## Forward Yield Curve

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FinPricing offers the following curves for various currencies via API:

- 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. Forward Curve Introduction**

A forward yield 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. For instance, one year three-month forward rate is the forward rate starting at one year and ending at one year and three months, in future.

Forward rate can be regarded as zero-coupon rate between two given future maturity dates. A term forward rate is the expectation of index rates under forward neutral probability measures.

A discount factor curve is the term structure of discount factors that is a plot of discount factors and their maturities.

A forward curve is commonly used for forecasting an asset value, while a discount curve is used for getting the present value of the asset. For example, the method is used to value a swap by first forecasting the unknown future cash flows using appropriate forward rates, and then discounting the flows as if they were known.

Forward curve is not market observable. It belongs to derived interest rate curves. One can derive forward curve from market traded instruments, such as bonds, deposit rates, futures, FRAs, and swap rates. Forward curve is generated by the bootstrapping algorithm

Spot curve, forward curve, and discount curve are economically related and financially equivalent.

**2. Forward Curve Data**

FinPricing provides more than 100 forward yield curves, including SOFR, LIBOR, ESTR, EURIBOR, SONIA forward curve, etc. Here are some most commonly used forward yield curves:

1-Month, 3-Month and 6-Month SOFR Forward Curves

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 |

1-Month, 3-Month and 6-Month LIBOR Forward Curves

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

2023-01-10 | 2023-02-10 | 4.662211338 | 4.928338517 | 5.228714648 |

2023-01-10 | 2023-03-10 | 4.922790158 | 5.107284596 | 5.238858681 |

2023-01-10 | 2023-04-10 | 4.967281718 | 5.151075234 | 5.249459795 |

2023-01-10 | 2023-05-10 | 5.096585315 | 5.18840892 | 5.262930024 |

2023-01-10 | 2023-06-10 | 5.236410033 | 5.209344317 | 5.277649954 |

2023-01-10 | 2023-07-10 | 5.020157995 | 5.19320702 | 5.295073672 |

2023-01-10 | 2023-08-10 | 5.030090122 | 5.152515076 | 5.184386358 |

2023-01-10 | 2023-09-10 | 5.060181643 | 5.091676013 | 5.052038771 |

2023-01-10 | 2023-10-10 | 4.889226151 | 4.999381915 | 4.894918234 |

2023-01-10 | 2023-11-10 | 4.86357621 | 4.883320696 | 4.718296391 |

2023-01-10 | 2023-12-10 | 4.851067361 | 4.75479716 | 4.522835234 |

EURIBOR Foward Curves

Valuation Date | Forward Date | 1M ESTR Forward Rate | 3M EURIBOR Forward Rate | M EURIBOR Forward Rate |
---|---|---|---|---|

2023-01-10 | 2023-02-10 | 2.381130941 | 2.286645643 | 2.86670652 |

2023-01-10 | 2023-03-10 | 2.709341383 | 2.371819598 | 2.971584594 |

2023-01-10 | 2023-04-10 | 2.883591091 | 2.487065223 | 3.099734207 |

2023-01-10 | 2023-05-10 | 3.17301405 | 2.577754492 | 3.254294944 |

2023-01-10 | 2023-06-10 | 3.304316227 | 2.666972235 | 3.43331074 |

2023-01-10 | 2023-07-10 | 3.391201111 | 2.753330965 | 3.636778128 |

2023-01-10 | 2023-08-10 | 3.457810158 | 2.842587936 | 6.576905634 |

2023-01-10 | 2023-09-10 | 3.4315437 | 2.930307397 | 9.783392309 |

2023-01-10 | 2023-10-10 | 3.433833607 | 3.018280854 | 12.89277258 |

2023-01-10 | 2023-11-10 | 3.420173376 | 3.03874356 | 15.94521806 |

2023-01-10 | 2023-12-10 | 3.360030235 | 3.043051119 | 19.05743439 |

The explicit modeling of market forward rates allows for a natural formula for interest rate option volatility that is consistent with the market practice of using the formula of Black for caps. It is generally considered to have more desirable theoretical calibration properties than short rate or instantaneous forward rate models.

To model the complete forward curve directly, it is essential to bring everything under a common measure. The terminal measure is a good choice for this purpose, although this is by no means the only choice.

Under terminal measure, the drifts of forward rate dynamics are stochastic and state-dependent that gives rise to sufficiently complicated non-lognormal distributions and has no analytic solution.

**3. Forward Curve Construction**

Normally, a hybrid of bootstrapping method is used to generate forward curve. Assuming cash Libor rates, Libor futures and Libor swaps as underlying instruments, the curve generation process can be described briefly as follows

Separate the underlying instruments into groups based upon the maturities of the instruments. Those with shorter maturities are classified as short to medium-term instruments, e.g. cash Libor rates, Libor futures and shortterm Libor swaps. Those with longer maturities are long-term instruments, e.g. medium to long term Libor swaps.

The bootstrapping method is employed to calculate discount factors at maturity dates of those cash Libor and Libor futures first. If there is a gap, the following calculated continuous compounding forward rate is used as the forward rate for this gap. For Libor futures, the expiration of the instrument (third Wednesday of the month) plus the tenor of the rate is used as the maturity of the underlying cash Libor. These maturity dates are adjusted based on the day roll convention, and recorded in the generated curve as anchor dates, together with the generated discount factors.

Forward rate curve can be used to value securities and derivatives, such as convertible bonds. A convertible bond
has an embedded call option that gives bondholders the right to convert their bonds into
equity at a given time for a predetermined number of shares in the issuing company. Whereas a reverse convertible bond
has an embedded put option that gives the issuer the right to convert the
bond’s principal into shares of equity at a set date.

Convertible bonds typically have lower yields than the yields on similar fixed rate bonds without
the convertible option. Reverse convertible bonds usually have shorter terms to maturity and higher yields than most
other fixed rate bonds.

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