## Spot, Zero, Forward Curves

### FinPricing offers:

#### Four user interfaces:

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

FinPricing offers the following curves for various currencies via API. All the interest rate curves have data points up to 50 years.

- OIS curves
- RFR (risk free rate) curves
- SOFR, €STR (ESTR, ESTER), SONIA, TONA, CORRA, AONIA, SARON curves
- IBORs (LIBOR, EURIBOR, TIBOR, CDOR, EONIA, etc.) fallback rate curves
- Swap rate curves
- Basis curves
- Spot rate or zero rate curves
- Forward rate curves
- Discount curves
- Inflation Swap rate (CPI, RPI, HICP) curves
- Nordic electricity futures curve
- VIX futures curve
- S&P 500 futures curve

**1. Spot or Zero Curve**

A spot rate curve or spot curve is the term structure of an interest rate curve that is defined as the relationship between spot rates and their maturities.

A zero rate curve or zero curve is the term structure of the yields-to-maturity of zero coupon bonds and maturities. Zero rate curve is the most commonly used spot rate curve. Other spot rate curves are treasury yield curves, bond yield curves, etc.

A forward rate curve is a scatter plot of forward interest rates and their start dates and forward periods. A forward rate is defined by forward start date and forward period. The popular forward periods are 1 month, 3 months, 6 months, and 12 months.

In general, spot rate, forward rate, discount factor curves are not market observable and need to be constructed from market observable curves, such as bond curves or swap curves. Given swap market is much more liquid than bond market with narrow bid-ask spreads and a wide selection of maturities, spot curves are mainly bootstrapped from swap curves.

Spot zero rate curve is widely regarded as the best proxy for risk-free curve and benchmark curve. In general, Spot curve (zero curve), forward curve, and discount curve are essential for financial valuation.

The shape of spot rate curve implies future interest rate expectation and economic forecasting. It helps market participants to understand market behavior, trends, and risk. It is also used as a funding curve among different counterparties in financial market.

In general, spot rate curves and yield curves are equivalent. When people talk about yield curves, they actually mean spot rate curves.

**2. Spot Rate Curve Data**

FinPricing offers more than 100 spot/zero rate curves. The most commonly used spot rate curves are SOFR, LIBOR, ESTR, EURIBOR, etc. These curves are displayed below:

USD SOFR Spot Curve:

Valuation Date | Curve Name | Maturity | Spot Rate |
---|---|---|---|

2023-01-16 | USD.SOFR | 2023-04-16 | 4.658072554 |

2023-01-16 | USD.SOFR | 2023-07-16 | 4.801509197 |

2023-01-16 | USD.SOFR | 2023-10-16 | 4.839151738 |

2023-01-16 | USD.SOFR | 2024-01-16 | 4.799990052 |

2023-01-16 | USD.SOFR | 2024-04-16 | 4.667001481 |

2023-01-16 | USD.SOFR | 2024-07-16 | 4.532976822 |

2023-01-16 | USD.SOFR | 2024-10-16 | 4.357655424 |

2023-01-16 | USD.SOFR | 2025-01-16 | 4.181896423 |

2023-01-16 | USD.SOFR | 2025-04-16 | 4.067678383 |

2023-01-16 | USD.SOFR | 2025-07-16 | 3.953885663 |

2023-01-16 | USD.SOFR | 2025-10-16 | 3.838838554 |

USD LIBOR 3-Month Spot Curve:

Valuation Date | Curve Name | Maturity | Spot Rate |
---|---|---|---|

2023-01-12 | USD.LIBOR.3M | 2023-04-12 | 4.741825556 |

2023-01-12 | USD.LIBOR.3M | 2023-07-12 | 4.907463547 |

2023-01-12 | USD.LIBOR.3M | 2023-10-12 | 4.976033286 |

2023-01-12 | USD.LIBOR.3M | 2024-01-12 | 4.956388255 |

2023-01-12 | USD.LIBOR.3M | 2024-04-12 | 4.858211761 |

2023-01-12 | USD.LIBOR.3M | 2024-07-12 | 4.706635307 |

2023-01-12 | USD.LIBOR.3M | 2024-10-12 | 4.536370855 |

2023-01-12 | USD.LIBOR.3M | 2025-01-12 | 4.372236227 |

2023-01-12 | USD.LIBOR.3M | 2025-04-12 | 4.236459316 |

2023-01-12 | USD.LIBOR.3M | 2025-07-12 | 4.101866016 |

2023-01-12 | USD.LIBOR.3M | 2025-10-12 | 4.003513286 |

EURIBOR 3-Month Spot Curve:

Valuation Date | Curve Name | Maturity | Spot Rate |
---|---|---|---|

2023-01-12 | EURIBOR.3M | 2023-04-12 | 2.226122768 |

2023-01-12 | EURIBOR.3M | 2023-07-12 | 2.332858843 |

2023-01-12 | EURIBOR.3M | 2023-10-12 | 2.447041622 |

2023-01-12 | EURIBOR.3M | 2024-01-12 | 2.561223411 |

2023-01-12 | EURIBOR.3M | 2024-04-12 | 2.630256155 |

2023-01-12 | EURIBOR.3M | 2024-07-12 | 2.697269804 |

2023-01-12 | EURIBOR.3M | 2024-10-12 | 2.765020271 |

2023-01-12 | EURIBOR.3M | 2025-01-12 | 2.832770335 |

2023-01-12 | EURIBOR.3M | 2025-04-12 | 2.839148032 |

2023-01-12 | EURIBOR.3M | 2025-07-12 | 2.842781558 |

2023-01-12 | EURIBOR.3M | 2025-10-12 | 2.846453997 |

SOFR 1-Month Forward Curve

Valuation Date | Forward Date | 1M SOFR Forward Rate | 3M SOFR Forward Rate | 6M SOFR Forward Rate |
---|---|---|---|---|

2023-01-10 | 2023-02-10 | 4.705220087 | 4.84080669 | 4.947190567 |

2023-01-10 | 2023-03-10 | 4.832302312 | 4.935521839 | 4.98313335 |

2023-01-10 | 2023-04-10 | 4.920093847 | 4.993891679 | 5.003309825 |

2023-01-10 | 2023-05-10 | 4.993361191 | 4.991191325 | 4.972992696 |

2023-01-10 | 2023-06-10 | 5.005813611 | 4.968930278 | 4.923641988 |

2023-01-10 | 2023-07-10 | 4.913189417 | 4.950983262 | 4.868067905 |

2023-01-10 | 2023-08-10 | 4.92784026 | 4.893234548 | 4.761388188 |

2023-01-10 | 2023-09-10 | 4.951233238 | 4.817519354 | 4.637925638 |

2023-01-10 | 2023-10-10 | 4.743212156 | 4.726173724 | 4.493697714 |

2023-01-10 | 2023-11-10 | 4.702485855 | 4.573138421 | 4.370045133 |

2023-01-10 | 2023-12-10 | 4.676737425 | 4.405419304 | 4.239637367 |

**3. Spot Rate Curve Construction and Bootstrapping**

Prior to the 2007 financial crisis, financial institutions performed valuation and risk management of any
interest rate derivatives on a given currency using a single-curve approach. This approach
consisted of building a unique curve and using it for both discounting and forecasting cash flows. However, after the
financial crisis, basis swap spreads were no longer negligible and the market was
characterized by a sort of segmentation. Consequently, market practitioners started to use a new valuation approach
referred to as multicurve approach, which is characterized by a unique discounting curve and multiple forecasting
curves

The current methodology in capital markets for marking to market securities and derivatives is to estimate and
discount future cash flows using rates derived from the appropriate term structure. The yield curve is increasingly
used as the foundation for deriving relative term structures and as a benchmark for pricing and hedging.

Yield curves are derived or bootstrapped from observed market instruments that represent the most liquid and
dominant interest rate products for certain time horizons. Normally the curve is divided into three parts. The short
end of the term structure is determined using LIBOR rates. The middle part of the curve is
constructed using Eurodollar futures or
forward rate agreements (FRA). The far end is derived using mid
swap rates.

The objective of the bootstrap algorithm is to find the zero yield or discount factor for each maturity point and
cash flow date sequentially so that all curve instruments can be priced back to the market quotes. All bootstrapping
methods build up the term structure from shorter maturities to longer ones.

First of all, one needs to have valuation models for each types of instruments. Given a Future price, the yield or zero rate can be directly calculated as

The swap pricing model is introduced at swap model.

Assuming that we have had all yields up to 4 years and now need to derive up to 5 years.

- Let x be the yield at 5 years.
- Use an interpolation method to get yields at 4.25, 4.5 and 4.75 years as Ax, Bx, Cx,Dx.
- Given the 5 year market swap rate, we can use a root-finding algorithm to solve the x that makes the value of the 5 year inception interest rate swap equal to zero.
- Now we obtain all yields or equivalent discount factors up to 5 years

Repeat the above procedure till the longest swap maturity.

There are two keys in yield curve construction: interpolation and optimization for root finding.

**4. Interpolation**

Most popular interpolation algorithms in curve bootstrapping are linear, log-linear and cubic spline. They can be
applied to either zero rates or discount factors.

Some critics argue that some of those simple interpolations cannot generate smooth forward rates and the others may
be able to produce smooth forward rates but fail to match the market quotes. Also they cannot guarantee the continuity
and positivity of forward rates.

The monotone convex interpolation is more rigorous. It meets the following essential criteria:

- Replicate the quotes of all input underlying instruments.
- Guarantee the positivity of the implied forward rates
- Produce smooth forward curves.

Although the monotone convex interpolation rule sounds almost perfectly, it is not very popular with market
practitioners.

**5. Optimization**

As described above, the bootstrapping process needs to solve a yield using a root finding algorithm. In other words,
it needs an optimization solution to match the prices of curve-generated instruments to their market quotes.

FinPricing employs the Levenberg-Marquardt algorithm for root finding, which is very common in curve construction.
Another popular algorithm is the Excel Solver, especially in Excel application.

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