Fixed Income Summary

Source and scope. Based on the five Fixed Income readings in CFA 2026 Level II SchweserNotes, Book 4, attached in CFA Lv2. This is an analytical study guide, not a reproduction of the textbook or its quizzes.

Conventions. Page references below are PDF viewer pages, counting the cover as page 1, rather than the book’s printed page numbers. Numerical examples are illustrative calculations, not market observations; unless stated otherwise, they use annual payments, annual compounding, and values per 100 of principal.

Reading approach. Begin with the economic intuition, derive the pricing relationship, and then identify the assumptions that make it valid. Practical qualifications are identified where a simplified curriculum statement should not be treated as a universal market rule.

Overview: How the five readings fit together

The logical progression is curve → contingent cash flows → default losses → transfer of credit exposure. Each reading adds another reason why a bond’s quoted yield is not enough to understand its value.

A useful conceptual decomposition is:

This is a guide to the economic components, not a universal instruction to price each component independently. Exercise, default, recovery, and interest rates can interact, so a complete model must treat those interactions consistently.

1. The Term Structure and Interest Rate Dynamics

Reading 23, Modules 23.1 to 23.6. Module 23.3, the swap rate curve, appears in the PDF body even though it is omitted from its opening contents list.

1.1 Four rate concepts answer four different questions

Spot rates value individual future payments. A three year spot rate is the annualized yield on a payment received only at the end of year three.

Par rates describe coupon rates that make newly issued benchmark bonds worth par. They summarize the discount curve through the pricing of coupon bonds, rather than through a single terminal payment.

Forward rates are rates implied today for borrowing or lending over a future interval. They arise from an equivalence between investing directly to a distant date and investing through successive periods.

Yield to maturity, or YTM, is the single internal rate of return that equates a bond’s promised cash flows to its price. For a coupon bond, it compresses information from several discount rates into one number.

Two bonds with the same maturity but different coupons can have different YTMs even when priced from the same spot curve. Their cash flows place different weights on different parts of that curve.

1.2 Discount factors are the most direct valuation language

Let $D_t$ be the current price of receiving one unit of currency at time $t$, and let $s_t$ be the corresponding annual spot rate.

For a bond with coupon payment $C$, principal $F$, and maturity $T$:

The essential idea is one payment date, one discount factor. Discounting every payment at the maturity matched spot rate generally gives the wrong value when the curve is not flat.

YTM is found only after the price is known, by solving:

A bond’s YTM is not a simple arithmetic average of spot rates. It is an implicitly determined rate whose relationship to the curve depends on the timing and size of all cash flows.

1.3 Bootstrapping: Recover the discount curve from par bonds

For a bond with principal normalized to one and annual par coupon $c_n$:

Therefore:

Bootstrapping works sequentially because the shorter dated cash flows can already be valued. Each new maturity supplies the final unknown discount factor in its own pricing equation.

Worked example. Suppose the one, two, and three year par rates are 3%, 4%, and 5%.

For example:

A three year bond paying a 4% annual coupon then has value:

It trades below par because its coupon is below the three year par rate. This is also a useful check on the direction of the calculation.

Exam trap. The two year par rate is 4%, but the two year spot rate is approximately 4.0202%. Treating them as interchangeable bypasses the effect of the first coupon.

1.4 Forward rates are pricing relationships before they are forecasts

Let $f_{j,k}$ denote the annualized forward rate for a loan beginning in $j$ years and lasting $k$ years.

Equivalently:

The forward price, payable at time $j$, of a zero coupon bond paying one at time $j+k$ is:

These equations follow from the law of one price. They do not require the assumption that investors correctly predict future rates.

Using the bootstrapped curve above, the one year forward rate beginning one year from now is:

The two year spot rate is approximately the geometric average of today’s one year rate and the one year forward rate. More generally, a long spot rate combines a sequence of shorter forward rates.

Interpretation. When a longer spot rate exceeds the relevant shorter spot rate, the forward rate spanning those dates exceeds the longer spot rate. The familiar ordering of forward, spot, and par curves describes a smoothly rising curve; it should not be applied mechanically to every irregular curve shape.

1.5 YTM, holding period return, and reinvestment risk

For a one year holding period with a coupon received at the horizon:

The realized return depends on the purchase price, cash flows actually received, reinvestment rates where relevant, and the sale price. A bond may have an attractive YTM yet produce a negative holding period return if market yields rise sufficiently.

The standard interpretation of YTM as an annualized compound return to maturity assumes timely payment of all promised cash flows and reinvestment of coupons at that same yield. For a zero coupon bond held to maturity without default, there is no coupon reinvestment assumption to make.

The benchmark for an active rate view is the forward curve, not simply today’s spot curve. Saying that rates will decline does not establish an opportunity if the market already prices an even larger decline.

For the two year zero coupon bond in the preceding example, the current price is 92.4197. The following are two different assumptions about next year’s curve.

If future spot rates realize today’s forward curve, the one year return on this benchmark zero equals today’s one year spot rate. The higher return in the unchanged curve scenario comes from a different rate outcome, not from a guaranteed free profit.

1.6 Rolling down the curve is a position in curve evolution

Rolling down the curve means buying a bond with a maturity longer than the intended holding period and selling it after its remaining maturity has shortened. With an upward sloping curve that remains unchanged, the shorter remaining maturity usually places the bond at a lower yield, creating a favorable price effect relative to an unchanged yield assumption.

The complete return also includes coupon income, pull toward par, reinvestment, and any movement in the curve itself. A premium bond can experience pull toward par that offsets some of the benefit from moving to a lower yield.

The strategy’s key assumption is not merely that the curve slopes upward today. The future curve must be sufficiently favorable relative to the forwards already embedded in the purchase price.

Borrowing at a short maturity to finance the position adds funding and rollover risk. A profitable looking carry position can lose money if long rates rise, funding costs increase, liquidity disappears, or leverage forces an early sale.

1.7 Swap curves and spread measures

In a plain interest rate swap, one party pays a fixed rate and receives a floating reference rate, while the other takes the opposite side. The par swap rate makes the two legs have equal present value at inception.

In the simplified single curve framework, with accrual fractions ${\alpha }_i$:

The denominator is the present value of a unit stream of fixed payments. Under annual payments and the curve used earlier, the three year par swap rate is 5%.

For example, a 4.20% swap rate against a 3.85% government yield produces a swap spread of 35 basis points. One basis point is 0.01 percentage point, or 0.0001 when the rate is expressed as a decimal.

The Z spread solves:

Practical qualification. The text retains some explanations rooted in unsecured interbank benchmarks while using the generic label MRR. A secured overnight reference rate does not carry the same bank credit exposure as an unsecured term rate, and modern swap spreads can be negative.

An OIS is a swap exchanging a fixed rate against a compounded overnight reference rate, not itself an unsecured overnight loan. In collateralized markets, projecting floating payments and discounting them may require different curves, so the single curve swap formula is a teaching framework rather than a universal implementation.

1.8 Term structure theories distinguish expectations from premiums

The most important conceptual distinction is between a forward rate implied by prices and an expected future rate under a particular economic theory. Expectations and risk compensation cannot generally be disentangled from the curve alone.

Under liquidity preference, an upward curve does not necessarily imply rising expected short rates. A downward curve is stronger evidence of expected declines because the expected decline must overcome a positive maturity premium.

1.9 Curve risk has more than one dimension

Level is a broad movement in yields. Slope is a change in the difference between longer and shorter yields, while curvature concerns the middle of the curve relative to its wings.

Effective duration is primarily a measure of sensitivity to a parallel benchmark curve shift. Two portfolios can have equal effective duration and still perform differently under a steepening, flattening, or butterfly movement.

Key rate durations isolate exposure to selected benchmark maturities:

Suppose key rate durations at two, five, and ten years are 1, 3, and 2, while those rates move by +20, +10, and −10 basis points.

The sum of key rate durations approximately equals effective duration when the bump definitions and curve reconstruction are consistent. The individual components reveal risks that the sum hides.

A level neutral steepening trade generally combines a long position at the short end and a short position at the long end, sized by rate sensitivity rather than equal principal. Its success depends on relative yield changes and is not assured merely because aggregate duration is near zero.

1.10 Economic forces influence different parts of the curve

The book’s historical statements about the proportion of yield variation explained by particular factors should be treated as sample dependent observations. They are not permanent coefficients to insert into a forecast.

Finally, distinguish yield volatility from price volatility. A shorter yield can fluctuate more than a longer yield while the longer bond remains more price sensitive because its duration is much higher.

2. The Arbitrage Free Valuation Framework

Reading 24, Modules 24.1 to 24.3.

2.1 No arbitrage imposes consistency, not a macroeconomic forecast

The law of one price requires identical future cash flows with identical relevant risks to have identical values. Otherwise, an investor could buy the cheaper payoff and sell the more expensive equivalent payoff.

Value additivity means the value of a complete bond equals the sum of the values of its coupon and principal components. If the bond is cheaper than its strips, buying and stripping it can exploit the discrepancy; if the strips are cheaper, buying and reconstituting them is the reverse trade.

Dominance means a security with at least as favorable cash flows in every state should not be cheaper, all else equal. Practical implementation still requires attention to transaction costs, financing, short selling, settlement, and matching the actual legal claims.

An arbitrage free model is therefore a model of relative price consistency. It does not prove that all securities are fundamentally attractive or that future realized returns will match model expectations.

2.2 Why fixed cash flows do not require an interest rate tree

A benchmark bond with known payments can be valued directly from spot discount factors. There is no need to predict which path future rates will follow in order to value cash flows that are already fixed.

An embedded option changes this. If rates fall, an issuer may call a bond, and the investor will no longer receive the originally scheduled later coupons.

The valuation model must therefore represent both future rate states and the cash flow decisions made in those states. This is the economic purpose of an interest rate tree.

2.3 What a binomial tree represents

A binomial tree assigns two possible next states at each node. In the book’s standard framework, upward and downward branches each have probability 0.5 for valuation purposes.

These are pricing probabilities, not a claim that actual market rates have a 50% chance of increasing each year. Pricing probabilities and modeled rates work together to reproduce current market prices.

The standard lognormal tree relates adjacent rates in a time slice by:

For annual steps, $\Delta t=1$, giving the book’s $e^{2\sigma }$ multiplier. This imposes proportional spacing, keeps rates positive in the basic model, and makes absolute rate dispersion greater when the level of rates is higher.

Terminology caution. The text sometimes calls node rates forward rates. They are state dependent future one period rates inside a pricing model, which is different from a single deterministic forward rate implied by today’s spot curve.

2.4 Backward induction: Value the future before the present

Let $V_{t,j}$ be a node value immediately after any coupon at that node. If the next coupon is $C_{t+1}$ and the node rate is $r_{t,j}$, then for an annual step:

At maturity, the final coupon and principal are included exactly once. The safest approach is to write down whether each node value is before or after the coupon before starting the calculation.

Worked example. Consider a two year bond with principal 100 and annual coupon 6. The current one year rate is 4%, and next year’s one year rate is either 3% or 7%, with equal pricing probabilities.

At the year one nodes, immediately after the first coupon:

The current price is:

The first coupon is added outside the average because it is the same in both states. The principal is not added again at year zero because it is already embedded in the future node values.

2.5 Calibration makes the tree consistent with the observed curve

A tree is calibrated when it reproduces the prices of the benchmark securities used to construct it. Choosing a volatility and drawing plausible rates is not sufficient.

The conceptual procedure is:

1. Set the initial short rate from the first benchmark maturity.
2. Choose the volatility assumption and establish the relative spacing of rates in the next time slice.
3. Adjust the level of that slice until the next benchmark bond is priced correctly.
4. Repeat for later maturities, preserving previously calibrated prices.

The middle of a time slice may be close to the corresponding implied forward rate, but simply setting it equal to that forward is not an exact calibration procedure. Nonlinear discounting matters.

The illustrative tree above implies a two year discount factor of 0.916083 and a two year spot rate of approximately 4.4798%. Discounting the bond’s fixed payments with that curve gives the same 102.8741 value as backward induction.

Important independence. This is the curve implied by the illustrative tree, not the different par curve used in Section 1. A tree and a spot curve can only be compared as alternative pricing methods when they are calibrated to each other.

2.6 Pathwise valuation and Monte Carlo simulation

Pathwise valuation discounts each payment along a complete rate path and then takes the probability weighted average across paths. Under the same assumptions, it gives the same value as backward induction.

For the book’s convention of $n$ annual discounting periods with the initial rate already known, there are $2^{n-1}$ rate paths. In a three period tree, the four complete paths have equal probability under symmetric branching, but the three final rate nodes have probabilities 0.25, 0.50, and 0.25.

Exam trap. Equally likely paths do not imply equally likely recombined nodes. The middle node can be reached by more than one path.

Monte Carlo simulation is especially useful when cash flows depend on the history of rates. For a mortgage pool, earlier refinancing opportunities change which borrowers remain and how likely they are to refinance later.

Two paths can reach the same current rate yet produce different future mortgage payments. A simple recombining tree that records only the current rate loses that information, while a simulation can retain the relevant history.

More simulated paths reduce sampling error. They do not fix an incorrect prepayment model, an inappropriate volatility assumption, or a poorly calibrated interest rate process.

2.7 Compare interest rate models by what they assume

For intuition, the two principal equilibrium models are:

The first term pulls the short rate toward a long run level $b$ at speed $a$. The second term determines random variation, with CIR making that variation depend on the current rate.

The key distinction is explaining rate dynamics from an equilibrium structure versus fitting current market prices through calibration. Neither label eliminates model risk.

3. Valuation and Analysis of Bonds With Embedded Options

Reading 25, Modules 25.1 to 25.8.

3.1 First identify who owns the option

An option has value to the party that can choose whether to exercise it. The investor’s position in that option determines whether it adds to or subtracts from the bond’s value.

For otherwise comparable bonds, the callable bond should be cheaper than the straight bond, while the putable bond should be more expensive. A larger coupon on a callable issue may simply compensate the investor for an unfavorable option position.

European exercise occurs on a specified date, American exercise is available throughout an eligible period, and Bermudan exercise occurs on a set of specified dates. Exercise dates and protection periods are part of the valuation, not administrative details.

3.2 The call and put rules modify continuation value

At an eligible exercise node, using a consistent coupon timing convention:

The issuer calls when continuing the bond would be more expensive than redeeming it. The investor puts when the redemption right is worth more than holding the bond.

Worked example using the Section 2 tree. Keep the coupon at 6 and make the bond callable or putable at 100 immediately after the first coupon.

Discounting these node values and the first coupon gives:

Relative to the straight bond value of 102.8741, the embedded call is worth 1.4003 and the embedded put is worth 0.4493. The exercise choices explain the value differences directly.

Practical qualification. A call price caps value at an eligible exercise decision under the specified convention. It is not necessarily a hard cap on the market price at every earlier date, especially during call protection or when coupons and exercise timing differ.

3.3 Volatility changes option value, not just the likely direction of rates

Holding today’s benchmark curve fixed and recalibrating consistently, greater interest rate volatility increases the value of both the issuer’s call and the investor’s put. The straight bond’s fixed payments still have the same benchmark present value.

The last two columns answer different questions. One changes the theoretical price while holding the spread fixed; the other keeps the observed price fixed and asks which spread now reproduces it.

3.4 The level and shape of rates affect exercise incentives

When rates decline, the issuer’s call becomes more attractive because the existing coupon is expensive relative to new financing. The callable investor participates in less of the straight bond’s price appreciation.

When rates increase, the investor’s put becomes more attractive because redeeming at the put price can be preferable to retaining a low coupon bond. The putable bond loses less value than its otherwise comparable straight bond.

In the book’s stylized comparison, flattening an upward sloping curve increases call value and reduces put value as the relevant future rate environment becomes lower. In practice, a flattening caused by rising short rates is not identical to one caused by falling long rates.

Always identify which maturities moved and how those movements change rates near the exercise dates. A slope label alone is not enough to price an option.

3.5 OAS asks how much spread remains after modeling the option

The option adjusted spread, or OAS, is the constant spread added to the benchmark tree’s short rates so that the modeled bond price equals its observed price, with exercise behavior included.

The calculation has three essential inputs: a benchmark curve, an interest rate volatility model, and the bond’s contractual cash flow and exercise terms. Different assumptions can produce different OAS values for the same market price.

A reliable procedure is:

1. Calibrate the benchmark tree to the selected curve.
2. Add a trial spread to the tree’s discount rates.
3. Recalculate continuation values and apply the exercise rules at every eligible node.
4. Adjust the spread until the resulting price matches the market price.

OAS is not a separate spread added after exercise has been determined once and then frozen. Exercise must remain consistent with the valuation being solved.

For a callable bond, the Z spread typically includes compensation for the issuer option as well as other risks. A useful spread based approximation is:

This is not an exact conversion of the option’s currency value into spread. It depends on the benchmark and valuation conventions.

A higher OAS can indicate better compensation relative to comparable bonds. It can also reflect weaker liquidity, different covenants, underestimated volatility, model error, or genuine credit risk.

Exam trap. If assumed volatility is too low, a callable bond’s call is undervalued, its theoretical bond value is too high, and the fitted OAS is too high. The bond may appear cheap when the apparent advantage is actually a modeling error.

3.6 Effective duration revalues changing cash flows

Modified duration treats promised cash flows as fixed. For an option bearing bond, a change in benchmark rates can also change exercise behavior, making effective duration the more appropriate measure.

Let $P_{down}$ be the model price after a downward parallel curve shift of size $\Delta y$, and $P_{up}$ the price after an upward shift.

The benchmark tree must be rebuilt or recalibrated for each shifted curve. Hold OAS and the other intended model assumptions constant while allowing exercise and cash flows to change.

Reestimating OAS to force each shocked price back to the original market price would defeat the purpose of measuring interest rate sensitivity. The analysis must distinguish benchmark rate risk from spread risk.

3.7 Asymmetric duration and negative convexity

Define the two directional measures:

For a callable bond near exercise, $D_{down}<D_{up}$. Falling rates provide less upside than the downside caused by a comparable rise in rates.

For a putable bond near exercise, $D_{down}>D_{up}$. The put limits losses when rates rise but leaves more participation when rates fall.

Worked example. Suppose a callable bond is worth 100, rises to 100.35 after a 10 basis point decline, and falls to 99.50 after a 10 basis point increase.

The effective duration is 4.25, down duration is 3.50, and up duration is 5.00. Effective convexity under the decimal yield convention is −1,500, reflecting a local exercise related bend rather than a reliable extrapolation for very large yield changes.

Callable bonds tend to behave more like straight bonds when rates are high and exercise is unlikely. As rates fall toward the exercise region, their price gains become constrained and convexity can turn negative.

This also explains extension risk. When rates rise, a previously likely call may become unlikely, extending the expected life and increasing duration just when the investor is already experiencing price losses.

3.8 Key rate duration follows the economically relevant payment date

For a callable bond with a low coupon, the call is less likely to be exercised, so the contractual maturity is often the dominant rate exposure. For a high coupon callable bond, the likely call date can become more important than final maturity.

For a putable bond, a high coupon makes exercise less likely, preserving maturity exposure. A low coupon makes the put more relevant and can concentrate exposure near the put date.

The book defines these sensitivities through bumps to benchmark par rates. Under that construction, a benchmark par bond has its exposure concentrated at the matched maturity, whereas a nonpar bond can have sensitivity at several points, including negative key rate durations.

Practical qualification. Those patterns depend on the curve parametrization. Bumping spot rates directly is not the same calculation as bumping par rates and rebuilding the spot curve.

3.9 Capped and floored floating rate bonds

An ordinary floater resets its coupon with a reference rate. When the coupon resets to the appropriate required market rate and credit conditions are unchanged, it is approximately worth par at a reset date.

For a coupon payment based on a reference rate fixed at the start of the accrual period:

Here $L$ is the reference rate, $m$ the contractual margin, and $\alpha$ the accrual fraction. This notation assumes the strike applies to the total coupon; actual contract terms must determine whether a cap or floor applies to the reference rate or the all in rate.

The investor is short the cap and long the floor. A floating coupon does not eliminate interest rate exposure once a cap or floor can bind.

Using the Section 2 tree with a floater paying the node reference rate, the first coupon is 4. With a 5% cap, the next coupon is limited to 5 in the 7% rate state, giving a bond value of 99.1014 and cap value of 0.8986.

With a 4% floor instead, the coupon rises from 3 to 4 in the low rate state. The floored bond is worth 100.4668, so the floor contributes 0.4668.

Timing trap. A coupon determined at the start of a period and paid at its end is paid in arrears. That does not mean the rate is also fixed at the end of that period; an actual in arrears reset is a different convention.

3.10 Convertible bonds combine credit and equity exposure

A convertible gives the investor the right to exchange the bond for a specified number of shares. The investor accepts a lower coupon in exchange for potential equity upside, while the issuer accepts possible future dilution.

For a simple immediately convertible bond without conflicting option restrictions, the standard valuation floor is:

The convertible generally trades above its immediate exercise value because retaining the conversion option preserves future choices. However, the straight bond floor is not a guaranteed redemption value; it changes with rates, credit spreads, and default expectations.

Worked example. A convertible has principal 1,000, conversion ratio 25, current share price 38, straight value 920, and market price 1,040.

At low stock prices, the convertible tends to behave more like risky debt. At high stock prices, it becomes more equity sensitive, while the middle region contains substantial option value.

The simple decomposition is a straight bond plus an equity conversion option, less any issuer call, plus any investor put. In a full valuation, those rights interact and should not be priced as unrelated vanilla options.

Contract terms can also change the conversion ratio for stock splits, extraordinary dividends, or a change of control. A cash put and a put that the issuer may settle in shares provide different forms of protection.

4. Credit Analysis Models

Reading 26, Modules 26.1 to 26.7.

4.1 Credit risk is broader than default

A bond can lose value without missing a payment. Wider credit spreads, a downgrade, reduced market liquidity, or a deterioration in expected recovery can all reduce its price before default occurs.

The central analytical distinction is between the probability of a loss event and the size of the loss if it occurs. A lower probability of default does not necessarily mean lower expected loss if the exposure or loss severity is much larger.

4.2 Exposure, recovery, and loss given default

Let $E_t$ be exposure measured at the potential default date, and let $R_t$ be the recovery fraction under the assumed recovery convention.

If recovery is supplied as a currency amount instead, LGD is exposure less that amount. Do not multiply exposure by a recovery amount as though it were a percentage.

In the book’s worked CVA framework, exposure at a future default date is based on the benchmark value at that date of the remaining promised payments. For coupon bonds, whether the coupon due on the default date is included depends on the assumed event timing.

Recovery convention matters. Recovery of par, recovery of market value, and recovery of the benchmark value of remaining payments are not interchangeable. Changing the convention changes the loss calculation and potentially the relationship between hazard rates and spreads.

4.3 Conditional, marginal, and cumulative default probabilities

Let $h_t$ be the conditional probability of default during period $t$, given survival to its start. Let $Q_t$ be the probability of surviving through period $t$.

The probability, measured today, of default specifically in period $t$ is:

The cumulative probability of default by time $T$ is:

With constant hazard, the hazard stays unchanged, marginal annual default probabilities decline because fewer issuers survive, and cumulative default probability rises. Confusing those three statements is one of the most common errors in this topic.

For a constant annual hazard of 3%, three year cumulative default probability is 8.7327%, not exactly 9%. An issuer that defaults in the first year cannot default again in the second or third year.

4.4 CVA converts expected credit losses into a present value deduction

Let $V_{ND}$ be the benchmark value of the promised payments assuming no default. Under the discrete framework used here:

For market valuation, probabilities should be consistent with risk neutral pricing. For forecasting actual credit losses or realized returns, physical probabilities are the relevant starting point; those are different tasks.

The simple expression also assumes the exposure, discount factors, and default probabilities can be combined as specified. With stochastic rates and dependence between default and exposure, the joint state dependent expectation must be modeled rather than multiplying unrelated averages.

4.5 Worked CVA example with an explicit recovery convention

Consider a three year zero coupon bond with principal 100, a flat benchmark rate of 4%, constant annual risk neutral hazard of 3%, and recovery of 40% of exposure at default. Default occurs at year end, and exposure is the benchmark value at that date of the remaining maturity payment.

For year one, exposure is $100/(1.04{)}^2=92.4556$. Loss severity is 60%, so the loss at default is 55.4734, weighted by a 3% default probability and discounted one year.

The promised YTM implied by this price is:

The YTM spread over the 4% benchmark is approximately 188.26 basis points. It is not the bond’s expected realized excess return, because the YTM calculation assumes the full principal will be paid.

As an independent valuation check, the probability weighted present value of recoveries in the three default states plus the surviving maturity payment also equals 84.2416. That is the same economics expressed through expected payments rather than a deduction from promised payments.

4.6 Physical and risk neutral probabilities answer different questions

A physical probability estimates how often default will actually occur. A risk neutral probability is a pricing input inferred under a valuation framework and reflects the compensation investors require for bearing priced risk.

Suppose a one year zero coupon bond promises 100, pays 40 if it defaults, is priced at 94, and the benchmark rate is 4%. Its implied pricing probability satisfies:

This gives $p=3.7333\%$. It is the probability consistent with that price, benchmark, and recovery assumption, not an independently established forecast of actual defaults.

At a fixed market price, assuming higher recovery requires a higher implied default probability to produce the same expected loss. Default probability and recovery cannot generally both be identified from a single bond price without additional assumptions.

A frequently useful approximation is:

Its precision depends on timing, recovery, term structure, and pricing assumptions. Observed bond spreads also contain liquidity and other components, so dividing a spread by loss severity is not a universal estimator of physical default risk.

4.7 Ratings, scoring, and notching

Credit scores are commonly used for individuals and smaller businesses, while credit ratings apply to issuers and debt instruments. Both are ordinal rankings, not equal interval numerical measures of risk.

Issue ratings can differ from issuer ratings because seniority, collateral, guarantees, and structural subordination change expected recovery. Notching reflects these differences across obligations of the same issuer.

A rating therefore provides useful comparative information but is not a substitute for a model of cash flows, leverage, debt service capacity, and recovery. Environmental, social, and governance issues matter when they change those economic inputs through fines, disruption, refinancing access, or unreliable reporting.

4.8 Migration can produce losses long before default

For a modest spread change with benchmark rates unchanged:

The book uses modified duration as the approximation for a plain bond’s sensitivity. More generally, spread duration is the direct measure, especially when separating benchmark rate and spread movements.

A duration of 4 and a 100 basis point widening imply an approximate 4% price decline. The return over a holding period must still include coupons, reinvestment where relevant, and any default payment.

Illustrative transition calculation. Suppose the price entering the horizon is 100, the coupon at the horizon is 5 if there is no default, and the unchanged rating horizon price is assumed to be 100. The upgrade narrows spreads by 40 basis points, the downgrade widens them by 100 basis points, and spread duration is 4.

The expected horizon value is 98.56 and the expected coupon is 4.90. The expected holding period return is therefore 3.46%, even though the contractual coupon is 5%.

The upgrade and downgrade prices are duration approximations. A more complete analysis would reprice the shorter remaining cash flows under each state’s curve and spread rather than assume a flat baseline horizon price.

4.9 Structural models explain default through the balance sheet

In the basic structural model, the firm has asset value $A_T$ and a single debt obligation with promised payment $K$ at maturity. Equity holders receive what remains after paying debt, but their limited liability prevents a negative equity payoff.

Equity resembles a call on the firm’s assets with strike equal to promised debt. Risky debt resembles a risk free debt claim minus a put on the firm’s assets.

This explains why greater asset volatility can hurt creditors: their upside is limited by the promised debt payment while they remain exposed to the downside. Under the basic model, the value of that short put corresponds to the credit deduction from risk free debt.

Strength. The model provides an economic explanation for default and links credit risk with leverage, asset volatility, and equity value. Limitation. Asset values and volatility are not directly observable in the same way as traded share prices, and real capital structures are much more complex than a single zero coupon debt claim.

The simplest maturity only model can also struggle to capture sudden default risk over very short horizons. More elaborate structural models can address some limitations, but the basic exam comparison concerns the simplified version.

4.10 Reduced form models explain default timing statistically

A reduced form model treats default as an event governed by an intensity process. The intensity may depend on leverage, coverage ratios, market information, and macroeconomic conditions without requiring the model to establish an explicit asset value default barrier.

In continuous time, a nonnegative default intensity $\lambda (u)$ gives survival probability:

An intensity is not literally the same object as a discrete annual default probability. With constant intensity over a year, that year’s conditional default probability is $1-e^{-\lambda }$.

Default being unexpected at the exact event time does not mean a reduced form model must assume no deterioration beforehand. The intensity can rise as conditions weaken, even though the final default event remains uncertain.

4.11 A spread curve combines credit expectations with market structure

Credit spreads compensate for more than default probability. Expected loss severity, risk aversion, liquidity, tax treatment, and technical supply and demand can all contribute relative to the chosen benchmark.

Higher expected default and lower expected recovery generally widen spreads. During a downturn, the two can deteriorate together, amplifying credit losses and spread volatility.

High quality spread curves are often flat or gently rising, while lower quality curves may be steeper because uncertainty accumulates with maturity. Distressed issuers can instead have inverted curves when immediate default risk dominates longer term concerns.

Practical qualification. A curve inversion can also reflect liquidity and the relative pricing of newly issued versus older debt. It should not automatically be interpreted as a clean forecast of improving issuer fundamentals.

A valid comparison should control for seniority, collateral, currency, embedded options, and benchmark selection. Otherwise, the apparent term structure partly reflects different securities rather than different maturities of the same risk.

4.12 Securitized debt requires analysis of the pool and the waterfall

Corporate credit generally concerns the issuer’s overall capacity to meet obligations. Securitized debt concerns a specified asset pool and the legal structure distributing its cash flows.

A granular pool contains many relatively small exposures; homogeneity means those exposures share relevant characteristics. Many similar short maturity loans can be analyzed statistically, while a small pool of large loans requires more attention to individual borrowers.

For revolving or changing pools, analyzing only the initial collateral is insufficient. Underwriting rules, replenishment criteria, and the incentives of the parties selecting future assets also matter.

Subordination redistributes losses rather than eliminating them. A senior tranche may initially be insulated from collateral defaults, but sufficiently severe or correlated losses can exhaust the protection beneath it.

A covered bond differs from a typical securitization because investors generally have recourse to both the issuer and a designated cover pool. Extension provisions can change payment timing, so credit quality and liquidity risk must be considered separately.

5. Credit Default Swaps

Reading 27, Modules 27.1 to 27.3.

5.1 Buying protection means being short credit risk

A CDS transfers specified credit event risk on a reference entity. The protection buyer pays premiums, while the protection seller makes the contractual payment if a qualifying credit event occurs.

Memorize the economic direction, not just the contract label. Buying a corporate bond and selling CDS protection are both broadly long credit risk, whereas buying CDS protection is the opposite direction.

A CDS resembles insurance economically, but its payout depends on contractual definitions and settlement procedures. It does not reimburse every possible decline in a bond’s market value.

5.2 Contract specification determines what is actually hedged

The key terms include the reference entity, reference obligation, seniority, notional, maturity, currency, credit event definitions, deliverable obligations, and settlement method. A contract on the same company but a different seniority is not automatically an appropriate hedge.

The book identifies bankruptcy, failure to pay, and restructuring as common credit events. Whether a particular event is covered depends on the agreement and applicable determination procedures.

Physical settlement generally exchanges an eligible obligation for its contractual notional amount. Cash settlement generally pays the loss fraction implied by the applicable recovery or auction price.

Worked example. A senior CDS has notional 10 million, and eligible senior obligations trade at 30% and 45% of par after a covered event. A subordinated obligation trades at 20% but is not eligible under the assumed senior contract.

Using the cheapest eligible senior obligation at 30%, the economic protection payment is 7 million. The lower price of the subordinated instrument does not make it deliverable.

Actual eligibility can involve more conditions than seniority alone. Always distinguish the book’s simplified cheapest eligible obligation example from the full deliverability terms of a real contract.

5.3 Premium and protection legs are both survival sensitive

The buyer pays premiums only while the contract remains outstanding, subject to accrued premium rules at default. The seller’s protection payment occurs only if there is a covered event.

Let $Q_i$ be survival probability to payment date $i$, $p_i$ the probability of default during its interval, and ${\alpha }_i$ its accrual fraction. A simplified risky premium annuity is:

The simplified protection leg is:

For a zero upfront fair spread $s$:

Actual pricing also handles accrued premium on default and more precise event timing. The simplified equations are useful because they reveal why default probabilities affect both legs, not only the protection payment.

A higher hazard or lower recovery generally increases the fair CDS spread. Raising hazard also shortens the expected life of premium payments, which is why treating the premium stream as a default free annuity is incorrect.

5.4 Fixed coupon and fair spread are different concepts

The book uses standardized running coupons of 100 basis points for investment grade examples and 500 basis points for high yield examples. The current fair spread need not equal that contractual coupon.

An upfront payment compensates for the difference. Define $U$ as positive when the protection buyer pays it to the seller.

Using the same risky annuity:

If the fair spread exceeds the fixed coupon, the buyer pays upfront because future coupons alone are insufficient. If the fixed coupon exceeds the fair spread, the seller pays the buyer upfront because the buyer is committing to overly large future premiums.

Worked example. For notional 10 million, fair spread 180 basis points, fixed coupon 100 basis points, and risky annuity 4.2:

The protection buyer pays 336,000 upfront. Under the book’s price convention of 100 less the upfront percentage, the associated quote is 96.64; it is not the price of purchasing the reference bond.

The fixed coupon changes the upfront amount required for a fair contract. It does not by itself change the reference entity’s underlying economic credit risk.

5.5 Spread widening creates value for an existing protection buyer

A buyer who obtained protection before a spread widening holds a contract whose terms are favorable relative to newly available protection. Its value can increase even if no default occurs.

For a small spread change:

The book expresses this approximation using CDS duration. The relevant sensitivity belongs to the CDS premium stream, not to the reference bond’s interest rate duration.

For notional 10 million, a 60 basis point spread widening, and remaining risky annuity approximated as 4.0, the buyer’s spread driven gain is approximately 240,000. This is not a complete realized profit calculation because coupons, passage of time, changes in annuity, discount rates, recovery assumptions, and transaction costs may also matter.

An offsetting contract can monetize or reduce the economic exposure. It does not necessarily legally terminate the original contract unless the parties arrange a termination, novation, or other appropriate closeout.

5.6 Index CDS scale the exposure across reference entities

A conventional index CDS covers a specified set of entities with specified weights. For an equally weighted index, the notional attributable to one name is total notional divided by the number of names.

Worked example. An equally weighted index has 100 names and notional 50 million. If one constituent defaults with a settlement recovery of 35%, its allocated notional is 500,000 and the protection payment is 325,000.

After that default is settled, the surviving reference notional is 49.5 million under this simplified equal weight setup. The rest of the index continues, rather than the whole contract paying its full notional and terminating.

Analytical qualification on correlation. The text associates higher constituent default correlation with a higher index spread. For an untranched index with additive payoffs, fixed marginal default curves, fixed recovery assumptions, and a common discounting setup, expected protection payments are the sum of constituent expected losses and do not mechanically rise merely because default correlation rises.

Correlation remains central to portfolio tail risk and to the allocation of losses across CDO tranches. In actual markets, correlation, marginal risk, liquidity, and risk premiums can change together, so their observed association should not be confused with a universal pricing identity.

5.7 Use CDS to express a specific credit view

For a curve trade, equal notionals do not imply equal spread exposure. Approximate level neutrality requires matching sensitivity, for example $N_{short}{A}_{short}\approx N_{long}{A}_{long}$.

Such a trade is not necessarily neutral to default itself. Matching spread sensitivity and matching the payment after a sudden default are different objectives.

5.8 Basis trades connect cash bonds with credit derivatives

Define the CDS bond basis as:

When the bond spread is 250 basis points and comparable CDS protection costs 180 basis points, the basis is −70 basis points. Buying the bond and buying protection appears to retain 70 basis points of gross spread before other costs and mismatches.

On 10 million of matched principal, that is 70,000 of annualized gross spread arithmetic. It is not a guaranteed profit or an all in return estimate.

The bond and CDS must be compared using consistent benchmarks, maturity, seniority, recovery, and cash flow assumptions. Funding cost, margin, transaction costs, liquidity, deliverability, and changes in the basis can prevent the position from behaving like textbook arbitrage.

Broader applications. A debt financed buyout may increase the stock price while worsening creditor protection, making a combined equity position and CDS protection economically coherent. Synthetic CDOs use CDS rather than cash bonds to obtain credit exposure, but differences from cash structures must be analyzed before treating a price disparity as arbitrage.

6. Integrated Analysis: Connect the Risks Before Choosing the Instrument

6.1 A corporate bond is not a single interest rate position

For a plain bond, a first order decomposition is:

For an option bearing bond, add the effects of changing volatility, exercise behavior, and nonlinear price sensitivity. For an actual holding period return, also include income, reinvestment, passage of time, and default payments.

This explains why government yields can decline while corporate bond prices also fall. A sufficiently large widening of credit spreads can overwhelm the gain from lower benchmark rates.

6.2 Scenario matrix

These are conditional scenarios, not unconditional forecasts. The analysis should specify what is held constant and what is allowed to change.

6.3 Three distinctions unify the whole topic

Pricing versus forecasting. A forward rate, implied hazard rate, or calibrated tree tells us what is consistent with prices under assumptions. It does not, by itself, establish the physical probability distribution of future outcomes.

Promised payments versus contingent payments. YTM values a promised schedule, while an option or default model asks which payments survive in each state. Much of Level II consists of identifying when the promised schedule is no longer sufficient.

Risk exposure versus compensation. Lower default risk is not automatically a better investment, and a wider spread is not automatically a bargain. Relative value requires comparing the compensation offered with the full set of risks and contractual differences.

7. Exam Application and Review

7.1 A disciplined sequence for valuation questions

1. Identify the claim. Determine coupon, principal, maturity, options, seniority, and whether cash flows depend on history.
2. Fix conventions. State compounding, payment frequency, rate units, default timing, coupon timing, and recovery basis.
3. Choose the benchmark. Distinguish government, swap, spot, par, and forward inputs.
4. Choose the valuation engine. Use discount factors for fixed payments, a calibrated tree for state dependent exercise, and an appropriate path model for history dependent cash flows.
5. Add credit consistently. Use a spread approach or a default and recovery approach without casually double counting the same compensation.
6. Compute the requested output. Price, option value, OAS, duration, CVA, expected return, and CDS upfront are different quantities.
7. Check the economic direction. Confirm who benefits from exercise, wider spreads, lower recovery, and higher volatility.

7.2 High value error checks

7.3 Active recall questions