• PV = Present Value
• r = the Return of Rate
From the convention 1.1 above, the present value now can rearrange as PV = FV / (1+r)n where r is referred to as the discount rate. The present value, as a result of the equation, can refer to the price of a bond and the future value is the expected cash flow i.e. the coupon payment.
However, bonds in reality have more than one cash flow and therefore each cash flow needs to be discounted in order to find the present value. The price at issue is give as follows (1.2) :
(1.2)
Where • P0 = the price of the fixed-interest security at time 0 (present)
• Cn = the coupon payment at time n
• M = redemption value of security
• i = the discount rate
And there must be two assumptions to set up the equation above:
- The bonds pay a single coupon annually.
- The next coupon is exactly one year away.
In the calculation of P0, it is assumed that last coupon payment has been just been made by the assumption, the next coupon is exactly one year away. However, bonds may be traded and priced at dates which lie between 2 coupon payments. So when a bond is bought or sold midway through a coupon period, a certain amount of coupon interest will have accrued. The coupon payment is always received by the person holding the bond at the time of the coupon payment, because the person may not hold the bond until the next coupon payment is made. Therefore, when a trade of a bond lied in the period of coupon payment is made, the buyer must compensate the seller for the fraction of the next coupon payment the seller is due but will not receive. This amount is called accrued interest.
In order to calculate the accrued interest, the number of days since the last coupon, coupon interest and the total number of days in the coupon period. The accrued interest will be calculated on the following basis (1.3):
Coupon Interest X Number of days since last coupon
Total days in coupon period
(1.3)
The dirty price is the bond price included the accrued interest and the clean price is the bond price excluded the accrued interest. Thus, Dirty Price = Clean Price + Accrued Interest and also Clean Price = Dirty Price – Accrued Interest.
In the UK bond Markets, the price quoted is the clean price, but the price actually paid is based on the dirty price.
- A 7% government bond is priced at £94.58 and matures in 8 years. What is its current yield? For the same bond what would the simple yield be? In which way does the simple yield differ to the current yield?
The formulae for the Current yield is:
Where • Yc = Current Yield.
• C = Coupon payment (%)
• Pb = Clean Price of Bond
(2.1)
Therefore, Yc = ( 7 / 94.58 ) * 100 = 7.4011 7.40(%)
The current yield of the bond is 7.40%
The Simple yield is on the basis of:
Where • Ys = Simple Yield
• C = Coupon payment (%)
• Pb = Clean Price of Bond
• t = years to maturity
(2.2)
Therefore, Ys = { ( 7 / 94.58 ) + ( 100 – 94.58 ) / ( 8 * 94.58 ) } * 100
= { 0.0740 + ( 5.42 / 756.64) } * 100
= ( 0.0740 + 0.0072 ) * 100
= 0.0812 * 100 = 8.12(%)
The simple yield of the bond is 8.12%
The simple yield is more sophisticated measure of return than current yield. Simple yield take into account capital gains or losses, resulting from the difference between the current price of the bond and its value upon maturity while current yield takes no account of the potential capital gains or losses. But simple yield does not take into account accrued interest as it uses the clean price in the equation. However, current yield can be used as a measure of value when the period of maturity is very long – when the coupon income will be more important in the total return than capital gains and losses and simple yield can be used in the case of what a bond in its final coupon period is directly comparable with a money market instrument.
Bibliography
Buckle, M. and Thomson, J. (1992) The UK Financial System, 3rd Edition, Manchester, Manchester University Press
Navy, N., ECS 1580 Lecture 6: Modern Financial Institutions and Markets, Middlesex University
[available at ]
Brown, P. (1998) Bond Markets : Structures and Yield Calculations, Gilmour Drummond Publishing
Website
ABN AMRO Private Banking
The bank of England http://www.bankofengland.co.uk/
