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Forward rates

The rate locked in today for borrowing between two future dates, implied by no-arbitrage from the spot curve.

The formulas

Implied forward
(1+st2)t2=(1+st1)t1(1+ft1,t2)t2−t1(1+s_{t_2})^{t_2} = (1+s_{t_1})^{t_1}\left(1+f_{t_1,t_2}\right)^{t_2-t_1}
One-period forward
1+ft,t+1=(1+st+1)t+1(1+st)t1 + f_{t,t+1} = \dfrac{(1+s_{t+1})^{t+1}}{(1+s_t)^{t}}
Spot as a geometric mean
(1+sn)n=∏t=0n−1(1+ft,t+1)(1+s_n)^n = \prod_{t=0}^{n-1}\left(1+f_{t,t+1}\right)

Where it comes from

  1. Two strategies must give the same time-t2t_2 value: invest at the long spot rate, or invest to t1t_1 and roll over at the agreed forward rate.
  2. Equating them and solving for the forward rate gives the formula; if it did not hold you could borrow one way and lend the other for a riskless profit.
  3. Chaining the one-period forwards reconstructs the spot rate as their geometric mean.

Worked example

With s₂ = 4.6% and s₃ = 5.1%, find the one-year forward rate from year 2 to year 3.

  1. 1+f2,3=(1.051)3(1.046)21 + f_{2,3} = \dfrac{(1.051)^{3}}{(1.046)^{2}}.
  2. (1.051)3=1.160935(1.051)^3 = 1.160935 and (1.046)2=1.094116(1.046)^2 = 1.094116.
  3. 1+f=1.0610711 + f = 1.061071, so f2,3=6.1071%f_{2,3} = 6.1071\%.
  4. The forward exceeds both spot rates, which is what an upward-sloping curve requires.

Answer: 6.107%

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Long spot = short spot × forward. Everything is one no-arbitrage equation.
  • A rising spot curve forces forwards ABOVE the spot rates; a falling curve pushes them below.

Traps

  • Taking the (t₂ − t₁)th root only when the gap is one period, or forgetting it when it is more.
  • Averaging spot rates arithmetically instead of chaining them geometrically.

Related

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