Learning Module 14: Credit Risk
Fixed Income
Expected Loss
\[ \text{EL} = \text{POD} \times \text{LGD} \tag{1} \]
Where:
- LGD = \(\text{EE} \times (1 - \text{RR})\)
- POD: Probability of Default
- LGD: Loss given Default
- EE: Expected Exposure
- RR: Recovery Rate
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### Expected Loss
$$
\text{EL} = \text{POD} \times \text{LGD} \tag{1}
$$
Where:
- LGD = $\text{EE} \times (1 - \text{RR})$
- POD: Probability of Default
- LGD: Loss given Default
- EE: Expected Exposure
- RR: Recovery Rate Credit Spread
\[ \text{Credit Spread} \approx \text{POD} \times \text{LGD} \tag{2} \]
Where:
- POD: Probability of Default
- LGD: Loss given Default
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### Credit Spread
$$
\text{Credit Spread} \approx \text{POD} \times \text{LGD} \tag{2}
$$
Where:
- POD: Probability of Default
- LGD: Loss given Default Price Impact from Spread Changes
\[ \% \Delta PV^{Full} = -AnnModDur \times \Delta Spread \tag{3} \]
Where:
- \(AnnModDur\) is the annualized modified duration
- \(PV^{Full}\) is the bond’s full price
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### Price Impact from Spread Changes
$$
\% \Delta PV^{Full} = -AnnModDur \times \Delta Spread \tag{3}
$$
Where:
- $AnnModDur$ is the annualized modified duration
- $PV^{Full}$ is the bond’s full price Price Impact from Larger Spread Changes
\[ \% \Delta PV^{Full} = -(AnnModDur \times \Delta Spread) + \tfrac{1}{2} AnnConvexity \times (\Delta Spread)^2 \tag{4} \]
Where:
- \(AnnModDur\) is the annualized modified duration
- \(PV^{Full}\) is the bond’s full price
- \(AnnConvexity\) is annualized convexity
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### Price Impact from Larger Spread Changes
$$
\% \Delta PV^{Full} = -(AnnModDur \times \Delta Spread) + \tfrac{1}{2}
AnnConvexity \times (\Delta Spread)^2 \tag{4}
$$
Where:
- $AnnModDur$ is the annualized modified duration
- $PV^{Full}$ is the bond’s full price
- $AnnConvexity$ is annualized convexity Annualized ModDur
\[ Annualized\ ModDur \approx \frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Yield) \times (PV_{0})} \tag{5} \]
- Here we apply the methodology from an earlier lesson to approximate modified duration and convexity. We estimated the modified duration by increasing and decreasing the yield-to-maturity by the same amount \((\Delta Yield)\) to calculate corresponding bond prices \(PV_{+}\) and \(PV_{-}\) for a given initial price \((PV_{0})\), as shown in Equation 5:
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### Annualized ModDur
$$
Annualized\ ModDur \approx \frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Yield)
\times (PV_{0})} \tag{5}
$$
- Here we apply the methodology from an earlier lesson to approximate modified
duration and convexity. We estimated the modified duration by increasing and
decreasing the yield-to-maturity by the same amount $(\Delta Yield)$ to
calculate corresponding bond prices $PV_{+}$ and $PV_{-}$ for a given
initial price $(PV_{0})$, as shown in Equation 5:Approximate Annualized Convexity (ApproxCon)
\[ ApproxCon = \frac{(PV_{-}) + (PV_{+}) - [2 \times (PV_{0})]}{(\Delta Yield)^{2} \times (PV_{0})} \tag{6} \]
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### Approximate Annualized Convexity (ApproxCon)
$$
ApproxCon = \frac{(PV_{-}) + (PV_{+}) - [2 \times
(PV_{0})]}{(\Delta Yield)^{2} \times (PV_{0})}
\tag{6}
$$