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Discuss the way that holding period, portfolio turnover and the use of leverage affect how risk is managed by asset managers.

FRM® Part II · Risk Management and Investment Management · VaR and Risk Budgeting in Investment Management · FRM2.IM.6.b

The idea

Three features of an investment operation decide how VaR should be used. Holding period: a pension fund holds positions for months or years and cannot unwind quickly, so it looks at weekly, monthly or longer horizons. A one-day VaR is scaled to a longer horizon by the square root of time only if positions stay fixed and returns are independent. Turnover: VaR from today's holdings describes today's portfolio only, so for a high-turnover manager it says little about risk over a long horizon, and monitoring must be frequent. Leverage: dollar VaR scales with total assets, so measured against equity capital it is multiplied by assets divided by equity. Let A be total assets and E equity capital. Equity VaR is the asset VaR multiplied by the leverage factor: Factor = A ÷ E. A common error is to leverage-adjust with debt over equity instead of assets over equity. Exam point: match the horizon to the time needed to change the position.

Formulas

Square-root-of-time scaling of VaR

VaR(T days) = VaR(1 day) × √T

VaR(1 day)
the one-day VaR
T
the horizon in days

Holds only if positions stay fixed and returns are independent, which fails for a high-turnover manager. Choose T as the time needed to unwind the position.

VaR as a share of equity capital for a leveraged fund

VaR on equity = z × σ × √T × (A ÷ E)

z
the standard normal quantile for the confidence level
σ
the annual volatility of asset returns
T
the horizon in years
A
total assets
E
equity capital

Normal returns and fixed positions. The leverage multiple is assets over equity, not debt over equity.

A practice question

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

Compared with a bank trading desk that uses a one-day VaR, a pension fund typically measures VaR over a longer horizon because:

  1. pension funds hold only assets whose returns are normally distributed, which is what permits the longer horizon.Why not: Normality is unrelated to the choice of horizon, and many pension assets are not normal.
  2. its positions are held for long periods and cannot be unwound quickly, so risk matters over months. - correct
  3. regulators forbid pension funds from calculating daily VaR.Why not: No such rule exists; the horizon follows from how fast positions can be changed.
  4. a longer horizon always produces a lower VaR than a one-day horizon.Why not: Wrong direction: under square-root-of-time scaling, VaR rises with the horizon.

The horizon should reflect the time needed to change or liquidate the position. For slow-moving, long-horizon investors this is much longer than the day used for trading books.

A harder one

A fund has equity capital of USD 200 million and borrows USD 300 million, so total assets are USD 500 million. Asset volatility is 10 percent a year with normal returns. The one-month 95 percent VaR as a percentage of equity capital is closest to:

  1. 4.7 percentWhy not: Reports VaR as a percentage of assets and ignores the leverage multiple of 2.5.
  2. 7.1 percentWhy not: Uses debt over equity (1.5) as the multiple instead of assets over equity (2.5).
  3. 11.9 percent - correct
  4. 16.8 percentWhy not: Uses the 99 percent quantile 2.326 instead of 1.645 for the 95 percent confidence level.

VaR on assets = 1.645 x 10% / sqrt(12) = 4.75 percent of assets; as a percentage of equity multiply by assets over equity, 500/200 = 2.5, giving 11.9 percent.

The hardest tier

A fund has equity of USD 100 million and total assets of USD 300 million. Asset returns are normal with volatility of 8 percent a year. The one-quarter 99 percent VaR as a percentage of equity capital is closest to:

  1. 55.8 percentWhy not: Uses the annual volatility for a quarter-year horizon, with no time scaling.
  2. 14.0 percentWhy not: Scales linearly by 0.25 instead of by the square root of 0.25.
  3. 19.7 percentWhy not: Uses the 95 percent quantile of 1.645 when the question specifies 99 percent.
  4. 27.9 percent - correct

Asset VaR = 2.326 x 8% x sqrt(0.25) = 9.3 percent of assets; times leverage of 300/100 = 3 gives 27.9 percent of equity.

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