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Class 4 – Bonds
Corporations and governments sometimes need to raise funds for various purposes, in
particular, in order to invest in capital projects - because internally generated funds
(profits in the case of corporations, and taxes in the case of governments) may be
insufficient, in a particular period.
Here we discuss the basic features of bonds.
A bond (also called debenture) is a legally binding agreement between a borrower (the
bond issuer) and a lender (the bondholder). The agreement specifies the principal amount
of the loan, the timing and amount of the cash flows, and any other provisions:
Other provisions include:
•call options, which allow the issuer to repurchase the bond at some point in
time prior to the maturity date
•put options, which allow the bond holder to sell the bond back to the issuer at a
designated price and time
•convertibility options, which allow the bond holder to convert the bond into a
certain specified number of shares of the company
A bond specifies a face amount (F), and a bond interest rate, also called the coupon rate.
The bond also specifies a maturity date, or term to maturity, during which the coupons
(the bond interest payments) are to be paid, and the redemption amount to be paid on
Bonds typically make coupon payments once per year (Europe) or semi annually (United
States, Canada, Japan)
P = C[1- (1+r)-n]/r + F/(1+r)n = CxPVIFA(n,r) + F/(1+r)n
This is the formula for the price of a bond with:
•either annual coupon payments of amount C for n years, where the effective
annual rate of interest is given by r
•semi annual coupon payments of C for n half years (i.e. n/2 years), where the
(effective) semi annual rate of interest is given by r
For example, a bond with face value $1,000, a 9% coupon rate, and semi annual
payments, pays $45 every 6 months until maturity.
Example: Consider a bond issued at par, with a face value of $1,000, 7% coupon, annual
payments, and 10 years to maturity.
Draw a diagram to illustrate this bond
0 1 2 3 4 5 6 7 8 9 10
Find the price (value) of this bond in exactly 2 years (just after the second coupon
payment of 70), if market interest rates for 8-year bonds of this type are (in exactly 2
i) 5%, answer: 1129.26 = 70xPVIFA(8,5%) + 1,000/(1.05)8
ii) 10% answer: 839.95 = 70xPVIFA(8,10%) + 1,000/(1.10)8
Repeat the question if the bond was originally a 20-year bond, and now has 19 years to
i) answer: 1241.71
ii) answer: 749.05
Note: A bond trades either:
•At a premium
•At a discount
Find the yield to maturity of a bond which pays semi annual coupons of 50, has a par
value of 1,000, has 15 years to maturity, and is priced at:
solve 900 = 50xPVIFA(30,r) + 1,000/(1+r)30 for r => r = .057036, etc.
i) 5.7036% per half year – which is 11.4072 annually – this is called the bond
ii) 4.3937% per half year – which is 8.7875% annually – this is called the bond