Title: What Is the Par Yield Curve — How It’s Constructed and How to Use It
Key takeaways
– The par yield curve is the schedule of coupon rates at which hypothetical coupon-bearing Treasuries of different maturities would trade at par (price = 100).
– Par yields are useful for setting coupon rates on newly issued bonds and for producing an arbitrage-free yield structure (via bootstrapping).
– The par curve is intimately related to the spot (zero-coupon) curve and forward rates: you can compute par yields from spot rates, and you can bootstrap spot rates from observed par yields.
– Practical uses: bond pricing, swap and curve construction, relative-value analysis. Limitations: market conventions, interpolation choices, and model assumptions (risk-free, frictionless markets).
Overview — what the par yield curve shows
A yield curve plots yields against maturity. The par yield curve specifically plots the coupon rate (annualized) that makes a bond of a given maturity trade at par (price = 100). For a coupon-bearing bond issued “at par,” the coupon rate equals the bond’s yield to maturity. The par curve therefore represents the coupon rates that would be charged on new issues so that they start life at par.
Because coupon-paying bonds have multiple cash flows, a par yield for an n-year bond is a weighted average of the discount factors for each cash flow. The par curve is distinct from the spot (zero-coupon) curve and the forward curve, but all three are mathematically linked.
How the par yield is defined (formula)
Assume:
– Face value F (typically 100),
– Annual coupon rate c (annualized),
– m coupon payments per year (m = 2 if semiannual),
– N = n × m total coupon periods,
– DF_i = discount factor for period i (present value of $1 paid at period i).
Price of the bond:
Price = (c/m) × F × SUM_{i=1..N} DF_i + F × DF_N
For a par-priced bond, Price = F, so solve for c:
1 = (c/m) × SUM_{i=1..N} DF_i + DF_N=> c = m × (1 − DF_N) / SUM_{i=1..N} DF_i
If DF_i are computed from spot rates s_i (with the same compounding convention as coupon payments), then you can compute the par yield directly from the spot curve.
Bootstrapping: deriving spot rates from observed par yields (step‑by‑step)
Observed market par yields (or market prices of coupon bonds) are often the starting point. Bootstrapping is the iterative method to extract the spot (zero-coupon) discount factors DF_i and spot rates from these par yields.
Practical step-by-step (semiannual coupons typical in US Treasuries)
1. Gather market par yields for successive maturities (e.g., 6 months, 1 year, 1.5 years, 2 years). Express each yield as an annual rate with the coupon frequency (e.g., semiannual).
2. Period convention: convert annual yields to per-period coupon amounts. For coupon rate c (annualized), coupon per period = (c / m) × F.
3. Start with the shortest maturity (first period). For a 6‑month (one‑period) bond:
– The only cash flow is coupon + principal at period 1.
– Set price = par and solve immediately to get DF_1. For example, if par yield = 2% (annualized, semiannual coupons), coupon = 1% of F per period, and DF_1 = (coupon + principal)/Price → DF_1 = (101)/100 inverted appropriately. More simply, with one period DF_1 = 1 / (1 + s_0.5/2), so s_0.5 equals the 6‑month par yield.
4. Move to the next maturity (two periods). Use the known DF_1 to solve for DF_2 from the par price equation:
– For a 1‑year par yield c1 (annual): Price = (c1/2)×F×DF_1 + (c1/2×F + F)×DF_2 = F.
– Rearrange to solve DF_2 = [F − (c1/2)×F×DF_1] / [F × (1 + c1/2)].
5. Convert DF_2 to the 1‑year spot rate s_1 (using the coupon compounding convention):
– If DF_2 = 1/(1 + s_1/2)^2, then s_1 = 2×(DF_2^(−1/2) − 1).
6. Repeat for longer maturities: for each new maturity N, plug known DF_1 … DF_{N−1} into the par-price formula and solve for DF_N:
– DF_N = [F − (c/ m) × F × SUM_{i=1..N−1} DF_i] / [F × (1 + c/m)]
– Then extract the annualized spot rate for maturity n from DF_N (apply the appropriate root for number of periods).
7. After you have DF_i for all periods, you can convert to continuously compounded spot rates, compute forward rates, or construct other curves as needed.
Worked numeric example (using the par yields cited)
Given (annualized par yields, semiannual coupon frequency m = 2):
– 0.5 year: 2.0% → coupon per period = 1.0
– 1.0 year: 2.3% → coupon per period = 1.15
– 1.5 year: 2.6% → coupon per period = 1.30
– 2.0 year: 3.0% → coupon per period = 1.50
Step 1 (0.5 year, N = 1):
DF1 = 1 / (1 + 0.02/2) = 1 / 1.01 = 0.990099
Step 2 (1.0 year, N = 2), solve for DF2 from:
100 = 1.15 × DF1 + 101.15 × DF2
=> DF2 = (100 − 1.15 × 0.990099) / 101.15 ≈ 0.97737
Convert to an annual spot rate s_1 using semiannual compounding:
s_1 = 2 × (DF2^(−1/2) − 1) ≈ 2.29%
Step 3 (1.5 years, N = 3), solve for DF3:
100 = 1.30 × (DF1 + DF2) + 101.30 × DF3
=> DF3 ≈ (100 − 1.30 × (0.990099 + 0.97737)) / 101.30 ≈ 0.96187
Convert to s_1.5: s_1.5 = 2 × (DF3^(−1/3) − 1) ≈ 2.606%
Step 4 (2.0 years, N = 4), solve for DF4:
100 = 1.50 × (DF1 + DF2 + DF3) + 101.50 × DF4
=> DF4 ≈ (100 − 1.50 × (0.990099 + 0.97737 + 0.96187)) / 101.50 ≈ 0.94244
Convert to s_2: s_2 = 2 × (DF4^(−1/4) − 1) ≈ 3.00%
These calculations illustrate the iterative bootstrap: each longer-maturity DF_i is solved using previously determined discount factors.
Using the par curve
– Pricing new issues: the par curve gives the coupon rate that makes a new bond trade at par.
– Benchmarking: par yields are commonly quoted in markets (e.g., par coupon curve used in swap markets).
– Curve construction: bootstrapping from par yields is a standard way to produce a consistent zero-coupon (spot) curve and forward curve for valuation and risk-management.
– Relative-value: compare par yields across sectors (e.g., Treasury vs. corporate) to study spread and liquidity effects.
Practical considerations and limitations
– Conventions matter: coupon frequency, day-count conventions, and compounding conventions must match when converting between par yields, discount factors, and spot rates.
– Market coverage and interpolation: par yields are available only at standard maturities; you must interpolate/extrapolate for off‑market maturities—results depend on interpolation method.
– Liquidity and taxes: quoted yields assume market liquidity and ignore taxes or transaction costs. Corporate and municipal curves add credit and tax adjustments.
– Model risk: small changes in inputs or conventions can change derived spot and forward rates; document conventions and procedures.
Bottom line
The par yield curve is a straightforward, practical curve showing coupon rates that produce par prices at different maturities. It is a key building block for bond issuance, valuation, and curve construction. By bootstrapping from observed par yields you can extract discount factors and spot rates; conversely, if you know spot rates you can compute par yields via the par-yield formula. Careful attention to compounding and market conventions is essential to produce internally consistent results.
Source
This article is based on concepts and examples described in Investopedia’s entry on the par yield curve (https://www.investopedia.com/terms/p/par-yield-curve.asp).