# The Futures-Implied Rate

### The FIR

At any given time, there is an implied borrowing rate associated with a liquid futures contract.
The spreads between this futures-implied rate (FIR) from the S&P500 futures contract and market interest rates such as US Treasury and Eurodollar rates have shown substantial regime
dependence since 1996, as shown in the charts below.

The charts show four regimes, which were:

1. From 1996 until the passage of the The Commodity Futures Modernization Act of 2000 (CFMA), which we set at the end of 2000.
2. After the CFMA but before the financial crisis, which we treated as beginning in July of 2007, the month in which “Bear Stearns disclosed that two of [its] subprime hedge funds had lost nearly all of their value amid a rapid decline in the market for subprime mortgage”(Wikipedia).
3. The financial crisis, which we treated as ending in March 2009, and
4. Recovery from the financial crisis.

### Defining and Estimating the FIR

We defined the (forward) FIR based on a no-arbitrage argument and estimated it as follows:

$\widehat{FIR}(t) = \frac{f(t,T_2)}{k_{div}(t) \times f(t,T_1)}$

where $$k_{div}(t)$$ denotes the implied dividend rate observed at time t in the options markets and $$f(t,T)$$ denotes the futures price observed at time t on a futures contract with maturity T.

$$T_2 > T_1$$ so the numerator (denominator) in this fraction reflects the price on the next (near) futures contract.

### Conclusions

1. The estimated FIR spread was volatile around a mean that varied between regimes.
2. The average estimated spread of the FIR to the forward Treasury rate was positive, as was the actual estimated spread almost all the time
3. The average estimated spread to the forward Eurodollar rate was generally negative.

Gunther et al Futures Working Paper