VaR on a Cargo

Question

You run a hedged crude position of 500,000 barrels. Because the flat price is hedged, the risk that matters is the basis (the differential to Brent), which has a daily standard deviation of 0.20 dollars per barrel. Using a normal-distribution Value at Risk with a 95% confidence factor of z = 1.65, what is the one-day 95% VaR of this position, in dollars? Explain why hedging the flat price makes this number far smaller than the VaR of the unhedged cargo.

Accepted Answer

Let's build Value at Risk (VaR) from scratch, because once you have the formula the arithmetic is one line.

What VaR is

VaR answers a single question: over a given horizon and confidence level, what is the most the position is likely to lose? A one-day 95% VaR of two million dollars means that on 95 days out of 100 the loss should not exceed two million (and on the other five it could be worse). It compresses a messy position into one number that management and lenders can track day to day.

The simplest version assumes the position's daily change in value is roughly normally distributed (bell-shaped) with a standard deviation sigma. The standard deviation is just a measure of how much the value typically wiggles in a day. For a normal distribution, a fixed multiple of sigma marks off any confidence level you like, and that multiple is called z:

 	about 1.65 standard deviations for 95% confidence,
 	about 2.33 standard deviations for 99% confidence.

So the formula is \begin{equation} 	ext{VaR} = z 	imes \sigma_{	ext{value}} \end{equation} where $\sigma_{	ext{value}}$ is the standard deviation of the position's value over the horizon. That is the whole model: pick your confidence level to get z, find how much the position's value moves in a typical day, and multiply.

Step 1: the per-barrel risk is the basis

The key insight is what is risky here. The cargo is worth a lot at flat price \begin{equation} 500{,}000\ 	ext{bbl} 	imes 80/	ext{bbl} = 40{,}000{,}000 \end{equation} but that flat price is hedged with short Brent futures, so day-to-day swings in Brent are offset by the futures leg. What is left exposed is the basis, the differential of your grade to Brent, and we are told its daily standard deviation is 0.20 dollars per barrel.

Step 2: scale up to the dollar standard deviation

The per-barrel basis sigma is 0.20 dollars. The position is 500,000 barrels, and the value moves by the basis move times the number of barrels, so the standard deviation of the position's value is \begin{equation} \sigma_{	ext{value}} = 0.20/	ext{bbl} 	imes 500{,}000\ 	ext{bbl} = 100{,}000 \end{equation} That is how much the hedged position's value typically swings in a single day.

Step 3: apply the VaR formula

With z = 1.65 for 95% confidence \begin{equation} 	ext{VaR} = 1.65 	imes 100{,}000 = 165{,}000 \end{equation} Equivalently, doing it all in one shot \begin{equation} 	ext{VaR} = 1.65 	imes 0.20/	ext{bbl} 	imes 500{,}000\ 	ext{bbl} = 165{,}000 \end{equation} So a normal bad day should not cost the hedged position more than about 165,000 dollars.

Why hedging shrinks the number

Notice how small that VaR is next to the forty million dollars of crude you are holding. That is the hedge working. VaR is measured on whatever risk actually remains, and here that is only the basis, with its modest 0.20-dollar daily sigma.

Now imagine you had not hedged. Then the risk is driven by Brent's own volatility, which on a typical day might be on the order of one to two dollars a barrel, far larger than the 0.20-dollar basis sigma. Take a daily flat-price sigma of, say, 1.20 dollars a barrel. The unhedged VaR would be \begin{equation} 	ext{VaR}_{	ext{unhedged}} = 1.65 	imes 1.20/	ext{bbl} 	imes 500{,}000\ 	ext{bbl} = 990{,}000 \end{equation} roughly a million dollars, six times the hedged figure, simply because the flat-price sigma is six times the basis sigma. The hedge deleted the big flat-price swings and left only the small differential, and that reduction shows up directly in the risk number. The whole point of hedging the flat price is visible right here: the VaR collapses from about a million to about 165,000.

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VaR on a Cargo

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