Financial Risk Manager Part 2

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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:

$UCVA = \sum_{t=1}^{n} (1 - RR_t) \times EE_t \times PD_t \times DF_t$

Where (at any time $t$ ):

$RR_t$ = recovery rate
$EE_t$ = expected positive exposure (non-discounted, net of collateral)
$PD_t$ = marginal probability of default
$DF_t$ = discount factor

First, we determine the net Expected Exposure ( $EE_t$ ): With a collateral posting of AUD 11 million against an EPE of AUD 14 million each year, the uncollateralized $EE_t$ is: $EE_t = 14 - 11 = \text{AUD } 3 \text{ million}$

Next, we extract the constant hazard rate ( $\lambda$ ) using the relationship $\lambda = \frac{\text{Spread}_t}{1 - RR_t}$ :

Year 1: $\lambda = \frac{200 \text{ bps}}{1 - 0.80} = \frac{0.02}{0.20} = 10\%$
Year 2: $\lambda = \frac{300 \text{ bps}}{1 - 0.70} = \frac{0.03}{0.30} = 10\%$
Year 3: $\lambda = \frac{400 \text{ bps}}{1 - 0.60} = \frac{0.04}{0.40} = 10\%$ The hazard rate is a constant $10%$.

Now, we calculate the marginal probability of default ( $PD_t$ ) for each year using $PD_t = e^{-\lambda(t-1)} - e^{-\lambda t}$ :

Year 1: $PD_1 = 1 - e^{-0.10 \times 1} = 0.09516$
Year 2: $PD_2 = e^{-0.10 \times 1} - e^{-0.10 \times 2} = 0.90484 - 0.81873 = 0.08611$
Year 3: $PD_3 = e^{-0.10 \times 2} - e^{-0.10 \times 3} = 0.81873 - 0.74082 = 0.07791$

Then, calculate the discount factors ( $DF_t$ ) at the risk-free rate of 3%:

$DF_1 = e^{-0.03 \times 1} = 0.9704$
$DF_2 = e^{-0.03 \times 2} = 0.9418$
$DF_3 = e^{-0.03 \times 3} = 0.9139$

Finally, we calculate the CVA components for each year and sum them up:

Year 1: $(1 - 0.80) \times 3 \times 0.09516 \times 0.9704 = 0.20 \times 3 \times 0.09516 \times 0.9704 = 0.05538$
Year 2: $(1 - 0.70) \times 3 \times 0.08611 \times 0.9418 = 0.30 \times 3 \times 0.08611 \times 0.9418 = 0.07299$
Year 3: $(1 - 0.60) \times 3 \times 0.07791 \times 0.9139 = 0.40 \times 3 \times 0.07791 \times 0.9139 = 0.08544$

$UCVA = 0.05538 + 0.07299 + 0.08544 = 0.21381 \text{ million (approx. AUD 0.214 million)}$ .

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:

$UCVA = \sum_{t=1}^{n} (1 - RR_t) \times EE_t \times PD_t \times DF_t$

Where (at any time $t$ ):

$RR_t$ = recovery rate
$EE_t$ = expected positive exposure (non-discounted, net of collateral)
$PD_t$ = marginal probability of default
$DF_t$ = discount factor

Next, we extract the constant hazard rate ( $\lambda$ ) using the relationship $\lambda = \frac{\text{Spread}_t}{1 - RR_t}$ :

Year 1: $\lambda = \frac{200 \text{ bps}}{1 - 0.80} = \frac{0.02}{0.20} = 10\%$
Year 2: $\lambda = \frac{300 \text{ bps}}{1 - 0.70} = \frac{0.03}{0.30} = 10\%$
Year 3: $\lambda = \frac{400 \text{ bps}}{1 - 0.60} = \frac{0.04}{0.40} = 10\%$ The hazard rate is a constant $10%$.

Now, we calculate the marginal probability of default ( $PD_t$ ) for each year using $PD_t = e^{-\lambda(t-1)} - e^{-\lambda t}$ :

Year 1: $PD_1 = 1 - e^{-0.10 \times 1} = 0.09516$
Year 2: $PD_2 = e^{-0.10 \times 1} - e^{-0.10 \times 2} = 0.90484 - 0.81873 = 0.08611$
Year 3: $PD_3 = e^{-0.10 \times 2} - e^{-0.10 \times 3} = 0.81873 - 0.74082 = 0.07791$

Then, calculate the discount factors ( $DF_t$ ) at the risk-free rate of 3%:

$DF_1 = e^{-0.03 \times 1} = 0.9704$
$DF_2 = e^{-0.03 \times 2} = 0.9418$
$DF_3 = e^{-0.03 \times 3} = 0.9139$

Finally, we calculate the CVA components for each year and sum them up:

Year 1: $(1 - 0.80) \times 3 \times 0.09516 \times 0.9704 = 0.20 \times 3 \times 0.09516 \times 0.9704 = 0.05538$
Year 2: $(1 - 0.70) \times 3 \times 0.08611 \times 0.9418 = 0.30 \times 3 \times 0.08611 \times 0.9418 = 0.07299$
Year 3: $(1 - 0.60) \times 3 \times 0.07791 \times 0.9139 = 0.40 \times 3 \times 0.07791 \times 0.9139 = 0.08544$

$UCVA = 0.05538 + 0.07299 + 0.08544 = 0.21381 \text{ million (approx. AUD 0.214 million)}$ .

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Question 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:

The investment bank and the counterparty have signed a credit support annex to cover this exposure, which requires collateral posting of AUD 11 million.

The current risk-free rate of interest is 3% and the term structure of interest rates remains flat over the 3-year horizon.

The collateral and the expected positive exposure values remain stable as projected over the 3-year life of the contract.

The expected positive exposure and the collateral are assessed by using the same discount factors.

The probability of default of the bank is 0% per year.

Given the information and the assumptions above, what is the correct estimate of the unilateral CVA for this position?

	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

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Last updated: July 5, 2026 at 07:28

AUD 0.214 million