ZLB Problem Flashcards
State several issues with the Taylor Rule
- Interest rates differ by maturity
2. i cannot fall below 0 because M pays at least 0
Define an equation to calculate the forward interest rate. How can the forward rate be used to derive i(2) assuming i(1) is known?
2i(2) - i(1) = f(1,2). Or the forward rate on a loan beginning in one year and ending 2 in years.
f(1,2) can be derived via the rearrangement of the above equation.
Define a general equation to calculate the forward interest rate
f(m-1,m) = mi(m) – (m-1) i(m-1)
Define the Expectations Hypothesis
That the expected returns from investing m years at i(m) and investing m-1 years at i(m-1) and then reinvesting at i(1) are equivalent.
Therefore, i(1) is the expected forward rate f(m-1,m)
ft (m-1,m) = Et(i t+m-1 (1))
What are the implications of a change in nom. interest rates that leaves forward rates unchanged?
Bonds of maturity m will fall or rise in price by -m delta i. (all bonds will change in price - the magnitude of the change depending on maturity)
Therefore, present consumption will either become more expensive or cheaper relative to consumption at all times greater than t + m.
For instance, if the Fed decides to decrease nom. interest rates specifically for bonds of maturity m, those bonds will face a price increase equivalent to -m delta i, rendering present consumption less expensive due to lower yields.
The Fed’s impact on consumption and prices depends on the maturity of its target interest rate. A change in the FFR would only cause an imperceptible change in consumption, since m = 1/365. (- delta i/365 is small)
However, since the FOMC only meets 8 times a year, it is implied that i(1/8) = i* so that in fact the price of consumption changes by -delta i/8.
How can the Fed push down nom. interest rates (in accordance with the benchmark taylor rule to move the real interest rate to its equilibrium level) via the manipulation of forward rates?
The Fed can change m such that -delta i1 * m = -delta i2*m2 (where -delta i * m is the price of consumption)
Therefore, the Fed can still achieve its desired stimulus by pegging the right security with the appropriate maturity i.e treasury bills with a 9 week maturity to a nominal interest rate that does not fall below 0.
For instance, if the Fed needs to decrease nom. interest rates to -4 from 4, it can directly engineer this change by targeting the security with maturity of 13 weeks, so that (8)/8 = (4)m, meaning m = 1/4.
Why can’t the nom. interest rate fall below 0?
Because this would lead to the theoretical complete withdrawal of deposits from banks, although in practice some depositors may still leave their deposits at a bank due to security.
