9 Fixed Income Yields

9.1 Introduction

Three questions animate the study of fixed income yields. First, what interest rate is embedded in an observed bond price — and how is that rate extracted from a security’s promised cash flows? Second, does the yield to maturity actually represent the return an investor will earn, and under what conditions do the two coincide? Third, what determines the return to holding a bond over a finite period that ends before maturity? These questions matter because fixed income securities constitute one of the largest asset classes in the global economy. The interest rates implied by bond prices serve as benchmarks for borrowing costs across the entire financial system, transmit the effects of monetary policy to households and firms, and anchor the discount rates used to value virtually every other financial asset. A precise command of these concepts is therefore foundational for anyone who studies or works in financial markets.

The first question leads to the concept of yield to maturity: the single discount rate that equates the present value of a bond’s entire stream of promised payments to its observed market price. This is not merely a computational convention. Yield to maturity is the language through which bond markets communicate value and make comparisons across instruments. Understanding how it is derived — and how it depends on whether the bond trades above or below par — provides the analytical core of fixed income valuation. The chapter covers both the bond equivalent yield, which uses simple annualization, and the effective annual yield, which accounts for compounding, and works through the relationship between them.

The second question exposes a critical limitation of the yield to maturity as a measure of investment return: it assumes that every coupon payment can be reinvested at exactly the yield prevailing at the time of purchase. In practice, interest rates change, and coupon income is reinvested at whatever rate the market offers at the time. When those reinvestment rates differ from the original yield, the realized compound return — the actual annualized gain from holding the bond to maturity — diverges from the quoted yield. This distinction is not merely theoretical; it has direct implications for how investors should evaluate and compare bond investments under different interest rate scenarios.

The third question introduces the holding period return, which is relevant whenever an investor sells a bond before it matures. Because bond prices move inversely with yields, a rise in market interest rates causes the price of an existing bond to fall, and a sale at that lower price generates a capital loss that offsets some or all of the coupon income earned during the holding period. Conversely, a decline in rates produces a capital gain. Together, the coupon income and the capital gain or loss determine the total holding period return, which can differ substantially from the yield to maturity depending on how rates evolve and how long the bond is held.

The chapter begins by mapping the landscape of fixed income securities, distinguishing money market instruments from longer-term bonds and surveying the main categories — Treasury securities, corporate bonds, and asset-backed securities — along with the role of credit ratings in explaining yield differences across issuers. It then derives the yield to maturity formally, verifies that a bond priced at par has a yield equal to its coupon rate (and conversely), establishes the intuitive rules linking price to yield, and illustrates the relationship between bond equivalent and effective annual yields through worked examples. The chapter closes by showing how the realized compound return depends on reinvestment rates and how capital gains and coupon income combine to determine the return over any finite holding period.

In the news

A bond’s price and its yield move in opposite directions, and nothing illustrates it like a hot inflation report. CNBC reported the 10-year Treasury yield climbing to about 4.46% after consumer prices rose to their highest level in nearly three years — as yields rose, existing bond prices fell, and the cost of mortgages and other loans rose with them. Computing exactly that relationship between price, coupon, and yield to maturity is the work of this chapter. Read it at CNBC.

9.2 Fixed Income Securities

The following definitions for fixed income securities are useful.

Maturity.: The date at which a fixed income security's payments expire. For traditional bonds this is the day that the principal (or face value--see below) is repaid.
Coupon payment.: A regular interest payment made on a fixed income security. Most securities that have coupon payments make them semiannually.
Face value (par value).: The maturity value of the bond. Par value and face value are generally interchangeable.
Bond indenture.: The contract issued between the issuer and the buyer in a bond transaction. It specifies the rights of the debt holder in the case of default and the actions (in terms of further debt issuance) what kind of debt can be issued in the future and so on.

Fixed income securities come in many varieties. Broadly this market can be divided into two groups: the money market and the bond market. Within the bond market there are several main classes. These include:

Treasury securities.: Various government securities including: bills (usually maturities of 28, 61 or 182 days--usually no coupons), notes (maturities of up to 10 years) and bonds (maturities greater than 10 years).
Corporate bonds.: Issued by individual corporations to finance investment and operations. These usually pay coupon payments and the coupon rate is usually set so that when first sold the bonds sell at par.
Asset backed securities.: Fixed income instruments that give the holder the right to receive payments from the asset that backs the security. A prominent example are mortgage backed securities that give to the holder the right to collect payments from a group of mortgage holders.

9.3 Credit Risk

Bonds provide a guaranteed source of fixed income, but this guarantee may not always be perfect. That is, there is some chance that a firm or issuing entity will go bankrupt and be unable to make payments on its outstanding debt. This risk is known as credit risk. Ratings agencies attempt to provide the market with information on the likelihood of a group defaulting. The common ratings agencies are Moody's Investor Services, Standard & Poor's Corporation and Fitch Investors Service. The ratings are

Moody's:
- Investment grade: Aaa, Aa, Baa
- High Yield (junk): Ba, B, Ca, C
S & P:
- Investment grade: AAA, AA, A, BBB
- High Yield (junk): BB, B, CCC, CC, C, etc.
Fitch:
- Investment grade: AAA, AA, A, BBB
- Speculative: BB, B, CCC, CC, C, etc.

In order for investors to purchase bonds that have a higher probability of default, they must be compensated in the form of higher returns. Therefore, bonds in lower ratings categories generally offer higher yields. In other words, investors are willing to pay less for them.

9.4 Yield to maturity

Suppose that a security pays no coupons, has a face value of $M$ and will mature in exactly 1 year. The observed price of the security is $P$. The bond's yield to maturity is then defined to be the discount rate $y$ that solves the following equation

\[P = \frac{M}{1 + y}\]

Yield to maturity.

The constant interest rate $y$ such that the present value of a security's future payments is exactly equal to its price. For a bond making $T$ coupon payments $b$ with face value $M$ it is the solution to the equation

\[P = \sum_{t = 1}^{T}\frac{b}{(1+y)^t} + \frac{M}{(1 + y)^T}\]

For a bond making semiannual payments, the annualized yield to the above would be $2y$.

Example.: Consider a bond with face value of $1000 with an 8% coupon rate paid in two semiannual payments of 4% each that matures in 5 years. What is the yield to maturity if the observed price is $1000? It is found by looking for the solution to the above equation. Note that there are 10 coupon payments, the first coming in 6 months.

\[1000 = \sum_{t = 1}^{10}\frac{40}{(1+y)^t} + \frac{1000}{(1 + y)^{10}}\]

Solving this for $y$ yields, $y = 0.04$, or the periodic yield to maturity is the value of the coupon rate, which implies that the annualized yield to maturity is exactly the coupon rate. This is not a coincidence. In general the yield to maturity of a coupon paying bond that is selling for par will be the coupon rate of the bond.

To verify this, write the per-period coupon as $b = cM$, where $c$ is the coupon rate, and evaluate the pricing equation at the candidate yield $y = c$. The coupon stream is an annuity, so applying the annuity factor $\frac{1}{c}\left(1 - (1+c)^{-T}\right)$ derived earlier,

\[P = \sum_{t=1}^{T}\frac{cM}{(1+c)^t} + \frac{M}{(1+c)^T} = cM\cdot\frac{1}{c}\left(1 - \frac{1}{(1+c)^T}\right) + \frac{M}{(1+c)^T}.\]

The two occurrences of $M(1+c)^{-T}$ cancel:

\[P = M\left(1 - \frac{1}{(1+c)^T}\right) + \frac{M}{(1+c)^T} = M.\]

So setting $y$ equal to the coupon rate prices the bond exactly at par. Because the pricing equation has a unique yield solution for a given price, the converse also holds: a bond priced at par ($P = M$) must have $y = c$. For this example 8% is the bond equivalent yield. This is the annualized yield to maturity when simple interest is used. The effective annual yield for this bond is then $(1.04)^2 - 1 = 0.0816$ or 8.16%.

Bond prices can be found in lots of places, including the Wall Street Journal Online.¹

Example.: Select a bond from wsj.com and calculate its YTM using the Excel spreadsheet YTM Calculation.xls. Notice that bond prices are usually expressed as a percent of the face value.

In general this equation cannot be solved algebraically if the bond is not selling at par. Excel and many other programs can be used to calculate it for you. In Excel the function that you want to use is YIELD( ). Some general rules apply however.

If the bond is selling above par then the YTM will be less than the coupon rate.
If the bond is selling below par then the YTM will be more than the coupon rate.

Yield to call.: Some bonds are callable before their maturity date. For such bonds, the yield to call can be calculated as the yield to maturity with the call date used instead of the maturity date.

9.5 Realized compound return

The yield to maturity gives the return that will be earned on the bond if all coupon payments can be reinvested at the current yield. It does not generally give the return to holding the bond.

Example.

Consider a 2 year bond paying coupon rate of 5% that is selling at par (for a face value of $1000). The bond makes annual coupon payments of $50 and the yield to maturity is $y = 0.05$ (from above). At the end of two years, the holder receives one coupon payment of $50 and the face value of the bond which is $1000. If the coupon payment received after the first year ($50) is reinvested at 5% then it will now be worth $50\cdot 1.05 = 52.50$. So the total amount of cash held at the end of the two years is $1000 + $50 + $52.50 = $1102.50. To find the effective annual yield, we must solve the equation $1000(1 + y)^2 = 1102.50$. Therefore the effective annual yield to holding this security is

\[y = \sqrt{\frac{1102.50}{1000}} - 1 = 0.05\]

Suppose however that in the first year the interest rate increases from 5% to 10%. Therefore the coupon payment that is received at the end of year 1 can be reinvested at 10%. What is the return to holding the security? At the end of the two years the cash in hand will be $1000 + $50 + $55 = $1105. This implies an effective annual yield of

\[r = \sqrt{\frac{1105}{1000}} - 1 = 0.0512\]

9.6 Holding period return

The price of bonds and thus the return to holding them will change over time. These capital gains work together with the coupon payments to give the return to holding the bond.

Example.

Consider a bond with maturity of ten years from now paying semiannual coupons of 6% (i.e. the annual coupon rate is 12%). Assume that the current 6-month periodic interest rate (risk adjusted) is 6%. Therefore the price of the bond will be 100% of the face value. Now, consider what happens if the moment after the bond is issued the 6-month periodic interest rate changes from 6% to 5% and you purchase the bond. The price that you pay for it is

\[P = \frac{6}{0.05} \left (1 - \frac{1}{1.05^{20}} \right ) + \frac{100}{(1.05^{20})} = \$112.462\]

At the end of the six months after you have received one coupon payment, the price of the bond will be given by

\[P = \frac{6}{0.05} \left (1 - \frac{1}{1.05^{19}} \right ) + \frac{100}{(1.05^{19})} = \$ 112.0853\]

The rate of return to holding the bond for those six months has been

\[R = \frac{112.0853 + 6 - 112.462}{112.462} = 0.050\]

there is a small difference due to roundoff error.

9.7 Application: An investor comparing a bond to cash

An individual investor with a lump sum to deploy is choosing between buying a ten-year Treasury note and leaving the money in a money-market fund. The pricing tools of this chapter are what let her compare the two on equal footing. By computing the bond’s yield to maturity she can see the annualized return she would lock in by holding to maturity, weigh it against the floating rate the money-market fund pays, and understand how the bond’s price would move if interest rates rose or fell before she sold. The relationship between price, coupon, and yield — abstract when written as a formula — becomes the concrete basis on which she decides where to put her savings and how much interest-rate risk she is willing to carry.

Further listening

Planet Money (NPR) — “Summer School 4: Bonds & Becky With The Good Yield”: how a bond’s price and yield are linked, and the default, inflation, and interest-rate risks a bondholder bears.

9.8 Homework problems

9.8.1 Conceptual

YLD-C1. A colleague quotes you the yield to maturity on a 20-year corporate bond and says, “This is the return you’ll earn if you buy the bond today and hold it to maturity.” State precisely what the yield to maturity $y$ is defined to be — the single discount rate that solves $P = \sum_{t=1}^{T}\frac{b}{(1+y)^t} + \frac{M}{(1+y)^T}$ — and explain the exact sense in which the colleague’s statement is and is not correct.

YLD-C2. Explain why the yield to maturity is a single discount rate applied to every promised payment, rather than a separate rate for each maturity. What does it mean to say that $y$ summarizes the entire price–cash-flow relationship in one number, and why is that number the natural way for bond markets to communicate value and compare instruments of different maturities and coupons?

YLD-C3. The chapter shows that a bond priced at par ($P = M$) has yield to maturity exactly equal to its coupon rate ($y = c$), and conversely. Reproduce the logic of the verification in words: starting from $b = cM$ and evaluating the pricing equation at $y = c$, explain why the coupon annuity and the discounted face value combine to give exactly $M$, and why uniqueness of the yield delivers the converse.

YLD-C4. State the two rules relating a bond’s price to its coupon rate: if the bond sells above par the YTM is below the coupon rate, and if it sells below par the YTM is above the coupon rate. Explain why each rule must hold, referring to the definition of $y$ as the discount rate that equates the present value of promised payments to the observed price $P$.

YLD-C5. The chapter distinguishes the quoted yield to maturity from the realized compound return. Suppose an investor buys a coupon bond at par and, over the life of the bond, market interest rates fall well below the original yield. Explain what happens to the realized compound return relative to the quoted yield to maturity, and identify the specific assumption behind the yield to maturity that is violated when rates change.

YLD-C6. “The yield to maturity assumes coupons are reinvested at the yield prevailing at purchase.” A student objects: “But I never reinvest my coupons — I spend them, so the reinvestment assumption is irrelevant to me.” Evaluate this objection. Under what circumstances does the reinvestment assumption matter for the return an investor realizes, and what does an investor who spends every coupon actually give up relative to the quoted yield when rates rise after purchase?

YLD-C7. An investor holds a long-maturity bond for six months and then sells it. Decompose the return over that period into its two components using the holding period return $R = \frac{P_1 + b - P_0}{P_0}$, and explain how each component behaves when market interest rates rise sharply just after purchase. Which component can turn a positive coupon return into a negative total return?

YLD-C8. Two investors both hold the same ten-year bond. One plans to hold it to maturity; the other plans to sell it after one year. Explain why a sudden increase in market interest rates affects these two investors differently, distinguishing between the effect on the price at which the second investor can sell and the effect on the reinvestment of coupons for the first investor.

YLD-C9. Bond prices and yields move in opposite directions. Explain why this inverse relationship follows directly from the pricing equation $P = \sum_{t=1}^{T}\frac{b}{(1+y)^t} + \frac{M}{(1+y)^T}$, and describe how the news item at the head of the chapter — a rising 10-year Treasury yield after a hot inflation report — illustrates the same mechanism for an investor already holding the bond.

YLD-C10. Two bonds have the same yield to maturity but different maturities: one matures in 2 years, the other in 20 years. Using the inverse price–yield relationship, explain which bond’s price changes by more (in percentage terms) for a given rise in the market yield, and why the pricing equation makes the longer bond more sensitive.

YLD-C11. Distinguish the current yield (annual coupon income divided by price) from the yield to maturity. Explain what the current yield omits, and give a scenario in which a bond with the higher current yield delivers the lower yield to maturity.

YLD-C12. Explain the difference between the bond equivalent yield (simple annualization of a semiannual periodic yield) and the effective annual yield (which accounts for compounding of the semiannual coupon). Why is the effective annual yield always at least as large as the bond equivalent yield, and when are the two equal?

9.8.2 Quantitative

YLD-Q1. A bond has face value $M = \$1000$, makes annual coupon payments of $b = \$70$, and matures in $T = 3$ years. The observed price is $P = \$1000$. Using $P = \sum_{t=1}^{T}\frac{b}{(1+y)^t} + \frac{M}{(1+y)^T}$, find the yield to maturity $y$, and explain how you can determine it here without solving the equation numerically.

YLD-Q2. A bond with face value $M = \$1000$ makes a single annual coupon of $b = \$60$ and matures in $T = 2$ years. Its observed price is $P = \$981.92$. Verify that the yield to maturity is $y = 0.07$ by substituting into $P = \sum_{t=1}^{2}\frac{b}{(1+y)^t} + \frac{M}{(1+y)^2}$ and confirming the present value equals the price. State whether this bond trades above or below par and check that the sign is consistent with the chapter’s rules.

YLD-Q3. Consider a 3-year bond with face value $M = \$1000$ paying an annual coupon of $b = \$80$, selling at par so its yield to maturity is $y = 0.08$. All coupons are reinvested. Suppose immediately after purchase the reinvestment rate rises to 10% and stays there for the life of the bond. Compute the total cash held at maturity and the realized compound return $r$ from $M(1+r)^T = \text{(final cash)}$.

YLD-Q4. A 2-year bond with face value $M = \$1000$ pays an annual coupon of $b = \$50$ and sells at par ($y = 0.05$). Suppose the first coupon is reinvested at 3% rather than 5%. Compute the final cash held at the end of two years and the realized compound return $r$ from $M(1+r)^2 = \text{(final cash)}$, and compare $r$ to the quoted yield to maturity.

YLD-Q5. A bond has maturity of ten years, pays semiannual coupons of $6$ per $100$ of face value (annual coupon rate 12%), and the current 6-month periodic interest rate is 6%, so the bond sells at par ($P = 100$). Immediately after issue the 6-month periodic rate falls to 4%. Compute the purchase price $P_0$ using $P = \frac{6}{y}\left(1 - \frac{1}{(1+y)^{20}}\right) + \frac{100}{(1+y)^{20}}$, then the price $P_1$ six months later (19 periods remaining), and finally the holding period return $R = \frac{P_1 + b - P_0}{P_0}$ over the first six months.

YLD-Q6. You buy a bond at $P_0 = \$980$, receive a coupon of $b = \$40$ over your holding period, and sell it at $P_1 = \$955$. Compute the holding period return $R = \frac{P_1 + b - P_0}{P_0}$, and separately report the coupon component and the capital-gain component of that return.

YLD-Q7. A bond with face value $M = \$1000$ has a coupon rate of 8% paid in two semiannual payments of $\$40$ each and matures in 5 years. Its periodic yield to maturity solves $1000 = \sum_{t=1}^{10}\frac{40}{(1+y)^t} + \frac{1000}{(1+y)^{10}}$, giving $y = 0.04$. Report the bond equivalent yield (simple annualization) and compute the effective annual yield using $(1+y)^2 - 1$.

YLD-Q8. A bond sells at par and has a semiannual periodic yield to maturity of $y = 0.035$. Report its bond equivalent yield (simple annualization) and compute its effective annual yield using $(1+y)^2 - 1$. By how many basis points does the effective annual yield exceed the bond equivalent yield?

YLD-Q9. A bond pays annual coupons of $b = \$65$ and its observed price is $P = \$928.57$, with face value $M = \$1000$. Compute the current yield (annual coupon income divided by price). Then, using the chapter’s rules relating price to par and coupon rate, state whether the yield to maturity is above or below the coupon rate for this bond.

YLD-Q10. A bond with face value $M = \$1000$ pays an annual coupon of $b = \$70$ and trades at a price of $P = \$1050$. Compute its current yield, and compute its coupon rate. State whether the bond trades above or below par, and rank the coupon rate, the current yield, and the yield to maturity from highest to lowest, justifying the ordering with the chapter’s rules.

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