# 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?

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