Shadow Rate – Purpose, Assumptions, Models

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Dan Buckley
Dan Buckley is an US-based trader, consultant, and part-time writer with a background in macroeconomics and mathematical finance. He trades and writes about a variety of asset classes, including equities, fixed income, commodities, currencies, and interest rates. As a writer, his goal is to explain trading and finance concepts in levels of detail that could appeal to a range of audiences, from novice traders to those with more experienced backgrounds.

The shadow rate is a concept in financial models, primarily used to measure the economy when nominal interest rates approach the zero lower bound.

It was introduced by Fischer Black (1938-1995) in his final paper, “Interest Rates as Options,” and has gained renewed interest following the Financial Crisis of 2007-2008 when nominal interest rates hit zero and policymakers moved to bond buying (quantitative easing).

This article will explore the purpose, assumptions, and models of the shadow rate.


Key Takeaways – Shadow Rate

  • The shadow rate is a concept developed by Fischer Black to address the limitations of nominal interest rates when they approach the zero lower bound.
  • It serves as an alternative measure to assess the level of monetary policy accommodation in such circumstances.
  • By treating nominal interest rates as options, the shadow rate provides a way to capture the effects of unconventional monetary policies that central banks may employ when traditional interest rate adjustments are no longer effective (e.g., QE).
  • It helps policymakers and economists gain insights into the dynamics of interest rates and their impact on economic conditions.
  • We briefly describe various types of shadow rate models.
  • We also describe the effect of the shadow rate on the prices of equities.


Purpose of the Shadow Rate

The shadow rate serves as a tool to help economists and policymakers understand the behavior of interest rates when nominal short-term rates are constrained by the zero lower bound.

It provides insights into the underlying economic conditions, even when nominal rates are not able to fully reflect them due to their zero(-ish) lower bound limitation. (Rates can go a little bit below zero in some circumstances.)


Assumptions Underlying the Shadow Rate

Currency as an Option

The core assumption behind the shadow rate is that currency is an option.

Individuals can either spend their money today or save it for future spending.

This concept implies that nominal short-term interest rates will always be greater than or equal to zero, as investors would choose not to lend their money if loans were to return less than the initial amount.

This kind of thought exercise is essentially the basis of all of finance (spend money now or spend what you expect to be more later).

The Zero Lower Bound

The zero lower bound assumption is important for the shadow rate concept. It suggests that nominal interest rates cannot fall much below zero.

This is because, as we’ve covered in other articles, central bankers are always going to target an inflation rate of at least zero.

And by extension, if the inflation rate is at least zero, and cash and bonds are yielding less than that, there’s a negative real return, which means purchasing power is being eroded.

In that case, there’s a lack of incentive to hold the debt, which can cause interest rates to rise (which typically isn’t wanted when economic conditions are weak).

This creates a constraint on how much they can decrease to stimulate economic activity during periods of deflation or recession.


Shadow Rate Models

Black Zero Lower Bound (ZLB) Model

Fischer Black’s model establishes a relationship between the shadow nominal short-term rate and the actual nominal short-term rate.

When the shadow rate is positive, both rates are equal.

However, when the shadow rate turns negative, the actual nominal short-term rate remains at or above zero, while the shadow rate can continue to decrease.

This model also demonstrates that long-term nominal interest rates can stay above zero even when short-term rates are constrained by the zero lower bound – because there’s a possibility that the shadow rate will become positive in the future.

The Shadow Real Rate

The shadow real short-term rate is another aspect of the shadow rate concept.

It is calculated by subtracting expected inflation from the shadow nominal short-term rate (basically what’s traditionally known as the Fisher equation).

This rate provides a more accurate representation of the real cost of borrowing, taking into account the expected change in the purchasing power of money over time.

Wu-Xia Shadow Rate

The Wu-Xia shadow rate is another attempt to measure and quantify the effects of unconventional monetary policy when interest rates are at or near zero.

This rate is named after its developers, economists Jing Cynthia Wu of the University of Notre Dame and Fan Dora Xia of Barclays Capital.

The Wu-Xia shadow rate was devised to encapsulate the impact of these unconventional policies into a single comparable rate.

It’s like an “equivalent” interest rate that could have the same effect on the economy as these unconventional methods.

So even when the nominal interest rate is at zero, the shadow rate can be negative, indicating the stimulative effects of unconventional policies.

This shadow rate allows economists and policymakers to better understand the true stance of monetary policy when traditional measures fall short, making it an important tool in economic analysis and research.

The Wu-Xia shadow rate is tracked by the Atlanta Federal Reserve.

Krippner Shadow Rate Model

Developed by Leo Krippner, this model is based on a yield curve model that can account for both negative and positive interest rates.

Bauer and Rudebusch (2016) Model

This model uses a nonlinear Gaussian term structure model with a shadow rate that can go below zero.

Christensen and Rudebusch (2015) Model

This two-factor shadow rate term structure model is used to analyze the term premium component of U.S. Treasury yields.

Ichiue and Ueno (2013) Model

This model accommodates the lower bound of interest rates by incorporating it into an affine term structure model.


These models use different mathematical and statistical techniques to model the shadow rate and capture the effects of unconventional monetary policy.

Each has its strengths and weaknesses, and the appropriate model to use depends on the specific economic circumstances and the goals of the analysis.


Why don’t interest rates often fall below zero? Why do policymakers use quantitative easing instead?

Interest rates can and sometimes do fall below zero in some economies.

This usually occurs when conventional monetary policy tools have become ineffective and central banks try to stimulate the economy.

Negative interest rates effectively mean that banks are charged for storing their money with the central bank, which encourages banks to lend more and individuals and businesses to invest or spend rather than save.

This theoretically can help to stimulate economic activity and combat deflation.

Several countries, like Japan and some in the Eurozone, have had periods of negative interest rates.

The US had them briefly on short-term Treasury bills (but has never gone below a 0.00-0.25% range for the fed funds rate).

However, there are several reasons why negative interest rates might be avoided:

  1. Bank Profits: Negative rates can squeeze banks’ net interest margins, reducing the profitability of lending and potentially making them less willing or able to provide loans.
  2. Savings Behavior: Negative rates could lead individuals to save more, rather than less, if they see it as a sign of significant economic problems. This could exacerbate economic downturns, the opposite of the intended effect.
  3. Cash Hoarding: In an environment with negative interest rates, businesses and individuals might choose to hoard cash, which doesn’t incur negative interest, rather than depositing it in banks or investing.
  4. Financial Stability: Negative rates could encourage excessive risk-taking, as investors search for yield, and this could potentially lead to asset bubbles and other financial stability concerns.

Because of these concerns, central banks might turn to other tools, such as quantitative easing (QE), when interest rates are close to or at zero.

QE involves the central bank creating new money and using it to buy assets like government bonds from financial institutions.

This increases the money supply and encourages lending and investment by lowering long-term interest rates, boosting asset prices, and improving market liquidity.

It can also signal to markets that the central bank is committed to stimulating the economy, which can have positive effects on confidence and expectations.

However, QE is not without its own potential downsides, including the risk of asset price bubbles, potential distortions to financial markets, and the challenge of eventually unwinding the policy.

Like all tools of monetary policy, its use involves a balancing act of potential benefits and risks.


Shadow Rate and Equity Valuations

The shadow rate, as a measurement of the stance of monetary policy even when interest rates are at or near zero, can have significant effects on various asset classes, including stocks.

Here’s how it typically works:

Equity Valuation

Lower interest rates, including negative shadow rates, often increase the valuation of equities.

Lower rates reduce the cost of borrowing, making it cheaper for companies to finance their operations, potentially increasing their profits.

Lower rates also reduce the discount rate used to calculate the present value of future cash flows, which can increase the valuation of equities.

Risk Appetite

Unconventional monetary policies reflected in the shadow rate can encourage investors to take on more risk.

When rates are low, safe investments (like cash and bonds) tend to provide lower returns, which can lead investors to riskier assets, like equities, in search of higher returns.

Moreover, when the shadow rate is very low there is often a lot of money and credit flowing into the system, which is going to make its way into assets.

This can push up equity prices.

Currency Effects

If the shadow rate in one country is lower than in other countries, it can lead to depreciation of the country’s currency.

This can make the country’s exports cheaper and increase the profits of companies with significant export operations, potentially increasing their equity prices.

Wealth Effect

When interest rates (including the shadow rate) are low, it can lead to an increase in asset prices, including home prices.

This can make people feel wealthier and encourage them to spend more, boosting the economy and potentially increasing corporate profits and equity prices.

These effects are not guaranteed, of course.

They depend on many other factors, including overall economic conditions, investor sentiment, and the specific circumstances of individual companies.

But the shadow rate can be an important indicator of the potential impact of monetary policy on equities and other asset classes.


FAQs – Shadow Rate

What is the primary purpose of the shadow rate in financial models?

The main purpose of the shadow rate is to provide insights into the underlying economic conditions when nominal interest rates are constrained by the zero lower bound.

This helps economists and policymakers better understand the behavior of interest rates during periods of deflation or recessionary periods when rates hit zero and make more informed decisions to stimulate growth and recovery.

The shadow rate emerged after the global financial crisis of 2008, during which many central banks, such as the US Federal Reserve, reduced their federal funds rate (the interest rate at which depository institutions lend reserve balances to other depository institutions overnight) to nearly zero.

This is often referred to as the zero lower bound. In such situations, conventional monetary policy tools become less effective as they cannot lower interest rates below zero to stimulate economic activity.

However, central banks still have methods to stimulate the economy, like quantitative easing (buying government bonds to inject money into the economy) and forward guidance (publicly communicating future monetary policy intentions).

These unconventional policies, while effective, are challenging to measure using traditional methods.

Why is the concept of currency as an option important for the shadow rate?

Currency as an option is a fundamental assumption in the shadow rate model.

It highlights that individuals can either spend their money today or save it for future spending, which implies that nominal short-term interest rates will always be greater than or equal to zero.

Can the shadow rate ever be negative, and what does it imply?

Yes, the shadow rate can be negative, particularly during periods of deflation or a recession with low inflation.

When the shadow rate is negative, the actual nominal short-term rate may remain at or above zero. In this case, the shadow rate may reflect the underlying economic conditions more accurately, providing insights into the potential impact of monetary policy even when nominal rates are constrained by the zero lower bound.

What is the Wu-Xia shadow rate model?

Developed by economists Jing Cynthia Wu and Fan Dora Xia, this model uses a no-arbitrage term structure model of Treasury yields that accommodates a lower bound on short rates.

How does the shadow rate model help us understand long-term nominal interest rates?

The shadow rate model demonstrates why long-term nominal interest rates might stay above zero even when short-term rates are constrained by the zero lower bound.

This is because long-term interest rates factor in the future and an economy isn’t likely to stay weak forever, and therefore the shadow nominal rate is likley to be positive over the long run.

Interest rates largely stay in line with nominal growth over the long term.

If inflation is at least zero and growth is at least zero (labor growth + productivity growth), then nominal interest rates should also be above zero.

How can the shadow rate concept help policymakers make better decisions during economic downturns?

By analyzing the shadow rate, policymakers can better understand underlying economic conditions that may not be apparent from nominal interest rates alone due to their zero lower bound constraint.

This information can be useful in shaping monetary policy, as it provides a clearer understanding of the potential effects of policy interventions on interest rates and the broader economy, even when nominal rates are close to zero.



The shadow rate, as originally introduced by Fischer Black, offers a valuable tool for understanding interest rate behavior when nominal rates are limited by the zero lower bound.

By considering the purpose, assumptions, models, and the shadow real rate, economists and policymakers can better comprehend the underlying economic conditions during periods of deflation and/or recession, and make better decisions to get the economy on more stable footing.