## Wednesday, October 7, 2015

### Interest rate parity and neo-Fisherism

Scott Sumner mentioned interest rate parity today which inspired me to see what the information equilibrium model has to say about it. The basic idea is that exchange rates and interest rates  between two countries should come to an equilibrium so that if the interest rate $r$ for a period $p$ rate (I used 3-month/90-day) and the (expected) exchange rate maintain the no-arbitrage condition

$$(1 + r_{1}) = \frac{X(t + p)}{X(t)} (1 + r_{2})$$

Now the exchange rate between two countries in the information equilibrium model, assuming money supply growth rate $\mu$ (over the same period p), is:

$$X(t) = \alpha \frac{(M_{1} e^{\mu_{1} t/p})^{(\kappa_{1} - 1)}}{(M_{2} e^{\mu_{2} t/p})^{(\kappa_{2} - 1)}}$$

We can show

$$\frac{X(t + p)}{X(t)} = \exp \left( \mu_{1} (\kappa_{1} -1 ) - \mu_{2} (\kappa_{2} -1 )\right)$$

Assuming these rates are small we can do a Taylor series and match up the terms in the no-arbitrage condition so that

$$r_{i} \approx \mu_{i} (\kappa_{i} -1 )$$

This is actually kind of a neo-Fisherite result (see Sumner's post) -- higher growth (more expansionary monetary policy) in the money supply means higher interest rates. Lower growth (less expansionary) in the money supply means lower rates. The interest rate parity argument also happens to be the one that Sumner says produces the neo-Fisherite result. The version here is actually the one that makes the most sense to me, but I'm biased. Putting this back in the equation for the exchange rate

$$\frac{X(t + p)}{X(t)} \approx \exp \left( r_{1} - r_{2} \right)$$

Is this true? Well, maybe ... if you squint really hard ...

Blue is the RHS of the equation above and yellow is the LHS. But overall it seems exchange rates are way too noisy for this to be useful.