**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**

- It is an over-the-counter agreement between parties.
- A certain rate of interest is agreed upon for a specified time in the future.
- The borrowing and lending is generally done at LIBOR rate.
- 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 T_{1} and T_{2}

R_{K}: The rate of interest agreed to in the FRA

R_{F}: The forward LIBOR interest rate as calculated today between the time period T_{1} and T_{2}

R_{M}: The actual LIBOR interest rate observed in the market at time T_{1} between time period T_{1} and T_{2}

P: The principal underlying the contract

The compounding frequency for FRAs is **(T _{2} – T_{1})**

Under normal conditions, company A would earn R_{M}

However, it will now earn **(R _{K} – R_{M})**

The cash flow is **P × (R _{K} – R_{M}) × (T_{2} – T_{1})**

As FRAs are settled at T_{1}, the payoff has to be discounted from T_{2} to T_{1}.

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**

V_{FRA} = P.V. of Cash flow at time T_{2}

Where R_{2 }is the continuously compounded risk less zero rate for a maturity of T_{2}

**Duration of a Bond**

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

If a bond has cash flows c_{i} at time t_{i} (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 t
_{i}being equal to the bond’s total present value provided by the cash flow at time t_{i}. - 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.