Charta is on Google Play: practise on the go. Get the app
Charta

Distinguish risk management from portfolio management and show how marginal VaR is put to work in managing portfolios.

FRM® Part II · Risk Management and Investment Management · Portfolio Risk: Analytical Methods · FRM2.IM.5.e

The idea

Risk management and portfolio management overlap but are not the same job. Risk management measures, reports and limits the risk being taken, and VaR is its tool. Portfolio management chooses positions to earn the best expected return for the risk, so it combines risk figures with return forecasts. Marginal VaR links the two. A manager computes, for each position, expected excess return divided by marginal VaR, increases positions where the ratio is high and cuts those where it is low, until the ratios are equal and the portfolio is optimal. Marginal VaR also sizes a trade that brings VaR back under a limit, so the required sale is Cut = excess VaR ÷ marginal VaR. A hedge is correctly sized when the hedge instrument's marginal VaR is zero. Component VaR, being additive, allocates a total VaR limit across desks. Remember that all of this is local, so after a large trade, recompute.

Formulas

Position cut that brings VaR back to its limit

required cut = excess VaR ÷ marginal VaR

excess VaR
portfolio VaR minus the VaR limit
marginal VaR
the change in portfolio VaR per unit of the position

A linear, local estimate, so after a large cut recompute VaR. Dividing by individual or average VaR per unit instead of marginal VaR gives the wrong size.

Short hedge that brings the hedge's marginal VaR to zero

hedge = ρ × (σP ÷ σH) × P

ρ
the correlation between the hedge and the portfolio
σP
the volatility of the portfolio
σH
the volatility of the hedge instrument
P
the value of the portfolio

The hedge is sized so its covariance with the hedged position is zero. Leaving out the volatility ratio, or inverting it, is the usual slip.

Component VaR of a position

component VaR = marginal VaR × position size

marginal VaR
the change in portfolio VaR per unit of the position
position size
the amount held in the position

Component VaRs sum exactly to the diversified portfolio VaR, which is why they allocate a limit without gaps or double counting.

A practice question

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

Which statement best distinguishes risk management from portfolio management?

  1. Risk management measures and controls the risk being taken, while portfolio management picks positions for the best expected return per unit of risk - correct
  2. Risk management seeks the highest expected return, while portfolio management seeks to minimize VaRWhy not: Reverses the roles: the return-seeking decision belongs to portfolio management.
  3. Risk management aims to eliminate all risk, while portfolio management accepts any risk the market offersWhy not: Risk management controls risk to appetite; it does not eliminate it, and portfolio management works within limits.
  4. Portfolio management uses risk figures but not expected returns, while risk management uses bothWhy not: Portfolio management needs expected returns to find the best return per unit of risk.

Risk management measures, reports and limits risk; portfolio management uses those measures with return forecasts to decide the positions. VaR serves both.

A harder one

A portfolio of USD 400 million has a VaR of USD 12 million against a limit of USD 10 million. Position A is USD 150 million with an individual VaR of USD 7.5 million and a marginal VaR of 0.025 per USD. Using marginal VaR, how much of A should be sold to return to the limit?

  1. USD 40 millionWhy not: Divides the excess by A's individual VaR per USD (0.05), which ignores diversification.
  2. USD 66.7 millionWhy not: Divides the excess by the portfolio's average VaR per USD (12 / 400 = 0.03) instead of A's marginal VaR.
  3. USD 80 million - correct
  4. USD 25 millionWhy not: Cuts A in proportion to the VaR reduction (2 / 12 of 150), as if A's VaR scaled with the whole portfolio.

Required fall in VaR = 12 - 10 = 2 million. Sale = 2 / 0.025 = USD 80 million. This is a linear estimate; since it is large, recompute marginal VaR afterwards.

The hardest tier

A manager holds USD 100 million of a portfolio with a daily volatility of 1.8%. A futures hedge H has a daily volatility of 1.5% and a correlation of 0.6 with the portfolio. How large a short position in H, in USD million, brings the marginal VaR of H to zero?

  1. 100Why not: A hedge of the full notional, which assumes a unit hedge ratio even though the volatilities and correlation differ.
  2. 60Why not: Multiplies the correlation by the notional (0.6 x 100), omitting the volatility ratio 1.8 / 1.5.
  3. 50Why not: Inverts the volatility ratio: 0.6 x 100 x 1.5 / 1.8.
  4. 72 - correct

Marginal VaR of H is zero when cov(H, P + hH) = 0. Short position = rho x (sigma_P / sigma_H) x 100 = 0.6 x (1.8 / 1.5) x 100 = USD 72 million.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Portfolio Risk: Analytical Methods