Capital Market Approaches for the Estimation of PD

Question 2

What is the formula for corporate bond spreads with simple compounding and a non-zero recovery rate?

Answer 2

(1+i) = (1-p) * (1+r) + p * k * (1+r)

Question 3

What is the formula for risk premium?

Answer 3

(r-i) = Φ = [(1+i)/(1 - p * (1-k))] - (1 + i)

Question 4

What is the formula for marginal and cumulative survival probability?

Answer 4

Marginal: ₀s₁ = 1 - ₀p₁
Cumulative: ₀s₂ = ₀s₁ * ₁s₂

Question 5

What are advantages for using corporate bond spreads for calculating PD?

Answer 5

1. Uses market data, hence objective
2. Forward looking, hence expected market default rates

Question 6

What are the weaknesses of using corporate bond spreads for calculating PD?

Answer 6

1. Assumption expectation theory is true: spreads reflect market expectations
2. Spreads may also reflect other factors: liquidity premia, transaction costs
3. Assumption of risk neutral investors
4. Only applicable to companies with bonds outstanding in the capital markets

Question 7

What is the formula for PD using corporate bond data with continuous compounding?

Answer 7

Continuous from yearly compounded: c = ln(1+y)
p = (1-e^-d_TT)/(1-k)

d is the spread

Question 8

What is the intuition behind merton’s model for estimating PD from share prices?

Answer 8

A company defaults when the value of its assets becomes lower than the value of its liabilities.

Question 9

What is the main formula behind the merton model for estimating PD?

Answer 9

P₀ = -N * (-d₁) * V₀ + Fe^{-r * T} * N(-d₂)

P = Payoff of long put option and loan at time 0
V= The value of company’s assets
F = Strike price equal to the debt repayment
T = Maturity of the loan and the option (equal)
P₀ + B₀ = Fe^{-r * T} (Market value of debt)

Question 10

What is the fomula for the probability of default from the Merton formula?

Answer 10

p = Pr(V_T < F) = 1 - N(d₂) = N(-d₂)

Question 11

What is the formula for calculating the spread from the merton model?

Answer 11

r-i = d = -1/T * ln[N(d₂) + (1/L) * N(-d₁)]

Where L is leverage

Question 12

What is the formula for d1 and d2 in the merton model?

Answer 12

d₁ = [ln(V/L) + (r+0.5 * σ²_V) * T]/[σ_V * sqrt(T)]
d₂ = d₁ - σ_V * sqrt(T)

Question 13

What are the advantages of the merton model?

Answer 13

1. It clearly identifies the key relevant variables that determin PD: (i) leverage, hence financial risk (ii) asset volatility (business risk)
2. It allows to estimate, through objective and clear criteria, the borrowing company PD and the risk premium a lender should demand for the credit risk it faces

Question 14

What are the disadvantages of the Merton model?

Answer 14

1. Just one debt (zero-coupon)
2. Only looks at default risk (no migration risk)
3. Some key variables cannot be empirically observed (asset value and volatility)
4. Constant risk free rate
5. Arbitrage-free logic, hence possbility to buy and sell the option’s underlying asset (company’s assets)
6. Liquidation costs are not accounted for
7. The normal distribution underestimates extreme events