Discount Factors from a Term Structure of Forward Rates

Summary

The question compares a single-rate discount factor with one that incorporates the shape of the yield curve through successive forward LIBOR rates. The proposed approach accumulates discounting across periods, using the applicable forward rate for each interval rather than extending the initial rate across the full horizon.

The accepted response states that the forward-rate product is correct, but offers no derivation or numerical evidence. The central implication is that discount factors should reflect the term structure when rates vary by maturity. Exact use depends on the period definitions, compounding convention, and how the forwards relate to the discount curve; these details are not discussed in the note. It also does not clarify differences between LIBOR-based projection and discounting frameworks.

Key ideas

• A single short rate applied across all periods does not reflect changes in the yield curve.
• Successive forward rates can be combined to account for the curve shape over a horizon.
• The accepted response endorses the proposed formula but does not explain conventions or derive it.

Tags

# Discount factor taking into account yield curve shape


# Discount factor taking into account yield curve shape












I have always been told that the discount factor formula is just: $$ DF(T) = \frac{1}{(1+L_{t_0})^T} $$ where $L_{t_0}$ is the LIBOR rate on one period (the first one I guess) and $T$ the number of periods.

But shouldn't we take into account the yield curve shape, and use instead forward rates: $$ DF(T)=\frac{1}{(1+L_{t_0})\times \prod_{i=1}^T[1+F(t_i,t_{i+1})]} $$ where $F(t_i,t_{i+1})$ is the forward LIBOR rate between start of period i $t_i$ and its end $t_{i+1}$?

Why haven't I encountered this? Is it wrong?

## Answer by Sithered (score 0, accepted)

https://quant.stackexchange.com/a/25929

After looking for an answer on another website, this formula is right.

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.