# Floating-Rate Note

A floating-rate note (FRN) or a floater is a bond whose coupon rate changes with changes in market interest rates. The coupon rate on an FRN has a floating component which is based on some reference rate such as LIBOR and a spread component which represents the credit risk of the issuer.

While a conventional bond has a fixed coupon rate and hence the bondholder knows the timing and amount of bond cash flows, the owner of a floating rate note, on the other hand, knows the timing of the cash flows but not the actual amount. His interest payment can go up or down depending on whether the overall interest rates go up or down. The coupon payment on an FRN equals the face value of the bond multiplied by the coupon rate which is reset after each coupon payment. This can be expressed written as follows:

$$ C_t=FV\times({\rm BM}_{t-1}+Spread) $$

Where C_{t} is the coupon rate for the period, FV is the face value of the bond, BM_{t-1} is the benchmark rate (i.e. reference rate) value at the start of the coupon payment period and spread (also called **quoted margin**, credit margin or default margin) represents a component of the coupon rate set at the time of issue of the bond.

FRNs typically have an interest rate **cap** and **floor** which means that the coupon rate will float between the lower value (floor) and upper value (cap).

## Example

Let’s say a government agency issues 10-year floating rate notes paying quarterly coupons based on 3-months LIBOR plus 40 basis points on 1 January 2018. Let’s assume that 3-months LIBOR has the following history:

Date | 3-month LIBOR |
---|---|

1 January 2017 | 0.88% |

1 April 2017 | 1.00% |

1 July 2017 | 0.95% |

1 October 2017 | 0.93% |

31 December 2017 | 0.88% |

The quarterly coupon payments on the bond be as follows:

Coupon Payment Date | 3-month LIBOR | Spread | Coupon Rate | Coupon Payment |
---|---|---|---|---|

1 April 2017 | 0.88% | 0.40% | =0.88%+0.40%=1.28% | $12.80 |

1 July 2017 | 1.00% | 0.40% | =1.00%+0.40%=1.40% | $14.00 |

1 October 2017 | 0.95% | 0.40% | =0.95%+0.40%=1.35% | $13.50 |

31 December 2017 | 0.93% | 0.40% | =0.93%+0.40%=1.33 | $13.30 |

## Pricing of floating-rate note

Since the interest rate on a floating-rate note is reset periodically, its price is expected to stay close to the par value unless there is major deterioration in its credit quality or the bond hits the cap or floor. Theoretically, the price of a floating-rate note should equal its par value at each reset date and any time before the next reset, the price equals the present value of the next coupon payment and par value.

Because coupon rate is updated after each payment, it has lower interest rate risk than conventional bonds. It is preferred by investors when they expect the interest rates to increase. However, this option of increased interest rate has a cost. The bond duration of a floating-rate note on the reset date equals the duration of a par bond with the same maturity as the next reset date of the FRN. The bond yield on FRNs is typically lower than the conventional fixed-rate bonds of the same maturity and credit quality.

A more complex analysis involves working out the forward interest rates that are expected to apply to each coupon date, calculating the expected coupon amounts based on the forward interest rates (inclusive of the FRN spread) and then discounting those cash flows to the valuation date. This can be expressed mathematically as follows:

$$ {\rm FRN}_{PV}=\sum_{i}^{n}{FV\times{\rm fr}_i\times{\rm TF}_i\times{\rm PVF}_i}+FV\times{\rm PFV}_n $$

Where FV is the face value of the FRN, fr_{i} is the forward interest rate for i^{th} period, TF_{i} is the time factor for the ith period represented number of days between the coupon period and valuation dated divided by 365, PVF_{i} is the present value factor based on market interest rates for each coupon date and PFV_{n} is the present value factor based on market interest rates that applies to the face value of the FRN. A more advanced may value the FRN by discounting the coupons based on spot interest rates applicable to each coupon payment.

## Inverse floater

An inverse floater or an inverse FRN is a bond which is indexed to a broad interest benchmark such that its coupon payment increases when the benchmark decreases and vice versa. An inverse FRN has a payoff pattern which is exactly opposite of a similar FRN, hence the name inverse-FRN.