Inflation DEFINITION

Inflation is a rise in prices from a set base year.

Annual Rate of Inflation DEFINITION

The annual rate of inflation is the additional proportionate cost of goods one year later. Goods that cost $1 in the base year will cost one year later.

EXAMPLE 12 Inflation

Suppose that there is constant 2% annual inflation from mid-2015 through mid-2020. What will be the projected price in mid-2020 of an item that costs $100 in mid-2015?

The compound interest formula applies with . The projected price is .

Suppose instead that inflation is 3% from mid-2015 to mid-2020. What will be the projected price in mid-2020 of an item that costs $100 in mid-2017?

Exponential Decay DEFINITION

Exponential decay is geometric growth with a negative rate of growth.

Purchasing Power RULE

The purchasing power of a dollar a year from now, relative to today, with annual inflation rate is

EXAMPLE 13 Deflated Dollars

Suppose that there is 25% annual inflation from mid-2015 through mid-2019. What would be the value of a dollar in mid-2019 in constant mid-2015 dollars? The inflation figure is unrealistic (we hope!), but it makes the calculations easy so that you can focus on the ideas.

We have . This—not 25%—is the negative interest rate, the rate at which the dollar is losing purchasing power. We have years, so the purchasing power of $1 four years from mid-2015, in terms of 2015 dollars, would be

For a more realistic rate of 3% annual inflation, we would have and negative interest rate . In 2019, the purchasing power of $1 would be

in 2015 dollars. Notice that “losing” 3% to inflation each year for 4 years amounts to losing almost—but not quite as much as—a total of . (This is just as with the previous inflation rate, where losing 25% per year for 4 years doesn’t completely reduce the value to 0.)

Suppose that there is 2% inflation from mid-2015 through mid-2020. What would be the value of a dollar in mid-2020 in constant mid-2015 dollars?

EXAMPLE 14 Depreciation of a Car

If you bought a car at the beginning of 2015 for $12,000 and its value in current dollars depreciates at a rate of 15% per year, what will be its value at the beginning of 2018 in current dollars?

We have . The compound interest formula gives

Suppose that the car cost $21,000 in mid-2015 and its value in current dollars depreciates 17% per year. What will be its value in current dollars in mid-2019, when you want to trade it in?

EXAMPLE 15 The Price of Our House and the Value of a Dollar

Where my family and I live, housing is relatively inexpensive. We bought our house in mid-1992 for $133,000 (close to the median price of U.S. housing then). What would be the equivalent cost in mid-2015 dollars?

We see from Table 21.5 that the CPI for 1992 is 140.3 and the CPI for 2015 is estimated to be 242.3. The table gives the average value for each year, which is very close to the value at mid-year. Month-by-month values are available at the Bureau of Labor Statistics Web site (data.bls.gov/pdq/).

Using the proportion, we have

or

892

so that

That’s what our house would sell for if its price exactly matched inflation. The ratio is the scaling factor for converting 1992 dollars to 2015 dollars. What we are observing is a proportion, or numerical similarity, between 1992 dollars and 2015 dollars, analogous to the geometric similarity discussed in Chapter 18 (page 738). To convert from 2015 dollars to 1992 dollars, we would multiply by .

What is the scaling factor for converting 2000 dollars to 2015 dollars?

Ways to save or invest offer different levels of risk:

Thanks to the Great Depression, when many banks failed and depositors lost everything, accounts in most banks and credit unions are now insured by the federal government for up to $250,000. If the bank or credit union fails, the government takes it over and depositors get their money back.

A bond is a loan at simple interest with repayment at the end of a fixed term. By buying bonds, investors loan to states, municipalities, and corporations for civic construction projects, company expansions, and other purposes. Bonds do not offer any guarantee: The bond issuer (the borrower) could default, meaning that the bond holder (the investor) would no longer receive interest payments and might lose the original investment.

Bonds usually can be bought and sold after their initial purchase. The selling price can vary with sentiment about creditworthiness of the issuer and with changes in interest rates.

Some bonds can be “called,” meaning that the issuer pays back the bond holder before the end of the bond term. Bonds tend to be called when interest rates fall: The issuer calls bonds previously issued at a high interest rate and then offers new bonds at a lower rate, thus saving on the cost of borrowing.

Stocks (also called securities or equities) are another way for a company to raise capital, and an opportunity for investors to share in the financial future of the company by owning shares in it. Usually, a part of the company’s profits is distributed each year to shareholders as dividends; if not, either because the company lost money, made little profit, or reinvested all of its profits, then the price of its shares may fall. The price of a company’s stock depends on “fundamentals” (measures of the “health” of the company), on the general state of the economy as a whole, and on current events.

An example of a derivative is shares in a mutual fund. A mutual fund holds an array of other companies’ stocks and bonds, perhaps concentrating on stocks of a particular kind (“green” companies, foreign companies, companies focused on growth, etc.). Investors in the mutual fund do not individually own any shares of the stocks that it invests in; the value of the mutual fund’s own shares derives from the investments that it holds. Hedge funds are basically mutual funds that are subject to less regulation than other mutual funds.

A popular kind of mutual fund is an index fund. An index fund selects investments to try to mirror the aggregate ups and downs of a particular stock exchange, as expressed by some “index” (summary) of it, and thus captures proportionate gains (but will also tend to suffer proportionate losses). Well-known indexes include the Dow Jones Industrial Average and the Standard and Poor’s (S&P) 500 Index. At another level removed, there are index funds of index funds.

Traditional derivatives include agricultural commodity futures (corn, soybeans, pork bellies), used by farmers and food processors to hedge against the risk of poor crops, as well as currency futures, used by companies to lock in a fixed exchange rate for future purchases of foreign goods and raw materials. There are even weather derivatives, whose price depends on the number of sunny days or amount of rainfall in a particular region! Derivatives arise also in the context of loans. A credit derivative bases its value not on the underlying loan, such as a mortgage, but on the risk that the loan will not be repaid. The buyer of the credit derivative does not own the mortgage loan itself, hence is not entitled to be paid by the homeowner.

A common form of credit derivative is a credit default swap, which currently amount to $26 trillion worldwide (almost $4,000 for every person on the globe). one party buys protection from a second against default on a debt owed by a third. The “protection buyer” pays a fee to the “protection seller” in return for the seller making good on the debt if the debtor defaults. The buyer thus “swaps” risk to the seller. But you aren’t going to want to believe this: The debtor can owe the debt to someone else entirely unrelated to the protection buyer or seller! In such a case, the protection buyer and seller are gambling together on whether the debtor will default. This has happened in connection with the “sovereign debt” of countries such as Greece and Ireland.

Where do you come into all this? A handful of investors made billions in the past few years through credit default swaps on “bundles” of home mortgages. Too many turned out to be subprime mortgages, that is, loans to people with poor credit ratings, whose mortgages consequently had higher interest rates. These homeowners defaulted in large numbers on their mortgage loans, in part because of the following:

The banks then foreclosed on the defaulted loans and forced the people out of their homes. (Spotlight 22.3, on page 922, looks in more detail at the mortgage crisis.)

The credit default swaps mentioned involved not the original mortgages, nor even the bundles of them—which themselves were derivatives of the underlying assets of the homes involved. The owners of the bundles had issued securities backed by the bundles—not by the homes—and the credit default swaps were on those securities. But the decline in housing prices reduced the value of the underlying asset of homes, hence of the bundles, hence of the derivatives based on the bundles. We are talking about derivatives of derivatives! The securities became “toxic assets": No one was willing to buy them at anywhere near the original price, and banks have continued to be unwilling to sell them at the big losses that would be involved.

Attempts by holders of the derivatives to foreclose on the homes have been to some degree thwarted by the fact that the derivative owners are not the direct owners of the mortgage loans. In some cases, there has been so much trading of loans and of derivatives on them, without sufficient documentation, that it is impossible to establish to whom a loan is owed.

Warren Buffett, one of the world’s richest men, was prescient in 2002 when he called financial derivatives “time bombs” and “financial weapons of mass destruction.”

Financial derivatives, combined with greed and exploitation, indeed destroyed the prosperity and optimism that the world enjoyed in the early years of the 21st century.

Real Rate of Growth RULE

The real (effective) annual rate of growth of an investment at annual interest rate with annual inflation rate is