Deriving Simple Libor Forward Rates from Spot Rates

Summary

The document gives a way to derive a simple Libor forward rate from spot rates by equating accumulation across consecutive periods with accumulation over the full term. Each rate is multiplied by its accrual fraction, and the forward rate is isolated using the short-period and longer-period spot rates. The approach reflects simple interest conventions rather than continuous compounding.

For USD Libor, the example uses an Actual/360 day-count basis: actual calendar days in each period are divided by 360. The response notes that some currencies use a different denominator, citing GBP Libor’s 365-day basis, so the applicable currency convention must be checked before calculating. The relationship assumes consistent rates, dates, and day-count treatment across the periods; the short explanation does not address payment timing, discounting conventions, or market-specific adjustments.

Key ideas

• Simple forward rates can be derived by equating compounded accrual across adjacent periods with accrual over the full term.
• The calculation uses both the short-term and long-term spot rates.
• Day-count fractions must match the currency’s quoting convention.
• The example uses Actual/360 for USD Libor and notes that some currencies use a 365-day basis.

Tags

# Relationship between simple Libor spot and forward rates


# Relationship between simple Libor spot and forward rates












How is the simple forward rate L(0,T,T+1) calculated given the spot rate L(0,T)?

## Answer by AlRacoon (score 2)

https://quant.stackexchange.com/a/71476

USD Libor rates are quoted on a Act/360 basis. You can determine USD Libor forward rates by application of the following formula.

$$ (1 + \text{SpotRate}(t) \times (\text{Act}(0,t)/360)) \times (1 + \text{FrdRate}(t,T) \times (\text{Act}(t,T)/360)) = (1 + \text{SpotRate}(T) \times (\text{Act}(0,T)/360) $$

where:

SpotRate(t) = the short term spot rate;

SpotRate(T) = the long term sport rate;

FrdRate(t,T) = forward rate from t to T;

Act(0,t) = Actual days from 0 to t;

Act(0,T) = Actual days from 0 to T;

Act(t,T) = Actual days from t to T

Note: This is for USD Libor and most other currency Libor rates. However, some currencies, such as GBP Libor are quoted on a 365 day basis. For these currencies you would substitute 365 for 360. Be sure to make sure you are using the appropriate daycount convention for the currency.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.