Building FRA Curve

## Curve Building With FRA

Interest rates curve is critical when a portfolio of interest products need to be revalued. Zero rates curve can be built using traded or observed futures prices and/or FRAs. Focus, however is given to deriving discount factors curve throughout the book. Zero rates curve can then be built by converting the discount factors into zero rates.

Discount factors curve can be generated from both observed prices of FRAs and 3M KLIBOR futures contracts using Equation 1-10 . For simpler generalisation the term $$\dfrac{t}{365}$$ will be replaced by $$dcf$$, the day count factor applicable for the tenor.

\begin{align} 1 + r_l.dcf_l &= (1+ r_s.dcf_s) (1+ r_f.dcf_f)\\ \textrm{where } r_l &= \textrm{ rate for the long period} \\ r_s &= \textrm{ rate for the short period} \\ r_f &= \textrm{ rate for the forward period} \\ dcf_l &= \textrm{ day count factor for the long period} \\ dcf_s &= \textrm{ day count factor for the short period} \\ dcf_f &= \textrm{ day count factor for the forward period} \\ \end{align}

Equation 2-5

Equation 2-5 is much more practical if one is building the curve using spreadsheet or programming languages as calculation of day count factors can be done separately. Inverting Equation 2-5;

\begin{align} \dfrac{1}{1+ r_l.dcf_l} &= \dfrac{1}{1+ r_s.dcf_s} \dfrac{1}{1+ r_f.dcf_f}\\ DF_l &= DF_s.DF_f\\ \textrm{where } DF_l &= \textrm{ discount factor for the long period} \\ DF_s &= \textrm{ discount factor for the short period} \\ DF_f &= \textrm{ discount factor for the forward period} \\ \end{align}

Equation 2-6

With a 3x6 FRA, Equation 2-6 would require $$r_s$$, $$dcf_s$$, $$r_f$$ and $$dcf_f$$ to generate $$DF_l$$. With 3x6 and 6x9 FRA, the new discount factor formula would require redefining the variables to be self explanatory as below:

\begin{align} DF_{0,n} &= DF_{0,1}.DF_{1,2}...........DF_{n-1,n}\\ \textrm{where } DF_{0,n} &= \textrm{discount factor for the period from time 0 to time } n\\ DF_{0,1} &= \textrm{discount factor for the period from time 0 to time 1} \\ DF_{1,2} &= \textrm{ discount factor for the period from time 1 to time 2} \\ DF_{n-1,n} &= \textrm{ discount factor for the period from time } n-1 \textrm{ to time } n\\ \end{align}

Equation 2-7

The above can be implemented to derive the discount factors from the 3- month money rate, 3 x 6 FRA and 6 x 9 FRA as follows:

\begin{align} DF_{0,9} &= DF_{0,3}.DF_{3,6}.DF_{6,9} \\ \textrm{where } DF_{0,9} &= \textrm{discount factor for the period from today to end of month 9} \\ DF_{0,3} &= \textrm{discount factor for the period from today 0 to end of month 3} \\ DF_{3,6} &= \textrm{discount factor for the period from end of month 3 to end of month 6} \\ DF_{6,9} &= \textrm{ discount factor for the period from end of month 6 to end of month 9} \\ \end{align}

The formula can also be generalised as below:

\begin{align} DF_{0,n.k} &= DF_{(i-1).k_{i-1},1.k_i} .DF_{(i-1).k_{i-1},2.k_i}...........DF_{(n-1).k_{n-1},n.k_n}\\ &= \prod_{i=1}^{n}{DF_{i-1,i.k_i}}\\ \textrm{where } i.k &= \textrm{end period of the discount factor} \\ (i -1).k &= \textrm{start period of the discount factor} \\ k_i &= \textrm{forward period of the discount factor} \\ \end{align}

Equation 2-8

For $$DF_{0,9}$$, $$k = 3$$ (months) for all $$k_i$$ and $$n=3$$ . When $$i = 1$$, the first term i.e. $$DF_{0,3}$$, will be the discount factor calculated from the cash rate. Equation 2-7 can also be used for a non-constant $$k$$ value.

Table 2-9 is an example of discount factors generated form FRAs using Table 2-1 and Table 2-2. Readers may change the rate to see changes in the discount factors.

Table 2-8

Generating the discount factors from futures contracts, however is not as simple and subject to the assumptions made. Our main assumption made with regards to the 3M KLIBOR futures contract is that it is a 3-month contract using month/months in a year (30/360) convention in calculating the tick value. Others may assumed that it is a 90-day contract using Actual/360 convention and still be able to derive a tick value of 25.

The next step is then to convert the futures prices to fra rates after taking into account convexity adjustment. The rates can then be tabulated as per table 2-9 and the discount factors can then be generated.

### Summary

The pricing and valuation mathematics for FRA and 3M KLIBOR futures contract use the basic mathematics in Chapter 1. Discount factors curve generation from these two products is introduced in this chapter as a prelude to other methods that will be discussed in Chapter 4, 5 and 6.