in the hot chocolate company! By investing in the hot chocolate you are providing yourself with what investors call a “hedge” – basically, insurance for yourself. If it’s a hot year, your water park will be profitable but your hot chocolate stock will be lower. But if it’s a cold year, then your water park will not be profitable, but your hot chocolate stock will make a lot of money. In this way, buying the hot chocolate stock instead of the ice cream stock reduces your overall exposure to risk. This is why people who work for Mercedes often buy stock in BMW – if something terrible happens at Mercedes, BMW stock would probably rise and so these workers are hedging their bets.
The overall point is that the risk associated with a financial investment depends not just on how volatile it is, but on how it interacts with other stocks in the market. A stock that tends to go up and down together with other stocks on the market is more risky – how much riskier depends on how closely it moves together with other stocks. And we know that riskier stocks have to pay higher returns. On the other hand, a stock that tends to go up when the rest of the market is down reduces risk for investors by providing a hedge. And stocks with lower risk can get away with paying lower returns on average.
Random Walk Theory
A completely different theory on stock prices says that stock prices and returns are unpredictable. If we let 𝑃𝑡 be the current price of the stock and we let 𝑃𝑡+1 be the price of the stock tomorrow, then according to the random walk theory:
𝑃𝑡+1 = 𝑃𝑡+ 𝜀
Where 𝜀 is random. In other words, the stock price might go up, down or stay the same. And what it does is random and unpredictable. Unlike CAPM, according to the random walk theory, changes in stock prices are unpredictable and are not systematic.
Economic Perspective on Financial Forecasting
Economists are very skeptical about trying to forecast financial variables, especially stock prices. No matter how much faith people put in their ability to predict stock prices, the simple fact is nobody has ever developed a technique for choosing stocks that can beat the average, overall market return. In other words, there is no strategy for choosing stocks that can consistently, over a long period of time, exceed the return you would get by just choosing a diverse portfolio of stocks or choosing them randomly. This is hard to believe given how people claim they can beat the market, but it is unequivocally true.
Here are two examples: First, monkeys choosing stocks randomly by throwing darts at the newspaper page consistently outperform financial forecasting experts. Second, over the decade 2000-2010, if every time financial experts recommended buying a stock, you instead sold the stock, you would have made more money than you made by following their advice.
To economists, the reason for this is simple economics: no strategy based on publicly available information can ever beat the market consistently. Suppose Apple announces the release of a new iPhone and you know that the new model is excellent and will cause the stock price to increase. But, if you know this, then so does everybody else! By the time you get around to buying Apple stock, the demand for the stock has already gone up, and this information is already built into the share price. This argument does not apply to private information. You can make profits by using insider information that other people don’t have, but that’s illegal.
