Credit Spread and Default Risk: Method

Credit Spread and Default Risk: How It Works

What it does

A bond that might default has to pay more than one that will not. The page takes that extra yield, the credit spread, and asks what chance of default would exactly justify it. It answers in three steps of rising accuracy, draws the implied survival curve to maturity and shows how far the answer moves with the assumed recovery rate. Advanced mode adds the Merton model, which reads default risk from the share price, the price of a coupon bond at a given hazard rate (or the hazard rate behind a price), and two accounting ratios.

The method

Symbols. y_b is the bond's yield and y_f the risk-free yield of the same maturity, both effective yearly rates used as decimals (6.5 % is 0.065); T the maturity in years; R_c the recovery rate, the share of the promised amount received after a default; L_gd = 1 − R_c the loss given default.

Spread. s_c = y_b − y_f. One basis point (bp) is 0.01 of a percentage point, so 0.025 is 250 bp.

Rule of thumb. The hazard rate h is the chance of default per year for a bond that has survived so far. Each year the holder expects to lose about h L_gd, and the spread pays for it: s_c ≈ h L_gd, so h_a = s_c / L_gd. Gibson (2004) uses exactly this pairing: a 60 bp spread with 40 % recovery and a hazard rate of 1 % a year.

Exact for one year. One unit in a one-year bond is promised 1 + y_b; with probability p₁ it pays R_c (1 + y_b) instead. Setting the expected payoff equal to the risk-free payoff:

1 + y_f = (1 + y_b) (1 − p₁ L_gd),   so   p₁ = (y_b − y_f) / [(1 + y_b) L_gd]

To maturity. For a bond paying everything at T, with recovery paid at maturity, the same argument over T years gives the cumulative default probability Q_T:

[(1 + y_f) / (1 + y_b)]^T = 1 − Q_T L_gd,   so   Q_T = {1 − [(1 + y_f) / (1 + y_b)]^T} / L_gd

With a constant hazard rate the chance of surviving to time t is S(t) = exp(−h t). Matching S(T) = 1 − Q_T gives h_c = −ln(1 − Q_T) / T, and the default probability within one year is p_a = 1 − exp(−h_c). At T = 1, Q_T equals p₁.

Break-even spread. Turning the last relation around, the spread that just pays for a hazard rate h at recovery R is s = (1 + y_f) / [1 − (1 − R)(1 − exp(−h T))]^1/T − 1 − y_f. The table uses the hazard rate found for the entered recovery rate.

Merton model (Advanced). The company's assets are worth V_A with yearly volatility σ_A; the debt is one payment D_f due in T_d years. Shareholders receive whatever the assets are worth above D_f, so the shares are a call option on the assets (Merton 1974). With N( ) the standard normal distribution function and r_c = ln(1 + y_f):

E₀ = V_A N(d₁) − D_f exp(−r_c T_d) N(d₂),   σ_E E₀ = N(d₁) σ_A V_A

d₁ = [ln(V_A / D_f) + (r_c + σ_A² / 2) T_d] / (σ_A √T_d),   d₂ = d₁ − σ_A √T_d

The share value E₀ and share volatility σ_E are entered; the two equations are solved for V_A and σ_A by Newton's method, and the iterations are tabulated. Then d₂ is the distance to default, N(−d₂) the risk-neutral probability of default at T_d, B_d = V_A − E₀ the value of the debt, y_D = −ln(B_d / D_f) / T_d its yield and s_m = y_D − r_c the implied spread.

Risky coupon bond (Advanced). Coupons C = F c / m are paid at t_k = k / m. Each is received with probability S(t_k) = exp(−h t_k); a default in period k, which has probability S(t_k−1) − S(t_k), pays R_c F at the end of that period; the face value is repaid with probability S(T). With DF(t) = (1 + y_f)^−t:

P = Σ_k [C S(t_k) + R_c F (S(t_k−1) − S(t_k))] DF(t_k) + F S(T) DF(T)

Given a price instead, the hazard rate is found by Newton's method inside a bracket.

Coverage and leverage (Advanced). Interest coverage = EBITDA / interest expense; leverage = debt / EBITDA.

What it assumes

Risk-neutral pricing: the whole spread is treated as pay for expected default loss. One recovery rate, known in advance and unrelated to when default happens. The same hazard rate every year. Yields compounded once a year. In Steps 3 and 4 a bond that pays everything at maturity, with recovery on the promised amount paid at maturity; in the coupon-bond section recovery on the face value, paid at the end of the coupon period of default, on a coupon date. In the Merton model: one zero-coupon debt payment, default possible only at its maturity, assets that follow a lognormal path with constant volatility, and a share volatility that describes the future.

When it misleads

It is not a forecast of defaults. Part of every spread is a premium for bearing risk, and part reflects how hard the bond is to sell and how it is taxed. The implied probability therefore overstates the frequency at which similar bonds have defaulted. Amato and Remolona (2005) report an average spread of about 182 bp on Baa-rated U.S. corporate bonds from 1997 to mid-2004 against an expected default loss of about 40 bp a year.

The recovery rate drives the answer. It is a guess before the event; the table across recovery rates shows the range. For bonds trading far below face value the choice of what is recovered (face value, promised amount or market value) changes the result as well.

The benchmark matters. The spread should be measured against a risk-free yield of the same maturity and currency. A callable bond, or one with other options, carries yield that has nothing to do with default.

The hazard rate is not constant. Markets usually price different default risk at different horizons; a single rate fitted to one bond is an average over its life.

The Merton model is a simplification. Real companies have many debts with different dates and can default before maturity, and share volatility changes. The model's spread and the market's often differ, as the page shows when both are entered.

A hazard rate may not exist for a price. A price above the default-free value, or at or below what the recovery alone is worth, cannot be explained by any hazard rate; the page says so. When the recovery amount is worth more than the bond's default-free value the price does not identify a hazard rate.

Worked example

The default case: a bond yielding 6.5 % against a risk-free 4.0 %, five years to maturity, recovery 40 %.

Spread: 0.065 − 0.040 = 0.025, or 250 bp. Loss given default: 1 − 0.40 = 0.60.

Rule of thumb: 0.025 / 0.60 = 0.0417, about 4.17 % a year.

One year, exact: 0.025 / (1.065 × 0.60) = 0.0391, or 3.91 %.

To maturity: (1.04 / 1.065)⁵ = 0.8880, so Q_T = (1 − 0.8880) / 0.60 = 0.1866: an 18.7 % chance of default within five years and 81.3 % of survival. Hazard rate: −ln(0.8134) / 5 = 0.0413. Within one year: 1 − exp(−0.0413) = 0.0405, or 4.05 %.

Break-even at 20 % recovery for the same hazard rate: 1 − 0.80 × 0.1866 = 0.8507; 1.04 / 0.8507^1/5 − 1.04 = 0.0342, or 342 bp.

The page's results carry full precision; the hand figures above are rounded, so a last digit can differ.

A calculator for education and analysis, not investment advice.