A bond is a security sold by governments or corporations in order to raise money from investors today in exchange for the promised future payment. The terms of a bond are described as part of the Bond Certificate, which indicated that the amounts and dates of all payments to be made. These payments are made until a final repayment date is known as the term of the bond.
Bonds are typically make two types of payments to their holders. The promised interest payments of a bond are called coupons. The Bond Certificate typically specifies that the coupons will be paid periodically until the maturity date of the bond. The principal or face value of a bond is the notional amount e use to compute the interest payments. Usually, the face value is repaid at maturity. It is generally denominated in standard increments such as $1000. A bond with a $1000 face value, for example, is often referred to as a “$1000 bond”.
The amount of each coupon payment is determined by the coupon rate of the bond. This coupon rate is set by the issuer and stated on the bond certificate. By convention, the coupon rate is expected as an APR s the amount of each coupon payment, CPN is
CPN = (Coupon Rate × Face value)/ Number of coupon payments per year
For example, a $1000 bond with a 10% coupon rate and semi annual payments will pay coupon payments of $1000 × 10%/2 = $50 every six months.
Zero Coupon Bonds:
The simplest type of bond is a zero-coupon bond, a bond that does not make coupon payments. The only cash payment the investor receives is the face value of the bond on the maturity date. Treasury Bills, which are US government bonds with a maturity of up to one year, are zero-coupon bonds. As the present value of a future cash flow is less that the cash flow itself, as a result, prior to its maturity date, the price of a zero-coupon bond is always less than its face value. That is, zero-coupon bonds always trade at a discount (a price lower than the face value), so they are also called pure discount bonds.
Suppose that a one-year, risk-free, zero-coupon bond with a $100000 face value had an initial price of $96618.36. If you purchased this bond and held it to maturity, you would have the following cash flows:
Although, the bond pays no interest directly, as an investor you are compensate for the time value of your money by purchasing the bond at a discount to its face value.
Yield to Maturity:
The IRR of an investment opportunity is the discount rate at which the NPV of the investment opportunity is equal to zero. The IRR of an investment in zero-coupon bond is the rate of return that investors will earn on their money if they buy the bond at its current price and hold it to maturity. The IRR of an investment in a bond is given a special name, the yield to maturity (YTM) or just the yield.
The yield to maturity of a bond is the discount rate that sets the present value of the promised bond payments to the current market price of the bond.
The yield to maturity for a zero-coupon bond is the return you will earn as an investor from holding the bond to maturity and receiving the promised face value payment.
Let’s determine the yield to maturity of the one year zero coupon bond. The yield to maturity of the one year bond solves the following equation:
96,618.36 = 100000/1+ YTM
In this case, 1+ YTM = 100000/ 96,618.36 = 1.035
That is, the yield to maturity for this bond is 3.5%. Because the bond is risk free, investing in this bond and holding it to maturity is like earning 3.5% interest on your initial investment. Thus, by the Law of One Price, the competitive market risk-free interest rate is 3.5%, meaning all one year risk-free investments must earn 3.5%.
Similarly, the yield to maturity for a zero-coupon bond with n periods to maturity, current price P and face value FV is
P = FV/ (1+ YTMn)n
Rearranging this expression, we get
Yield to Maturity of an n Year Zero Coupon Bond
YTMn = (FV/P)1/n –1
Like Zero-coupon bonds, Coupon Bonds pay investors their face value at maturity. In addition, these bonds make regular coupon interest payments. Two types of US Treasury coupon securities are currently traded in financial markets. Treasury Notes, which have original maturities from one to ten years and Treasury Bonds, which have original maturities of more than ten years.
We can compute the yield to maturity of a coupon bond. The yield to maturity for a bond is the IRR of investing in the bond and holding it to maturity; it is the single discount rate that equates the present value of the bond’s remaining cash flows to its current prices,
CPN + FV
Because the coupon payments represent an annuity, the yield to maturity is the interest rate y that solves the following equations:
Yield to Maturity of a Coupon Bond
P = CPN × 1/y(1-1/(1+y)n) + FV/(1+y)N
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