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Deriving a Forward LIBOR Rate from Discount Factors

Article Quant Q&A · Author: Lost1
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Summary

The document resolves a confusion about the time-zero value of a LIBOR rate for a future accrual period. It defines the simple rate to maturity through the price of a zero-coupon bond, then compares two ways of valuing the future-period loan. The proposed subtraction of discount factors represents the present value of the interest payment, rather than the forward rate itself.

The accepted explanation divides that present value by the discount factor for the period's end date. This gives the forward rate as the ratio of the two maturity discount factors minus one, matching the alternative compounding argument. The result assumes the stated simple-rate convention, with the rate to a maturity equal to the reciprocal bond price minus one. It does not address day-count fractions or other market quoting conventions, which may change the formula's details.

Key ideas

  • A discount-factor difference gives the present value of the forward-period interest payment.
  • Recovering the forward rate requires dividing that value by the discount factor at the period end.
  • The forward rate can be expressed as the ratio of maturity discount factors minus one.
  • The derivation assumes simple rates defined from unit-face zero-coupon bond prices.
  • Day-count and market quoting conventions are not discussed.

Tags

Fixed incomeDerivatives pricing
Full text
# What is time 0 price of Libor starting t for the period $t$ to $t+\delta t$


# What is time 0 price of Libor starting t for the period $t$ to $t+\delta t$












I was asked this in an interview.

The correct answer, I was told, follow from this argument

Let $L_0[0,t]$ denote the time 0 price of Libor for period $0$ to $t$.

Let $L_0[t,t+\delta_t]$ denote the time 0 price of Libor for period $t$ to $t+\delta t$

$(1+L_0[0,t])^{-1}$ dollar today is worth $1$ dollar at time $t$ hence $1+L_0[t,t+\delta_t]$ dollar at time $t+\delta t$

So the correct answer should be $(1+L_0[0,t])^{-1}-(1+L_0[0,t+\delta t])^{-1}$ by discounting the extra 1 dollar back to time 0.

I understand this, but I thought

$1$ dollar today is worth $1+L_0[0,t]$ dollar at time $t$, hence $(1+L_0[0,t])(1+L_0[t,t+\delta_t])$ dollar at time $t+\delta t$

That should be the same as investing 1 dollar for period 0 to $t+\delta t$. This should be $1+L_0[0,t+\delta t]$. Equating these give a different answer.

$(1+L_0[0,t])(1+L_0[t,t+\delta_t])= 1+L_0[0,t+\delta t]$

This gives $L_0[t,t+\delta_t] = (1+L_0[0,t+\delta t])(1+L_0[0,t])^{-1} - 1$

This is out by a factor of $1+L_0[0,t+\delta t]$. why is that the case? what did I miss?

## Answer by Gordon (score 0, accepted)

https://quant.stackexchange.com/a/30518

The answer $$(1+L_0[0,t])^{-1}-(1+L_0[0,t+\delta t])^{-1}$$ is incorrect.

Note that, since $(1+L_0[0,t])^{-1}$ dollar toady worth $1+L_0[t, t+\delta t]$ at time $t+\delta t$, and $(1+L_0[0,t+\delta t])^{-1}$ dollar today worth 1 dollar at time $t+\delta t$, therefore, $$(1+L_0[0,t])^{-1}-(1+L_0[0,t+\delta t])^{-1}$$ dollar today worth $L_0[t, t+\delta t]$ at time $t+\delta t$, then \begin{align*} (1+L_0[0,t])^{-1}-(1+L_0[0,t+\delta t])^{-1} &= L_0[t, t+\delta t] P_0(t+\delta t)\\ &=L_0[t, t+\delta t] (1+L_0[0,t+\delta t])^{-1}. \end{align*} That is, \begin{align*} L_0[t, t+\delta t] = (1+L_0[0,t+\delta t])(1+L_0[0,t])^{-1} - 1, \end{align*} which is the same as your answer in your second part.

> EDIT:

Here, it is assumed that $L(0, t) = \frac{1}{P_0(t)}-1$, where $P_0(t)$ is the price at time $0$ of a zero-coupon bond with maturity $t$ and unit face value.

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.