BUS FINANCE 826

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BUS FINANCE 826
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Overview

• Interest rate risk associated with financial
  intermediation
• Federal Reserve policy
• Repricing model
• Maturity model
• Duration model
• Term structure of interest rate risk
• Theories of term structure of interest rates

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Central Bank Policy and Interest Rate Risk

• Japan March 2001 announced it would no longer
  target the uncollateralized overnight call rate.
• New target Outstanding current account balances
  at BOJ
• Targeting of bank reserves in U.S. proved
  disastrous

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Central Bank and Interest Rate Risk

• Effects of interest rate targeting.
• Lessens interest rate risk
• October 1979 to October 1982, nonborrowed
  reserves target regime.
• Implications of return to reserves target policy
• Increases importance of measuring and managing
  interest rate risk.

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Web Resources

• For information related to central bank policy,
  visit
• Bank for International Settlements www.bis.org
• Federal Reserve Bank www.federalreserve.gov
• Bank of Japan www.boj.or.jp

Web Surf
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Repricing Model

• Repricing or funding gap model based on book
  value.
• Contrasts with market value-based maturity and
  duration models recommended by the Bank for
  International Settlements (BIS).
• Rate sensitivity means time to repricing.
• Repricing gap is the difference between the rate
  sensitivity of each asset and the rate
  sensitivity of each liability RSA - RSL.

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Maturity Buckets

• Commercial banks must report repricing gaps for
  assets and liabilities with maturities of
• One day.
• More than one day to three months.
• More than 3 three months to six months.
• More than six months to twelve months.
• More than one year to five years.
• Over five years.

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Repricing Gap Example

• Assets Liabilities Gap Cum. Gap
• 1-day 20 30 -10 -10
• gt1day-3mos. 30 40 -10
  -20
• gt3mos.-6mos. 70 85 -15
  -35
• gt6mos.-12mos. 90 70 20
  -15
• gt1yr.-5yrs. 40 30 10
  -5
• gt5 years 10 5 5
  0

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Applying the Repricing Model

• DNIIi (GAPi) DRi (RSAi - RSLi) Dri
• Example
• In the one day bucket, gap is -10 million. If
  rates rise by 1,
• DNIIi (-10 million) .01 -100,000.

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Applying the Repricing Model

• Example II
• If we consider the cumulative 1-year gap,
• DNIIi (CGAPi) DRi (-15 million)(.01)
• -150,000.

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Rate-Sensitive Assets

• Examples from hypothetical balance sheet
• Short-term consumer loans. If repriced at
  year-end, would just make one-year cutoff.
• Three-month T-bills repriced on maturity every 3
  months.
• Six-month T-notes repriced on maturity every 6
  months.
• 30-year floating-rate mortgages repriced (rate
  reset) every 9 months.

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Rate-Sensitive Liabilities

• RSLs bucketed in same manner as RSAs.
• Demand deposits and passbook savings accounts
  warrant special mention.
• Generally considered rate-insensitive (act as
  core deposits), but there are arguments for their
  inclusion as rate-sensitive liabilities.

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CGAP Ratio

• May be useful to express CGAP in ratio form as,
• CGAP/Assets.
• Provides direction of exposure and
• Scale of the exposure.
• Example
• CGAP/A 15 million / 270 million 0.56, or
  5.6 percent.

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Equal Changes in Rates on RSAs and RSLs

• Example Suppose rates rise 2 for RSAs and RSLs.
  Expected annual change in NII,
• ?NII CGAP ? R
• 15 million .01
• 150,000
• With positive CGAP, rates and NII move in the
  same direction.

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Unequal Changes in Rates

• If changes in rates on RSAs and RSLs are not
  equal, the spread changes. In this case,
• ?NII (RSA ? RRSA ) - (RSL ? RRSL )

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Unequal Rate Change Example

• Spread effect example
• RSA rate rises by 1.2 and RSL rate rises by 1.0
• ?NII ? interest revenue - ? interest expense
• (155 million 1.2) - (155 million 1.0)
  
• 310,000

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Restructuring Assets and Liabilities

• The FI can restructure its assets and
  liabilities, on or off the balance sheet, to
  benefit from projected interest rate changes.
• Positive gap increase in rates increases NII
• Negative gap decrease in rates increases NII

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Weaknesses of Repricing Model

• Weaknesses
• Ignores market value effects and off-balance
  sheet cash flows
• Overaggregative
• Distribution of assets liabilities within
  individual buckets is not considered. Mismatches
  within buckets can be substantial.
• Ignores effects of runoffs
• Bank continuously originates and retires consumer
  and mortgage loans. Runoffs may be rate-sensitive.

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The Maturity Model

• Explicitly incorporates market value effects.
• For fixed-income assets and liabilities
• Rise (fall) in interest rates leads to fall
  (rise) in market price.
• The longer the maturity, the greater the effect
  of interest rate changes on market price.
• Fall in value of longer-term securities increases
  at diminishing rate for given increase in
  interest rates.

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Maturity of Portfolio

• Maturity of portfolio of assets (liabilities)
  equals weighted average of maturities of
  individual components of the portfolio.
• Principles stated on previous slide apply to
  portfolio as well as to individual assets or
  liabilities.
• Typically, MA - ML gt 0 for most banks and thrifts.

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Effects of Interest Rate Changes

• Size of the gap determines the size of interest
  rate change that would drive net worth to zero.
• Immunization and effect of setting
• MA - ML 0.

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Maturity Matching and Interest Rate Exposure

• If MA - ML 0, is the FI immunized?
• Extreme example Suppose liabilities consist of
  1-year zero coupon bond with face value 100.
  Assets consist of 1-year loan, which pays back
  99.99 shortly after origination, and 1 at the
  end of the year. Both have maturities of 1 year.
• Not immunized, although maturities are equal.
• Reason Differences in duration.

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Duration

• The average life of an asset or liability
• The weighted-average time to maturity using
  present value of the cash flows, relative to the
  total present value of the asset or liability as
  weights.

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Term Structure of Interest Rates

• YTM

YTM
Time to Maturity
Time to Maturity
Time to Maturity
Time to Maturity
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Unbiased Expectations Theory

• Yield curve reflects markets expectations of
  future short-term rates.
• Long-term rates are geometric average of current
  and expected short-term rates.
• _ _
• RN (1R1)(1E(r2))(1E(rN))1/N - 1

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Liquidity Premium Theory

• Allows for future uncertainty.
• Premium required to hold long-term.
• Market Segmentation Theory
• Investors have specific needs in terms of
  maturity.
• Yield curve reflects intersection of demand and
  supply of individual maturities.

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Pertinent Websites

• Bank for International Settlements www.bis.org
• Federal Reserve www.federalreserve.gov
• Bank of Japan www.boj.or.jp

Web Surf
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Overview

• Market value-based model for assessing and
  managing interest rate risk
• Duration
• Computation of duration
• Economic interpretation
• Immunization using duration
• Problems in applying duration

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Price Sensitivity and Maturity

• In general, the longer the term to maturity, the
  greater the sensitivity to interest rate changes.
  
• Example Suppose the zero coupon yield curve is
  flat at 12. Bond A pays 1762.34 in five years.
  Bond B pays 3105.85 in ten years, and both are
  currently priced at 1000.

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Example continued...

• Bond A P 1000 1762.34/(1.12)5
• Bond B P 1000 3105.84/(1.12)10
• Now suppose the interest rate increases by 1.
• Bond A P 1762.34/(1.13)5 956.53
• Bond B P 3105.84/(1.13)10 914.94
• The longer maturity bond has the greater drop in
  price because the payment is discounted a greater
  number of times.

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Coupon Effect

• Bonds with identical maturities will respond
  differently to interest rate changes when the
  coupons differ. This is more readily understood
  by recognizing that coupon bonds consist of a
  bundle of zero-coupon bonds. With higher
  coupons, more of the bonds value is generated by
  cash flows which take place sooner in time.
  Consequently, less sensitive to changes in R.

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Price Sensitivity of 6 Coupon Bond
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Price Sensitivity of 8 Coupon Bond
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Remarks on Preceding Slides

• The longer maturity bonds experience greater
  price changes in response to any change in the
  discount rate.
• The range of prices is greater when the coupon is
  lower.
• The 6 bond shows greater changes in price in
  response to a 2 change than the 8 bond. The
  first bond has greater interest rate risk.

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Duration

• Duration
• Weighted average time to maturity using the
  relative present values of the cash flows as
  weights.
• Combines the effects of differences in coupon
  rates and differences in maturity.
• Based on elasticity of bond price with respect
  to interest rate.

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Duration

• Duration
• D Snt1Ct t/(1r)t/ Snt1 Ct/(1r)t
• Where
• D duration
• t number of periods in the future
• Ct cash flow to be delivered in t periods
• n term-to-maturity r yield to maturity (per
  period basis).

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Duration

• Since the price of the bond must equal the
  present value of all its cash flows, we can state
  the duration formula another way
• D Snt1t ? (Present Value of Ct/Price)
• Notice that the weights correspond to the
  relative present values of the cash flows.

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Duration of Zero-coupon Bond

• For a zero coupon bond, duration equals maturity
  since 100 of its present value is generated by
  the payment of the face value, at maturity.
• For all other bonds
• duration lt maturity

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Computing duration

• Consider a 2-year, 8 coupon bond, with a face
  value of 1,000 and yield-to-maturity of 12.
  Coupons are paid semi-annually.
• Therefore, each coupon payment is 40 and the per
  period YTM is (1/2) 12 6.
• Present value of each cash flow equals CFt (1
  0.06)t where t is the period number.

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Duration of 2-year, 8 bond Face value
1,000, YTM 12
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Special Case

• Maturity of a consol M ?.
• Duration of a consol D 1 1/R

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Duration Gap

• Suppose the bond in the previous example is the
  only loan asset (L) of an FI, funded by a 2-year
  certificate of deposit (D).
• Maturity gap ML - MD 2 -2 0
• Duration Gap DL - DD 1.885 - 2.0 -0.115
• Deposit has greater interest rate sensitivity
  than the loan, so DGAP is negative.
• FI exposed to rising interest rates.

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Features of Duration

• Duration and maturity
• D increases with M, but at a decreasing rate.
• Duration and yield-to-maturity
• D decreases as yield increases.
• Duration and coupon interest
• D decreases as coupon increases

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Economic Interpretation

• Duration is a measure of interest rate
  sensitivity or elasticity of a liability or
  asset
• dP/P ? dR/(1R) -D
• Or equivalently,
• dP/P -DdR/(1R) -MD dR
• where MD is modified duration.

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Economic Interpretation

• To estimate the change in price, we can rewrite
  this as
• dP -DdR/(1R)P -(MD) (dR) (P)
• Note the direct linear relationship between dP
  and -D.

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Semi-annual Coupon Payments

• With semi-annual coupon payments
• (dP/P)/(dR/R) -DdR/(1(R/2)

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An example

• Consider three loan plans, all of which have
  maturities of 2 years. The loan amount is 1,000
  and the current interest rate is 3. Loan 1, is
  an installment loan with two equal payments of
  522.61. Loan 2 is a discount loan, which has a
  single payment of 1,060.90. Loan 3 is
  structured as a 3 annual coupon bond.

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Duration as Index of Interest Rate Risk
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Immunizing theBalance Sheet of an FI

• Duration Gap
• From the balance sheet, EA-L. Therefore,
  DEDA-DL. In the same manner used to determine
  the change in bond prices, we can find the change
  in value of equity using duration.
• DE -DAA DLL DR/(1R) or
• DE -DA - DLkA(DR/(1R))

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Duration and Immunizing

• The formula shows 3 effects
• Leverage adjusted D-Gap
• The size of the FI
• The size of the interest rate shock

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An example

• Suppose DA 5 years, DL 3 years and rates are
  expected to rise from 10 to 11. (Rates change
  by 1). Also, A 100, L 90 and E 10. Find
  change in E.
• DE -DA - DLkADR/(1R)
• -5 - 3(90/100)100.01/1.1 - 2.09.
• Methods of immunizing balance sheet.
• Adjust DA , DL or k.

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Immunization and Regulatory
Concerns

• Regulators set target ratios for a banks capital
  (net worth)
• Capital (Net worth) ratio E/A
• If target is to set ?(E/A) 0
• DA DL
• But, to set ?E 0
• DA kDL

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Limitations of Duration

• Immunizing the entire balance sheet need not be
  costly. Duration can be employed in combination
  with hedge positions to immunize.
• Immunization is a dynamic process since duration
  depends on instantaneous R.
• Large interest rate change effects not accurately
  captured.
• Convexity
• More complex if nonparallel shift in yield curve.

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Convexity

• The duration measure is a linear approximation of
  a non-linear function. If there are large changes
  in R, the approximation is much less accurate.
  All fixed-income securities are convex. Convexity
  is desirable, but greater convexity causes larger
  errors in the duration-based estimate of price
  changes.

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Convexity

• Recall that duration involves only the first
  derivative of the price function. We can improve
  on the estimate using a Taylor expansion. In
  practice, the expansion rarely goes beyond second
  order (using the second derivative).

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Modified duration

• DP/P -DDR/(1R) (1/2) CX (DR)2 or DP/P
  -MD DR (1/2) CX (DR)2
• Where MD implies modified duration and CX is a
  measure of the curvature effect.
• CX Scaling factor capital loss from 1bp rise
  in yield capital gain from 1bp fall in yield
• Commonly used scaling factor is 108.

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Calculation of CX

• Example convexity of 8 coupon, 8 yield,
  six-year maturity Eurobond priced at 1,000.
• CX 108DP-/P DP/P
• 108(999.53785-1,000)/1,000
  (1,000.46243-1,000)/1,000)
• 28.

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Duration Measure Other Issues

• Default risk
• Floating-rate loans and bonds
• Duration of demand deposits and passbook savings
• Mortgage-backed securities and mortgages
• Duration relationship affected by call or
  prepayment provisions.

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Contingent Claims

• Interest rate changes also affect value of
  off-balance sheet claims.
• Duration gap hedging strategy must include the
  effects on off-balance sheet items such as
  futures, options, swaps, caps, and other
  contingent claims.

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Pertinent Websites

• Securities Exchange Commission www.sec.gov

Web Surf