Calculating the Yield to Maturity using the Bond Price

The yield to maturity is the discount rate that returns the bond's market price: YTM = [(Face value/Bond price)1/Time period]-1.

Key Points

• To achieve a return equal to YTM (i.e., where it is the required return on the bond), the bond owner must buy the bond at price P0, hold the bond until maturity, and redeem the bond at par.

• If a bond's coupon rate is less than its YTM, then the bond is selling at a discount. If a bond's coupon rate is more than its YTM, then the bond is selling at a premium. If a bond's coupon rate is equal to its YTM, then the bond is selling at par.

• Formula for yield to maturity: Yield to maturity(YTM) = [(Face value/Bond price)1/Time period]-1.

Terms

• The interest rate used to discount future cash flows of a financial instrument; the annual interest rate used to decrease the amounts of future cash flow to yield their present value.

Figures

1. USD Yield Curve

2005 USD yield curve

YTM

The yield to maturity is the discount rate which returns the market price of the bond. YTM is the internal rate of return of an investment in the bond made at the observed price. Figure 1

To achieve a return equal to YTM (i.e., where it is the required return on the bond), the bond owner must buy the bond at price P0, hold the bond until maturity, and redeem the bond at par.

If the yield to maturity for a bond is less than the bond's coupon rate, then the (clean) market value of the bond is greater than the par value (and vice versa). If a bond's coupon rate is less than its YTM, then the bond is selling at a discount. If a bond's coupon rate is more than its YTM, then the bond is selling at a premium. If a bond's coupon rate is equal to its YTM, then the bond is selling at par.

Calculating YTM

Formula for yield to maturity: Yield to maturity(YTM) = [(Face value/Bond price)1/Time period]-1

As can be seen from the formula, the yield to maturity and bond price are inversely correlated.

Consider a 30-year, zero-coupon bond with a face value of $100. If the bond is priced at an annual YTM of 10%, it will cost$5.73 today (the present value of this cash flow, 100/(1.1)30 = 5.73). Over the coming 30 years, the price will advance to $100, and the annualized return will be 10%. What happens in the meantime? Suppose that over the first 10 years of the holding period, interest rates decline, and the yield-to-maturity on the bond falls to 7%. With 20 years remaining to maturity, the price of the bond will be 100/1.0720, or$25.84. Even though the yield-to-maturity for the remaining life of the bond is just 7%, and the yield-to-maturity bargained for when the bond was purchased was only 10%, the return earned over the first 10 years is 16.25%. This can be found by evaluating (1+i) from the equation (1+i)10 = (25.842/5.731), giving 1.1625.

Over the remaining 20 years of the bond, the annual rate earned is not 16.25%, but rather 7%. This can be found by evaluating (1+i) from the equation (1+i)20 = 100/25.84, giving 1.07. Over the entire 30 year holding period, the original $5.73 invested increased to$100, so 10% per annum was earned, irrespective of any interest rate changes in between.

Key Term Glossary

bond
A documentary obligation to pay a sum or to perform a contract; a debenture.
Appears in these related concepts:
Bond
A bond is an instrument of indebtness of the bond issuers toward the bond holders.
Appears in these related concepts:
cash flow
The sum of cash revenues and expenditures over a period of time.
Appears in these related concepts:
discount
To find the value of a sum of money at some earlier point in time. To find the present value.
Appears in these related concepts:
discount rate
The interest rate used to discount future cash flows of a financial instrument; the annual interest rate used to decrease the amounts of future cash flow to yield their present value.
Appears in these related concepts:
interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed.
Appears in these related concepts:
interest rate
The percentage of an amount of money charged for its use per some period of time. It can also be thought of as the cost of not having money for one period, or the amount paid on an investment per year.
Appears in these related concepts:
investment
A placement of capital in expectation of deriving income or profit from its use.
Appears in these related concepts:
market value
The total value of the company as traded in the market. Calculated by multiplying the number of shares outstanding by the price per share.
Appears in these related concepts:
maturity
Date when payment is due.
Appears in these related concepts:
par
Equal value; equality of nominal and actual value; the value expressed on the face or in the words of a certificate of value, as a bond or other commercial paper.
Appears in these related concepts:
par value
the stated value or amount of a bill or a note
Appears in these related concepts:
period
The length of time during which interest accrues.
Appears in these related concepts:
the price above par value at which a security is sold
Appears in these related concepts:
Present value
Present value, also known as present discounted value, is the value on a given date of a payment or series of payments made at other times.
Appears in these related concepts:
Present Value
Also known as present discounted value, is the value on a given date of a payment or series of payments made at other times. If the payments are in the future, they are discounted to reflect the time value of money and other factors such as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or could have been) earning interest in the intervening time. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.
Appears in these related concepts:
required return
the minimum gain expected by investors
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return
Gain or loss from an investment.
Appears in these related concepts:
Yield
In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return. Yield applies to various stated rates of return on stocks (common and preferred, and convertible), fixed income instruments (bonds, notes, bills, strips, zero coupon), and some other investment type insurance products
Appears in these related concepts:
yield to maturity
The internal rate of return on a bond held to maturity, assuming scheduled payment of principal and interest.
Appears in these related concepts:
Yield to maturity
The yield to maturity (YTM) of a bond or other fixed-interest security, such as gilts, is the internal rate of return (IRR, overall interest rate) earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity and that all coupon and principal payments will be made on schedule.
Appears in these related concepts:
Zero coupon bonds
A zero-coupon bond (also called a discount bond or deep discount bond) is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity.