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Describe the VaR of linear derivatives and work it out.

FRM® Part I · Valuation and Risk Models · Calculating and Applying VaR · FRM1.VR.2.d

The idea

A linear derivative such as a forward or a futures contract has a value that moves one for one, or in a fixed proportion, with its underlying. Under normal returns, its VaR can therefore be computed by the same method as for the underlying. A long forward on a non-dividend stock equals a long position in the stock plus borrowing the present value of the delivery price. Its exposure is therefore the stock position. For an equity index future, Exposure = contracts × multiplier × futures price. Then VaR = z × σ × exposure. A currency forward has an exposure equal to the foreign amount discounted at the foreign rate and converted at spot. A short position has the same VaR magnitude as a long one. In a multi-asset book the signs enter the correlation term. Do not use the delivery price or margin as the exposure.

Formulas

VaR of an equity index futures position

VaR = z × σ × contracts × multiplier × F

z
the standard normal quantile
σ
the volatility of the index return
contracts
the number of futures contracts
multiplier
the currency amount per index point
F
the futures price

Exact under normal returns because the futures is linear. The exposure is the notional value, not the margin.

Exposure of a currency forward

exposure = Q × e^(−rf × T) × S

Q
the foreign currency amount to be delivered
rf
the foreign continuously compounded interest rate
T
the years to delivery
S
the spot exchange rate, domestic per foreign

Discount at the foreign rate and convert at spot, then VaR = z × σ × exposure. Using the delivery price or the undiscounted amount is the trap.

A practice question

From the question bank, with the reason each wrong option is wrong.

A long forward contract on a non-dividend-paying stock is economically equivalent, for risk purposes, to:

  1. A long position in the stock with no associated financing.Why not: Ignores the borrowing of the present value of the delivery price, which is why the forward costs nothing at inception.
  2. A long call option on the stock with a strike equal to the delivery price of the forward.Why not: A call has a kinked payoff and is non-linear; a forward has a linear payoff.
  3. A zero-coupon bond that pays the delivery price at maturity.Why not: The forward's value depends on the stock price; a bond on the delivery price is not exposed to it.
  4. A long stock position financed by borrowing the present value of the delivery price. - correct

The forward is worth S - K e^(-rT). The first term is a stock position and the second a fixed borrowing, so the price risk equals that of the stock position.

A harder one

A trader is long 30 index futures contracts. The index is at 4,500, the contract multiplier is USD 50 per index point, and daily index volatility is 1.2%. The one-day 99% VaR (z = 2.326) is closest to:

  1. USD 133,200Why not: Uses the 95% quantile of 1.645 instead of the 99% quantile.
  2. USD 188,400 - correct
  3. USD 3,800Why not: Forgets the USD 50 contract multiplier, so the exposure is understated.
  4. USD 595,900Why not: Applies a ten-day square-root scaling to a one-day question.

Exposure = 30 x 50 x 4,500 = USD 6.75 million. VaR = 2.326 x 1.2% x 6.75 million = USD 188,400. The exposure is the notional value of the futures, not the margin.

The hardest tier

A portfolio has a long equity forward with a delta-equivalent exposure of USD 8 million to stock A, and a short position of USD 5 million in stock B. Daily volatilities are 1.5% (A) and 1.1% (B) and the correlation is 0.60. The ten-day 99% VaR (z = 2.326, square-root-of-time) is closest to:

  1. USD 1,171,000Why not: Treats the short position as if it were long, adding the correlation term instead of subtracting it.
  2. USD 478,000Why not: Assumes a correlation of one, which makes the long and short exposures offset almost fully.
  3. USD 227,000Why not: Gives the one-day figure and does not scale to ten days.
  4. USD 717,000 - correct

Dollar volatilities: A = 120,000, B = -55,000 (short). Variance = 120,000² + 55,000² - 2 x 0.6 x 120,000 x 55,000 = 9.505 x 10^9, sigma = 97,500. One-day VaR = 226,800; ten-day = 226,800 x sqrt(10) = USD 717,000.

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