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Deriving Forward Discount Factors and Simple Forward Rates

Article Quant Q&A · Author: user2967440
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Summary

The document explains how to obtain a forward discount factor and forward reference rate from discount factors observed from today to two future dates. For dates t1 and t2, it uses the multiplicative relationship between discount factors: the factor from t1 to t2 is the today-to-t2 factor divided by the today-to-t1 factor. This applies directly to the second floating cashflow period in the swap example once its endpoint discount factors have been interpolated.

It then derives a simple-compounded forward rate from the ratio of those discount factors and the year fraction between the dates. The relation can be checked by reconstructing the discount factor to t2 from the earlier factor and the forward rate. The document also distinguishes annual from continuous compounding when converting a zero-coupon discount factor into a zero rate. The rate formulas depend on the chosen day-count period and compounding convention, which the example does not specify in detail.

Key ideas

  • Forward discount factors follow from the ratio of discount factors to the two endpoint dates.
  • A simple forward rate can be derived from that ratio and the accrual period.
  • The forward rate formula is consistent with discounting first to the start date, then across the forward period.
  • Converting discount factors to zero rates depends on the compounding convention.

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Fixed incomeDerivatives pricingStatistics
Full text
# Calculate forward discount factors and forward reference rate when discount factors are known


# Calculate forward discount factors and forward reference rate when discount factors are known












I am trying to learn how to value interest rate swap through portfolio of FRA's(forward rate agreement).But I have got stuck in calculation of floating leg.

Here is the scenario as given below for which I need help.

- The swap starts at 05-Jan-19 for which the zero coupon discount factor is 1.The 1st cashflow period is from 05-Jan-19 to 05-Jul-19.

- The start date and end date of cashflow for 2nd period is from 05-Jul-19 to 05-Jan-20.

- By linear interpolation (zero coupon discount factors are given);I got zero coupon discount factors at 05-Jul-19 and 05-Jan-20 (2nd period). Assume these zero coupon discount factors to be df1 and df2 respectively.

Questions - How can I find forward discount factor for this 2nd period(05-Jul-19 to 05-Jan-20).Also how can I find forward reference rate for this 2nd period.

## Answer by Magic is in the chain (score 2, accepted)

https://quant.stackexchange.com/a/45355

Let $df\left(t_1, t_2\right)$ represent the discount factor between the two periods. You then have:

$df\left(t_0, t_2\right) =df\left(t_0, t_1\right) \,df\left(t_1, t_2\right) $

So

$df\left(t_1,t_2\right) =\frac{df\left(t_0, t_2\right)}{df\left(t_0, t_1\right) }$

The forward rate between the two periods as at time 0 is as follows:

$F\left(0, t_1,t_2\right)=\frac{1}{t_2-t_1} \left(\frac{df\left(t_0, t_1\right)}{df\left(t_0, t_2\right) }-1\right)$

Which you can easily verify by noting that:

$df \left(t_0,t_2\right)=\frac{df \left(t_0, t_1\right)}{1+\left(t_2-t_1\right)\, F\left(0, t_1,t_2\right)}$

Re-zero rate comment below, if you assume annual compounding then the discount factor for t years is:

$df(t)=\frac{1}{\left(1+r\right)^t}$

Which means

$r=\left(df(t)\right)^{\frac{1}{t}}-1$

And if you assume continuous compounding then

$df(t)=e^{-r\,t} \Rightarrow r=-\frac{1}{t} \ln df(t)$

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.