Calculating Forward Contract CVA with Conditional Default Probabilities
Summary
The document works through a credit valuation adjustment (CVA) question for a long gold forward with possible counterparty default at two future dates. The attempted calculation estimates positive exposure at each date using call-option values on the forward price, applies the loss given default from the recovery assumption, and weights those losses by default probabilities. An answer points out that the later default probability must be conditional on survival through the earlier date, so the later contribution is also multiplied by the probability of surviving the first year.
A second answer notes a separate discrepancy: the stated answer's exposures and CVA appear consistent with a risk-free rate of 2.5%, rather than the 2% in the question. Thus the exchange identifies both a survival-probability adjustment and a possible inconsistent input in the supplied solution. It does not fully reconcile every convention in the exposure calculation, so the numerical figures should be read in light of the input ambiguity.
Key ideas
- CVA weights positive exposure at each default date by loss given default and the probability of default.
- A later default contribution must account for survival through earlier dates when probabilities are conditional.
- For the later date, the example's answer applies the stated default probability only after adjusting for first-year survival.
- The provided exposure figures may reflect a risk-free rate different from the one stated in the question.
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# How to calculate the CVA of a forward contract?
# How to calculate the CVA of a forward contract?
I am having trouble calculating the CVA of a forward contract. The question is presented below
Question:
> There exists a long forwards position underlying on gold with 2 years remaining. The counterparty can only default either at the end of years 1 or 2. The default probabilities for these points in time are 1% and 4% respectively. A recovery rate of 40% is assumed with a risk free rate of 2% The forward contract was entered at $1,400 and a 2-year gold forward currently has a forwards price of 1,445 with expected volatility of 19%. Calculate the CVA of this contract.
Answer:
> Exposures at $t=1$ and $t=2$ are \$125.23 and \$167.01 respectively with a total CVA of $4.760.
My Attempt:
The general intuition is to first calculate the exposure (potential loss to the investor due to counterparty risk) at times $t=1$ and $t=2$. We will let
- $v_{t}$ be the value of exposure at time $t$.
- $c(T,K)$ be the value of a European call option underlying on the forward price maturing at time $T$ with strike price $K$.
Now, the first exposure occurs at $t=1$ which is in fact related to the value of the forwards contract. In addition, the forward is assumed to be settled at maturity. Thus the exposure of the contract at $t=1$ is
$$\begin{align} v_{1} =& \mathrm{max}((F_{1}-1400)e^{-2\%\cdot1},0) \\ =& e^{-2\%\cdot 1} \mathrm{max}(F_{1}-1400,0) \\ =& e^{-2\%\cdot 1} c(1,1400) \end{align}$$
That is, the exposure at time $t=1$ is $e^{-2\%\cdot 1}$ times a European call option with maturity in $1$ year and strike price $1,400$. The Black-Scholes Merton valuation of this is \$129.04. The exposure is therefore
$$v_{1} = e^{-2\%\cdot 1}\times\$129.04. $$ $$v_{1} = \$126.48 $$
A similar construction can be made for the exposure at $t=2$. That is
$$\begin{align} v_{2} =& \mathrm{max}(F_{2}-1400,0) \\ =& c(2,1400) \end{align}$$
I.e. European call option with maturity in $2$ years and strike price $1,400$. The call has a price of \$168.69 and is the exposure at $t=2$.
Since the recovery rate is 40%, the loss rate is 60%. Thus we expect to lose
$$\begin{align} \mathrm{Loss}(v_{1}) =& 60\%\times126.48 = \$75.89 \\ \mathrm{Loss}(v_{2}) =& 60\%\times168.69 = \$101.21 \\ \end{align}$$
Finally we consider the probabilities in each year and arrive at the CVA. That is,
$$\begin{align} \mathrm{CVA} =& 1\%\times75.89 + 4\%\times101.21 \\ \mathrm{CVA} =& 4.81 \\ \end{align}$$
In conclusion my answers are
- $v_{1} = 126.49$
- $v_{2} = 168.69$
- CVA = 4.81
This differs from the disclosed answers of
> Exposures at $t=1$ and $t=2$ are \$125.23 and \$167.01 respectively with a total CVA of $4.760.
I am not sure if the disclosed answers are correct. However, if someone could go through my working and spot an error or provide an explanation, it would be highly appreciated
For support: Risk Management and Financial Institutions 4ed. John C. Hull, P.g. 440 shows a similar example.
## Answer by oscar (score 3, accepted)
https://quant.stackexchange.com/a/30217
Actually the problem is that the probability of default the second year is conditioned by the default the first year. So you have to multiply 4%*101.21*99%, because 1% of the times it has already defaulted the first year
## Answer by SkorohodAlex (score 1)
https://quant.stackexchange.com/a/66529
The calculations are correct; I played a little with the parameters and it seems that the risk free rate is not 2%, but 2.5%. Given this input, the exposures are 125.23 and 167.01, as given in the answer provided to you. Also the CVA matches, with these exposures it becomes 4.76.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.