we use present value of $1 and present value of an annuity formulas to determine the present
value of a fixed income security – bond.
Bonds: Interest Rates and Bond Prices
The value of a
pure discount bond
(also called a zero coupon bond, or simply a zero) is:
PV = F/(I+R)
The bond promises a single payment at fixed future date.
level payment bonds
, the coupon payments form an annuity, and the principal repayment is a lump sum.
The value of this type of bond is simply the sum of the values of its two parts. The value of a bond then is
PV = C x A
+ $1,000 / (1+R)
Remember that when we solved bond prices in Excel we used the present value of an annuity for the interest
payments, and the present value of a single sum for the face, as shown in the par and discount examples below:
= PV (interest/coupons payments) + PV (principal/par repayment)
$1,000 face, 8% coupon, 30 Year, semi annual. ($40 paid semi annually). Market Interest rate = 8%. We have
a par bond = PV = $1,000.
$1,000 face, 8% coupon, 30 Year, semi annual. ($40 paid semi annually). Market Interest rate = 10%.
PV of Coupons
PV of Par
$810.71 Sells at discount
Another important concept is
yield to maturity
, which is the single rate that discounts the payments on the bond
to its purchase price.
The PRICE & YIELD functions in Excel may be helpful in solving bond problems. Click this link:
Example Using Excel Functions
to see how these functions work.
YTM is the rate implied by the bond’s price. Essentially, it is the rate required to discount the bond’s cash flows
to the current market price. It is a useful habit to assess the relations between coupon rate and yield to maturity
= we have a discount bond; if
= we have a premium bond.
Simply put, there is an inverse relation between