Fixed-Income Securities: Bond Pricing, Yield Curves, Duration, and Credit Risk

A bond is a contract for dated payments. To understand its value, ask three questions: What payments are promised? What discount factors price those dates? What could change the payments or their value? The first two questions lead to bond pricing; the third leads to interest-rate and credit risk.

Learning goals and conventions

You should finish able to price zero-coupon and coupon bonds, recover spot and forward rates, distinguish YTM from expected return, calculate duration and convexity, and explain what immunization can protect.

The basic calculations assume annual effective rates, annual end-of-year coupons, positive fixed cash flows, and settlement immediately after a coupon payment. Thus there is no accrued interest in the examples. Ignore taxes, trading costs, default, and embedded options until explicitly introduced. Write 5% as 0.05 and measure time in years.

For preparation, review the Present Value Relations lesson in Learn. Keep its central rule: rates must match cash-flow timing, currency, and risk.

Symbol	Meaning
$F$	Face value repaid at maturity
$C$	Coupon amount per payment date
$T$	Maturity in years in the annual examples
$P_0$	Price today, including any accrued interest when applicable
$r_{0,t}$	Annual effective spot rate from today to date $t$
$D(0,t)$	Price today of one unit paid at date $t$
$y$	Yield to maturity under the stated compounding convention

1. Lecture 4: what belongs to fixed income?

Fixed-income securities are debt claims with contractual payment rules. The label does not mean every payment or market price is guaranteed or constant: floating-rate notes reset their coupons, borrowers may default, and mortgages can repay early.

Market segment	Basic claim	A risk to investigate
U.S. Treasuries	Obligations of the U.S. government	Interest-rate and inflation exposure
Corporate debt	Obligations of a company	Default, recovery, and credit-spread changes
Municipal bonds	Debt of public issuers or specified projects	Issuer or project credit risk; tax treatment varies
Mortgage-backed securities	Claims linked to pools of mortgages	Prepayment and changing payment timing
Asset-backed securities	Claims backed by other receivables or loans	Collateral performance and payment priority

New securities are sold in the primary market; existing securities trade in secondary markets, often through dealers or electronic platforms. A quoted price is not necessarily an executable price for every trade size. Bid–ask spreads and liquidity matter when turning theoretical replication into a trade.

Contract terms to read first

Identify the issuer, currency, face value, coupon schedule, maturity, seniority, collateral, and any call or put provision. Also distinguish clean price, excluding accrued coupon interest, from dirty price, the invoice price including accrued interest. Cash-flow PV formulas price the full claim at settlement; comparing them with an unadjusted clean quote can create a false discrepancy.

2. Pure discount bonds and spot rates

A pure discount or zero-coupon bond makes one payment, $F$, at maturity $T$, with no interim coupons:

Date              0                 1                 2
                   |-----------------|-----------------|
Investor flow     -P                 0                +F

For a default-free payment and matching spot rate:

$$P_0=\frac{F}{(1+r_{0,T})^T}=FD(0,T), \qquad D(0,T)=(1+r_{0,T})^{-T}.$$

The spot rate is the rate for investing from today until a particular future date. The two-year spot rate applies over the full two-year interval; it is not simply the rate earned in the second year.

Worked example: price and implied spot rate

A two-year zero pays 1,000, and its annual effective spot rate is 5%:

$$P_0=\frac{1{,}000}{1.05^2}\approx907.03.$$

Conversely, given price and face value:

$$r_{0,T}=\left(\frac{F}{P_0}\right)^{1/T}-1.$$

Using the unrounded price recovers 5%.

A zero trades below face value when its applicable yield is positive. At zero yield it trades at face value; with a negative yield above -100% under this convention it trades above face value. Having no coupon does not by itself guarantee a discount. For risky zeros, the contractual payment is promised rather than certain.

3. Lecture 5: coupon bonds are collections of zeros

For annual coupon rate $c$, the coupon is $C=cF$. A three-year bond with face value 1,000 and coupon rate 5% promises:

Date              0                 1                 2                 3
                   |-----------------|-----------------|-----------------|
Investor flow     -P                +50               +50              +1,050

The final payment includes both coupon and principal. With date-specific discount factors:

$$P_0=\sum_{t=1}^{T}C D(0,t)+F D(0,T) =\sum_{t=1}^{T-1}\frac{C}{(1+r_{0,t})^t} +\frac{C+F}{(1+r_{0,T})^T}.$$

Replication, stripping, and no arbitrage

Buy a zero paying 50 at year 1, another paying 50 at year 2, and another paying 1,050 at year 3. Together they reproduce the coupon bond’s payments. If the claims have the same credit, legal, and trading characteristics, the law of one price requires equal prices in a frictionless market.

If the coupon bond were overpriced relative to its replicating zeros, an idealized arbitrage would sell the bond and buy the zeros. The future payments cancel while the initial price difference is received today. Real short-sale constraints, financing costs, and bid–ask spreads can prevent small discrepancies from being exploitable.

U.S. Treasury STRIPS allow eligible Treasury interest and principal components to trade separately as zero-coupon claims. Treating a corporate bond mathematically as a bundle of payments does not mean every corporate bond can be operationally stripped in this program.

Worked example: two spot rates

A two-year bond pays 50 after one year and 1,050 after two. Let $r_{0,1}=4\%$ and $r_{0,2}=5\%$:

$$P_0=\frac{50}{1.04}+\frac{1{,}050}{1.05^2} \approx48.08+952.38=1{,}000.46.$$

Each payment uses the rate for its own maturity. Do not discount the second payment using $1.04\times1.05$: the 5% rate already describes the entire two-year investment.

Bootstrapping a discount curve

Suppose a one-year zero paying 100 costs 96.153846, so $D(0,1)=1/1.04$. A two-year annual 5% coupon bond with face value 100 costs 100.045788. Its price satisfies

$$100.045788=5D(0,1)+105D(0,2).$$

Therefore

$$D(0,2)=\frac{100.045788-5D(0,1)}{105}\approx0.907029,$$

and $r_{0,2}=D(0,2)^{-1/2}-1\approx5\%$. This sequential extraction is called bootstrapping. Actual curve construction must also handle irregular dates, interpolation, day-count rules, and instrument differences.

4. Yield to maturity: one number summarizing a price

YTM is the single rate $y$ solving

$$P_0=\sum_{t=1}^{T}\frac{CF_t}{(1+y)^t}.$$

For a fixed-rate annual coupon bond:

$$P_0=\frac{C}{y}[1-(1+y)^{-T}]+\frac{F}{(1+y)^T},$$

with the zero-rate case evaluated by its limit. YTM is generally found numerically. With positive future payments, price falls monotonically as $y$ increases, which makes the economically relevant solution well behaved.

Worked example: our reference bond

Take $F=1{,}000$, $C=50$, $T=3$, and $y=6\%$:

$$P_0=\frac{50}{1.06}+\frac{50}{1.06^2}+\frac{1{,}050}{1.06^3} \approx973.2699.$$

The bond is below par because its coupon rate is below YTM. Under these timing conventions, coupon equal to yield gives par, and coupon above yield gives a premium.

Do not confuse three rates:

• Coupon rate: annual contractual coupon divided by face value, here 5%.
• Current yield: annual coupon divided by price, here approximately 5.1373%; it omits the capital gain or loss to maturity.
• YTM: the internal rate implied by price and all promised cash flows, here 6%.

YTM is not automatically a forecast or a guaranteed realized return. Receiving all promised payments and reinvesting interim coupons at YTM is required to turn it into the same compounded terminal-wealth return. Selling before maturity introduces an uncertain sale price.

For semiannual coupons, use $C=cF/2$, $2T$ periods, and the appropriate semiannual rate. A nominal annual YTM compounded twice yearly gives a periodic rate $y/2$; an annual effective yield gives $(1+y)^{1/2}-1$ instead.

5. Term structure and forward rates

The term structure describes rates across maturities at a given observation date. A yield curve can be upward sloping, inverted, flat, or humped. Specify what is plotted: spot rates, par yields, and individual coupon-bond YTMs are different curves.

An upward curve may reflect expected increases in short rates, term premia, or maturity-specific demand and supply. It does not prove short rates will rise. Liquidity, risk compensation, and market segmentation complicate the interpretation.

Deriving the forward rate

Compare two ways of investing from today to $t_2$: lock in the $t_2$ spot investment, or invest to $t_1$ and simultaneously lock a forward investment from $t_1$ to $t_2$. Under consistent annual effective compounding:

$$(1+r_{0,t_1})^{t_1}(1+f_{t_1,t_2})^{t_2-t_1} =(1+r_{0,t_2})^{t_2}.$$

Thus

$$f_{t_1,t_2}=\left[\frac{(1+r_{0,t_2})^{t_2}} {(1+r_{0,t_1})^{t_1}}\right]^{1/(t_2-t_1)}-1.$$

Equivalently, $(1+f_{t_1,t_2})^{t_2-t_1}=D(0,t_1)/D(0,t_2)$.

For one-year spot 4% and two-year spot 5%:

$$f_{1,2}=\frac{1.05^2}{1.04}-1\approx6.0096\%.$$

Check: $1.04\times1.060096\ldots=1.1025=1.05^2$. The forward rate is implied by today’s prices. It is not necessarily the expected future short rate or the rate that will actually prevail next year.

6. Lecture 6: interest-rate risk and duration

For fixed positive payments, increasing the discount yield reduces price. Differentiate the annual price formula:

$$\frac{dP}{dy}=-\sum_{t=1}^{T}\frac{tCF_t}{(1+y)^{t+1}}<0.$$

Macaulay duration is the average payment time weighted by each payment’s share of price:

$$w_t=\frac{CF_t/(1+y)^t}{P},\qquad \sum_t w_t=1,\qquad D_M=\sum_{t=1}^{T}tw_t.$$

A zero-coupon bond has $D_M=T$ because all its value arrives at maturity. A positive-coupon bond has duration below maturity because some value arrives earlier.

Modified duration translates that timing measure into price sensitivity:

$$D_{\mathrm{mod}}=-\frac{1}{P}\frac{dP}{dy} =\frac{D_M}{1+y},\qquad \frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta y.$$

Here $\Delta y$ is a decimal change: a 100-basis-point increase means 0.01, not 1. Under a nominal annual yield compounded $m$ times yearly, with $D_M$ measured in years, the denominator becomes $1+y/m$.

Worked example: duration of the reference bond

For the three-year 5% coupon bond at 6% YTM:

Year	Cash flow	PV at 6%	PV weight
1	50	47.1698	0.04847
2	50	44.4998	0.04572
3	1,050	881.6002	0.90581

Using unrounded values gives

$$D_M\approx2.857347\text{ years},\qquad D_{\mathrm{mod}}\approx2.695611.$$

For a rise from 6% to 7%, duration predicts a price change of approximately -2.6956%, or -26.24, giving a price near 947.03. Exact repricing gives 947.5137, a change of approximately -2.6464%. The difference comes from curvature.

A useful scale is DV01, the approximate money loss for a one-basis-point yield rise:

$$DV01\approx PD_{\mathrm{mod}}\times0.0001\approx0.26236$$

per 1,000 face value for this bond. A large position scales this amount proportionally.

7. Convexity improves the local approximation

Normalized convexity under annual compounding is

$$\mathcal{C}=\frac{1}{P}\frac{d^2P}{dy^2} =\frac{1}{P}\sum_{t=1}^{T}\frac{t(t+1)CF_t}{(1+y)^{t+2}}.$$

A second-order Taylor approximation gives

$$\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta y +\frac12\mathcal{C}(\Delta y)^2.$$

For the reference bond, $\mathcal{C}\approx10.004464$. With $\Delta y=0.01$, the convexity correction is approximately 0.0500 percentage points. The combined predicted change is -2.6456%, giving a price near 947.52, close to exact repricing at 947.51.

Method	Price after yield rises to 7%
Duration only	947.03
Duration plus convexity	947.52
Exact cash-flow repricing	947.51

This remains a local approximation. Larger yield changes increase the importance of higher-order terms. Plain fixed positive cash flows have positive convexity; callable bonds and mortgage-backed securities can exhibit negative effective convexity because their expected payment timing changes with rates.

A single yield is only one risk factor

Modified duration measures sensitivity to the bond’s YTM. Applying it to a portfolio under a parallel yield shift is a simplification. Curves can steepen, flatten, or twist. Key-rate durations measure sensitivity to selected parts of a curve; credit-spread changes are another source of price movement. Option-bearing securities often require effective duration from a model that re-estimates cash flows after a rate shift.

8. Immunization: match liabilities, not just asset durations

Suppose a liability requires a payment in the future. A rate fall raises the value of that liability, but it also raises the value of fixed-income assets. Immunization aims to offset these changes.

Under a common small parallel yield shift, let $A$ and $L$ be asset and liability PVs. The surplus $S=A-L$ changes approximately by

$$\Delta S\approx(-AD_{A,\mathrm{mod}}+LD_{L,\mathrm{mod}})\Delta y.$$

To remove first-order sensitivity, match dollar duration:

$$AD_{A,\mathrm{mod}}=LD_{L,\mathrm{mod}}.$$

If $A=L$, this reduces to equal modified durations. Matching duration alone while leaving asset and liability PVs unequal generally fails to immunize the surplus.

Worked example: a single future liability

A certain liability of 1,000 is due in three years. At a flat 5% annual rate, its PV is $1{,}000/1.05^3\approx863.84$ and Macaulay duration is three years.

Buying a default-free zero paying exactly 1,000 at that same date matches the liability cash flow directly. A rate move changes both market values identically, assuming the same discount curve and no other differences. This is cash-flow matching.

A portfolio of coupon bonds can instead match PV and duration initially. Classical local immunization also considers asset versus liability convexity, commonly requiring sufficient asset convexity under its assumptions. It needs monitoring and rebalancing as time passes and rates change. Duration matching does not remove default risk, inflation mismatch, liquidity needs, or exposure to nonparallel curve changes. Price risk and coupon reinvestment risk can offset locally around a planned horizon; they do not disappear in every scenario.

9. Corporate bonds: promised yield and expected return

A corporate bond’s contractual cash flows may not all be paid. Recovery depends on the issuer’s assets, debt seniority, collateral, and restructuring outcome. Credit deterioration can lower price before an actual default occurs.

Use promised YTM for the internal rate computed from the full contractual schedule. Expected return must account for possible default and recovery. Calling the latter “expected YTM” can be misleading: expected realized returns, the IRR of expected payments, and the expectation of scenario-specific IRRs need not be equal for multi-period bonds.

Worked example: a one-year risky zero

A bond costs 95 and promises 100 in one year. Suppose the real-world default probability is 2%, with recovery of 40 paid at year-end in default. The alternative risk-free annual rate is 3%.

$$y_{\mathrm{promised}}=\frac{100}{95}-1\approx5.2632\%.$$

The expected payoff and expected one-year holding return are

$$\mathbb{E}[CF_1]=0.98(100)+0.02(40)=98.8,$$ $$\mathbb{E}[R]=\frac{98.8}{95}-1=4\%.$$

For this one-period example, we can decompose the spread exactly:

Component	Calculation	Value
Expected-loss component, called default premium in the outline	Promised yield minus expected return	1.2632 percentage points
Expected excess return, called risk premium here	Expected return minus risk-free return	1.0000 percentage point
Total promised spread	Promised yield minus risk-free return	2.2632 percentage points

The first component equals expected payoff loss divided by price, $1.2/95$. The second compensates investors in this example for holding the risky claim. Real observed credit spreads may also reflect liquidity, taxes, optionality, and market conditions. These labels are a teaching decomposition, not a universally standardized separation observable directly from a quoted multi-year YTM.

Expected default losses are not the same as compensation for bearing systematic risk. A model may use expected cash flows with an appropriate risk-adjusted discount rate, or risk-neutral expected cash flows with consistent default-free discounting. Do not deduct expected losses from cash flows and then add the same expected-loss adjustment again in the discount rate.

Ratings and their limits

The broad long-term rating categories are:

Category	Moody’s scale	S&P / Fitch scales
Highest investment grade	Aaa	AAA
Other investment-grade categories	Aa, A, Baa	AA, A, BBB
Speculative or high yield	Ba and below	BB and below

Including the finer modifiers, the usual lowest investment-grade ratings are Baa3 and BBB-. The next grades, Ba1 and BB+, are speculative. The SEC’s credit-ratings bulletin explains that ratings are opinions about credit risk, can change, and do not cover every investment risk. A high rating is not a promise of stable price or freedom from default.

10. Practice questions

Use annual effective rates and year-end payments unless specified otherwise.

1. A three-year default-free zero pays 1,000 at maturity. Price it at a 4% spot rate. What happens to its price if the spot rate is negative?
2. A two-year bond pays 40 in year 1 and 1,040 in year 2. With one-year spot 3% and two-year spot 4%, find price.
3. Using the spot rates in question 2, find the one-year forward rate starting one year from today.
4. A bond has price 980 and modified duration 4.2. Estimate its price change for a 25-basis-point yield increase, ignoring convexity.
5. A bond has modified duration 5 and convexity 30. Estimate the percentage price change for a 100-basis-point yield fall, including convexity.
6. Assets have PV 900 and modified duration 5. Liabilities have PV 1,000 and modified duration 5. Is the surplus immunized against a common small parallel yield change?
7. A one-year bond costs 90, promises 100, and has a 5% default probability with recovery 40. Calculate promised yield and expected return.
8. Explain why a 6% forward rate does not establish that next year’s short rate will be 6%, and why a bond’s YTM need not equal its expected return.

11. Answers and checks

1. $P=1{,}000/1.04^3\approx889.00$. A negative annual effective spot rate greater than -100% gives a price above face value.
2. $P=40/1.03+1{,}040/1.04^2\approx1{,}000.37$.
3. $f_{1,2}=1.04^2/1.03-1\approx5.0097\%$. Check by multiplying $1.03(1+f)$ to recover $1.04^2$.
4. $\Delta P/P\approx-4.2(0.0025)=-1.05\%$. Thus $\Delta P\approx-10.29$ and the estimated new price is 969.71.
5. $\Delta y=-0.01$, so $\Delta P/P\approx-5(-0.01)+\tfrac12(30)(0.01)^2=5.15\%$. Convexity adds 0.15 percentage points.
6. No. Asset dollar duration is 4,500 and liability dollar duration is 5,000. The surplus derivative is $-4{,}500+5{,}000=500$ per unit yield change. Equal percentage sensitivities do not offset unequal PVs.
7. Promised yield is $100/90-1\approx11.1111\%$. Expected payoff is $0.95(100)+0.05(40)=97$, so expected one-year return is $97/90-1\approx7.7778\%$.
8. A forward rate is fixed by today’s no-arbitrage price relationships and may include risk premia; future realized rates are uncertain. YTM uses promised payments, while expected return accounts for default, recovery, sale prices, and reinvestment outcomes as relevant.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Fixed-Income Securities lecture slides: the course framework for pricing and interest-rate risk.
• MIT OpenCourseWare, Fixed-Income Securities recitation: supplementary problems on spot rates, YTM, duration, convexity, and immunization.
• U.S. Treasury, History of STRIPS: how eligible Treasury principal and coupon claims trade separately.
• SEC Investor.gov, What Are Corporate Bonds?: corporate bond terms and risks.
• SEC Investor.gov, The ABCs of Credit Ratings: rating categories and their limitations.

Feynman study lab

Explain it simply

Price a coupon bond as a collection of dated payments rather than one undated total.

Concrete example. A bond paying 5 after year one and 105 after year two is worth 5 / 1.04 + 105 / 1.04², about 101.89, with a flat 4% annual curve.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why is adding 5 and 105 insufficient to find today’s price?

Check your explanation after you try

Payments arrive at different dates. Discount each at the relevant maturity’s rate; credit and contract terms may also matter.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.