# 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.

## 2 Answers

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 $$G_{T-2}\approx F_{T-2,T}+Cov_{T-2}[R_{T-2,T-1}R_{T-1,T},S_T]$$

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.