Exam FMCash flow analysisFree to read

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: $(1 + s_{t_{2}})^{t_{2}} = (1 + s_{t_{1}})^{t_{1}} (1 + f_{t_{1}, t_{2}})^{t_{2} - t_{1}}$
One-period forward: $1 + f_{t, t + 1} = \frac{( 1 + s _{t + 1} ) ^{t + 1}}{( 1 + s _{t} ) ^{t}}$
Spot as a geometric mean: $(1 + s_{n})^{n} = t = 0 \prod n - 1 (1 + f_{t, t + 1})$

Where it comes from

Two strategies must give the same time- $t_{2}$ value: invest at the long spot rate, or invest to $t_{1}$ and roll over at the agreed forward rate.
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.
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.

$1 + f_{2, 3} = \frac{( 1.051 ) ^{3}}{( 1.046 ) ^{2}}$ .
$(1.051)^{3} = 1.160935$ and $(1.046)^{2} = 1.094116$ .
$1 + f = 1.061071$ , so $f_{2, 3} = 6.1071%$ .
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.

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Forward rates

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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