Cash flow additivity, no arbitrage and forward rates

In short

• Cash flow additivity: the PV of a stream = the sum of the PVs of its parts, once all are measured at the same date.
• No arbitrage: prices must not allow a riskless profit with zero net investment (ignoring transaction costs).
• Two strategies are economically equivalent when the PV of their cash flow differences is zero.
• The implied forward rate solves $(1+r_2)^2 = (1+r_1)(1+F_{1,1})$; it is the breakeven reinvestment rate.
• If a quoted forward rate differs from the implied one, borrow at the cheap rate and lend at the dear one.
• Upward-sloping spot rates: $F_{1,1} > r_2 > r_1$. A rise in $F_{1,1}$ signals expected future rate rises.
• Longer horizons chain the same way: $(1+r_3)^3 = (1+r_1)(1+F_{1,1})(1+F_{2,1}) = (1+r_1)(1+F_{1,2})^2$.