CHAPTER 8-1
Interest Rate Risk 1, Ch 8 Appendix B
© McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.
1
Overview
This chapter discusses the interest rate risk
associated with FIs:
• Federal Reserve monetary policy.
• Repricing, or funding gap, model.
Appendix 8A:
• The Maturity Model.
• www.mhhe.com/saunders11e.
© McGraw Hill LLC
2
Level and Movement of Interest
Rates
Federal Reserve: U.S. central bank.
• Open market operations influence money supply,
inflation, and interest rates.
• Actions of Fed (December 2008) in response to
economic crisis.
• Target rate between 0.0 and 0.25 percent.
• Fed eventually raised interest rates (for the first
time in 10 years) in December 2015.
• Increased rates six more times by June 2018.
© McGraw Hill LLC
3
U.S. T-Bill Rate, 1965 to 2021
FIGURE 8-1 Interest Rate on U.S. Three-month Treasury Bills, 1965 to 2021.
Source: Federal Reserve Board website, various dates. www.federalreserve.gov or Federal Reserve
Bank of St. Louis website research.stlouisfed.org/fred2/graph/?id-TB3Ms.
© McGraw Hill LLC
Access the text alternative for slide images.
4
Central Bank and Interest Rates
Actions mostly target short term rates.
• Focus on federal funds rate, in particular.
Interest rate changes and volatility
increasingly transmitted from country to
country due to increased globalization of
financial markets.
• Statements by Jerome Powell can have dramatic
effects on world interest rates.
© McGraw Hill LLC
5
Appendix 8A
Term Structure of Interest Rates
• Term structure of interest rates compares the
market yields (or interest rates) on securities,
assuming that all characteristics (default risk,
coupon rate, etc) except maturity are the same.
• Maturity Premium (MP): the difference in yields
between long-term bonds and short-term bonds
with same characteristics. It can be positive,
negative, or zero.
© McGraw Hill LLC
Yield Curve
• A graphical representation of yields (or interest
rates) of bonds that have same characteristics
but different maturity
© McGraw Hill LLC
Yield Curve
• A yield curve can be upward-sloping
• Positive maturity premium
© McGraw Hill LLC
Yield Curve
• A yield curve can be downward-sloping
• Negative maturity premium
© McGraw Hill LLC
Yield Curve
• A yield curve can be flat
• Zero maturity premium
© McGraw Hill LLC
Theories of Term Structure of
Interest Rates
• Unbiased expectations theory
• Liquidity premium theory
• Market segmentation theory
© McGraw Hill LLC
Unbiased Expectations Theory
• At a given point in time, the yield curve reflects
the market’s current expectations of future shortterm rates.
• The interest rate on a long-term bond equals the
average of the expected present and future
short-term interest rates over the lifetime of the
bond.
• Assumption: bond holders consider bonds with
different maturities to be perfect substitutes.
© McGraw Hill LLC
Unbiased Expectations Theory
• In equilibrium, the return to holding a four-year
bond to maturity should equal the expected
return to investing in four successive one-year
bond.
• Long-term rates are a geometric average of
current and expected short-term interest rates.
© McGraw Hill LLC
Unbiased Expectations Theory
• Long-term rates are a geometric average of
current and expected short-term interest rates.
(1+1R1)(1+E[2r1])(1+E[3r1])(1+E[4r1]) = (1+1R4)4
•
1RN : actual current interest rates on securities purchased today with
a maturity of N years.
• trN : estimates of future N-year interest rates starting t years into the
future
© McGraw Hill LLC
Unbiased Expectations Theory
Example. Given
1R1 = 2.94%
E[2r1] = 4.00%
E[3r1] = 4.74%
E[4r1] = 5.10%
Find 1R2, 1R3, 1R4
© McGraw Hill LLC
Unbiased Expectations Theory
• Upward-sloping yield curve: the market expects
that short-term rates will rise throughout the
relevant time period.
• Downward-sloping yield curve: the market
expects that short-term rates will fall throughout
the relevant time period.
• Flat yield curve: the market expects that shortterm rates will remain constant throughout the
relevant time period.
© McGraw Hill LLC
Unbiased Expectations Theory
• Consider the assumption again
• bond holders consider bonds with different maturities
to be perfect substitutes.
• Is it realistic? Are investors really risk-neutral?
• In reality, are forward rates perfect predictor of
future interest rates?
• No.
• Because of future uncertainty of returns, there is a
risk in holding long-term securities, and the risk
increases with the security’s maturity.
• Investors are risk-averse
© McGraw Hill LLC
Liquidity Premium Theory
• Bonds of different maturities are imperfect
substitutes.
• Investors prefer to hold shorter-term securities
because they are more liquid with little risk of a
capital loss. Investors must be offered a liquidity
premium to buy longer-term securities that have
higher risk of capital losses.
• The longer the maturity the greater its risk. The
liquidity premium increases as maturity
increases.
© McGraw Hill LLC
Liquidity Premium Theory
• The interest rate on a long-term bond equals the
geometric average of the current and expected
short-term interest rates over the lifetime of the
bond plus a liquidity premium.
1RN = (1+1R1)(1+E[2r1]+L2) … (1+E[Nr1]+LN)
© McGraw Hill LLC
1/N – 1
Liquidity Premium Theory
Example. Given
1R1 = 2.94%
E[2r1] = 4.00%
E[3r1] = 4.74%
E[4r1] = 5.10%
and L2 = 0.10%, L3 = 0.20%, L4 = 0.30%,
Find 1R2, 1R3, 1R4
© McGraw Hill LLC
Liquidity Premium Theory
Example. Given
1R1 = 2.94%
E[2r1] = 4.00%
E[3r1] = 4.74%
E[4r1] = 5.10%
and L2 = 0.10%, L3 = 0.20%, L4 = 0.30%,
Find 1R2, 1R3, 1R4
© McGraw Hill LLC
Market Segmentation Theory
• Bonds of different maturities are not perfect
substitutes. Individual investors have specific
maturity preferences.
• For example, banks might prefer to hold shortterm U.S. Treasury bills because of the shortterm nature of their deposit liabilities, while
insurance companies might prefer to hold longterm U.S. Treasury bonds because of the longterm nature of their life insurance contractual
liabilities.
© McGraw Hill LLC
Market Segmentation Theory
• Therefore, interest rates are determined by
distinct supply and demand conditions within a
particular maturity bucket or market segment.
© McGraw Hill LLC
Market Segmentation Theory
• Therefore, interest rates are determined by
distinct supply and demand conditions within a
particular maturity bucket or market segment.
© McGraw Hill LLC
Forecasting Interest Rates
• As interest rates change, so do the values of
financial securities. The ability to predict or
forecast interest rates is critical to the profitability
of Fis.
• The unbiased expectations theory can be used
to forecast (short-term) interest rates in the
future.
• A forward rate is an expected rate on a shortterm security that is to be originated at some
point in the future.
© McGraw Hill LLC
Forecasting Interest Rates
• To find a forward rate on a one-year security to
be issued one year from today, 2f1, we rewrite
the unbiased expectations theory equation.
(1+1R1)(1+2f1) = (1+1R2)2
Solving for 2f1, we get
2 / (1+ R ) – 1
f
=
(1+
R
)
2 1
1 2
1 1
In general
N / (1+ R ) N – 1 – 1
f
=
(1+
R
)
N 1
1 N
1 1
© McGraw Hill LLC
Forecasting Interest Rates
• Example
Given
1R1 = 2.423%
1R2 = 2.641%
1R3 = 2.757%
1R4 = 2.814%
Find 2f1, 3f1, 4f1.
© McGraw Hill LLC
CHAPTER 8-2
Interest Rate Risk 1
© McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.
1
Repricing Model
1
Repricing, or funding gap, model is a simple
model used by small FIs in the U.S.
• Essentially a book value accounting cash flow
analysis of the repricing gap between the
interest income earned on an FI’s assets and the
interest expense paid on its liabilities over a
particular period of time.
Contrasts with market value-based maturity
and duration models in appendix.
© McGraw Hill LLC
2
Repricing Model
2
• Rate sensitivity means repricing at (or
near) current market interest rates within a
specified time horizon.
• Repricing gap is the difference between
rate-sensitive assets (RSAs) and ratesensitive liabilities (RSLs).
• Refinancing risk.
• Reinvestment risk.
© McGraw Hill LLC
3
Maturity Buckets
Commercial banks report quarterly repricing
gaps for assets and liabilities with maturities
of:
• One day.
• More than one day to three months.
• More than three months to six months.
• More than six months to twelve months.
• More than one year to five years.
• More than five years.
© McGraw Hill LLC
4
Repricing Gap: Example
(1)
Assets
(2)
Liabilities
(3)
Gap
(4)
Cumulative
Gap
$ 20
$ 30
−$10
−$10
2. More than 1 day to 3 months.
30
40
−10
−20
3. More than 3 months to 6 months.
70
85
−15
−35
4. More than 6 months to 12 months.
90
70
+20
−15
5. More than 1 year to 5 years.
40
30
+10
−5
6. Over 5 years.
10
5
+5
0
260
260
1. One day.
© McGraw Hill LLC
5
Applying the Repricing Model
1
ΔNlli = ( GAPi ) ΔR i = ( RSA i − RSLi ) ΔR i
• Example 1:
In the one-day bucket, gap is −$10 million.
If rates rise by 1%,
ΔNlli = ( −$10 million )  .01 = −$100,000
© McGraw Hill LLC
6
Applying the Repricing Model
2
• Example 2:
If we consider the cumulative 1-year gap,
ΔNlli = ( CGAP ) ΔR i = ( −$15 million )(.01)
= −$150,000
© McGraw Hill LLC
7
Balance Sheet Example
• Example 2:
If we consider the cumulative 1-year gap,
ΔNlli = ( CGAP ) ΔR i = ( −$15 million )(.01)
= −$150,000
© McGraw Hill LLC
8
Rate-Sensitive Assets
Examples from hypothetical balance sheet:
• Short-term consumer loans: Repriced at yearend, 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.
© McGraw Hill LLC
9
Rate-Sensitive Assets
Examples from hypothetical balance sheet:
• Total one-year rate-sensitive assets (RSAs) of
$155 million.
• The remaining $115 million of assets are not rate
sensitive over the one-year repricing horizon.
© McGraw Hill LLC
10
Rate-Sensitive Liabilities
1
RSLs bucketed in same manner as RSAs.
Demand deposits warrant special attention.
• Generally considered rate-insensitive (act as
core deposits).
• Not included as RSLs, but arguments can be
made for their inclusion as RSLs.
© McGraw Hill LLC
11
Rate-Sensitive Liabilities
2
Liability items that fit the one-year rate or
repricing sensitivity test.
• Three-month CDs.
• Three-month bankers acceptances.
• Six-month commercial paper.
• One-year time deposits.
© McGraw Hill LLC
12
Demand Deposits as RSLs
Against Inclusion.
For Inclusion.
• Explicit interest rate on
demand deposits is zero
by regulation.
• Pay implicit interest
because FIs do not
charge fees that fully
cover their costs for
checking services.
• Rates paid by FIs do not
fluctuate directly with
changes in the general
level of interest rates.
• Risk of runoff.
• Act as core deposits.
© McGraw Hill LLC
13
Balance Sheet Example
Cumulative one-year repricing gap (CGAP)
CGAP = One-year RSAs – One-year RSLs
= $155 million – $140 million
= $15 million
© McGraw Hill LLC
14
GAP Ratio
May be useful to express interest rate
sensitivity in ratio form as CGAP/Assets,
referred to as “gap ratio.”
• Provides direction and scale of exposure.
Example:
• Gap ratio = CGAP/A = $15 million / $270 million
= 0.056, or 5.6 percent.
© McGraw Hill LLC
15
Equal Rate Changes on RSAs,
RSLs
• Example 8-1: Suppose rates rise 1% for RSAs
and RSLs. Expected annual change in NII,
Nll = CGAP   R
= $15 million  .01
= $150,000
• CGAP is positive, change in NII is positively
related to change in interest rates.
• CGAP is negative, change in NII is negatively
related to change in interest rates.
© McGraw Hill LLC
16
Unequal Changes in Rates
• If changes in rates on RSAs and RSLs are
not equal, the spread changes.
• In this case,
Nll = ( RSA  R RSA ) − ( RSL  R RSL )
© McGraw Hill LLC
17
Unequal Changes in Rates
• If changes in rates on RSAs and RSLs are
not equal, the spread changes.
• In this case,
Nll = ( RSA  R RSA ) − ( RSL  R RSL )
© McGraw Hill LLC
18
Unequal Rate Change Example
• Example 8-2:
RSA rate rises by 1.2% and RSL rate rises
by 1.0%.
Nll =  interest revenue −  interest expense
= ( $155 million  1.2% ) − ( $155 million  1.0% )
= $310,000
© McGraw Hill LLC
19
Weaknesses of Repricing Model
Ignores market value effects of interest rate
changes.
• The present values of the cash flows on assets and
liabilities change as interest rate changes. Thus, the
market value of the assets and liabilities change.
Overaggregation.
• Distribution of assets and liabilities within individual
buckets is not considered.
• Mismatches within buckets can be substantial.
© McGraw Hill LLC
20
1
Weaknesses of Repricing Model
2
Ignores effects of rate-insensitive runoffs.
• Bank continuously originates and retires consumer and
mortgage loans.
• Runoff of rate-insensitive asset/liability is rate-sensitive.
Off-balance-sheet items are not included
when considering cash flows.
• Hedging effects of off-balance-sheet items not
captured.
• Example: Futures contracts.
© McGraw Hill LLC
21
End of Main Content
© McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.
22
CHAPTER 8-3
Interest Rate Risk 1 – Appendix 8A
© McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.
1
Weaknesses of Repricing Model
• Repricing model uses book value
accounting.That is, the assets and liabilities are
recorded at historic values.
• It ignores market value effects of interest rate
changes.
• The present values of the cash flows on assets
and liabilities change as interest rate changes.
Thus, the market value of the assets and
liabilities change.
© McGraw Hill LLC
2
1
Market Value Approach
• Assets and liabilities should be revalued to
reflect current market conditions.
• Market value accounting approach, used in
maturity model (Ch 8 appendix A) as well as in
duration model (Ch 9), the effects of interest rate
changes on the market values of assets and
liabilities are explicitly taken into account.
© McGraw Hill LLC
3
Interest Rate and Bond Price
• FI holds a bond that has
• One year to maturity
• Face value (F) of $100
• One single annual coupon rate of 0.10 (reflecting the
current interest rate
• The fair market price of the one-year bond, P1B,
is equal to the present value of the cash flows on
the bond
P1B = (F+C) / (1+R) = ($100+$10)/(1+0.10) = $100
© McGraw Hill LLC
4
Interest Rate and Bond Price
• If the central bank increases the interest to 0.11
• P1B = (F+C) / (1+R) = ($100+$10)/(1+0.11) =
$99.1
• Capital loss (Δ P1B) = - $0.90
• Percent change in price = - 0.90%
© McGraw Hill LLC
5
Interest Rate and Bond Price
• Therefore, there is an inverse relationship
between price of a bond and the yield of the
bond. (ΔP/ΔR < 0)
• For an FI,
• Rising interest rates reduces the market value of the
assets. It’s a bad news
• Rising interest rates reduces the market value of the
liabilities. It’s a good news
• Falling interest rates have the reverse effect: They
increase the market value of both assets and
liabilities.
© McGraw Hill LLC
6
Interest Rate and Bond Price
• FI holds a bond that has
• Three year to maturity
• Face value (F) of $100
• Annual coupon rate of 0.10 (reflecting the current interest rate
• Find the change in the market value,P3B, of the bond if the central
bank increases the interest rate from 0.10 to 0.11
• With R = 0.10
• P3B = ($10)/(1+0.10) + ($10)/(1+0.10)2 + ($100+$10)/(1+0.10)3 = $100
• With R = 0.11
• P3B = ($10)/(1+0.11) + ($10)/(1+0.11)2 + ($100+$10)/(1+0.11)3 = $97.56
• Δ P1B = -$2.44
• %Δ P1B = -2.44%
© McGraw Hill LLC
7
Interest Rate and Bond Price
Lesson learned:
1. A rise (fall) in interest rates generally leads to a fall
(rise) in the market value of an asset or liability
2. The longer the maturity of a fixed-income asset or
liability, the larger the fall (rise) in market value for any
given interest rate increase (decrease).
3. The fall in the value of longer-term securities increases
at a diminishing rate for any given increase in interest
rates.
© McGraw Hill LLC
8
Interest Rate and Bond Price
© McGraw Hill LLC
9
Maturity Model
• MA: the weighted average maturity of an FI’s assets
• MA = WA1MA1 + WA2MA2 + … + WANMAN
• ML: the weighted average maturity of an FI’s liabilities
• ML = WL1ML1 + WL2ML2 + … + WLNMLN
where
Mi = weighed-average maturity of an FI’s assets or liabilities (i = A or L)
Wij = importance of each asset (liability) in the portfolio as measured by
the market value of that asset (liability) position relative to the market
value of all the assets (liabilities).
Mij =Maturity of the jth asset (or liability), j = 1, … , N
© McGraw Hill LLC
10
Maturity Model
Same three principles apply in a portfolio context
1. A rise (fall) in interest rates generally leads to a fall
(rise) in the market values of an asset or liability
portfolios
2. The longer the maturity of the asset or liability portfolio,
the larger the fall (rise) in market value for any given
interest rate increase (decrease).
3. The fall in the value of the asset or liability portfolio
increases with its maturity at a diminishing rate for any
given increase in interest rates.
© McGraw Hill LLC
11
Maturity Model
Same three principles apply in a portfolio context
1. A rise (fall) in interest rates generally leads to a fall
(rise) in the market values of an asset or liability
portfolios
2. The longer the maturity of the asset or liability portfolio,
the larger the fall (rise) in market value for any given
interest rate increase (decrease).
3. The fall in the value of the asset or liability portfolio
increases with its maturity at a diminishing rate for any
given increase in interest rates.
© McGraw Hill LLC
12
Maturity Model
• Maturity gap, MA – ML, is positive.
• This is the case of most commercial banks.
• E=A–L
• ΔE = ΔA – ΔL
© McGraw Hill LLC
13
Maturity Model
• Coupon rate = 0.10 for both A and L
• Interest rate = 0.10 before.
• Interest rate changes to 0.11
• A falls by 2.44, L falls by 0.81, thus E falls by
1.63 (That’s 16.3% decrease in FI’s net worth!)
© McGraw Hill LLC
14
Maturity Model
• Interest rate changes to 0.17 this time
• A falls by 15.47, L falls by 5.38, thus E falls by
10.09 (The FI becomes insolvent!)
© McGraw Hill LLC
15
Maturity Model
• Extreme maturity mismatch (30 yrs vs 1 yr)
• A small increase in interest rate (by 1.5%) makes
the FI completely insolvent
© McGraw Hill LLC
16
Maturity Model - Weaknesses
The weaknesses of the maturity model
1. Does not account for the degree of leverage in the FI’s
balance sheet
2. Ignores the timing of the cash flows from the FI’s assets
and liabilities
© McGraw Hill LLC
17