Chapter Eight

1
Chapter Eight

• Interest Rate Risk I

2
Overview

• This chapter discusses the interest rate risk
  associated with financial intermediation
• Federal Reserve monetary policy
• Repricing model
• Maturity model
• Duration model
• Term structure of interest rate risk
• Theories of the term structure of interest rates

3
Central Bank Interest Rate Risk

• Federal Reserve Bank U.S. central bank
• Open market operations influence money supply,
  inflation, and interest rates
• Targeting of bank reserves in U.S. proved
  disastrous
• Oct-1979 to Oct-1982, nonborrowed reserves target
  regime.
• Implications of reserves target policy
• Increases importance of measuring and managing
  interest rate risk.

4
Central Bank and Interest Rate Risk

• Effects of interest rate targeting.
• Lessens interest rate risk
• Greenspan view Risk Management
• Focus on Federal Funds Rate
• Simple announcement of Fed Funds increase,
  decrease, or no change.

5
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.
• Refinancing risk

6
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.

7
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

8
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,
• DNII(1) (-10 million) .01 -100,000.

9
Applying the Repricing Model

• Example II
• If we consider the cumulative 1-year gap,
• DNII (CGAPone year) DR (-15 million)(.01)
• -150,000.

10
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.

11
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.

12
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.

13
Equal Rate Changes on RSAs, 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.
• Change proportional to CGAP

14
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 )

15
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

16
Restructuring Assets 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
• Example State Street Boston
• Good luck?
• Or Good Management?

17
Weaknesses of Repricing Model

• Weaknesses
• Ignores market value effects and off-balance
  sheet (OBS) 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.

18
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.

19
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, maturity gap, MA - ML gt 0 for most
  banks and thrifts.

20
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.

21
Maturities 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 maturity gap equals zero.
• Reason Differences in duration
• (See Chapter 9)

22
Maturity Model

• Leverage also affects ability to eliminate
  interest rate risk using maturity model
• Example
• Assets 100 million in one-year 10-percent
  bonds, funded with 90 million in one-year
  10-percent deposits (and equity)
• Maturity gap is zero but exposure to interest
  rate risk is not zero.

23
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.

24
Term Structure of Interest Rates

• YTM

YTM
Time to Maturity
Time to Maturity
Time to Maturity
Time to Maturity
25
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

26
Liquidity Premium Theory

• Allows for future uncertainty.
• Premium required to hold long-term.

27
Market Segmentation Theory

• Investors have specific needs in terms of
  maturity.
• Yield curve reflects intersection of demand and
  supply of individual maturities.

28
Pertinent Websites

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

29
Chapter Nine

• Interest Rate Risk II

30
Overview

• This chapter discusses a market value-based model
  for assessing and managing interest rate risk
• Duration
• Computation of duration
• Economic interpretation
• Immunization using duration
• Problems in applying duration

31
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.

32
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.

33
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.

34
Price Sensitivity of 6 Coupon Bond
35
Price Sensitivity of 8 Coupon Bond
36
Remarks on Preceding Slides

• In general, 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.

37
Extreme examples with equal maturities

• Consider two ten-year maturity instruments
• A ten-year zero coupon bond
• A two-cash flow bond that pays 999.99 almost
  immediately and one penny, ten years hence.
• Small changes in yield will have a large effect
  on the value of the zero but essentially no
  impact on the hypothetical bond.
• Most bonds are between these extremes
• The higher the coupon rate, the more similar the
  bond is to our hypothetical bond with higher
  value of cash flows arriving sooner.

38
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.

39
Duration

• Duration
• D Snt1CFt t/(1R)t/ Snt1 CFt/(1R)t
• Where
• D duration
• t number of periods in the future
• CFt cash flow to be delivered in t periods
• n term-to-maturity
• R yield to maturity.

40
Duration

• Since the price (P) 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 CFt/P)
• Notice that the weights correspond to the
  relative present values of the cash flows.

41
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

42
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.

43
Duration of 2-year, 8 bond Face value
1,000, YTM 12
44
Special Case

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

45
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.

46
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

47
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.

48
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.

49
Semi-annual Coupon Payments

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

50
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 a two-payment loan with two equal
  payments of 522.61 each.
• Loan 2 is structured as a 3 annual coupon bond.
  
• Loan 3 is a discount loan, which has a single
  payment of 1,060.90.

51
Duration as Index of Interest Rate Risk
52
Immunizing the Balance 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))

53
Duration and Immunizing

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

54
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.

55
Immunization and Regulatory Concerns

• Regulators set target ratios for an FIs 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

56
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.

57
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.

58
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).

59
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.

60
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.

61
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.

62
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.

63
Pertinent Websites

• Bank for International Settlements www.bis.org
• Securities Exchange Commission www.sec.gov
• The Wall Street Journal
• www.wsj.com