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Outline duration gap management and use it to safeguard a bank's net worth.

FRM® Part II · Liquidity and Treasury Risk Measurement and Management · Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques · FRM2.LTR.17.c

The idea

Duration gap management protects the market value of equity rather than reported income. DA and DL are the durations of assets and liabilities, L is total liabilities, A is total assets and y is the yield. The leverage ratio L ÷ A matters because liabilities fund only part of the assets. Duration gap is defined as DG = DA − (L ÷ A) × DL. For a change in yield dy, the approximate change in the market value of equity is dE = −DG × A × dy ÷ (1 + y). A positive gap means equity falls when rates rise. A negative gap means equity rises. To immunize, set DG to zero, either by restructuring the balance sheet or by using derivatives, for instance selling futures when the gap is positive. Matching DA to DL is not enough unless liabilities equal assets. Because equity is a thin slice of assets, a small asset move is a large percentage change in equity. On the exam, state the leverage adjustment and divide by one plus the yield.

Formulas

Duration gap

DG = DA − (L ÷ A) × DL

DA
the duration of assets
DL
the duration of liabilities
L
total liabilities
A
total assets

Weight liability duration by L ÷ A, not A ÷ L. Matching asset and liability duration does not immunise unless liabilities equal assets.

Change in the market value of equity

ΔE = −DG × A × Δy ÷ (1 + y)

DG
the duration gap
A
total assets
Δy
the change in yield
y
the starting yield

A positive gap loses value when yields rise. Divide by one plus the yield, and express the result against equity, not assets, to see how leverage magnifies it.

A practice question

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

Which expression gives a bank's duration gap, where D_A and D_L are the durations of assets and liabilities, A is total assets and L is total liabilities?

  1. D_A - D_LWhy not: Ignores leverage: liabilities only fund part of the assets, so D_L must be scaled by L / A.
  2. D_A - (A / L) x D_LWhy not: Inverts the leverage ratio; the weight on liabilities is L / A, which is below 1.
  3. D_A - (L / A) x D_L - correct
  4. (L / A) x D_A - D_LWhy not: Applies the leverage weight to the wrong duration.

Equity change is the change in A minus the change in L. Dividing by A, DG = D_A - (L/A) x D_L. This weights liability duration by the share of assets that liabilities fund.

A harder one

A bank has total assets of USD 1,000 million, liabilities of USD 920 million, asset duration of 4.2 years and liability duration of 2.1 years. Market yields are 6.00% and rise by 1.00 percentage point. The estimated change in the market value of equity is closest to:

  1. Decrease of USD 22.7 millionWhy not: Omits the division by (1 + yield) = 1.06.
  2. Decrease of USD 21.4 million - correct
  3. Increase of USD 21.4 millionWhy not: Wrong sign: a positive duration gap loses value when yields rise.
  4. Decrease of USD 19.8 millionWhy not: Uses a duration gap of 4.2 - 2.1 = 2.1 without the leverage weight of 0.92.

DG = 4.2 - 0.92 x 2.1 = 2.268. Change in E = -2.268 x 1,000 x 0.01 / 1.06 = -21.4 million.

The hardest tier

A bank has total assets of USD 3,000 million and a duration gap of +1.9 years. It will hedge by selling Treasury futures whose price is 108 per 100 face, with a contract size of USD 1,000 per point so that one contract is worth USD 108,000. The futures' duration is 6.4 years. Ignoring yield adjustments, the contracts needed are closest to:

  1. Buy 8,247 contractsWhy not: Wrong direction: a positive gap loses when yields rise, so the bank needs a short futures position.
  2. Sell 17,361 contractsWhy not: Puts a duration of 4.0 years in the numerator in place of the 1.9-year duration gap.
  3. Sell 8,906 contractsWhy not: Uses a USD 100,000 face amount instead of the USD 108,000 market value of a contract.
  4. Sell 8,247 contracts - correct

Contracts = DG x A / (D_F x contract value) = 1.9 x 3,000 million / (6.4 x 108,000) = 8,247 sold. Futures gain when yields rise and offset the equity loss.

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