Market-Implied Credit Risk

A higher yield on corporate bonds indicates the presence of default risk.

If we assume that the spread represents the product of default probability and loss given default:

$$s \approx \lambda (1 – R)$$then:$$\lambda \approx \frac{s}{1 – R}$$where:

• $s$ — credit spread over the risk-free rate $$s = Y_{\text{corporate}} – Y_{\text{risk-free}}$$
• $R$ — recovery rate in the event of default
• $\lambda$ — conditional default probability (hazard rate / default intensity)

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For example, if:$$s = 2\%, \quad R = 40\%$$

then:$$\lambda = \frac{2\%}{60\%} = 3.33\%$$

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A More Exact Calculation

Now let’s look at a more detailed approach.

Assume we have a bond where the expected loss, implied by market prices, is:

$$8.75 = 104.09 – 95.34$$

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Building the Expected Loss Table

We construct a table of expected losses across time.

The logic is as follows:

• Default can occur at discrete time intervals (first column)
• Q is the unconditional probability of default per year
• Recovery is $40 for a $100 bond
• The risk-free value column represents the present value of the remaining bond cash flows, discounted at the risk-free rate
• The final columns calculate the present value of expected losses, also discounted at the risk-free rate

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Solving for Default Probability

Summing the table gives:$$288.48 Q$$

Setting this equal to the observed expected loss:

$$288.48 Q = 8.75$$$$Q = 3.03\%$$

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Key Result

• Approximate method → 3.33% (conditional hazard rate)
• Exact method → 3.03% (discrete unconditional probability)

These are consistent, but the second is more precise.

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Historical vs Market-Implied Default Risk

When we compare historical default probabilities with market-implied ones, the difference is significant.

The ratio column in the table shows that:

Market-implied default risk is several times higher than historical default risk.

This requires explanation.

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Why the Difference?

Because in our calculation, we discounted expected losses using the risk-free rate, we obtained not the real-world (physical) probability, but a risk-neutral probability.

To understand this, consider the following.

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Decomposition of Bond Spreads

The table shows how much of the bond spread (in basis points) exceeds Treasury yields.

Here, the risk-free rate is defined as:$$\text{Risk-free rate} \approx \text{Swap rate} – 10 \text{bps}$$Risk-free rate≈Swap rate−10bps

This adjustment is necessary because Treasury yields include:

• liquidity premium
• safe-haven demand

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What Explains Excess Return?

The remaining spread (excess return) is compensation for:

• Liquidity risk
• Systematic risk
• Default contagion (diversification is difficult)

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Key Insight

If these risks were incorporated directly into the discount rate, the excess return would disappear.

Instead, Hull’s framework keeps the discount rate risk-free and reflects these risks through higher implied default probabilities.

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Which Default Probability Should We Use?

• For derivatives pricing → use risk-neutral default probability
• For credit analysis / lending decisions → use real-world (physical) default probability

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Excel File

Market Implied Credit Risk Model

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Source

Options, Futures, and Other Derivatives — John C. Hull