FRA

# Chapter 2: Forward Rate Agreements and Short Dated Futures

## Introduction

Chapter 1, is an introduction to the basic concept of present value whereby settlements of transactions are for value same day or two business days from deal date. In this chapter, the concept is extended to include forward dated instruments namely forward rate agreements and deposit futures. These instruments are well developed and actively traded in developed market such as the US, Europe and Japan. Unfortunately, FRAs and futures markets in Malaysia are not as active nor developed.

## Forward Rate Agreements(FRAs)

### Description

By definition FRA is an agreement to borrow or lend a specified sum of money for a specified period starting from a future date which is after the standard delivery date (either value today or spot date, depending on market). Diagrammatically; Diagram 2-1

Delivery of the transactions, however will not take place on the borrowing/lending date. Instead, the party benefited from the transaction will be compensated. The Interbank Offer Rate (IBOR) such as LIBOR (London Interbank Offer Rate) are often used as the reference rate in computing compensation amount due.

### The Mechanics

As a standard, the forward borrowing/lending period in most market is three (3) months. Simple FRA quotations are provided in Table 2-1.

Table 2-1

The terms ‘3 x 6’ should be read as three by six and referred to the number of months. The term ‘3’ refers to the starting date of the forward borrowing/lending while the term ‘6’ refers to the maturity of the borrowing or lending. The difference between the term ‘6’ and the term ‘3’ is the forward borrowing period. Diagramatically; Diagram 2-2

A 3 x 6 FRA is quoted as 2.90% / 2.80%. The price provider is willing to lend for three (3) month starting three months from today at a rate of 2.90%. At 2.80%, the price provider will be willing to borrow for three (3) months starting three (3) months from today.

#### Example 2-1

FRAs prices are quoted by banks through licensed brokers. For illustration, prices in Table 2-1, is assumed to be provided by Bank A. Notice that the FRA rates quoted are bigger on the left hand side and is the reverse of the money market quotation in Malaysia. It is best to remember that a user will always lend at the lower rate and borrow at a higher rate.

Bank B is keen to borrow for the three (3) months starting three (3) months from today. The applicable FRA is the ‘3 x 6’ and Bank B will be able to enter into an FRA transaction with Bank A at the rate of 2.90%. Table 2-2 contains relevant information for confirmation letter of the transaction. FpML (Financial product Markup Language - the open source XML standard for electronic dealing and processing of OTC derivatives) provides standardised format for various purposes.

Table 2-2

FRA expiry is the date which the contract ceases to exist. On this date, the contracted rate will be compared to the ‘Reference Rate’ and the settlement amount will be calculated and paid on settlement date. Assuming settlement is for value same day (today), expiry and settlement date will be the same. USD-based FRAs, for instance are settled for value spot (two business day).

Lets assume that the 3M KLIBOR is at 3.00% on expiry date of the FRA. The borrower i.e Bank B has benefitted from the FRA transaction because the FRA allows Bank B to borrow MYR10,000,000 at 2.90% instead of 3.00%. Unfortunately, FRA agreements are non deliverable and Bank B has to be compensated for the difference of 0.10%.

The rationale behind the compensation is simple. Bank B entered into the FRA contract to borrow MYR10 million at 2.90% for three months. On expiry of the FRA, 3M KLIBOR is fixed at 3.00% and Bank B has to borrow from the interbank market at the higher rate. The difference therefore has to be compensated by Bank A, the counterparty to the transaction.

### Settlement Calculation

A borrowing MYR10 million at 2.90% for 3 months (or 91 days) will have a maturity amount of:

\begin{align} \textrm{Maturity Amount} &= FV\left(1+\dfrac{rt}{365}\right)\\ &=10,000,000\left(1+\dfrac{0.029\textrm{ x }91}{365}\right)\\ &=10,072,301.37 \end{align}

The same borrowing at the rate of 3.00% on the other hand will have a maturity amount of:

\begin{align} \textrm{Maturity Amount} &= FV\left(1+\dfrac{rt}{365}\right)\\ &=10,000,000\left(1+\dfrac{0.03\textrm{ x }91}{365}\right)\\ &=10,074,794.52 \end{align}

The difference between the two amount is MYR2,493.15. The amount however is not the settlement amount because the difference is based on the maturity amount - a future value. It needs to be discounted to the current date at the settlement rate:

\begin{align} \textrm{Settlement Amount} &= \dfrac{2,493.15}{\left(1+\frac{rt}{365}\right)}\\ &=\dfrac{2,493.15}{\left(1+\frac{0.03\textrm{ x }91}{365}\right)}\\ &=2,474.64 \end{align}

The above calculation can be represented by the following formula:

\begin{align} \textrm{Settlement Amount} &= \dfrac{{FV\left(1+\frac{r_s t_f}{365}\right)} + FV{\left(1+\frac{r_f t_f}{365}\right)} } {\left(1+\frac{r_s t_f}{365}\right)} \\ &=FV\dfrac{(r_s-r_f)\frac{t_f}{365}}{\left(1+\frac{r_f t_f}{365}\right)} \\ &=FV\dfrac{(r_s-r_f)t_f}{365+r_s t_f}\\ \textrm{where }FV &= \textrm{face value or notional principal of the FRA} \\ r_s&=\textrm{FRA settlement rate}\\ r_f&=\textrm{contracted FRA rate}\\ t_f&=\textrm{FRA borrowing/lending period} \end{align}

Equation 2-1

A positive settlement amount from using Equation 2-1 indicates that the borrower has benefitted from the FRA. Hence, the borrower will receive the settlement amount from the counterparty. On the other hand, if the settlement amount is negative, the borrower has to pay the amount to the counterparty.

Equation 2-1 is the settlement amount of an FRA for value settlement date. Valuation of an FRA for value date prior to the settlement date can be with minor modification to the equation. Simply present value (discount) the amount obtained from the equation as shown below:

\begin{align} \textrm{Settlement Amount} &= FV\dfrac{(r_s-r_f)t_f}{\left(365+r_s t_f\right)\left(1+\frac{r_o t_o}{365}\right)}\\ \textrm{where }r_o&=\textrm{rate for the period from value date to settlement date}\\ t_o&=\textrm{number of days from value date to settlement date}\\ \end{align}

Equation 2-1a

Alternatively, Equation 2-1 can be generalised into:

\begin{align} \textrm{Settlement Amount} &= \dfrac{{FV\left(1+\frac{r_s t_f}{365}\right)} + FV{\left(1+\frac{r_f t_f}{365}\right)} } {\left(1+\frac{r_m t_m}{365}\right)} \\ &=FV\dfrac{(r_s-r_f)\frac{t_f}{365}}{\left(1+\frac{r_m t_m}{365}\right)}\\ \textrm{where }r_m&=\textrm{rate for the period from value date to the theoretical maturity date}\\ t_m&=\textrm{number of days from value date to the theoretical maturity date }\\ \end{align}

Equation 2-1b

CALCULATOR: FRA SETTLEMENT

When value date is the same as the settlement date, $$t_m$$ will be equal to $$t_f$$ and $$r_m$$ will be equal to $$r_f$$ and Equation 2-1b will revert to Equation 2-1.

The following is a FRA settlement calculator for Example 2-1. User can change the 'Value Date', $$r_f$$ ('FRA Rate (%)'), $$r_o$$ ('Rate to Settle. Date (%)') and $$r_s$$('Settlement Rate (%)') as simulations to the settlement amount.