FRA Pricing

Pricing And Valuation

Equation 1-10 is the basic formula for the pricing of a forward borrowing and lending which relies on the concept of compounding - reinvestment of principal and interest into the next investment period. The equation is applicable to FRAs and short-term deposit futures. Table 2-2 is provided for the purpose of illustration of the pricing of both products.

Table 2-3

User can change the value in the 'Quotes (%)' field for simulation. This not only change the discount factor for the respective row, but may also affect the valuation of FRA in Table 2-3 (only changes to the 3M and 6M rates will affect our FRA valuation example).

Recall that in Example 1-2, a deposit was terminated and Equation 1-10 was introduced to calculate the breakeven rate of the termination.

\( \left(1+\dfrac{r_{l}t_{l}}{365}\right)=\left(1+\dfrac{r_{s}t_{s}}{365} \right)\left(1+\dfrac{r_{f}t_{f}}{365}\right) \)

The equation suggest that a deposit for a period of \(t_l\) at the rate of \(r_l\), is equivalent to two consecutive deposits for a period of \(t_s\) and \(t_f\) at the rates of \(r_s\) and \(r_f\), respectively, with the interest reinvested as well. This often referred compounding whereby the principal and interest are reinvested at \(r_f\) for a period of \(t_f\). Equation 1-10 is the no arbitrage pricing or condition for FRA and short term deposit futures. Deviation from the pricing derived from the formula will lead to arbitraging subject to costs involved.

In the context of a 3 x 6 FRA, \(t_s\) is three (3) months while \(t_l\) is six (6) months. \(t_f\) will be the difference between \(t_l\) and \(t_s\) which is three months. \(r_s\), the rate applicable for \(t_s\) is the 3-month KLIBOR i.e. 2.75%. \(r_l\), the rate applicable for \(t_l\) on the other hand is the 6-month KLIBOR i.e. 2.78%. \(r_f\), the forward rate of the rate applicable to \(t_f\). will have to be calculated. Rearranging Equation 1-10 to solve for \(r_f\) will yield the following formula:

\( r_f= \left(\dfrac{\left(1+ \frac{r_lt_l}{365}\right)}{\left(1+ \frac{r_st_s}{365}\right)} -1\right)\dfrac{365}{t_f} \)

Equation 2-2

Equation 2-2 can be simplified further into:


\( \begin{align} r_f= \left(\dfrac{DF_s}{DF_l}-1\right)\dfrac{365}{t_f}\\ \textrm{where }DF_s &= \textrm{discount factor applicable for } t_s\\ DF_l &= \textrm{discount factor applicable for } t_l \end{align} \)

Equation 2-3

Equation 2-3 will be used extensively in later chapters. For now, we will utilise Equation 2-2 as we will be working with rates provided provided in Table 2-2. Calculating the 3 x 6 forward rate yields:

\( \begin{align} r_f&= \left(\dfrac{\left(1+ \frac{0.028 \textrm{ x }183}{365}\right)} {\left(1+ \frac{0.0275 \textrm{ x }92}{365}\right)} -1\right) \dfrac{365}{91} \\ &=2.8309\textrm{%} \end{align} \)

Reader can use the calculator below to calculate \(r_f\), \(r_s\) and \(r_f\) in Equation 2-2 bycusing the rates provided in Table 2-3. Rates in the table can be changed by user. Additional flexibility is given through the calcuator. User can calculate 'Rate to Settle Date (%)', 'Rate to Maturity (%)' and 'FRA Rate (%)' by filling the date fields - 'Settlement Date' and 'Maturity Date'.

The current value or the marked to market value of the contract can be found using the simple concept of present value of the cash flows as described in Chapter 1. Table 2-4 mapped the theoretical cash flow for Example 2-1 to enable mark-to-market process.

Table 2-4

On the \(92^{\textrm{nd}}\) day (Sep 17, 2020), MYR10 million will be borrowed and a cash inflow is expected into the account. After three months i.e. on the \(183^{\textrm{rd}}\) day (Dec 17, 2020), the borrowed fund is expected to be returned with the interest due at the rate of 2.90% p.a. amounting to an outflow of MYR10,072,301.37. These cash flows are then discounted at the relevant rate in Table 2-2.

Discount factors in the ‘Discount Factor’ column are calculated using Equation 1-8. The ‘Present Value’ column is ‘Cash Flows' column multiplied by the ‘Discount Factor’ column and the sum of the present value is the value of the FRA. In other words, the marking to market recalculates the cash flows using market rate, discount the cash flows to the present and sums them up to arrive at the value of the FRA.

The contracted FRA rate in Example 2-1, for instance, is higher than the calculated forward rate. Due to the spread between bid and offer, it may not possible to arbitrage the FRA market against the cash market. If it is possible to lend at the FRA rate of 2.90% however, an arbitrage can be undertaken by doing the following:

  1. Borrow MYR10 million for 6-month at the rate of 2.80% as per rate Table 2-2;
  2. Lend MYR10 million at the rate of 2.75%p.a. as per rate in Table 2-2;
  3. Lend MYR10 million and interest on maturity of (ii) at a forward rate of 2.90% via FRA at 2.90%;
  4. Upon maturity of (ii) and expiry of (iii) lend the net amount from (ii) and (iii).

Table 2-4

The result of the above transaction is tabulated as Table2-4 assuming the settlement rate of the FRA is 2.50%.A net profit of 1,664.91 is recorded at the end of the arbitrage exercise.

Values in the yellow coloured column can be change by readers to simulate rate scenarios in for each row. Rate last row (iv. Lend) is also taken to be the FRA settlement rate for the calculation of settlement amount. By changing the rate the arbitrage remains sufficiently stable. At settlement rate of 5.00%, for instance, the arbitrage profit remains profitable at 2,096.94 despite massive loss incurred by the FRA.