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Use the Merton model to obtain default probability and distance to default, and set out where the model falls short.

FRM® Part II · Credit Risk Measurement and Management · Introduction to Credit Risk Modeling and Assessment · FRM2.CR.5.e

The idea

Merton views the firm as financed by equity and one zero-coupon debt of face value D due at T. Equity is a European call on the firm's assets V with strike D, so default occurs if V is below D at T. Let σ be the asset volatility and r the risk-free rate. The standardised term is d2 = (ln(V ÷ D) + (r − σ²/2) × T) ÷ (σ × √T). The risk-neutral default probability is then N(−d2). Distance to default is the number of standard deviations the log asset value sits above the default point. In the real-world version r is replaced by the asset's expected return μ. Risky debt equals V minus the call, which gives the credit spread. Higher asset volatility raises equity value but lowers distance to default. Shortcomings are default only at maturity, one zero-coupon liability, constant volatility, normal returns, and asset value and volatility that are unobservable and inferred from equity. Do not confuse N(−d2) with N(−d1).

Formulas

Merton d₂ for a firm with one zero-coupon debt

d₂ = [ln(V ÷ D) + (r − σ² ÷ 2) × T] ÷ (σ × √T)

V
the market value of the firm's assets today
D
the face value of the single zero-coupon debt
r
the continuously compounded risk-free rate
σ
the volatility of the asset value
T
the time to the debt's maturity, in years

The risk-neutral version, with the risk-free rate as the drift. Dropping the σ² ÷ 2 term, or forgetting √T in the denominator, are the usual slips.

Risk-neutral probability of default in the Merton model

PD = N(−d₂)

N
the standard normal cumulative distribution function
d₂
the Merton d₂, using the risk-free rate in the drift

Default means assets end below the debt's face value at maturity. N(d₂) is the probability of no default, and N(−d₁) is not the PD.

Real-world distance to default

DD = [ln(V ÷ D) + (μ − σ² ÷ 2) × T] ÷ (σ × √T)

V
the market value of the firm's assets today
D
the default point, the face value of the debt
μ
the expected return on the assets, continuously compounded
σ
the volatility of the asset value
T
the horizon, in years

Counts the standard deviations between log asset value and the default point. Using r instead of μ gives the risk-neutral distance, which is smaller.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the Merton model, the equity of a firm financed by a single zero-coupon bond is best described as:

  1. A European put option on the firm's assets with a strike equal to the face value of the debtWhy not: Reverses the option type; it is the debt holders who are short a put on the assets.
  2. A European call option on the debt with a strike equal to the asset valueWhy not: Swaps the underlying and the strike; the underlying is the firm's assets and the strike is the debt face value.
  3. A risk-free zero-coupon bond plus a long position in the assetsWhy not: Mixes up the payoffs; shareholders receive the residual above the debt face value, which is option-like.
  4. A European call option on the firm's assets with a strike equal to the face value of the debt - correct

Shareholders receive max(V - D, 0) at maturity, the payoff of a call on assets V with strike D. Debt holders hold risk-free debt less a put on the assets.

A harder one

Holding all other Merton inputs constant, an increase in the volatility of the firm's assets will:

  1. Reduce both the distance to default and the value of equityWhy not: Misses that equity is a call option, whose value rises with the volatility of the underlying.
  2. Reduce the distance to default and increase the value of equity - correct
  3. Increase both the distance to default and the value of equityWhy not: A higher volatility widens the distribution of asset values, so the default point is closer in standard-deviation terms.
  4. Increase the distance to default and reduce the value of equityWhy not: Gets both effects backwards: higher volatility raises default probability and raises option value.

Higher volatility raises the chance assets end below the debt, so distance to default falls and PD rises. The call option value (equity) rises with volatility; debt holders, who are short the put, lose.

The hardest tier

A firm has asset value USD 150 million, asset volatility 20 percent and zero-coupon debt of USD 120 million due in two years. The risk-free rate is 3 percent and the expected return on assets is 8 percent (both annual, continuously compounded). The real-world distance to default, in standard deviations, is closest to:

  1. 0.86Why not: Uses the risk-free rate in place of the expected asset return, giving the risk-neutral distance.
  2. 1.35Why not: Omits the sigma² / 2 correction to the drift.
  3. 1.21 - correct
  4. 1.72Why not: Divides by sigma instead of sigma x sqrt(T), ignoring the two-year horizon in the standard deviation.

DD = [ln(150/120) + (0.08 - 0.02) x 2] / (0.20 x sqrt 2) = (0.2231 + 0.12) / 0.2828 = 1.21.

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