| Maturity (years) | Yield (%) | Typical source |
|---|---|---|
| 0.25 | 11.80 | T-bill / money market |
| 1.00 | 12.30 | Short-term note |
| 5.00 | 12.85 | Medium-term bond |
| 10.00 | 13.05 | Long-term bond |
- Paste your points as maturity in years and yield in percent.
- Select a curve method to interpolate or fit smooth rates.
- Choose grid step to control table granularity and chart smoothness.
- Pick compounding for discount factor calculations.
- Generate to show results above the form instantly.
- Download CSV or PDF to save tables and share outputs.
Market inputs and units
Enter maturities in years and spot yields in percent. A quarterly bill is 0.25, a 6‑month bill is 0.50, and a 10‑year bond is 10.00. Keep units consistent across markets. If you have day counts, convert to year fractions first. When rates are negative, keep the sign; the discount factor formula still works.
Curve construction choices
Linear interpolation preserves local shape between quotes and is easy to audit. Natural cubic spline adds smooth first and second derivatives, reducing kinks that can create unrealistic forwards. Nelson–Siegel is a parametric fit that summarizes level, slope, and curvature, and the calculator reports β0, β1, β2, τ, and the fit SSE for transparency.
Discount factors for valuation
The calculator converts each spot yield y(t) to a discount factor DF(t). With m‑times compounding: DF(t)=(1+y/m)^(−m·t). With continuous compounding: DF(t)=e^(−y·t). For example, with y=12% and t=2, semiannual DF≈(1+0.12/2)^(−4)≈0.790. Discount factors are then used to price coupons, swaps, and project cash flows.
Forward rates and curve health
Forward rates are derived from discount factors over each grid step: f=(ln DF(t1)−ln DF(t2))/(t2−t1). A smaller grid step, such as 0.25 years, gives a smoother Plotly line but can amplify micro‑noise in forwards. A larger step, such as 0.50, is more stable for sparse curves. Large spikes often indicate inconsistent quotes or an over‑smooth fit.
Interpreting slope and curvature
An upward‑sloping curve typically implies higher expected short rates or term premia at longer maturities. An inverted curve often signals tighter policy expectations or recession concerns. Curvature highlights “humps” where intermediate maturities price differently than short and long ends. Track changes in 2s–10s and 3m–10y spreads to summarize slope shifts.
Exporting for reporting
After calculation, the table and Plotly chart can be exported. CSV is ideal for spreadsheets, scenario engines, and model inputs. PDF captures method, grid settings, and the computed curve table for audit trails and stakeholder review. Save multiple runs to compare curve shifts across dates, issuers, or stress scenarios. For risk work, export two curves and compute changes in spot, DF, and forwards by maturity. Pair the PDF with source quotes to document assumptions. Consistent naming of files by date and market helps reproduce results during reviews and supports quick backtesting of curve-building choices internally.
1) What data should I paste into the points box?
Paste maturity in years and the corresponding spot yield in percent, one line per point. Use commas or spaces. Keep maturities strictly positive and cover the curve range you need.
2) When should I use spline instead of linear interpolation?
Use spline when you want a smoother curve and your points are reasonably consistent. If the input set has jumps or scarce quotes, spline can overshoot, so compare forwards to validate stability.
3) What does Nelson–Siegel add?
Nelson–Siegel summarizes the term structure with a small parameter set (level, slope, curvature). It is useful for scenario comparison and for producing stable curves when you prefer a parametric shape.
4) How are discount factors computed here?
Discount factors are computed from spot yields using your chosen compounding convention. The calculator supports annual, semiannual, quarterly, monthly, and continuous discounting. Yields are interpreted as nominal spot rates.
5) Why can forward rates look volatile?
Forwards are sensitive to small differences in discount factors, especially with small grid steps. Try a larger step, add more input points, or switch methods. Volatility can also reflect genuine market discontinuities.
6) What is the best export format for audits?
Use PDF to preserve settings, method, and the full curve table in a single artifact. Pair it with the original quote source. Use CSV when you need to reuse the curve as model input.