Finance Formulas

Fixed Income

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Implied Forward Rate

Builds onFuture Value of a Single Sum · Bond Price & Yield to Maturity — if this page feels steep, start there.

(1+z2)2=(1+z1)(1+f1,1)(1+z_2)^2 = (1+z_1)(1+f_{1,1})

Reading the notation

z1, z2z_1,\ z_2
spot rates: today's rates for money locked up 1 and 2 years respectively
f1,1f_{1,1}
the forward rate: a 1-year rate, starting 1 year from now, agreed today
(1+z2)2(1+z_2)^2
what $1 grows to down the two-year road
(1+z1)(1+f1,1)(1+z_1)(1+f_{1,1})
what $1 grows to rolling one year, then the forward year

Why it must be true

Two roads to the same destination must cost the same. Investing for two years at the two-year spot rate z2z_2 must end with exactly what you'd get investing one year at z1z_1 and rolling into a one-year rate agreed today for next year — the forward rate f1,1f_{1,1}. If the two roads differed, you'd borrow down one and lend down the other for free money.

So forwards aren't predictions someone makes — they are arithmetic already hiding inside the spot curve. An upward-sloping curve implies higher forward rates, mechanically.

The derivation

Grow $1 down each road and set the outcomes equal (no-arbitrage):

(1+z2)2⏟two-year road=(1+z1)(1+f1,1)⏟one year, then the forward\underbrace{(1+z_2)^2}_{\text{two-year road}} = \underbrace{(1+z_1)(1+f_{1,1})}_{\text{one year, then the forward}}

Solve for the forward:

f1,1=(1+z2)2(1+z1)−1f_{1,1} = \frac{(1+z_2)^2}{(1+z_1)} - 1

The same logic generalizes: (1+zn)n=(1+zm)m(1+fm,n−m)n−m(1+z_n)^n = (1+z_m)^m (1+f_{m,n-m})^{n-m} — spot factors multiply into forward factors.

When to reach for it

Given two points on the spot curve, find the rate the curve implies for the gap between them — or check a quoted forward against no-arbitrage.

Listen for

“implied forward rate”“1y1y / the rate one year forward”“spot rates of … and …”“rate implied by the yield curve”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

  • ≈

    Doubling-back approximation: f₁,₁ ≈ 2·z₂ − z₁. Spots of 4% and 5% imply a forward near 6%. The exact answer is a few basis points below this — bracket with it.

  • ≈

    Slope reading: upward curve → forward ABOVE both spots; flat curve → forward = spots; inverted → below. Any answer between the two spots on a sloped curve is wrong.

Traps in applying it

  • ✗Averaging the spots — the forward is roughly twice the long spot MINUS the short one, on the far side, not in between.
  • ✗Mismatched compounding: all rates must share the same periodicity before the factors multiply.
  • ✗Off-by-one on periods: (1+z₂)² has TWO years of growth against one year of z₁ plus one of forward.

Limits & criticisms

The identity is pure no-arbitrage arithmetic — always true of the curve — but reading forwards as forecasts loads in the expectations hypothesis, which empirically fails: forwards systematically overshoot realized rates because they also carry term premia (compensation for duration risk). Forwards are the market's break-even rates, not its predictions.

Where it came from

The idea that the yield curve embeds its own future rates crystallized in the 1930s: John Hicks (Value and Capital, 1939) and Friedrich Lutz formalized the expectations theory of the term structure, making the forward rate the curve's built-in "market forecast." Trading desks turned the identity into products — FRAs and the vast interest-rate swap market are all priced off implied forwards.

Today central-bank watchers read implied forwards as the market's policy-rate path, and every swap desk on earth bootstraps forward curves before lunch.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Forward from the curve

f1,1=(1+z2)21+z1−1f_{1,1} = \frac{(1+z_2)^2}{1+z_1} - 1

The workhorse: divide the two-year growth factor by the one-year factor — what's left is the rate the curve charges for year two.

Drill this face →

Spot from a forward quote

z2=(1+z1)(1+f1,1)−1z_2 = \sqrt{(1+z_1)(1+f_{1,1})} - 1

Building the curve the other way: chain the short spot and the forward, then take the root — exactly how swap desks bootstrap.

Drill this face →

On the BA II Plus

Worked example: Government spot rates are z₁ = 5% and z₂ = 3.62%. Compute the implied 1y1y forward rate.

  1. 1.1.0362 [yˣ] 2 [=]the two-year growth factor (1+z₂)²
  2. 2.[÷] 1.0500 [=]divide out the first year
  3. 3.[-] 1 [=]back to a rate

→ 2.25%

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