The internal rate of return approach can be used to obtain r. Since the current price is higher than $100, r must be lower than 7%.

Initially, try 5% as r:

$7 x 4.3295 [5%, five - year annuity] + $100 x 0.7835 [PV 5%, five - year] = $30.31 + $78.35 = $108.66

Try 6% as r:

$7 x 4.2124 [6%, five - year annuity] + $100 x 0.7473 [PV 6%, five - year] =

$29.49 + $74.73 = $104.22

Yield = 5% + (108.66 – 106.62 / 108.66 – 104.22) x 1% = 5.46%

The 5.46% is the yield to maturity (YTM) (or redemption yield) of the bond. The YTM is the rate of return at which the sum of the present values of all future income streams of the bond (interest coupons and redemption amount) is equal to the current bond price. It is the average annual rate of return the bond investors expect to receive from the bond till its redemption. YTMs for bonds are normally quoted in the financial press, based on the closing price of the bond. For example, a yield often quoted in the financial press is the bid yield. The bid yield is the YTM for the current bid price (the price at which bonds can be purchased) of a bond.

**Term structure of interest rates and the yield curve**

The yield to maturity is calculated implicitly based on the current market price, the term to maturity of the bond and amount (and frequency) of coupon payments. However, if a corporate bond is being issued for the first time, its price and/or coupon payments need to be determined based on the required yield. The required yield is based on the term structure of interest rates and this needs to be discussed before considering how the price of a bond may be determined.

It is incorrect to assume that bonds of the same risk class, which are redeemed on different dates, would have the same required rate of return or yield. In fact, it is evident that the markets demand different annual returns or yields on bonds with differing lengths of time before their redemption (or maturity), even where the bonds are of the same risk class. This is known as the term structure of interest rates and is represented by the spot yield curve or simply the yield curve.

For example, a company may find that if it wants to issue a one - year bond, it may need to pay interest at 3% for the year, if it wants to issue a two - year bond, the markets may demand an annual interest rate of 3. 5%, and for a three-year bond the annual yield required may be 4.2%. Hence, the company would need to pay interest at 3% for one year; 3.5% each year, for two years, if it wants to borrow funds for two years; and 4.2% each year, for three years, if it wants to borrow funds for three years. In this case, the term structure of interest rates is represented by an upward sloping yield curve.

The normal expectation would be of an upward sloping yield curve on the basis that bonds with a longer period of maturity would require a higher interest rate as compensation for risk. Note here that the bonds considered may be of the same risk class but the longer time period to maturity still adds to higher uncertainty.

However, it is entirely normal for yield curves to be of many different shapes dependent on the perceptions of the markets on how interest rates may change in the future. Three main theories have been advanced to explain the term structure of interest rates or the yield curve: expectations hypothesis, liquidity-preference hypothesis and market-segmentation hypothesis. Although it is beyond the remit of this article to explain these theories, many textbooks on investments and financial management cover these in detail.

**Valuing bonds based on the yield curve**

Annual spot yield curves are often published by the financial press or by central banks (for example, the Bank of England regularly publishes UK government bond yield curves on its website). The spot yield curve can be used to estimate the price or value of a bond.

Example 3

A company wants to issue a bond that is redeemable in four years for its nominal value or face value of $100, and wants to pay an annual coupon of 5% on the nominal value. Estimate the price at which the bond should be issued.

The annual spot yield curve for a bond of this risk class is as follows:

One-year 3.5%

Two-year 4.0%

Three-year 4.7%

Four-year 5.5%

The four-year bond pays the following stream of income:

Year 1 2 3 4

Payments $5 $5 $5 $105

This can be simplified into four separate bonds with the following payment structure:

Year 1 2 3 4

Bond 1 $5

Bond 2 $5

Bond 3 $5

Bond 4 $105

Each annual payment is a single payment in that particular year, much like a zero-coupon bond, and its present value can be determined by discounting each cash flow by the relevant yield curve rate, as follows: