Yield Curve and Forward Rates: Method

Yield Curve and Forward Rates: Method

What it does

A yield curve is published as par yields: for each maturity, the coupon rate at which a bond would be worth exactly its face value. That is a convenient quotation, but it is not the rate at which a single future payment should be discounted, because a coupon bond is many payments at many dates. The page converts a par curve into the three quantities that are used for that purpose: discount factors (the value today of 1 paid at a future date), spot rates (the yield of a single payment at that date, also called zero-coupon rates) and forward rates (the rates for future periods that today's curve implies).

The method

Symbols: m is the number of coupon payments and compounding periods per year; the coupon dates are t_k = k/m years, k = 1, 2, … ; c_k is the par yield at t_k, d_k the discount factor, z_k the spot rate and f_k the forward rate for the period that ends at t_k. All rates are per year, compounded m times a year.

1. Interpolation. The entered points are joined by straight lines to give c_k at every coupon date. Before the first entered maturity the first par yield is used.

2. Bootstrapping. A par bond maturing at t_k pays c_k/m at each coupon date and 1 at maturity, and is worth 1:

1 = (c_k/m)(d₁ + d₂ + … + d_k) + d_k

For k = 1 there is one payment, so d₁ = 1 / (1 + c₁/m). For each later date all earlier factors are already known; with A_k−1 = d₁ + … + d_k−1,

d_k = (1 − (c_k/m) A_k−1) / (1 + c_k/m)

3. Spot rates. The spot rate is the rate that compounds to the discount factor over k periods: d_k = (1 + z_k/m)^−k, so z_k = m (d_k^−1/k − 1).

4. Forward rates. Investing to t_k must give the same result as investing to t_k−1 and reinvesting for one period at the forward rate: (1 + f_k/m) = d_k−1/d_k, with d₀ = 1. For a period that runs from a years to b years and holds j coupon periods,

f(a, b) = m [(d_a/d_b)^1/j − 1]

5. Slope. The 10-year par yield minus the 2-year, and minus the 3-month, in percentage points. A curve whose short yields exceed its long yields is called inverted.

6. Pricing a bond (Advanced). A bond with annual coupon rate c_b maturing at t_n is worth, per 100 of face value, P = 100 [(c_b/m) A_n + d_n]. Its yield to maturity y solves P = Σ (100 c_b/m)(1 + y/m)^−k + 100 (1 + y/m)⁻ⁿ. Parallel shift: the same number of basis points (1 bp = 0.01 percentage point) is added to every par yield and steps 1 to 4 are repeated.

What it assumes

Coupons are paid at equal intervals of 1/m year and today is a coupon date, so there is no accrued interest and no day-count convention. Par yields between the entered maturities lie on straight lines. Par yields at maturities shorter than one coupon period are used only for the interpolation and for the 3-month slope measure. The curve is one issuer's: Treasury par yields describe U.S. government debt, and a bond with credit risk is worth less than this curve gives.

When it misleads

Forward rates are not forecasts. They are the break-even rates contained in today's prices. Studies of Treasury data (Fama and Bliss, 1987) find that forward rates include a premium for holding longer maturities, which varies over time, so a forward rate above today's short rate does not by itself say that rates are expected to rise.

Forward rates magnify the interpolation. A forward rate is a difference of nearby points of the curve, so straight-line interpolation of par yields gives a forward curve with kinks and steps at the entered maturities, most visibly between widely spaced points such as 10, 20 and 30 years. The Treasury itself fits a monotone convex spline to forward rates, and the Federal Reserve's published zero-coupon curve (Gürkaynak, Sack and Wright) fits a smooth Nelson–Siegel–Svensson function instead; neither is implemented here. Spot rates and discount factors are far less sensitive to the choice.

A par curve is itself an estimate. The Treasury derives its curve from indicative bid quotations of the most recently auctioned securities, which often trade at a small premium to older issues. An actual bond can therefore trade at a price different from the one this curve gives.

Worked example

Treasury par yield curve of 12 November 2026, semi-annual (m = 2): 6 months 4.42 %, 1 year 4.65 %, 2 years 4.86 %.

Date 1 (0.5 year): d₁ = 1 / (1 + 0.0442/2) = 0.978378; z₁ = 4.420 %.

Date 2 (1 year): d₂ = (1 − 0.0465/2 × 0.978378) / (1 + 0.0465/2) = 0.955048; z₂ = 2 (0.955048^−1/2 − 1) = 4.653 %. Forward rate for the second half-year: f₂ = 2 (0.978378/0.955048 − 1) = 4.886 %.

Date 3 (1.5 years): the par yield is halfway between 4.65 and 4.86, c₃ = 4.755 %; A₂ = 0.978378 + 0.955048 = 1.933426; d₃ = (1 − 0.04755/2 × 1.933426) / (1 + 0.04755/2) = 0.931877. Date 4 (2 years): d₄ = 0.908301, z₄ = 4.867 %.

One-year rate, one year forward: 2 [(0.955048/0.908301)^1/2 − 1] = 5.082 %. Check: (1 + 0.04653/2)² × (1 + 0.05082/2)² = 1.10096 = (1 + 0.04867/2)⁴.

Slope: 10-year minus 2-year = 5.10 − 4.86 = 0.24 points; 10-year minus 3-month = 5.10 − 4.32 = 0.78 points.

Daily par yields are published by the U.S. Department of the Treasury under "Daily Treasury Par Yield Curve Rates"; type the maturities in years and the yields in percent into the table.

A calculator for education and analysis, not investment advice.