### What Are Interest Rates?

In order to understand negative interest rates, it's important to take a step back and think about interest rates more generally. Simply put, an interest rate is a rate of return on savings. For example, at a 5% per year interest rate, $1 saved today will return $1.05 one year from now. Some other relevant points about interest rates are as follows:

- Interest rates can take a nominal or a real form. Nominal interest rates specify the dollar amount returned after a defined period of time and do not take inflation into account. Real interest rates, on the other hand, take inflation into account and reflect by how much purchasing power increases as a result of savings. In other words, a real interest rate can be thought of as a return on savings in terms of stuff rather than money directly.
- To a close approximation, nominal interest rates are equal to their corresponding real interest rates plus the rate of inflation. Alternatively, real interest rates are equal to their corresponding nominal interest rates minus the rate of inflation.
- Interest rates, like most other quantities in a free-market economy, are determined by the forces of supply and demand. Real interest rates are determined by supply and demand in the market for loanable funds, and nominal interest rates are determined by the supply and demand of money.
- Interest rates can be used to mitigate business cycles. Specifically, lower interest rates stimulate economic activity and have an expansionary effect, whereas higher interest rates rein in the economy and have a contractionary effect.

### How do Negative Interest Rates Work?

Mathematically speaking, negative interest rates work in exactly the same fashion as their more commonplace positive equivalents. To see how let's look at a few examples:

Assume that a nominal interest rate is equal to 2% per year. In this case, $1 saved today will return $1 * (1 + .02) = $1.02 one year from now.

Now assume that a nominal interest rate is equal to -2% per year. In this case, $1 saved today will return $1 * (1 + -.02) = $0.98 one year from now.

Easy, right? We can do the same thing with real interest rates.

Assume that a real interest rate is equal to 3% per year. In this case, $1 saved today will be able to purchase 3% more stuff next year (i.e. one will have 1.03 times as much purchasing power).

Now assume that a real interest rate is equal to -3% per year. In this case, $1 saved today will be able to purchase 3% less stuff next year (i.e. one will have 0.97 times as much purchasing power).

It is also the case that the nominal interest rate is equal to the real interest rate plus the rate of inflation, regardless of whether the underlying interest rates are positive or negative.

### Negative Real Interest Rates

Conceptually speaking, negative real interest rates make more sense than negative nominal interest rates, since they simply amount to a decrease in purchasing power. For example, if nominal interest rates are at 2% and inflation is at 3%, then the real interest rate is equal to -1%. The money that investors put in the bank does grow in a nominal sense, but inflation more than eats away at the nominal return in terms of purchasing power.

### Negative Nominal Interest Rates

Negative nominal interest rates, on the other hand, take a little getting used to. After all, a nominal interest rate of -2% per year means that a saver who deposits $1 in a bank will get back 98 cents after one year. Who would do that when they could keep cash under their mattress instead and have $1 after one year instead?

The simple answer in most cases is that there are logistical costs associated with keeping cash under one's mattress -- most obviously, one would be wise to purchase a safe for the cash, which has costs of its own. By this logic, it stands to reason that negative nominal interest rates wouldn't automatically cause all savers to take their cash out of banks and put it under their (real or metaphorical) mattresses. Large institutional clients, in particular, likely wouldn't want to take the trouble to figure out what to do with physical delivery of large sums of cash. That said, the incentive to clear these logistical hurdles increases as nominal interest rates get more negative. Furthermore, negative nominal interest rates sometimes happen implicitly via the imposition of bank fees without causing all customers to run away.

The above scenario refers to a situation where negative interest rates are set directly. It should be noted that negative nominal interest rates could also arise indirectly if bond prices rise to levels high enough to result in negative yields. (The logistical differences arises mainly from the fact that bond yields are determined largely in secondary markets.)

### Negative Nominal Interest Rates and Monetary Policy

When considering only nonnegative interest rates, monetary policy faces an important limitation -- if lowering nominal interest rates acts as an economic stimulus, then what is a central bank to do when nominal interest rates hit zero? In this nonnegative world, a central bank must resort to other means of monetary stimulus -- perhaps quantitative easing, which aims to change a different set of interest rates than traditional monetary policy. Alternatively, an economy is left with fiscal stimulus as it only means of trying to help an economy in recession, which comes with its own set of difficulties.

### Examples of Negative Interest Rates

Up until the recent past, negative nominal interest rates were, not surprisingly, basically uncharted territory, and even some central bank leaders are unsure about how introducing negative nominal interest rates will play out. Despite these concerns, several central banks have implemented negative nominal interest rates, and even Federal Reserve chair Janet Yellen said that she would consider such a strategy if it were deemed necessary.

Below is a list of examples of economies that have implemented negative nominal interest rates:

- In January 2016, the Bank of Japan adopted a negative nominal interest rate policy. As one would expect, this decision was made in an attempt to revive the sluggish Japanese economy. Specifically, the Bank of Japan said it will apply a rate of negative 0.1 percent to excess reserves that financial institutional place at the bank, effective February 16. In addition, the Bank of Japan did not set a lower bound on the yields of Japanese government bonds, meaning that longer-maturity Japanese interest rates may also end up in negative territory,
- In February 2016, Sweden lowered nominal interest rates to negative 0.5 percent from negative 0.25%. (This implies to some degree that Sweden didn't find negative nominal interest rates to be catastrophic!)
- In February 2016, the European Central Bank set a negative nominal interest rate of negative 0.3%. Part of the stated reason for this policy was to convince banks to increase lending rather than to hold onto excess reserves.
- Even Switzerland, which is a major banking center, shows yields on bonds that run as low as negative 1.12 percent.

As far as is currently known, none of these policies result in a mass exodus of cash from banking systems in these countries. (To be fair, most negative interest rate policies are implemented so as to target commercial banks rather than bank customers directly, but different interest rates tend to be highly correlated.) Market reactions to interest rates going negative are somewhat mixed (though lower interest rates generally trigger a positive market reaction). In addition, negative nominal interest rates can also result in inflation and currency depreciation, but this is actually the desired goal of the negative nominal interest rate policy in some cases.

### (Unintended) Consequences of Negative Nominal Interest Rates

The implementation of negative nominal interest rates could result in changes in behavior that extend far beyond the banking sector itself. Secondary considerations include things such as the following:

- Would people start trying to prepay bills in order to avoid holding their money and having it be subject to a negative nominal interest rate? (What would my landlord think if I tried to give him a year's rent up front?) Would the companies refuse to accept early payment because then they would be left holding the cash that declines in value? (Companies currently act like it's doing others a favor by offering credit- would they start acting like it's doing a favor to allow prepayment or prompt payment?)
- Would governments incur significantly increased costs of currency printing? After all, storing cash under mattresses requires actual physical currency in a way that cash in a bank account does not.
- Would fireproof safes and safe-deposit boxes suddenly become very lucrative businesses?

### The Ethics of Negative Interest Rates

Not surprisingly, negative nominal interest rates are not without their critics. On a basic level, some assert that negative interest rates are contrary to the fundamental notion of saving and the role that saving plays in an economy. Some, such as Bill Gross, even claim that negative nominal interest rates are a threat to the very idea of capitalism itself. In addition, countries such as Germany assert that the business models of their financial institutions depend critically on positive nominal interest rates, especially when products such as insurance are considered.

In addition, the legality of negative nominal interest rates is questioned in some jurisdictions. In the United States, for example, it is not obvious whether the Federal Reserve Act allows for such a policy to be implemented directly