All securities can be evaluated in terms of the characteristics common to all assets: value, return, risk, maturity, marketability, liquidity, and taxability. In this and the next three articles (Bond Price Relations; Pricing Bonds with Different Cash Flows and Compounding Frequencies; The Yield to Maturity and Bond Equivalent Yields), we will analyze debt securities in terms of these characteristics. In this article, we look at how debt instruments (which we will usually refer to here as bonds) are valued and how their rates of return are measured. It should be noted that this article is very technical, entailing a number of definitions. Understanding how bonds are valued and their rates determined, though, is fundamental to being able to evaluate and select bonds.
An investor who has purchased a bond can expect to earn a possible return from the bond's periodic coupon payments; from capital gains (or losses) when the bond is sold, called, or matures; and from interest earned from reinvesting coupon payments. Given the market price of the bond, the bond's yield is the interest rate that makes the present value of the bond's cash flow equal to the bond price. This yield takes into account these three sources of return. In the article titled The Yield to Maturity and Bond Equivalent Yield, we will discuss how to solve for the bond's yield given its price. Alternatively, if we know the rate we require to buy the bond, then we can determine its value.
Like the value of any asset, the value of a bond is equal to the sum of the present values of its future cash flows:
where Vob is the value or price of the bond, CFt is the bond's expected cash flow in period t, including both coupon income and repayment of principal, R is the discount rate, and M is the term to maturity on the bond. The discount rate is the required rate; that is, the rate investors require to buy the bond. This rate is typically estimated by determining the rate on a security with comparable features: same risk, liquidity, taxability, and maturity.
Many bonds pay a fixed coupon interest each period, with the principal repaid at maturity. The coupon payment, C, is often quoted in terms of the bond's coupon rate, CR. The coupon rate is the contractual rate the issuer agrees to pay on the bond. This rate is often expressed as a proportion of the bond's face value (or par) and is usually stated on an annual basis. Thus, a bond with a face value (F) of $1,000 and a 10% coupon rate would pay an annual coupon of $100 each year for the life of the bond: C = CRF = (.10)($1,000) = $100.
The value of a bond paying a fixed coupon interest each year (annual coupon payment) and the principal at maturity, in turn, would be
Where M = Number of years to maturity
With the coupon payment fixed each period, the C term in Equation 2.2 can be factored out and the bond value can be expressed as:
The term 1/(1+R)t is the present value of $1 received each period for N periods (M years in the above case). It is defined as the present value interest factor (PVIF). The PVIF for different terms and discount rates can be found using PVIF tables found in many finance text books. It also can be calculated using the following formula:
Thus, if investors require a 10% annual rate of return on a 10-year, investment-grade corporate bond paying a coupon equal to 9% of par each year and a principal of $1,000 at maturity (M = 10 years), then they would price the bond at $938.55:
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.