Why Futures and Forward Prices Differ When Rates and Prices Correlate
Summary
The discussion addresses the futures–forward price difference when interest rates and the underlying price move together. It attributes the difference to daily futures settlement: gains and losses arrive over time and can be reinvested or financed at rates that vary with market conditions. The forward instead settles its accumulated result at maturity.
The pricing argument expresses the futures value in terms of expected future spot value and risk-free returns, then identifies a covariance term between returns and the underlying. Under the stated approximation, that covariance explains the futures–forward adjustment. A second explanation emphasizes that a fair comparison must match equity exposure; comparing one futures contract directly with one forward can mislead. The examples are brief and rely on simplifying assumptions about rates and contract scaling, so the precise relationship depends on the payoff conventions and market model.
Key ideas
- Daily settlement makes futures cash flows occur at different times from forward settlement.
- The covariance between interest rates and the underlying price can contribute to the futures–forward price difference.
- When underlying prices and rates are positively correlated, futures may be more valuable than comparable forwards.
- A valid comparison must match the contracts' equity exposure rather than assume equal contract counts.
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# Falling Futures prices positively correlated with interest rates
# Falling Futures prices positively correlated with interest rates
I'm having trouble understanding how Futures are worth more than Forwards when price and interest rates are positively correlated but both declining.
For instance, a Future with losses of -5 at T(n-1) and -5 at T(n) vs. a Forward with losses of -10 at T(n). How is the Future more valuable? Most textbooks I've read explain it as "losses derived from a falling futures price can be financed at a lower interest rate", but you're still having to borrow 5 at T(n-1) and paying back 5(1+r) at T(n). So total losses for the Future are 5+5(1+r) which is greater than 10 for the forward.
## Answer by fni (score 0)
https://quant.stackexchange.com/a/30829
Call $G_t$ the price of the future, and $F_{t,T}$ the price of a forward at time t with maturity T. You know that by NA $F_{t,T}=R_{t,T}E_t[S_T]=R_{t,T} S_t$ where $R_{t,T}$ is the gross return on a risk free bond with maturity T and $E_t[ \cdot ]$ is a risk neutral expectation. One day before maturity the price of the future will be such that $$G_{T-1}=R_{T-1,T}E_{T-1}[S_T]$$ and two days before maturity $$G_{T-2}=R_{T-2,T-1}E[G_{T-1}]=E_{T-2}[R_{T-2,T-1}R_{T-1,T}S_T]=\\E_{T-2}[R_{T-2,T-1}R_{T-1,T}]E_{T-2}[S_T]+Cov_{T-2}[R_{T-2,T-1}R_{T-1,T},S_T]$$ You can also rewrite $E_{T-2}[S_T]=\frac{F_{T-2,T}}{R_{T-2,T}}$ and if $\frac{R_{T-2,T-1}R_{T-1,T}}{R_{T-2,T}}\approx 1$ then $$G_{T-2}\approx F_{T-2,T}+Cov_{T-2}[R_{T-2,T-1}R_{T-1,T},S_T]$$
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/31302
Your comparison is not quite apples to apples. You should compare one futures contract at (n-2) with (1+r) forward contracts. That makes them equal in terms of equity exposure. If you do this, you find that you borrow 5 and pay back 5(1+r1) at n where r1 is less than r if rates have gone down. hence the futures is better if this correlation holds.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.