Explanation:
A is correct. To derive the unilateral credit valuation adjustment (UCVA), we take the PD of the bank to be equal to 0% and use the standard formula:
where (at any time t):
EPE = expected positive exposure (non-discounted)
PD = probability of default
DF = discount factor
The discount factor (DFₜ) is determined from the risk-free rate of 3%. For year 1, 2, and 3, they are exp(-0.03)=0.9704, exp(-0.032)=0.9418, and exp(-0.033)=0.9139, respectively.
The hazard rate is constant over the 3 years, and λ = spread/(1 − RR) = 10%.
Therefore:
Year 1 cumulative probability of default = 1 − exp(-0.1*1) = 9.52% (marginal probability (PD₁))
Year 2 cumulative probability of default = 1 − exp(-0.1*2) = 18.13%; thus, marginal probability (PD₂) = 18.13 − 9.52 = 8.61%.
Year 3 cumulative probability of default = 1 − exp(-0.1*3) = 25.92%; thus, marginal probability (PD₃) = 25.92 − 18.13 = 7.79%.
Collateral amounts of AUD 14 million for each of the years 1, 2 and 3 are considered. Therefore, the rest of the derivation becomes:
| Year 0 | Year 1 | Year 2 | Year 3 | |
|---|---|---|---|---|
| Marginal probability of default [PD(t)] | 9.52% | 8.61% | 7.79% | |
| Discount factor (DF) | 0.9704 | 0.9418 | 0.9139 | |
| Recovery rate (RR) | 80% | 70% | 60% | |
| Expected exposure (EE) (AUD million) | 14 | 14 | 14 | |
| Collateral (C) (AUD million) | 11 | 11 | 11 | |
| EE’ (netted) (AUD million) | 3 | 3 | 3 | |
| (1−RR)*(EE’)PD(t)(DF) (AUD million) | 0.0554 | 0.0730 | 0.0854 |
B is incorrect. AUD 0.2527 million is the result obtained when the hazard rate of 10% is used as the marginal default probability for each of the 3 years.
C is incorrect. AUD 0.5201 million is the result obtained when the recovery rate and not the LGD is used.
D is incorrect. AUD 0.9980 million is the result obtained when collateral is not considered.
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A CRO at an investment bank has asked the risk department to evaluate the bank’s derivative position with a counterparty over a 3-year period. The risk department assumes that the counterparty’s default probability follows a constant hazard rate process. The table below presents trade and forecast data on the CDS spread, the expected exposure, and the recovery rate of the counterparty:
| Year 1 | Year 2 | Year 3 | |
|---|---|---|---|
| Expected positive exposure (AUD million) | 14 | 14 | 14 |
| CDS spread (bps) | 200 | 300 | 400 |
| Recovery rate (%) | 80 | 70 | 60 |
Additionally, the CRO has presented the risk team with the following set of assumptions to use in conducting the analysis:
Given the information and the assumptions above, what is the correct estimate of the unilateral CVA for this position?
A
AUD 0.214 million
B
AUD 0.253 million
C
AUD 0.520 million
D
AUD 0.998 million