The shadow rate is an

interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

in some

financial models Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fin ...

. It is used to measure the economy when

nominal interest rate In finance and economics, the nominal interest rate or nominal rate of interest is the rate of interest stated on a loan or investment, without any adjustments or fees. Examples of adjustments or fees # An adjustment for inflation(in contrast with ...

s come close to the

zero lower bound The Zero Lower Bound (''ZLB'') or Zero Nominal Lower Bound (''ZNLB'') is a macroeconomic problem that occurs when the short-term nominal interest rate is at or near zero, causing a liquidity trap and limiting the central bank's capacity to stimulate ...

. It was created by

Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Background Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard ...

in his final paper, "Interest Rates as Options". The shadow rate derives from Fischer Black's insight that currency is an option. If someone has money, they can either (1) spend it today or (2) not spend it and have money tomorrow. Thus, when loans would return less money than was initially loaned out, investors will choose to "exercise the option" and not loan their money. Thus, the nominal short-term interest rate is always greater than or equal to zero. In Black's model, the shadow nominal short-term rate is what the nominal short-term rate would be if it was allowed to go below the zero lower bound. When the shadow nominal short-term rate is positive, the nominal short-term rate is equal to the shadow rate. But when the shadow short-term rate is negative — such as during

deflation In economics, deflation is a decrease in the general price level of goods and services. Deflation occurs when the inflation rate falls below 0% (a negative inflation rate). Inflation reduces the value of currency over time, but sudden deflation ...

or a bad recession with low inflation — the nominal short-term rate will diverge and stay above zero. In Black's model, even when nominal short-term interest rates stay close to zero, the long-term nominal interest rates can be well above zero. This is because nominal interest rates behave like options and there is some chance that the shadow short-term rate becomes positive in the future. There is also a shadow real rate. The shadow real short-term rate is equal to the shadow nominal short-term rate minus expected inflation. Fischer Black published his paper in 1995 and mentioned that the most recent time that the USA had experienced the zero lower bound was

the Great Depression The Great Depression (19291939) was an economic shock that impacted most countries across the world. It was a period of economic depression that became evident after a major fall in stock prices in the United States. The economic contagio ...

. Shadow rate models got renewed interest with the

Financial Crisis of 2007–2008 Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fi ...

when interest rates plunged to near zero and, even, below zero in some instances.

Models

Due to the option effect, the shadow short-term rate cannot be observed directly in the market. Economists use models to infer its value from its effect on longer-term interest rates in the

yield curve In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...

. The value of the shadow short-term rate depends on assumptions about how interest rates move, so different models might calculate different values for it. Jing Cynthia Wu and Fan Dora Xia's models were published in "Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound" and in "Negative Interest Rate Policy and Yield Curve". Their rates are also available at the

Federal Reserve Bank of Atlanta The Federal Reserve Bank of Atlanta, (informally referred to as the Atlanta Fed and the Bank), is the sixth district of the 12 Federal Reserve Banks of the United States and is headquartered in midtown Atlanta, Georgia. The Atlanta Fed cover ...

. Leo Krippner's model was published in the book ''Zero Lower Bound Term Structure Modeling: A Practitioner’s Guide''. His initial models were done while at the

Reserve Bank of New Zealand The Reserve Bank of New Zealand (RBNZ, mi, Te Pūtea Matua) is the central bank of New Zealand. It was established in 1934 and is constituted under the Reserve Bank of New Zealand Act 1989. The governor of the Reserve Bank is responsible for N ...

References

{{Reflist Mathematical finance Monetary policy