# Forward Rate Agreements

Forward Rate Agreements

A Forward rate agreement (FRA) is an over-the-counter agreement between parties that a certain rate of interest will apply to either lending or borrowing at a specified time in future.

Main Features of a FRA

1. It is an over-the-counter agreement between parties.
2. A certain rate of interest is agreed upon for a specified time in the future.
3. The borrowing and lending is generally done at LIBOR rate.
4. Interest is paid at the end of the period, but present value is typically paid at the beginning of the period.

Now consider a forward rate agreement where company A has agreed to lend money to company B for the period of time T1 and T2

RK: The rate of interest agreed to in the FRA

RF: The forward LIBOR interest rate as calculated today between the time period T1 and T2

RM: The actual LIBOR interest rate observed in the market at time T1 between time period T1 and T2

P: The principal underlying the contract

The compounding frequency for FRAs is (T2 – T1)

Under normal conditions, company A would earn RM

However, it will now earn (RK – RM)

The cash flow is P × (RK – RM) × (T2 – T1)

As FRAs are settled at T1, the payoff has to be discounted from T2 to T1.

The payoff for company A is Example 4.3 from ‘Options, futures and other Derivatives by John C. Hull

Suppose that a company enters into an FRA that specifies it will receive a fixed rate of 4% on a principal of \$100 million for a 3-month period starting in 3 years. If 3-month LIBOR proves to be 4.5% for the 3-month period what is the cash flow to the lender?

The cash flow at 3.25 years to the lender will be

100,000,000 × (0.04 – 0.045) × (0.25) = ─\$125,000

The cash flow at the 3-year point will be

─\$125,000/ (1 + 0.045 × 0.25) = ─\$123,609

Valuation of an FRA

VFRA = P.V. of Cash flow at time T2 Where R2 is the continuously compounded risk less zero rate for a maturity of T2

Duration of a Bond

1. A zero-coupon bond with a maturity of n years has a duration of n years.
2. A coupon bearing bond has a duration of less than n years as the holder receives some cash.

If a bond has cash flows ci at time ti (1≤i≤n), the relationship between the bond price and bond yield is The duration D is • The bond price is the PV of all the payments.
• The duration is the weighted average of the times when the payments are made, with the weight applied to time ti being equal to the bond’s total present value provided by the cash flow at time ti.
• All the discounting is done with the bond yield rate.
 Time (years) Cash Flow Present value Weight weight X time 0.5 5 4.709 0.0500 0.025 1.0 5 4.435 0.0471 0.047 1.5 5 4.176 0.0443 0.066 2.0 5 3.933 0.0417 0.083 2.5 5 3.704 0.0393 0.098 3.0 105 73.256 0.7776 2.333 Price 94.213 Duration 2.653

Duration provides the information about the effect of a small parallel shift in the yield curve on the value of the bond portfolio. The percentage decrease in the value of the portfolio equals the duration of the portfolio multiplied by the amount by which the interest rates are increased in the small parallel shift. (only when the parallel shifts are small)

Duration is a measure of risk because it has a direct relationship with price volatility. The greater duration of the bond, the greater its percentage price volatility.

The graph of the relationship between price and yield is convex for any bond. The degree to which the graph is curved shows how much a bond’s yield changes in response to a change in price.