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Credit spreads compensate more than expected default

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Start with the idea

Credit risk combines how often default occurs, how much is lost if it occurs and how much exposure exists at that moment. Market spreads include more than this average loss: compensation for uncertainty, liquidity and funding can be substantial.

Symbols, units & horizon

EL: expected currency credit loss over the stated horizon
PD: default probability over that horizon
LGD: fraction lost conditional on default
EAD: currency exposure at default
s: annual decimal credit spread in a stylised model
λ: constant annual hazard intensity, not a regression penalty
ℛ: recovery fraction
≈: approximation excluding liquidity and risk premia

When and why to use this

Use expected loss in scenario budgets and the spread approximation as a diagnostic for what assumptions are embedded in a quote.

E L = P D \times L G D \times E A D

Algebra and arithmetic

Take the expectation of default loss

Under fixed LGD and EAD, loss is $1_{d e f a u l t} L G D E A D$ . The expectation of the indicator is PD.
Therefore EL=PD×LGD×EAD. More generally average the conditional exposure and severity jointly rather than multiplying unconditional estimates.

Work it by hand

PD=.02,LGD=.60,EAD=$5m → EL=$60,000 over the stated horizon.

Expected loss over a defined horizon combines default probability, loss given default, and exposure at default. It does not describe the tail distribution or loss correlation. Recovery can fall at the same time defaults cluster, which is precisely when a constant-recovery model is weakest.

s \approx λ (1 - ℛ)

Small-time approximation

Balance a stylised spread with hazard loss

Over a small interval dt, default probability is approximately λdt. Loss per exposure is approximately λ(1−ℛ)dt.
Equating this to spread income sdt and cancelling dt gives $s \approx λ (1 - ℛ)$ . Invert with $λ \approx \frac{s}{(1 - ℛ)}$ .

Work it by hand

s=.012 and recovery=.4 imply hazard≈.02/year under this stripped-down model. This is not a direct estimate of physical default probability.

This simplified continuous-time spread relation uses constant hazard λ and recovery fraction ℛ. It omits liquidity, risk premia, contract features, and technical demand. Observed spreads should not be inverted into literal physical default probabilities without addressing those components.

Decompose bond risk into rate duration, spread duration, default exposure, optionality, and liquidity.
For a credit hedge, test basis risk: the cash bond and derivative can diverge because of funding or deliverability.
Scenario-test downgrades, correlated defaults, recovery deterioration, and frozen refinancing markets.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import exp

def expected_credit_loss(default_probability, loss_given_default, exposure):
    if not 0 <= default_probability <= 1 or not 0 <= loss_given_default <= 1:
        raise ValueError("PD and LGD must be fractions in [0,1]")
    return default_probability*loss_given_default*exposure

def hazard_model(hazard, recovery, years):
    """Constant intensity; spread approximation is not a full bond price."""
    return 1-exp(-hazard*years), hazard*(1-recovery)

print(expected_credit_loss(.02,.6,1e6), hazard_model(.03,.4,1))

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