Bond pricing, yield measures, term structure, duration and convexity, credit risk analysis, and securitization for CFA Level I.
Definition first
This guide is designed for first-pass understanding. Start with core terms, then apply the framework in your own account workflow.
Fixed Income is one of the most mathematically rigorous topics on the CFA Level I exam, carrying approximately 10–12% of the total weight. If you understand bonds, you understand the foundation of all financial pricing — because the time value of money concepts you learn here apply to every asset class. This guide covers bond features, pricing mechanics, yield measures, the term structure of interest rates, duration and convexity, credit risk, and securitization. Master these concepts and you'll have a significant advantage not just on the exam but in understanding how financial markets actually work.
Bond Features & Terminology
A bond is a debt instrument that obligates the issuer (borrower) to make specified payments to the bondholder (lender) over a defined period. Before diving into pricing and risk, you need to understand the basic features:
Par value (face value): The principal amount repaid at maturity, typically $1,000 per bond. Bond prices are quoted as a percentage of par — a price of 98.5 means the bond trades at $985.
Coupon rate: The annual interest rate paid on the par value. A 5% coupon on a $1,000 bond pays $50 per year ($25 semi-annually, since most US bonds pay coupons twice per year).
Maturity date: When the issuer repays the par value. Bonds can be short-term (1–5 years), intermediate (5–12 years), or long-term (12+ years).
Currency denomination: The currency in which coupon and principal payments are made. This matters for international bonds and introduces currency risk for foreign investors.
Issuer: Governments (sovereign bonds, municipal bonds), corporations (corporate bonds), or supranational organizations (World Bank bonds). The issuer determines credit risk.
Embedded Options
Many bonds include provisions that give either the issuer or the bondholder the right to take certain actions before maturity. These embedded options significantly affect pricing and risk:
Callable bonds: The issuer can redeem the bond before maturity at a specified call price (typically par or a small premium). Issuers call bonds when interest rates fall — they can refinance at lower rates, just like refinancing a mortgage. Callable bonds offer higher yields than non-callable bonds to compensate investors for call risk. Call protection periods (during which the bond cannot be called) are common.
Putable bonds: The bondholder can sell the bond back to the issuer at par before maturity. This protects investors when rates rise (they can sell the bond and reinvest at higher rates). Putable bonds offer lower yields because the put option is valuable to the bondholder.
Convertible bonds: Can be converted into a specified number of common shares — a feature that connects to options pricing concepts. Convertibles offer lower yields because the conversion option has value. If the stock price rises above the conversion price, the bondholder can convert and participate in the equity upside.
The general principle: options favorable to the issuer (calls) increase the yield investors demand. Options favorable to the bondholder (puts, conversion rights) decrease the yield investors require. This makes intuitive sense — you need to be compensated for bearing risk, and you'll accept less return when you have protective options.
Bond Pricing & Yield Measures
A bond's price is the present value of its expected future cash flows (coupon payments plus par value at maturity), discounted at the required yield:
Price = ∑ [C / (1 + r)^t] + [Par / (1 + r)^n]
Where C = coupon payment, r = discount rate per period, and n = number of periods to maturity. The relationship between price and yield is inverse: when yields rise, prices fall, and vice versa. This is the most fundamental relationship in fixed income.
Premium, Par & Discount
When the coupon rate equals the market yield, the bond trades at par (100% of face value). When the coupon rate exceeds the market yield, the bond trades at a premium (above par) — investors are willing to pay more for above-market coupon payments. When the coupon rate is below the market yield, the bond trades at a discount (below par). As a bond approaches maturity, its price converges to par regardless of premium or discount — this is called "pull to par."
Yield Measures
The CFA exam tests several yield measures, and you must know what each one means, how it's calculated, and when it's most useful:
Yield Measure
Definition
Key Assumptions
Current yield
Annual coupon / Current market price
Ignores capital gains/losses and time value of money
Yield to maturity (YTM)
Discount rate that equates PV of all cash flows to current price
Assumes bond is held to maturity; coupons reinvested at YTM
Yield to call (YTC)
YTM assuming the bond is called at the first call date
Relevant only for callable bonds trading above the call price
Yield to worst (YTW)
Lowest of YTM, YTC, and yield to any other call date
Conservative measure for bonds with embedded options
Yield spread
Bond yield minus benchmark (typically government) yield
Measures credit and liquidity risk premium
Option-adjusted spread (OAS)
Spread after removing the value of embedded options
Allows comparison of bonds with and without options
YTM is the most widely used yield measure, but it has an important limitation: it assumes all coupon payments are reinvested at the YTM. If rates change after you buy the bond, your actual return will differ from YTM. This is reinvestment risk, and it's more significant for bonds with higher coupons and longer maturities.
The Term Structure of Interest Rates
The term structure (yield curve) shows the relationship between yields and maturities for bonds of the same credit quality. The most commonly referenced yield curve is for US Treasury securities, which are considered free of credit risk.
Yield Curve Shapes
Normal (upward-sloping): Longer maturities have higher yields. This is the most common shape and reflects the term premium — investors demand higher yields for locking up money for longer periods due to greater uncertainty.
Inverted (downward-sloping): Shorter maturities have higher yields than longer maturities. This typically signals that the market expects interest rates to fall, often because of an anticipated economic recession. An inverted yield curve has preceded every US recession in the last 50 years — a connection to the business cycle theory covered in the economics curriculum.
Flat: Yields are similar across all maturities. This is a transitional shape, often occurring when the curve is moving from normal to inverted or vice versa.
Humped: Intermediate-term yields are higher than both short-term and long-term yields. This shape is uncommon but can occur during periods of monetary policy transition.
Theories of the Term Structure
Three theories explain why the yield curve takes its shape:
Pure expectations theory: Forward rates are unbiased predictors of future spot rates. A normal yield curve means the market expects rates to rise; an inverted curve means rates are expected to fall. Under this theory, there is no term premium — the curve shape is entirely driven by rate expectations.
Liquidity preference theory: Investors prefer short-term bonds (more liquid, less price risk), so they demand a premium for holding longer-term bonds. This term premium causes the yield curve to be upward-sloping even when rate expectations are flat. This theory is generally considered the most realistic.
Market segmentation theory: Different investor groups operate in different maturity segments. Pension funds buy long-term bonds to match their long-term liabilities; money market funds buy short-term securities. Supply and demand within each segment determines yields, with limited substitution across segments.
Spot Rates, Forward Rates & Bootstrapping
Spot rates and forward rates are essential for precise bond valuation and for understanding the yield curve. A spot rate is the yield on a zero-coupon bond for a specific maturity. It represents the pure time value of money for that period with no reinvestment risk.
A forward rate is the implied future spot rate derived from the current term structure. For example, the "1-year rate, 2 years from now" (denoted 1f2) is the rate the market expects to prevail for a 1-year investment starting in 2 years. Forward rates are calculated from spot rates using the no-arbitrage principle:
(1 + S2)^2 = (1 + S1) x (1 + 1f1)
Where S1 and S2 are 1-year and 2-year spot rates, and 1f1 is the 1-year forward rate one year from now. If S1 = 3% and S2 = 3.5%, then 1f1 = [(1.035)^2 / (1.03)] − 1 = 4.0%. The forward rate of 4.0% exceeds both spot rates, which makes sense — in a normal (upward-sloping) yield curve, forward rates exceed spot rates.
Bootstrapping
Bootstrapping is the process of deriving spot rates from the prices of coupon-bearing bonds. You start with the shortest maturity (where the spot rate equals YTM for a zero-coupon bond), then use each successive coupon bond to solve for the next spot rate.
For example, if you know the 1-year spot rate is 3% and you have a 2-year bond with a 4% coupon trading at par, you can solve for the 2-year spot rate:
1,000 = 40 / (1.03) + 1,040 / (1 + S2)^2
Solving: S2 = 4.02%. You continue this process for longer maturities, using previously derived spot rates to discount earlier cash flows. Bootstrapping is important because it provides arbitrage-free discount rates for pricing any fixed-income security.
Interest Rate Risk: Duration & Convexity
Interest rate risk is the risk that bond prices will decline when interest rates rise. It's the most important risk for most fixed-income investors, and the CFA exam tests it extensively. Duration and convexity are the tools used to measure and manage interest rate risk.
Macaulay Duration
Macaulay duration is the weighted average time to receive a bond's cash flows, where the weights are the present values of each cash flow as a proportion of the bond's total price. For a zero-coupon bond, Macaulay duration equals maturity (you receive all cash at maturity). For a coupon bond, Macaulay duration is less than maturity because you receive some cash before maturity.
Macaulay duration is measured in years and has an intuitive interpretation: it's the point in time at which reinvestment risk and price risk exactly offset each other. If you hold a bond for exactly its Macaulay duration, your realized return will approximately equal YTM regardless of interest rate changes (this is the basis of immunization strategies).
Modified Duration
Modified duration measures the sensitivity of a bond's price to changes in yield:
Where n = number of coupon periods per year. The price change for a given yield change is:
% Price Change ≈ −Modified Duration x ΔYield
A bond with a modified duration of 7 will decline approximately 7% in price if yields rise by 1 percentage point (100 basis points). This is a linear approximation that works well for small yield changes but becomes less accurate for larger changes.
Effective Duration
For bonds with embedded options (callable, putable), modified duration is inappropriate because the bond's cash flows change when yields change (the issuer may call the bond if rates fall). Effective duration accounts for this by calculating the price sensitivity numerically:
Effective Duration = (P− − P+) / (2 x P0 x Δy)
Where P− = price if yields decline, P+ = price if yields rise, P0 = current price, and Δy = the yield change used in the calculation. Effective duration uses a pricing model (typically an interest rate tree) to estimate how the bond's price changes when yields shift, accounting for the possibility that options will be exercised.
Key Duration Relationships
Understanding what affects duration is critical for both the exam and practical risk management:
Higher coupon = lower duration. Coupon payments return cash sooner, reducing the weighted average time to receive cash flows.
Longer maturity = higher duration. Cash flows are received further in the future, making the bond more sensitive to rate changes.
Higher yield = lower duration. At higher discount rates, distant cash flows have lower present values, reducing their weight in the duration calculation.
Zero-coupon bond duration = maturity. All cash flow comes at maturity, so there's no coupon effect.
Convexity
Duration provides a linear approximation of the price-yield relationship, but the actual relationship is curved (convex). Convexity measures this curvature and improves the accuracy of price change estimates for large yield changes:
% Price Change ≈ −Modified Duration x ΔYield + 0.5 x Convexity x (ΔYield)^2
Convexity is always positive for option-free bonds (the price-yield curve is always convex). Positive convexity is beneficial: when yields fall, the bond price rises more than duration alone would predict; when yields rise, the price falls less than duration predicts. Higher convexity is always preferable, all else equal.
Callable bonds have negative convexity at low yields. When yields fall below the coupon rate, the likelihood of the bond being called increases, capping the price upside. The price-yield curve bends the wrong way (becomes concave), which is unfavorable for the bondholder. This is one of the key risks of callable bonds.
Credit Risk & Credit Analysis
Credit risk is the risk that the bond issuer will fail to make scheduled payments. While interest rate risk affects all bonds, credit risk primarily affects corporate and lower-quality sovereign bonds (government bonds from developed nations are generally considered credit-risk-free).
Components of Credit Risk
Default probability: The likelihood that the issuer will fail to make a scheduled payment. Higher-rated issuers have lower default probabilities. Historical data shows that BBB-rated bonds have a 5-year cumulative default rate of roughly 2%, while B-rated bonds have rates around 15–20%.
Loss given default (LGD): The portion of the investment lost if default occurs. LGD = 1 − Recovery Rate. Senior secured bonds have higher recovery rates (lower LGD) than subordinated unsecured bonds because they have priority claims on assets.
Expected loss = Default Probability x LGD. This drives the credit spread — the additional yield investors demand above the risk-free rate to compensate for credit risk.
Credit Spreads
The credit spread is the difference between a bond's yield and the yield on a comparable-maturity government bond. Credit spreads compensate investors for default risk and tend to be cyclical:
Spreads widen during recessions and periods of financial stress as default risk increases and investors flee to safety (government bonds). The 2008 financial crisis saw investment-grade spreads widen from roughly 100 bps to over 600 bps.
Spreads narrow during economic expansions as default risk declines and investors become more willing to take credit risk.
Spread changes have a significant impact on bond prices. For a bond with a modified duration of 6, a 50 bps widening in spreads causes approximately a 3% price decline. Credit spread risk is therefore a major concern for corporate bond investors, separate from interest rate risk.
Credit Ratings
Credit rating agencies (S&P, Moody's, Fitch) assign ratings that reflect their assessment of an issuer's creditworthiness. The basic scale (using S&P notation) runs from AAA (highest quality, lowest default risk) down to D (in default). The critical dividing line is between BBB− (lowest investment-grade) and BB+ (highest speculative-grade or "high yield" or "junk").
This boundary matters enormously because many institutional investors (pension funds, insurance companies, banks) are restricted to investment-grade bonds by regulation or their own investment policies. When a bond is downgraded from BBB− to BB+ (a "fallen angel"), forced selling by these institutions can cause a sharp price decline beyond what the credit deterioration alone would justify.
Rating agencies have been criticized for being too slow to downgrade (they missed the deterioration of subprime mortgage securities before 2008) and for potential conflicts of interest (issuers pay for ratings). Despite these criticisms, credit ratings remain the most widely used measure of credit quality and are deeply embedded in financial regulation and investment mandates.
Securitization: MBS, ABS & CDOs
Securitization is the process of pooling financial assets (mortgages, auto loans, credit card receivables) and issuing securities backed by those asset pools. It transforms illiquid, individual loans into tradeable securities. Securitization is both a critical financing mechanism and a source of systemic risk, as the 2008 financial crisis demonstrated.
Mortgage-Backed Securities (MBS)
MBS are backed by pools of residential or commercial mortgages. The most basic form is the pass-through security: mortgage payments (principal and interest) from the pool are "passed through" to investors after deducting servicing fees. Agency MBS (issued by Ginnie Mae, Fannie Mae, Freddie Mac) carry a government or quasi-government guarantee against credit risk; non-agency MBS do not.
The key risk of MBS is prepayment risk. Homeowners can prepay their mortgages at any time (typically by refinancing when rates fall). This creates uncertainty about the timing and amount of cash flows. When rates fall, prepayments accelerate (homeowners refinance), and MBS investors receive their principal back early — precisely when reinvestment rates are low. When rates rise, prepayments slow, and investors are stuck with below-market coupon rates for longer than expected.
Collateralized Mortgage Obligations (CMOs) are structured MBS that redistribute prepayment risk among different tranches. Sequential-pay CMOs direct all principal payments to the first tranche until it's retired, then to the second, and so on. This gives short-term investors more certainty about timing (the first tranche) while concentrating extension risk in the later tranches.
Asset-Backed Securities (ABS)
ABS are backed by pools of non-mortgage assets: auto loans, credit card receivables, student loans, equipment leases, and more. The structure is similar to MBS but with some differences:
Auto loan ABS have shorter maturities and more predictable prepayment behavior than MBS (people refinance cars less often than homes).
Credit card ABS are revolving structures with a revolving period (new receivables replace paid ones) followed by an amortization period.
ABS typically use credit enhancement structures (overcollateralization, subordination, reserve accounts, excess spread) to achieve higher ratings than the underlying assets would merit on their own.
Collateralized Debt Obligations (CDOs)
CDOs are backed by pools of bonds, loans, or other debt instruments. They played a central role in the 2008 financial crisis. The pool is divided into tranches with different risk levels: senior tranches (AAA-rated, first claim on cash flows), mezzanine tranches (BBB to A-rated), and equity tranches (unrated, absorb first losses but receive highest potential returns).
The pre-crisis problem was that CDOs backed by subprime mortgages were given AAA ratings on their senior tranches based on the assumption that default correlations within the pool would remain low. When the housing market turned and defaults became highly correlated, the senior tranches suffered massive losses despite their high ratings. This revealed a fundamental flaw in the rating methodology and the underestimation of systemic risk.
Putting Fixed Income Into Practice
Fixed income knowledge is essential for any investor who holds bonds — which should be virtually everyone with a diversified portfolio. Understanding duration helps you assess how sensitive your bond holdings are to interest rate changes, which is central to portfolio risk management. Credit analysis helps you evaluate whether the extra yield from corporate bonds adequately compensates for default risk. Yield curve analysis provides insights into the market's expectations for future interest rates and economic conditions.
When you see the yield curve invert, you know the bond market is pricing in a potential recession. When credit spreads widen, you know the market is pricing in higher default risk. These signals are valuable for asset allocation decisions across your entire portfolio, not just the fixed-income portion.
Clarity helps you track your bond holdings alongside your equities, crypto, and other investments, giving you a complete picture of your portfolio's risk exposure. Whether you hold individual bonds, bond funds, or Treasury securities, understanding the concepts in this guide helps you make informed decisions about duration, credit quality, and yield.
This guide, combined with equity investments, financial statement analysis, and corporate issuers, covers the core analytical toolkit that every CFA candidate — and every serious investor — needs to master. The mathematical rigor of fixed income may feel daunting, but the concepts are logical and build on each other. Once you internalize the relationship between price, yield, duration, and convexity, you'll see bonds in an entirely new light.