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Use the Merton model to work out the value of a firm's debt and of its equity, together with the volatility of firm value and of firm equity.

FRM® Part II · Credit Risk Measurement and Management · Estimating Default Probabilities · FRM2.CR.9.l

The idea

The Merton model treats a firm's assets as following geometric Brownian motion, with debt as a single zero-coupon claim of face value D maturing at T. Equity holders hold a call option on the assets with strike D, so their value is: E0 = V0 × N(d1) − D × e^(−rT) × N(d2). The first input to the two normal terms, with asset volatility σ, is: d1 = (ln(V0 ÷ D) + (r + σ²÷2) × T) ÷ (σ × √T). The second input is the first, reduced by the asset volatility over the horizon: d2 = d1 − σ × √T. Debt is worth what is left, which equals the present value of D at the risk-free rate minus a put option on the assets: D0 = V0 − E0. Equity volatility follows from the equity as a function of the assets: σE × E0 = N(d1) × σV × V0. The usual errors are forgetting to discount D, omitting N(d1) in the volatility link, and using equity volatility as the asset volatility. Equity is more volatile than the assets because of leverage.

Formulas

Value of equity in the Merton model

E0 = V0 × N(d1) − D × e^(−r × T) × N(d2)

V0
the market value of the firm's assets today
N
the standard normal cumulative distribution function
d1
(ln(V0 ÷ D) + (r + σ² ÷ 2) × T) ÷ (σ × √T), with σ the asset volatility
D
the face value of the zero-coupon debt
r
the continuously compounded risk-free rate
T
the time to the debt's maturity, in years
d2
d1 − σ × √T

Black-Scholes for a call on the assets with strike D. Debt is then V0 − E0; the trap is leaving D undiscounted, which understates equity and overstates debt.

The first normal input of the Merton equity call

d1 = (ln(V0 ÷ D) + (r + σ² ÷ 2) × T) ÷ (σ × √T)

V0
the market value of the firm's assets today
D
the face value of the zero-coupon debt
r
the continuously compounded risk-free rate
σ
the volatility of the asset value, not of the equity
T
the time to the debt's maturity, in years

d2 = d1 − σ × √T. Use asset volatility here; plugging in equity volatility, which leverage inflates, is the classic slip.

Equity volatility from asset volatility in the Merton model

σE = N(d1) × σV × V0 ÷ E0

N(d1)
the delta of the equity call with respect to the asset value
σV
the volatility of the firm's assets
V0
the asset value today
E0
the equity value today

Holds instantaneously, since equity is a function of the assets. Leaving out N(d1) overstates equity volatility badly; solving it backwards gives σV = σE × E0 ÷ (N(d1) × V0).

A practice question

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

In the Merton model, how are a firm's equity and debt represented?

  1. Equity is a put option on the firm's assets with strike equal to the debt's face value.Why not: Equity holders gain when assets exceed debt, which is a call payoff; the put belongs to the debt.
  2. Equity is a risk-free bond and debt is a call option on the firm's assets.Why not: Confuses the claims; equity is the option holder.
  3. Equity and debt are both calls on the assets, with different strikes.Why not: Debt has a capped payoff and is not a call on the assets.
  4. Equity is a call option on the firm's assets with strike equal to the debt's face value; debt is the assets minus that call. - correct

At maturity equity pays max(V - D, 0), a call; debt pays min(V, D), the assets minus the call.

A harder one

A firm has asset value 120 and zero-coupon debt with a face value of 100 due in one year. The risk-free rate is 4% (continuous) and asset volatility is 20%. In the Merton model N(d1) = 0.8872 and N(d2) = 0.8441. What is the value of the debt?

  1. 96.08Why not: Gives the present value of the face value at the risk-free rate and ignores the default put.
  2. 94.64 - correct
  3. 97.95Why not: Does not discount the strike when computing the equity call.
  4. 25.36Why not: Gives the equity value instead of the debt value.

E0 = 120 x 0.8872 - 100 x exp(-0.04) x 0.8441 = 25.36; debt = 120 - 25.36 = 94.64.

The hardest tier

A firm has assets worth 140 and zero-coupon debt of face value 100 due in three years. The risk-free rate is 3% (continuous) and asset volatility is 25%. In the Merton model N(d1) = 0.8852 and N(d2) = 0.7789. What is the credit spread, the continuously compounded yield on the debt minus the risk-free rate?

  1. 1.54% - correct
  2. 4.54%Why not: Reports the yield on the debt and forgets to subtract the risk-free rate.
  3. 1.87%Why not: Uses a simple annual yield instead of the continuously compounded yield.
  4. 1.25%Why not: Divides the percentage discount from face value by the horizon, instead of using the log of the price ratio.

E0 = 140 x 0.8852 - 100 x exp(-0.09) x 0.7789 = 52.74; debt = 87.26. Yield = -ln(87.26/100)/3 = 4.54%; spread = 1.54%.

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