4. Measurement of Probability of Default from Bond Prices
4. Measurement of Probability of Default from Bond Prices
4.1 Risk Neutral Probability of Default
Payoff =100*RR
Payoff=100
P = 100 1 + YTM = ( 100 1 + R f ) × ( 1 − PD ) + ( 100 × RR 1 + R f ) × PD P=\cfrac{100}{1+\text{YTM}}=\left(\cfrac{100}{1+R_f}\right)\times(1-\text{PD})+\left(\cfrac{100\times \text{RR}}{1+R_f}\right)\times \text{PD} P=1+YTM100=(1+Rf100)×(1−PD)+(1+Rf100×RR)×PD
→ P D = 1 1 − R R ( 1 − 1 + R f 1 + YTM ) = 1 LGD ( YTM − R f 1 + YTM ) \to PD=\cfrac{1}{1-RR}\left(1-\cfrac{1+R_f}{1+\text{YTM}}\right)=\cfrac{1}{\text{LGD}}\left(\cfrac{\text{YTM}-R_f}{1+\text{YTM}}\right) →PD=1−RR1(1−1+YTM1+Rf)=LGD1(1+YTMYTM−Rf)
→ YTM − R f ≈ P D × L G D → YTM ≈ R f + P D × L G D \to \text{YTM}-R_f\approx PD\times LGD\to \text{YTM}\approx R_f+PD\times LGD →YTM−Rf≈PD×LGD→YTM≈Rf+PD×LGD
- YTM \text{YTM} YTM: market determined yield
- PD \text{PD} PD: risk neutral probability of defualt
- RR \text{RR} RR: recovery rate
- LGD \text{LGD} LGD: loss given default
4.2 Objective Probability of Default
Payoff =100*RR
Payoff=100
P ∗ = 100 1 + YTM = ( 100 1 + R f + RP ) × ( 1 − PD ∗ ) + ( 100 × RR 1 + R f + RP ) × PD ∗ P^*=\cfrac{100}{1+\text{YTM}}=\left(\cfrac{100}{1+R_f+\text{RP}}\right)\times(1-\text{PD}^*)+\left(\cfrac{100\times \text{RR}}{1+R_f+\text{RP}}\right)\times \text{PD}^* P∗=1+YTM100=(1+Rf+RP100)×(1−PD∗)+(1+Rf+RP100×RR)×PD∗
→ YTM − R f − RP = PD ∗ ( 1 − RR ) \to \text{YTM}-R_f-\text{RP}=\text{PD}^*(1-\text{RR}) →YTM−Rf−RP=PD∗(1−RR)
- PD ∗ \text{PD}^* PD∗ is the real-world probability of default
- RP \text{RP} RP is the risk premium (liquidity premium and tax effects)
4.3 Spread Risk
4.3.1 Spread Measure
| Spread Measure | Definition |
|---|---|
| Yield spread | YTM risky bond - YTM benchmark government bond |
| I-spread | YTM risky bond - linearly interpolated YTM on benchmark government bond or swap rate |
| Z-spread | Basis points added to each spot rate on a benchmark curve |
| Asset-swap spread | Spread on floating leg of asset swap on a bond |
| CDS spread | Market premium of CDS of issuer bond |
| Option adjusted spread(OAS) | Z-spread adjusted for optionality of embedded options |
| Discount margin | Fixed spread above current LIBOR needed to price bond correctly |
4.3.2 Influencing factors of Credit spread
Credit spreads are reduced during times of economic recovery. This phenomenon is called flight to quality.
Economy ↑ → \uparrow\;\to ↑→ Buy risky bonds ↑ → \uparrow\;\to ↑→ Risky bond value ↑ → \uparrow\;\to ↑→ Risky bond yield ↓ → \downarrow\;\to ↓→ Credit spread ↓ \downarrow ↓
The credit spread of callable bonds (redeemable bond) widens (narrows) as volatility of interest rate increases(decreases).
σ interest \sigma_{\text{interest}} σinterest ↑ → \uparrow\;\to ↑→ call option value ↑ → \uparrow\;\to ↑→ callable bond value ↓ → \downarrow \; \to ↓→ Credit spread ↑ \uparrow ↑
The credit spread of puttable bonds narrows (widens) as volatility of interestrate increases (decreases).
4.3.3 Spread’ 01 (DVSC)
The spread’ 01 is analogous to the D V 01 DV01 DV01. It measures the price change implied by a one basis point change in the credit spread.
The smaller the z-spread, the larger the effect on the bond price (i.e., the greater the credit spread sensitivity). The DVCS exhibits convexity.
4.3.4 The Credit Spread Curve
The first step to creating the curve is to plot the most liquid credit spreads observable in the market, generally from CDS premiums or bond spreads.
Plotting the curve is further complicated by the choice of reference.
An alternative method uses the credit spread around a single, liquid observation (e.g., credit spread with five years to maturity) to map the entire curve.