# What is a central bank's shadow rate

I was reading a WSJ article about the European Central Bank shadow rate, which is -5.1% at the moment.

The article says about the shadow rate that "Calculated with the rates on longer-dated credit instruments, this gauges where the ECB's benchmark rate might be if it could be set meaningfully below 0".

So what is this shadow rate and how is it connected to the interest rate?

• If overnight nominal (EONIA @ -.47bps) is an option on an "underlying" shadow at -510bps, then methinks the option is SO in-the-money that it blows up the affine term structure? Implicitgly, the implicit "delta" on the option is so low, any estimation error is levered up accordingly? If delta is 1% expected versus 0.5%/2% actual, then the estimate of spot could be 50% or 200% of estimated... [conceptually speaking]? – demully Mar 17 at 9:08

It looks like it's referring to Wu and Xia (2016) shadow rates. Some more media coverage is here. The core idea of a shadow rate goes back at least to Fischer Black.

### Black (1995)

Fischer Black's idea was that the nominal short rate $$r_t$$ is an option. One can either:

• Invest and earn the real shadow rate $$s_t$$, which is based on the investment opportunity set, plus expected inflation.
• Hold currency and earn 0.

Thus the shadow rate $$s_t$$ can go negative while the short rate $$r_t$$ observed in markets is always non-negative.

### Wu and Xia (2016)

Wu and Xia (2016) take that idea and estimate a shadow federal funds rate.

The short rate is maximum of a lower bound $$\underline{r}$$ and the shadow rate $$s_t$$:

$$r_t = \max(\underline{r}, s_t)$$

In a Gaussian affine term structure model (GATSM), the forward rates are affine in state variables $$X_t$$:

$$f^{GATSM}_{n, n+1, t} = a_n + b_n'X_t$$

In Wu and Xia (2016), their non-linear model implies the forward rate can be approximated by: $$f^{SRSTM}_{n, n+1, t} = \underline{r} + \sigma^\mathbb{Q}_n g\left(\frac{a_n + {b_n^\mathbb{Q}}'X_t - \underline{r}}{\sigma^\mathbb{Q}_n}\right)$$

They linearize $$g$$ around the current estimate and apply a Kalman filter to estimate. You'll want to read their paper to see precisely what they do and it's possible I'm butchering something (this isn't my area).

### References:

Black, Fischer, "Interest Rates as Options," Journal of Finance, 1995

Wu, Cynthia Jing and Fan Dora Xia, "Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound," Journal of Money, Credit, and Banking, 2016

• So according to Fischer Black, which is the difference between the nominal short rate and the real shadow rate? Also, if the real shadow rate doesn't take into account inflation, then why is it called "real" and not "nominal"? – octavian Jul 29 '17 at 6:49
• @octavian The sense in which Black used it did take into account inflation. My reading is that Black's shadow rate isn't the exact same as Wu and Xia's though. One interpretation of this shadow rate business is that it's a clever modification on affine term structure models. With an affine model, the nominal short rate may go negative, which arguably can't happen because investors can always hold cash. What if we say shadow rates follow the affine model and the observed interest rate is an option on the shadow rate? The shadow short rate can go negative and the nominal rate would still be +. – Matthew Gunn Jul 31 '17 at 14:30
• I'm honestly not up to date on the term-structure and monetary policy literature, and I'm reluctant (without reading much more) to give a strong interpretation as to what this "shadow rate" precisely represents in these models. – Matthew Gunn Jul 31 '17 at 14:32

The shadow rate is what the interest rate would be if money did not behave like an option.

The concept was created by Fischer Black and his insight was that money acts like an option. Someone with a dollar can either (1) buy something today or (2) not spend the dollar and have a dollar tomorrow. When the economy is good, an investor can loan money and, in the future, get back more money than they lent out. However, when the economy is bad, many of the loan opportunities will not return more money to an investor. So, when the economy is bad, investors "exercise the option" of money to not give loans and just holds cash until later.

In an economy without money --- a barter economy --- if the investor wanted to preserve their investment until later, they could not do nothing. They would have to do "real" investment. That is, they would have to loan their (perishable) thing of value to a company and, in the future, get back something physical. Even if it is of less value.

The relationship of the shadow rate to the nominal interest rate is pretty simple when the shadow real rate plus inflation rate is positive. In that case, the nominal interest rate is equal to the shadow real rate plus the inflation rate. This is the "economy is good" case above, where borrowers can return more money than they borrow.

When the shadow rate plus inflation rate is negative, the short-term nominal interest rate will be zero, as investors exercise the option. (Think of it as a loan to a company that does nothing with the money and just returns the cash when the loan's time is up.) Actually, the short-term nominal interest rate will be slightly above zero, because of possible interest rate moves in the future, as explained in Fischer Black's paper.

It is important to keep in mind that the "shadow rate" is part of a model of interest rates. The model is certainly a simplification and may not match reality in many ways. (For example, actual interest rates have gone negative at times.) Even assuming the model is accurate, it is difficult to calculate the current shadow rate. It has to be inferred from long-term interest rates and assumptions about inflation and how the short-term shadow rate can change. So, if an article says the economy's (not central bank's) shadow rate is -5.1%, there are a lot of assumptions built into that number. I would not trust it to the .1% accuracy that it claims.

I believe the shadow rates model is important, particularly for its definition of money. It deserves more research. The model can help us understand interest rates approaching zero and why banks are holding cash instead of loaning it.