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Give examples of linear portfolios and of non-linear portfolios, and explain what separates them.

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

The idea

A portfolio is linear when its change in value is a proportional function of the changes in the underlying risk factors, so doubling a market move doubles the profit or loss and a rise and a fall of equal size give mirror-image results. Cash equities, currency positions, futures and forwards are linear. A forward is worth the spot price less the discounted delivery price, which moves one for one with spot. Non-linear portfolios contain options or similar payoffs, or positions such as bonds measured against yield. Their value is a curved or kinked function of the risk factor, and gamma or convexity matters. The usual mistake is to call every derivative non-linear, or to think a delta-neutral option book is linear. Delta only describes the slope at today's price. The exam expects you to separate linear from non-linear by the shape of the value function.

A practice question

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

Which of the following portfolios is a linear portfolio with respect to its risk factors?

  1. A portfolio of cash equity positions and equity index futures. - correct
  2. A portfolio of long call and long put options on an equity index.Why not: Options have kinked payoffs, so their value is not a proportional function of the underlying.
  3. A portfolio of delta-hedged options on one equity index.Why not: Delta hedging removes only the first-order exposure at today's price; the gamma term still makes the value non-linear.
  4. A bond portfolio valued as a function of its yield to maturity.Why not: Bond price is a convex function of yield, so it is not linear in that risk factor.

Shares and futures change in value in proportion to the underlying price, so doubling the move doubles the profit or loss. Options and bonds measured against yield have curved or kinked value functions.

A harder one

A desk holds four positions on the same stock. Which of them is the only non-linear position?

  1. A short position in 10,000 shares held outright.Why not: A short stock position has a negative but constant exposure, so its value is still linear in the share price.
  2. A long forward to buy 10,000 shares in one year's time.Why not: A forward is worth S minus the discounted delivery price, which is linear in S despite being a derivative.
  3. A long position in one-year calls on 10,000 shares. - correct
  4. A short futures position on 10,000 shares of the stock.Why not: A futures contract has a payoff proportional to the price change, so it is linear in the underlying.

The call has a kinked payoff and non-constant delta. Stock, forwards and futures all have exposures that are fixed multiples of the underlying price, whether long or short.

The hardest tier

Revaluing four portfolios, each built on 1,000 units of one stock priced at 100, gives the following profit and loss (USD thousand) for stock price shocks of -10%, -5%, +5% and +10%. Portfolio 1: -10.00, -5.00, +5.00, +10.00. Portfolio 2: -4.73, -2.61, +3.07, +6.56. Portfolio 3: +4.73, +2.61, -3.07, -6.56. Portfolio 4: +0.53, -0.23, +1.15, +3.12. Which classification is correct?

  1. Only portfolio 1 is linear; portfolios 2, 3 and 4 are non-linear. - correct
  2. Portfolios 1 and 2 are linear because both lose when the stock falls and gain when it rises.Why not: Looks only at the direction of P&L; linearity needs P&L proportional to the shock, and portfolio 2 is not.
  3. Portfolios 2 and 3 are linear because they are exact mirror images of one another.Why not: Mirror symmetry between two portfolios says nothing about whether each is proportional to the shock; neither is.
  4. Only portfolio 4 is non-linear; the other three are linear because their P&L changes sign with the shock.Why not: Checks the sign of the P&L rather than proportionality; a 10% shock gives 6.56 but a 5% shock gives 3.07 or 2.61, not half.

Portfolio 1 gives exactly 5.00 per 5% and 10.00 per 10%, symmetric and proportional. In portfolios 2 and 3 the up and down moves are asymmetric and not proportional (6.56 against 3.07; -4.73 against -2.61), and portfolio 4 gains on both sides. Portfolio 1 is a stock or forward position; 2 a long call; 3 a short call; 4 a long straddle.

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