Economic Modelling · Simple models for credit risk

Credit Spreads and Pricing Credit Risky Bonds

Updated 11 October 2026 · Fact-checked

A credit spread is the extra yield a defaultable bond pays over a risk-free bond of the same term. In simple models, spread ≈ λ(1 − R), where λ is the annual default probability and R the recovery rate. Observed spreads are usually larger because they also include liquidity and risk premiums.

Understand Credit Spreads and Pricing Credit Risky Bonds

A government bond is treated as risk-free. A corporate bond can default, so investors want a higher yield. The gap between the two yields is the credit spread. It is the price you pay, in yield terms, for taking credit risk.

The spread must cover the expected loss. Expected loss each year is the chance of default times the loss if default happens. The loss given default is (1 − R), where R is the recovery rate, the fraction of face value you get back after default. So expected annual loss ≈ λ(1 − R), with λ the annual default probability (or hazard rate). This is the core link between spread, default and recovery.

For a zero-coupon bond with face value 1 and maturity t, the risky price is the discounted expected payoff. If default can happen at any time with constant hazard λ and recovery is a fraction R of face value paid at maturity, then the price is approximately e^(−rt) × [e^(−λt) + R(1 − e^(−λt))]. If R = 0, this is e^(−(r+λ)t), so the risky yield is r + λ and the spread is exactly λ.

In practice, observed spreads are higher than spreads implied by historical default losses. This is the credit spread puzzle. Two main reasons are given. First, liquidity: corporate bonds trade less often and with wider dealer margins than government bonds, so buyers want compensation. Second, a risk premium: defaults cluster in bad economic times, and investors dislike losses then. They demand extra return for bearing this systematic risk and for uncertainty in the default rate. Tax and other factors can also play a part.

A useful split is: observed spread = expected default loss + risk premium for default risk + liquidity premium (+ other factors). Pricing with real-world default rates alone will therefore value a bond too high. Pricing with risk-neutral default probabilities, backed out of market prices, absorbs the premiums. These risk-neutral probabilities are higher than real-world ones.

Key rules to remember

Credit spread (definition): s = y − r; y is the yield on the risky bond and r is the yield on the risk-free bond of the same term and cash flows.
Approximate spread from default and recovery: s ≈ λ(1 − R); λ is the annual default probability or hazard rate, R the recovery rate. It is an approximation that works best for small λ. It gives the expected-loss part only.
Risky zero-coupon price, recovery of face value at maturity: P = e^(−rt) × [R + (1 − R) e^(−λt)]; Constant hazard rate λ, continuous compounding. It assumes recovery is a fraction R of face value paid at maturity. Equivalent to e^(−rt) × [e^(−λt) + R(1 − e^(−λt))].
Zero recovery case: P = e^(−(r + λ)t), so s = λ; Spread equals the hazard rate exactly when R = 0 and compounding is continuous.
Spread from prices (continuous compounding): s = −(1/t) × ln(P_risky ÷ P_riskfree); Use when you have risky and risk-free zero-coupon prices for the same maturity.
Spread decomposition: Observed spread = expected loss + risk premium + liquidity premium (+ other); Used to explain why observed spreads exceed spreads implied by historical defaults.
CDS premium, rough approximation: CDS spread ≈ λ(1 − R); Under risk-neutral λ. A fair CDS premium equals the present value of expected protection payments divided by the risky annuity of premiums.

How to solve Credit Spreads and Pricing Credit Risky Bonds questions

Use this order for any question on spreads and risky bond pricing. It keeps the compounding basis and the probability measure clear.

1Identify what is given and what is asked: prices, yields, λ, R, or the spread.
2Note the compounding basis (continuous or annual effective) and whether the question gives real-world or risk-neutral default probabilities.
3Write the risk-free price or yield first, using the same term and cash flows.
4Write the risky price or yield. Use the pricing formula if you have λ and R, or the yield if given.
5Compute the spread as y − r, or use s = −(1/t) ln(P_risky ÷ P_riskfree) for zero-coupon bonds.
6If asked for default probability, rearrange s ≈ λ(1 − R) or the exact price formula to solve for λ.
7If asked to explain a gap, split the spread into expected loss, risk premium and liquidity premium, and say which is residual.
8State assumptions and check that the answer is sensible: spread positive, λ between 0 and 1, price below the risk-free price.

Quickest way: Spread, λ and R shortcut

When to use it: Use in multiple-choice questions or to check a long calculation, when the default rate is small and the question allows an approximation.

Take s = y − r in percentage points.
Apply λ ≈ s ÷ (1 − R) to get the risk-neutral hazard rate.
For zero-coupon prices, use s = −ln(P_risky ÷ P_riskfree) ÷ t.
If R = 0, spread equals λ with no division.
Check the answer: a higher R means a higher λ is needed for the same spread.

Common mistakes in Credit Spreads and Pricing Credit Risky Bonds

Treating the whole spread as default compensation.
Students read s ≈ λ(1 − R) as the full spread.
Fix: Say that this is the expected-loss part only. Observed spread also contains risk and liquidity premiums.
Using s = λ when recovery is not zero.
The zero-recovery case is the easiest to remember.
Fix: Always divide by (1 − R) when finding λ from s: λ ≈ s ÷ (1 − R).
Mixing compounding bases, such as subtracting an annual effective rate from a continuously compounded rate.
Yields are quoted in different forms across questions.
Fix: Convert both yields to the same basis before subtracting.
Using the real-world default probability to price the bond and expecting it to match market prices.
Students forget that investors demand a premium for default risk.
Fix: Explain that market prices imply risk-neutral probabilities, which are higher than real-world ones.
Calling the recovery rate the loss rate.
The two terms are close in the wording.
Fix: Loss given default = 1 − R. Write it out before using it.
Attributing the credit spread puzzle to liquidity alone.
Liquidity is the easiest reason to recall.
Fix: Give at least both: liquidity premium and a risk premium for systematic or uncertain default risk. Mention tax or other factors if the question asks for more.

Worked examples

Example 1

A one-year zero-coupon corporate bond trades at ₹92 per ₹100 face value. A one-year government zero-coupon bond trades at ₹95 per ₹100. Using continuous compounding, find the credit spread. Then estimate the risk-neutral annual default probability if the recovery rate is 40%.

Show the solution

Risk-free yield r = ln(100 ÷ 95) = ln(1.052632) = 0.051293, or 5.129%.
Risky yield y = ln(100 ÷ 92) = ln(1.086957) = 0.083382, or 8.338%.
Spread s = 0.083382 − 0.051293 = 0.032089, or about 3.21%.
Check with s = −ln(92 ÷ 95) = ln(1.032609) = 0.032089. This matches.
Loss given default = 1 − 0.40 = 0.60.
λ ≈ s ÷ (1 − R) = 0.032089 ÷ 0.60 = 0.05348, or about 5.35%.

Answer: The credit spread is about 3.21% a year. The implied risk-neutral default probability is about 5.35% a year.

Example 2

A two-year zero-coupon bond has face value ₹1,000. The continuously compounded risk-free rate is 6% a year. The hazard rate is 2% a year and recovery is zero. (a) Find the price. (b) Find the spread. (c) The historical default rate is 2% but the market spread is 3.5%. Give two reasons for the difference.

Show the solution

(a) With R = 0, P = 1,000 × e^(−(0.06 + 0.02) × 2) = 1,000 × e^(−0.16).
e^(−0.16) = 0.852144, so P = ₹852.14.
(b) The risky yield is r + λ = 8%, so the spread is 2% a year, which equals λ.
(c) The market spread exceeds expected loss by 1.5 percentage points.
Reason 1: a liquidity premium, because corporate bonds are harder to trade than government bonds.
Reason 2: a risk premium, because defaults cluster in bad times and the default rate is uncertain, so investors demand extra return.

Answer: (a) ₹852.14. (b) 2% a year. (c) The extra 1.5% reflects a liquidity premium and a risk premium for systematic and uncertain default risk.

Exam tips

Always state the compounding basis and whether λ is real-world or risk-neutral. Examiners give marks for stated assumptions.
In discussion questions, split the spread into expected loss, risk premium and liquidity premium, then explain each in a sentence.
Show s ≈ λ(1 − R) as an approximation, and give the exact price formula if the numbers are not small.
In Paper B or R-based work, set up risk-free and risky prices as vectors, compute spreads with log ratios, and label the units.

Practice questions from Simple models for credit risk

Credit Spreads and Pricing Credit Risky Bonds in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Credit Spreads and Pricing Credit Risky Bonds: frequently asked questions

How do I calculate credit spread from bond prices?

Find the yield of the risky bond and the yield of the risk-free bond of the same term and cash flows, then subtract. For zero-coupon bonds with continuous compounding, s = −(1/t) ln(P_risky ÷ P_riskfree).

How is credit spread linked to default probability?

Approximately, spread = λ(1 − R), where λ is the annual default probability and R the recovery rate. Rearranged, λ ≈ s ÷ (1 − R). The λ you get from market prices is risk-neutral, so it is higher than the historical default rate.

Why are observed credit spreads higher than expected default losses?

Spreads include more than expected loss. They include a liquidity premium for less tradable bonds and a risk premium because defaults rise in bad economic times and are uncertain. This gap is called the credit spread puzzle.

What is the basic idea of credit default swap pricing?

A CDS pays the buyer the loss on default in return for regular premiums. The fair premium makes the present value of expected protection payments equal to the present value of premiums. Roughly, the CDS spread is about λ(1 − R) under risk-neutral probabilities.