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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.

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    $\begingroup$ Can you explicitly spell out formulas for $c(1, 1400)$ and $c(2, 1400)$? $\endgroup$ – Gordon Jun 21 '16 at 14:29
  • $\begingroup$ Hi, @Gordan. The formula for $c(T,K)$, a European call option underlying on a forward contract with maturity $T$ and strike price $K$ is given by $$\begin{align} c(T,K) = F_{0}e^{-rT}N(d_{1})-Ke^{-rT}N(d_{2}) \end{align}$$ Where $$\begin{align} d_{1} =& \frac{\ln\left(\frac{F_{0}}{K}\right)+\frac{\sigma^2}{2}T}{\sigma\sqrt{T}} \\ d_{2} =& d_{1} - \sigma\sqrt{T}\\ \end{align}$$ $\endgroup$ – Gustavo Louis G. Montańo Jun 21 '16 at 14:42
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    $\begingroup$ Your results are close to the disclosed answer. I think they are acceptable. The differences may be caused by the approximation for the cumulative distribution function. $\endgroup$ – Gordon Jun 21 '16 at 17:01
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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

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    $\begingroup$ Actually, this is not correct, one should not use conditional default probabilities (i.e. conditional on survival) for the CVA calculation but unconditional ones!!! See for example formula (7) here: papers.ssrn.com/sol3/papers.cfm?abstract_id=1782063 $\endgroup$ – byouness May 14 '18 at 17:58

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