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Bond price relations

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Relation between Coupon Rate, Required Rate, Value, and Par Value

The value of a bond - in our example here, a corporate bond with a face value of $1,000 and a coupon equal to 9% of par each year, for an investor who requires a 10% annual return - is not equal to its par value. This can be explained by the fact that the discount rate and coupon rate are different.

  • Specifically, for investors in the above case to obtain the 10% rate per year from a bond promising to pay an annual rate of CR = 9% of par, they would have to buy the bond at a value, or price, below par: The bond would have to be purchased at a discount from its par, Vob < F.
  • In contrast, if the coupon rate is equal to the discount rate (i.e., R = 9%), then the bond's value would be equal to its par value, Vob = F. In this case, investors would be willing to pay $1,000 for this bond, with each investor receiving $90 each year in coupons.
  • Finally, if the required rate is lower than the coupon rate, then investors would be willing to pay a premium over par for the bond, Vob > F. This might occur if bonds with comparable features were trading at rates below 9%. In this case, investors would be willing to pay a price above $1,000 for a bond with a coupon rate of 9%.

Thus, the first relationship to note is that a bond's value (or price) will be equal to, greater than, or less than its face value depending on whether the coupon rate is equal to, less than, or greater than the required rate.

Bond - Price Relation 1

If CR = R => Vb = F: Bond valued at par

If CR < R => Vb < F: Bond valued at par

If CR > R => Vb > F: Bond valued at par

 

In addition to the preceding relations, the relation between the coupon rate and required rate also explains how the bond's value changes over time.

  • If the required rate is constant over time, and if the coupon rate is equal to it (i.e., the bond is priced at par), then the value of the bond will always be equal to its face value throughout the life of the bond. This is illustrated in Figure 1 by the horizontal line that shows the value of the 9% coupon bond is always equal to the par value. Here investors would pay $1,000 regardless of the terms to maturity.
  • On the other hand, if the required rate is constant over time and the coupon rate is less (i.e., the bond is priced at a discount), then the value of the bond will increase as it approaches maturity; if the required rate is constant and the coupon rate is greater (i.e., the bond is priced at a premium), then the value of the bond will decrease as it approaches maturity. These relationships are also illustrated in Figure 1.

Figure 1 The Value over Time of an Original 10-Year, 9% Annual Coupon Bond Selling at Par, Discount, and Premium


70_1_1

Relation between Value and Rate of Return

Given known coupon and principal payments, the only way an investor can obtain a higher rate of return on a bond is for its price (value) to be lower. In contrast, the only way for a bond to yield a lower rate is for its price to be higher. An inverse relationship exists between the price of a bond and its rate of return. Thus, the second bond relationship to note is that there is an inverse relationship between the price and rate of return on a bond.

Bond - Price Relation 2

If R Increases => VoDecreases 

If R Decreases => Vob  Increases 

The inverse relation between a bond's price and rate of return is illustrated by the negatively sloped price-yield curve shown in Figure 2. The curve shows the different values of a 10-year, 9% annual coupon bond given different rates. As shown, the 10-year bond has a value of $938.55 when R = 10% and $1,000 when R = 9%. In addition to showing a negative relation between price and yield, the price-yield curve is also convex from below (bow shaped). This convexity implies that for equal increases in yields, the value of the bond decreases at a decreasing rate (for equal decreases in yields, the bond's price increases at an increasing rate). The bow shape of a bond's price-yield curve is referred to as the bond's convexity. It has important implications related to bond investment and management that we will examine in more detail later.

Figure 2 Price-Yield Relation, 10-Year, 9% Annual Coupon Bond

The Relation between a Bond's Price Sensitivity to Interest Rate Changes and Term to Maturity

The third bond relationship to note is the relation between a bond's price sensitivity to interest rate changes and its maturity. Specifically:

Bond - Price Relation 3

The greater the bond's term to maturity, the greater its price sensitivity to a given change in interest rates.

This relationship can be seen by comparing the price sensitivity to interest rate changes of the 10-year, 9% coupon bond in our above example with a one-year, 9% coupon bond. As shown in Figure 3, if the required rate is 10%, then the 10-year bond would trade at $938.55, whereas the one-year bond would trade at $990.91 ($1,090/1.10). If the interest rate decreases to 9% for each bond (a 10% change in rates), both bonds would increase in price to $1,000. For the 10-year bond, the percentage increase in price would be 6.55% [($1,000 - $938.55)/$938.55], whereas the percentage increase for the one-year bond would be only 0.9%. Thus, the 10-year bond's price is more sensitive to the interest rate change than the one-year bond. In addition, the greater price sensitivity of longer maturity bonds also implies that their price-yield curves are more convex than the price-yield curves for smaller maturity bonds.

Figure 3 Bond Relation 3

The Relation between a Bond's Price Sensitivity to Interest Rate Changes and Coupon Payments

Consider two 10-year bonds, each priced at a discount rate of 10% and each paying a principal of $1,000 at maturity, but with one bond having a coupon rate of 10% and priced at $1,000, and the other having a coupon rate of 2% and priced at $508.43:

 
70_4


Now suppose that the rate required on each bond decreases to a new level of 9%. The price on the 10% coupon bond, in turn, would increase by 6.4% to equal $1,064.18, whereas the price on the 2% coupon bond would increase by 8.3% to $550.76:

 
70_5


In this case, the lower coupon bond's price is more responsive to a given interest rate change than the price of the higher coupon bond. Thus: 

Bond - Price Relation 4 

The lower a bond's coupon rate, the greater its price sensitivity to changes in discount rates.


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Article copyright 2011 by R. Stafford Johnson. Reprinted and adapted from Bond Evaluation, Selection, and Management, 2nd Edition with permission from John Wiley & Sons, Inc. The statements and opinions expressed in this article are those of the author. Fidelity Investments® cannot guarantee the accuracy or completeness of any statements or data. This reprint and the materials delivered with it should not be construed as an offer to sell or a solicitation of an offer to buy shares of any funds mentioned in this reprint.
The data and analysis contained herein are provided "as is" and without warranty of any kind, either expressed or implied. Fidelity is not adopting, making a recommendation for or endorsing any trading or investment strategy or particular security. All opinions expressed herein are subject to change without notice, and you should always obtain current information and perform due diligence before trading. Consider that the provider may modify the methods it uses to evaluate investment opportunities from time to time, that model results may not impute or show the compounded adverse effect of transaction costs or management fees or reflect actual investment results, and that investment models are necessarily constructed with the benefit of hindsight. For this and for many other reasons, model results are not a guarantee of future results. The securities mentioned in this document may not be eligible for sale in some states or countries, nor be suitable for all types of investors; their value and the income they produce may fluctuate and/or be adversely affected by exchange rates, interest rates or other factors.

In general, the bond market is volatile, and fixed income securities carry interest rate risk. (As interest rates rise, bond prices usually fall, and vice versa. This effect is usually more pronounced for longer-term securities). Fixed income securities also carry inflation risk, liquidity risk, call risk and credit and default risks for both issuers and counterparties. Lower-quality fixed income securities involve greater risk of default or price changes due to potential changes in the credit quality of the issuer. Foreign investments involve greater risks than U.S. investments, and can decline significantly in response to adverse issuer, political, regulatory, market, and economic risks. Any fixed-income security sold or redeemed prior to maturity may be subject to loss.

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