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Chapter 2

Interest calculations

and yield curves

Interest is the price paid for borrowing money. For the calculation of interest

amounts, different agreements apply. One of those agreements concerns the deter-

mination of the number of interest days in an interest period. Another agreement

concerns the number of interest payments that take place during the period of the

contract. These agreements are also important for calculating the future value of an

amount after an investment period and for the calculation of the present value of a

future cash flow.

Different interest rates apply for different periods. The relationship between the

term and the corresponding interest rate is represented by a yield curve. The shape

of a yield curve provides information about the market perception regarding the in-

terest rate development. The market perception is implied by forward yields.

2.1 Calculation of interest amounts

Interest amounts are normally paid out in arrears and are calculated using the fol-

lowing equation:

Interest amount = Principal x interest rate x daycount fraction.1

Because interest rates are always presented per annum, an adjustment factor is ap-

plied to bring the interest rate in line with the maturity period. This adjustment fac-

tor is called the daycount fraction.

1 The equation to calculate coupon amounts should be entered in a HP financial calculator

as follows: COUP = NOM x C% x D / B. If, in an equation, a % character is added to a varia-

ble, this variable should be entered as a percentage: e.g. 4% = 0.04.

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The equation to calculate the daycount fraction is:

Daycount fraction = number of days in coupon period (tenor)

----------------------------------------------y---e---a----r-----b---a----s---i--s------------------------------------------------

2.1.1 The duration of the coupon period

The start date of a coupon period is normally a fixed number of days later than the

(re)fixing date of the interest rate. The first coupon period, for instance, of a loan

starts when the nominal amount is transferred from the lender to the borrower.

With money market deposits, this is normally two working days after the closing of

the loan agreement (t+2). The exception is Great Britain, where the coupon period

starts on the trading date. With exchange traded bonds, normally after three work-

ing days (t+3). With an interest rate derivative the first coupon period also normally

starts on t+2. If the interest rate is fixed periodically during the term of a contract,

the coupon period starts two working days after the fixing date, except for Great

Britain.

The end date of a coupon period is called the coupon date. On this date the coupon

is paid. The coupon dates of regular periods (1, 2, 3 months etcetera) normally fall on

the same day in the month as the start date. There are, however, exceptions to this

rule.

If the coupon date falls in a weekend or on a bank holiday, the coupon cannot be

paid on this date. This is because the central bank's payment system is not opera-

tional on these days. The coupon date will then be adjusted to the previous or the

next business day according to the convention agreed upon in the market or in the

specific contract.

The most used conventions are `following' and `modified following'. With the con-

vention following, the coupon date will be postponed to the next business day. This

is also the case with the convention modified following with one exception, how-

ever. If the adjusted coupon date would fall in the next month, the coupon date is

set on the previous business day. In the money market, the modified following con-

vention is normally used. This is also the case in ISDA agreements.

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Below are the maturity dates of the regular periods for trading day 13 April 2009.

spot / / Wed

m / / Fri

m / / Mon

m / / Wed

m / / Mon / is Sat

m / / Tue

m / / Thu

If the spot date falls on a month ultimo date, i.e the last trading day of a month, all

regular dates will in principle be set on a month ultimo date too. Additionally, the

modified following convention is applied. In this case, the convention is referred to

as end-of-month convention (EOM). The table below shows the maturity dates for

several regular periods for trading day 28 April 2009.

spot / / Thu

m / / Fri / is Sun

m / / Tue

m / / Fri note st

m / / Mon note st

m / / Wed

m / / Fri / is Sat

If the coupon date of a long term contract is adjusted, the question is whether the

coupon term should be adjusted too. Again, two different conventions may be used:

adjusted and unadjusted. When the convention `adjusted' is used, the number of

the interest days is adjusted to the new coupon date. When the convention `unad-

justed' is used, the number of interest days stays unchanged.

The number of days in a coupon period is calculated by including the start date and

excluding the end date.

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What's the difference between APR and APY? APR vs. APY: The Big Difference is Compounding Years Invested APR (Simple Interest) APY (Annual Compounding) APY (Daily Compounding) End of Year 1 $10,500.00 $10,500.00 $10,512.67 End of Year 2 $11,000.00 $11,025.00 $11,051.63 End of Year 3 $11,500.00 $11,576.25 $11,618.22

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