Amir Sadr - Mathematical Techniques in Finance

While economics as a social science studies the behavior of economic agents in the generation, acquisition, and expenditure of goods and services, finance is focused on the acquisition and management of capital in financial markets.

Focusing on the end user, finance can be divided into personal finance, corporate finance, and government finance. Savings, investments, and loans, such as credit card, student, automobile, and home mortgage, insurance products, and estate planning are examples of personal finance. The raising of capital by borrowing and debt or selling shares and equity by a company and the management of a company's funds are the focus of corporate finance. Monetary policy, central banking, tax systems, and the oversight of the banking sector and financial markets fall under government finance.

Using the reductive definition of finance as the study of money, we follow the money to get our bearings. In accounting, a balance sheet is a snapshot of an entity's (person, corporation, country) net worth or equity: assets minus liabilities equals equity. Table 1.1shows a snapshot of the net worth of the three dominant players in the U.S. economy: households, firms, and government.

As the table shows, households hold the largest amount of equity, followed by firms, while the U.S. government runs a negative balance and is in debt. Indeed, the U.S. government is the world's biggest borrower and routinely borrows money to finance its expenditures. The breakdown of households' net worth is shown in Table 1.2.

Bonds, stocks, foreign exchange, commodities, and their derivatives are the major sectors of financial markets. Tables 1.3through 1.5show the market size as of year end 2020.

TABLE 1.1 Balance sheet of the United States at 2020 year end.

Source: U.S. Federal Reserve Z.1 Statistical Release.

TABLE 1.2 Households sector balance sheet as of 2020 year end.

TABLE 1.3 Market size ($Trillions).

Sources: World Bank, BIS, SIFMA.

TABLE 1.4U.S. bond market ($Trillions).

Source: SIFMA.

TABLE 1.5 Global derivatives market size ($Trillions).

Source: BIS.

Households typically earn wages and receive salary from firms, while firms earn income when households consume their goods and services. The government collects taxes from households and firms for its expenditures for defense, government services, infrastructure, public health, and transfer payments such as social security and Medicare. Banks and financial intermediaries facilitate the transfer of funds between these three sectors: households and firms deposit their excess funds in banks and earn interest, and banks avail these funds in the form of consumer and corporate loans. Other financial intermediaries such as investment banks, insurance companies, and investment companies provide capital and financial services to firms and individuals.

The capital markets and financial instruments facilitate the flow of funds between different sectors of the economy. Focusing on the United States, the bond market is the largest market with capitalization of $50 trillion at the end of 2020. The U.S. government routinely borrows by issuing debt in the form of coupon bonds. Similarly corporations finance their growth by issuing debt in the form of corporate coupon bonds. States and municipalities also raise capital by issuing debt for infrastructure and other projects.

TABLE 1.6 Market participants and financial products.

Banks and other financial institutions provide home mortgage, auto, and student loans in the form of level pay loans. These loans and receivables are in turn bought and securitized as mortgage‐backed and asset‐backed securities by companies originally set up by the U.S. government to promote home ownership and student loans, prominent among them are Fannie Mae (Federal National Mortgage Association), Freddie Mac (Federal Home Loan Mortgage Corporation), and Sallie Mae (Student Loan Marketing Association).