**interest rate calculations**. The methods used to calculate interest due can be conveniently divided into three areas:

**(i)**

**discount**methods, as per

**banker's acceptances**,

**bills of exchange**,

**commercial paper**, and

**treasury bills**;

**(ii)**

**yield**-based money market calculations, such as bank

**deposits**,

**certificates of deposit**, some kinds of

**commercial paper**, and short-dated

**bonds**with one or more

**coupons**to

**maturity**; and

**(iii)**bond market calculations.

The money markets use both methods (i) and (ii) to calculate interest due. Discount methods were favoured given the complications of adding small amounts of interest to a short-term instrument. Selling the instrument below par, using a simple formula was both easier and facilitated repayment at maturity. The discount represented the buyer's interest on his investment. Method (iii) is favoured by the bond markets, although some hybrid instruments such as

**floating rate notes**use method (ii).

The difference in approach arises from the

**tenor**or maturity of the instruments. It was not quite so important to calculate interest to an exact fraction when the bond has a maturity of ten years, or more. The major complication arises in the exact method used to establish the number of elapsed days of interest in any period, both for determining the amount of coupon, or interest due, and also for secondary market transactions when

**accrued interest**has to be included in the

**settlement**or

**transaction**price. There are a number of ways of accruing interest used by both money markets and the bond markets, but they can be grouped into three methods: an actual day count method, an actual day count method divided by a notional number of days in a calendar year (known as the

**basis**), and a fixed day count divided by the basis.

**1.**

*Money markets*: The principal day count methods are: actual/360 and actual/365 (fixed). The basis for calculating interest is therefore a year of either 360 days, or 365 days. The effect of a 360-day year is to increase the amount of interest paid by a factor of 1.013889 over that quoted. The following summarizes the situation in different countries. Basically, the UK and the old Commonwealth or Empire countries use the 365-day basis, the rest of the world, with the exception of Belgium, the 360-day basis (see Table).

Country | Interest basis calculation |

Australia | Actual/365 |

Austria | Actual/360 |

Belgium | Actual/365 |

Canada | Actual/365 |

Denmark | Actual/360 |

France | Actual/360 |

Germany | Actual/360 |

Ireland | Actual/365 |

Italy | Actual/360 |

Japan | Actual/360 |

The Netherlands | Actual/360 |

New Zealand | Actual/365 |

Norway | Actual/360, under 1 month; 30/360 over 1 month; also Actual/365 on some money market instruments |

Saudi Arabia | Actual/360 |

Sweden | 30/360 |

Switzerland | Actual/360; federal bills: 30/360 |

United Kingdom | Actual/365 |

United States | Actual/360 |

In the international markets, the domestic market interest calculation methods have been followed, with again the exception of Belgium, where Actual/360 is sometimes used, although Actual/365 is also quoted (cf.

**London interbank offered rate**).

The Actual/360 calculation also known as

*bank interest basis, certificate of deposit basis, Euro basis*, or

*money market basis*. Whatever the maturity, the convention is to quote the interest rate as if for a year and the amount payable is calculated on the actual number of days elapsed in the calculation period multiplied by the interest rate and divided by the number of days in the computation year, which is known as the basis:

For example, to calculate the interest on a three month (90 actual days) deposit of 5.75% of US$1,000,000 for the period starting on 28 January and ending on 28 April (for calculation purposes, interest is charged from the day following the settlement date up to and including the day on which the payment is due). In the above example this would be:

For the Actual/365 calculation, the interest is also quoted as an annual rate, regardless of the tenor of the transaction, and is calculated on the actual number of days in the period times the rate of interest over a basis of 365. Thus a £ 5 million sterling certificate of deposit at 6.75% issued on 24 February and maturing on 24 August would pay the following coupon interest:

If it had been a leap year the interest would have been slightly different, since one day would have been added to the calculation:

Note that the interest rate for Actual/Actual (see below) is often used, with some confusion, for Actual/365.

**2.**

*Bond markets*: The principal methods of working out interest are: (i) Fixed Coupon; (ii) Bond basis (also called eurobond basis); and (iii) Actual/365 or Actual/Actual.

*Fixed Coupon*. This method of calculating interest divides the

**coupon rate**by the frequency of payments, even though this may lead to differences in the number of elapsed days over which the interest is actually received due to weekends, holidays, or leap years. The interest rate is quoted as an annual rate and is calculated on the years in the period instrument over 100(%).

For example a bond with a fixed coupon of 10% paid quarterly in March, June, September, and December with principal of DM10,000,000, with an interest period running from, say, 31 March to 30 June would pay a coupon of:

Fixed Coupons are used for

**eurobonds**(note that the vast majority of eurobonds pay annually).

*Bond basis*(sometimes notated as

*30/360*or

*30E/360*). This is the method of day count used in the secondary market to calculate the accrued interest on certain types of

**fixed rate**bonds, most notably those traded in the

**eurobond market**. The accrued interest is calculated on a computational year of 360 days with each calendar month treated as one-twelfth of 360 days, that is 30 days. For interest purposes each period from a date in one month to the same date in the next or subsequent months is also considered to be 30 days, or multiples thereof. Accrued interest is calculated from, and including the date of, the last coupon date or, in the case of a new issue, from the day interest starts to accrue (the

**dated date**), up to, but excluding, the

**value date**of the transaction. This is different from the conventions used in the money markets, where the first day is ignored, but the last day is included. Examples of the method showing the difference in day count between actual days, Actual/365, and Actual/Actual are given in the Table.

Interest accrues from coupon date | Value date | Number of days accrued interest | ||

Bond basis (eurobond; 30E/360) | Actual/365, normal | Actual/Actual, leap year | ||

Within the year: | ||||

1 Jan. | 28 Feb. | 57 | 58 | 58 |

1 Jan. | 29 Feb. | 58 | — | 59 |

1 Jan. | 1 Mar. | 60 | 59 | 60 |

1 Jan. | 3 Mar. | 62 | 61 | 62 |

1 Jan. | 30 Mar. | 89 | 88 | 89 |

1 Jan. | 31 Mar. | 89 | 89 | 90 |

15 Jan. | 28 Feb. | 43 | 44 | 44 |

15 Jan. | 29 Feb. | 44 | — | 45 |

15 Jan. | 1 Mar. | 46 | 45 | 46 |

15 Jan. | 3 Mar. | 48 | 47 | 48 |

1 Feb. | 28 Feb. | 27 | 27 | 27 |

1 Feb. | 29 Feb. | 28 | — | 28 |

1 Feb. | 1 Mar. | 30 | 28 | 29 |

1 Feb. | 3 Mar. | 32 | 30 | 31 |

15 Feb. | 28 Feb. | 13 | 13 | 13 |

15 Feb. | 28 Feb. | 14 | — | 14 |

15 Feb. | 1 Mar. | 16 | 14 | 15 |

15 Feb. | 3 Mar. | 18 | 16 | 17 |

28 Feb. | 29 Feb. | 1 | — | 1 |

28 Feb. | 1 Mar. | 3 | 1 | 2 |

28 Feb. | 3 Mar. | 5 | 3 | 4 |

28 Feb. | 5 Mar. | 7 | 5 | 6 |

28 Feb. | 30 Mar. | 32 | 30 | 31 |

28 Feb. | 31 Mar. | 32 | 31 | 32 |

The following year: | ||||

30 Nov. | 28 Feb. | 88 | 90 | 90 |

30 Nov. | 29 Feb. | 89 | — | 91 |

30 Nov. | 1 Mar. | 91 | 91 | 92 |

30 Nov. | 3 Mar. | 93 | 93 | 94 |

30 Nov. | 30 Mar. | 120 | 120 | 121 |

30 Nov. | 31 Mar. | 120 | 121 | 122 |

31 Dec. | 28 Feb. | 58 | 59 | 59 |

31 Dec. | 29 Feb. | 59 | — | 60 |

31 Dec. | 1 Mar. | 61 | 60 | 61 |

31 Dec. | 3 Mar. | 63 | 62 | 63 |

31 Dec. | 30 Mar. | 90 | 89 | 90 |

31 Dec. | 31 Mar. | 90 | 90 | 91 |

28 Feb. | 27 Feb. | 359 | 364 | 365 |

28 Feb. | 28 Feb. | nil | nil | nil |

Note that there are two different forms of the bond basis (

*30/360*or sometimes

*360/360*, and

*30E/360*) which are due to differences in the way the number of days from the last coupon payment to the settlement date are calculated. For 30/360, the 30-day month method involves taking the two dates for accrued interest purposes by assuming 30 days in each month using a formula, where the first date is (MM

_{1}- DD

_{1}- YY

_{1}) and the later date (MM

_{2}- DD

_{2}- YY

_{2}). If DD

_{1}is a 31-day month, change to 30; if DD

_{2}is 31, change to 30 if DD

_{1}is 30; otherwise leave as 31 (this gives a maximum of 30 days in a month). The number of elapsed days is calculated by:

{(YY

_{2}- YY

_{1})} × 360) + {(MM

_{2}- MM

_{1}) × 30)} + (DD

_{2}- DD

_{1})

For 30Esol;360, the calculation is the same as the above, but involves changing DD

_{1}to 30, if the calendar month has 31 days, and changing DD

_{2}to 30 days, if 31 days.

For example, a holding of US$500,000 of a eurobond with an annual coupon of 8%, sold for delivery on 3 March , with a coupon payment date of 15 February , would have accrued eighteen days interest:

Note that 30/360 is used in the US domestic market for some US government

**agency**, municipal, and corporate bonds; 30E/360 for the vast majority of eurobonds (but excluding floating rate notes, where the accrued interest is calculated on a

**money market basis**), German government and agency bonds, and European currency

**cross-currency swaps**and

**interest rate swaps**which include a fixed rate payment.

*Actual/365*or

*Actual/Actual*. This method calculates accruing interest by multiplying the principal times the interest rate times the number of days, all divided by 365. With a leap year, it is adjusted so that it is the number of actual days in the period divided by 366. Note that the calculation method takes the day from 31 December to 1 January as part of the earlier year.

Take as an example an 8% coupon US$50,000 Treasury Bond. The period 30 November to 31 March in a leap year would be 122 days. For interest assessment purposes, because US Treasury issues are calculated on an Actual/Actual basis, the calculation has to be split between the normal and the leap year:

The Actual/Actual accrual method is used for US

**treasury notes**and

**US treasury bonds**and some interest rate swaps denominated in US dollars where interest is paid semi-annually.

*Coupon or interest frequency*: The great majority of

**fixed rate bonds**pay interest (coupons) either once or twice a year (known as

*annual*or

*semiannual*basis). Note also that for simplicity of trading, the vast majority of securities issued in a particular market will tend to follow the market's accepted convention for interest frequency. Occasionally, specialized securities may use different methods; for example,

**collateralized mortgage obligations**and floating rate notes, which may pay interest quarterly or even monthly.

**3.**

*Yield calculations*. (For money market calculations which use

**simple interest**, see

**bank basis**;

**discount**;

**money market basis**.)

*Simple, flat*, or

*income yield*:

Using the US Treasury example given above and if the quoted full price were 98

^{1}/

_{4}, we would have:

*4*.*Redemption yield*(also known as

*yield-to-maturity*;

*yield-to-call*;

*yield-to-put*, and so forth). This is the rate of interest at which the total of the discounted values of the future payments of interest and capital are equal to the current price. This is equivalent to:

where

*C*is the coupon;

*P*the principal amount at redemption, call, or put;

*i*the interest rate as an annual rate;

*f*the coupon frequency;

*t*

_{sc}the number of days, expressed as a fraction, from the settlement date to the next coupon date; and

*n*

_{cm}the number of periods from the next coupon date to redemption, maturity, call, or put.

The

**clean price**is arrived at by deducting the accrued interest.

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