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Gap management

Jamal Munshi, Sonoma State Univesity, 1992
All rights reserved

The duration of a stream of cash flows is approximately the term of an equivalent zero coupon bond. It is a weighted average number of years over which the funds represented by the cash flows are recovered.
How is it computed?

    Example: Cash flows of $100 1 year from now and $100 3 years from now with k = 8% have present values given by 100/1.08 = $92.59 and 120/1.08^3 = $79.38. The weighted average number of years = (1*92.59 + 3*79.38)/(92.59+79.38) = 330.73/171.97 = 1.923. The duration of these cash flows is 1.923 years. Note that the present value of a zero coupon bond paying $200 1.923 years from now is $172, the same as the two $100 payments.
How are durations of many items combined into an average duration?

    Durations may be linearly combined using the equation Dp = Sum(Wi*Di) where Dp is the portfolio duration, Wi is the portion of the portfolio in asset i and Di is the duration of asset i. This also works for the liability side. Example:
What is elasticity?

    Elasticity is the ratio of percent changes of two related variables. The elasticity of X with respect to Y is the percent change in X for a 1% change in Y and it is written as (dX/X)/(dY/Y) where the little "d" indicates an incremental change. For example a man who weighs 150 lbs and drinks 12 bottles of beer a week finds that a 2 bottle a week increase in beer consumption results in a 5 lb increase in weight. His elasticity of weight (W) with respect to beer (B) is (dW/W)/(dB/B) = (5/150)/(2/12) = 0.2. This means that a 1% increase in beer consumption results in a 0.2% increase in weight. A relationship like this considered "elastic" if the elasticity is high.
What is negative elasticity?

    A negative elasticity indicates an inverse relationship - one in which Y falls when X rises and vice versa. For example in the above example, an inverse relationship between weight and beer would mean that you lose weight when you increase your beer consumption. This may not be true of beer but it is true of the relationship between the value of a debt contract such as a bond or a loan and prevailing interest rates.
What is a useful property of duration?

    The present value of a future cash payment or stream of fixed cashflows changes with the discount rate. This is called interest rate risk because the value of a bond or loan portfolio falls when the interest rate rises. Duration is approximately equal to the elasticity of value to (1+k) where k is the discount rate. This means that,,

    (dP/P) / (dK/K) = - D

    where P = value of the debt contract, dP = change in value, K = 1+k, dK = change in k, and D = the duration of the debt contract. The minus sign in front of D reminds us that the relationship between P and k is inverse. If k increases P falls and if k decreases P rises. Duration is a measure of interest rate risk faced by holders of debt contracts.

Example computation

    Elasticity = duration = 4.5, k=12%, P=$50. What is the value of dP if k rises by 1%? Set dK=0.01, K = 1+.12=1.12, P=50. Solving for dP we get
    dP = -4.5*100*.01/1.12 = -2

    So a rise in rates of 1% will result in a loss of approximately $2 in the value of the debt contract.

What is the duration gap of a bank?

    The assets of banks consist primarily of a portfolio of debt contracts. The sensitivity of asset value to rate changes may be estimated with the weighted average duration of the assets. But things aren't as bad as the asset duration may indicate because, as it turns out, bank assets are funded primarily with debt. So when rates rise, the market value of their assets fall but those of their debt obligations also fall and offset some of the loss in asset value. An ideal condition in which the reduction in asset value is exactly offset by reduction in liabilities, called "immunization", is not possible for bankers because their assets are long term (high D) and their liabilities are short term (low D) and also because some of their assets are funded by bank capital. The positive difference between asset side and liability side durations exposes banks to interest rate risk and this difference is called the duration gap. Bank managers try to keep the gap as small as possible by using methods and techniques called gap management.
How do these changes affect net bank capital?

    Net bank capital, or "net worth", is the difference between assets and liabilities and so the change in bank capital may be written in terms of this equation as follows:

    E = TA - L
    TA = EA + NEA
    dE = dEA - dL
    dE/E = percent change in net worth

    where E = equity, TA = total assets, EA = earning assets, NEA = non-earning assets, L = liability, and d indicates change. Only earning assets are subject to interest rate risk.

    Example: TA = 100, EA = 85, L = 90, E = 10. There is a 3% loss in EA and a 1% loss in L. What is the impact on E? Loss in EA = 0.03*85 = $2.55, loss in L = 0.01*90 = $0.90. Net loss = dE = 2.55 - 0.90 = $1.65. dE/E = 1.65/10 = 16.5% loss of equity.

How can duration be used to estimate the sensitivity of E to k?

  • Step 1: compute Di, duration of each asset item
  • Step 2: compute Dx, the average duration of the earning assets using Dx = Sum(Wi*Di) where Wi = $value of asset i divided by total earning assets
  • Step 3: compute Dj, duration of each liability item
  • Step 4: compute Dy, the average duration of the liabilities using Dy = Sum(Wj*Dj) where Wj = $value of liability j divided by total liabilities
  • Step 5: solve for dEA by setting -Dx = (dEA/EA)/(dK/K). Recall that K = 1+k and k is the interest rate
  • Step 6: solve for dL by setting -Dy = (dL/L)/(dK/K)
  • Step 7: compute dE = dEA - dL
  • Step 8: compute the percentage loss/gain in bank capital as dE/E
Can we combine some of these steps to simplify the computation?

  • Yes. The 8-step procedure above is algebraically identical to a simpler computation.
  • We compute the duration gap G, and use that to get dE directly as follows:
  • Step 1: G = Sum(Wi*Di) where Wi = value of each asset and liability item divided by total assets and Di is the corresponding duration. Wi is positive for assets and negative for liabilities.
  • Step 2: solve for dE by setting G = (dTA/TA)/(dK/K)
  • Step 3: set dE = dTA and compute the percentage loss/gain in bank capital as dE/E
  • This procedure is computationally more efficient but harder to understand.
Example problem

  • A bank with TA = 100 is capitalized at 10%, i.e., E = 10. The bank's assets are: [Description, $, Di] =
  • [Securities,10,1], [Securities,10,7], [VRM,11,0.5], [FRM,11,6], [Loans,25,1], [Loans,25,4].
  • Its liabilities are: [Description, $, Dj] =
  • [Checking,20,0.5], [Savings,20,2], [CDs,20,1], [CDs,10,2], [Debt,10,1], [Debt,10,3].
  • What is the percent loss in E for a 100 bp increase in rates which are currently at 8%? (bp = basis point = 0.01%.)
Solution

  • EA = 10+10+11+11+25+25 = 92, L = 20+20+20+10+10+10= 90, E = 100-90=10, NEA = 100-92=8
  • Asset side: Sum(Wi*Di)=(10/92)*1+(10/92)*7+(11/92)*0.5+(11/92)*6+(25/92)*1+(25/92*4) = -3.005
  • Liability side: Sum(Wi*Di)=(20/90)*.5+(20/90)*2+(20/90)*1+(10/90)*2+(10/90)*1+(10/90 *3) = 1.444
  • (dEA/EA)/(dK/K) = -3.005, solve for dEA
  • (dEA/92)/(.01/1.08) = -3.005: dEA = 3.005*.01*92/1.08 = -$2.56
  • (dL/L)/(dK/K) = 1.444, solve for dL
  • (dL/90)/(.01/1.08) = -1.444: dL = 1.444*.01*90/1.08 = -$1.20
  • dE = dEA - dL = -2.56 + 1.20 = -$1.36
  • dE/E = -1.36/10 = -13.6%
  • A 1% rate increase will result in a 13.6% loss in net worth and a 1% decrease in rates will result in a 13.6% increase in net worth (approximately)
Shortcut solution using G

  • G = Sum(Wi*Di) for assets and liabilities with assets as positive and liabilities as negative
  • G = (10*1 +10*7 +11*0.5 +11*6 +25*1 +25*4 -20*0.5 -20*2 -20*1 -10*2 -20*1 -20*3)/100
  • G = 1.465
  • (dTA/TA)/(dK/K) = -1.465, solve for dTA
  • (dTA/100)/(.01/1.08) = -1.465, dTA = 1.465*.01*100/1.08 = $1.356
  • dE = dTA = -1.356
  • dE/E = 1.356/10 = -13.56%
  • A 1% rate increase will result in a 13.56% loss in net worth and a 1% decrease in rates will result in a 13.56% increase in net worth (approximately)
  • Note: the relationship between interest rate and value is inverse. When rates rise value falls and when rates fall value rises. We use the minus sign to remind ourselves of the inverse nature. Alternatively we could drop the minus sign and simply remember that a rise in rates results in a loss in value.
What are some gap management techniques?

    Note that bank equity value is very sensitive to interest rate changes. This sensitivity is worrisome to regulators because a loss of net worth may lead to insolvency; and to shareholders because their wealth depends on net worth. Bank managers use gap management techniques to keep G and interest rate sensitivity as low as possible. If the sensitivity is too high they may take action that reduces the duration of asset side items, increases the duration of liability side items, increases the ratio of L/EA, or make use of derivatives. Derivative contracts that may be used to hedge interest rate risk include interest rate futures, options, options on futures, swaps, and swaptions. The same contracts may also be used to speculate. The difference between a speculator and a hedger is that the speculator makes a forecast of future interest rates and implements a derivative strategy to profit from the forecast. Hedgers do not have a forecast and use derivative strategies for protection agains rate changes in either direction.
What are some problems with gap management?

    There are four:1. immunization works as a hedge against parallel shifts in the yield curve but does not protect against other kinds of changes in the interest rate structure; 2. most duration figures provided by bankers are gross estimates because of prepayment risk; 3. after a single interest rate change the durations also change and the portfolio must be rebalanced and re-immunized; and 4. duration gap analysis is a computational convenience that is unneccesary since most bankers have access to computers and specialized simulation and scenario analysis software that are better at assessing the interest rate exposure of the bank. Also read this document on the "myth" of gap management.