Upload
student
View
6
Download
2
Tags:
Embed Size (px)
DESCRIPTION
aete5e5ateatatae4t
Citation preview
Lecture 5: Interest Rate Risk (Part I)
Dr Lixiong Guo
Semester 2, 2015
Topics for Today
What affect the level and movement in interest rate?
Repricing model.
– Equal change in interest rates.
– Unequal change in interest rates.
Duration model.
2
The Level and Movement of Interest Rates
While many factors influence the level and movement of interest
rates, it is the central bank’s monetary policy strategy that most
directly underlies the level and movement of interest rates.
While central bank’s actions are targeted mostly at short-term
rates, changes in short-term rates usually feed through to the
whole term structure of interest rates.
The increased financial market integration over the last decade
has also affected interest rates. Financial market integration
increases the speed with which interest rate changes and
associated volatility are transmitted among countries, making the
control of interest rates by the central bank more difficult and less
certain than before.
3
Interest Rate on U.S. 91-Day T-Bills, 1965-2012
4
Interest Rate Risk of FIs
Asset transformation naturally results in mismatch in asset &
liability maturities.
If interest rate is constant over time and deposits can be rolled
over at the same rate, this is no risk to the bank.
– The interest spread is locked in.
However, when interest rate changes over time, the bank is
exposed to interest rate risk.
– Net interest income (NII) is affected
• Measured by the Repricing Model
– Net worth (market value of equity) is affected.
• Measured by the Duration Model
5
Refinancing Risk
Refinancing risk
Suppose the cost of funds is 9% per year and the return on
assets is 10% per year. Over the first year, the FI locks in a profit
spread of 1%. However, its profits for the second year are
uncertain. If interest rates were to rise and the FI can only borrow
new one-year liabilities at 11% in the second year, its profit
spread in the second year would be negative 1% and the FI
would lose 0.01 × $100𝑚 = $1 𝑚 in the second year.
6
0 1
Liabilities, $100
1 2 0
Assets, $100 𝑅𝐴1
𝑅𝐴2
𝑅𝐿1
𝑅𝐿2
Reinvestment Risk
Reinvestment risk
Suppose the cost of funds is 9% per year and the return on
assets is 10% per year. Over the first year, the FI locks in a profit
spread of 1%. However, its profits for the second year are
uncertain. If interest rates were to all and the FI can only invest in
new one-year assets at 8% in the second year, its profit spread in
the second year would be negative 1% and the FI would lose
0.01 × $100𝑚 = $1 𝑚 in the second year.
7
1 2 0
Liabilities $100
0 1
Assets, $100
𝑅𝐿1
𝑅𝐿2
𝑅𝐴1
𝑅𝐴2
Repricing Model
Repricing gap is the difference between the amounts of assets
and liabilities whose interest rates will be repriced or changed
over some future period.
– The assets are called Risk-sensitive assets (RSA)
– The liabilities are called Risk-sensitive liabilities (RSL)
Repricing can be the result of
– A rollover of an asset or liability, e.g. a loan is paid off at or prior to
maturity and the funds are used to issue a new loan at current
market rates.
– Or it can occur because the asset or liability is a variable-rate
instrument, e.g. a variable rate mortgage whose interest rate is
reset every quarter based on movements in a prime rate.
8
Repricing Model (cont.)
Repricing gap provides a measure of an FI’s net interest income
exposure to interest rate changes in different maturity buckets.
Assume equal changes in interest rates on RSAs and RSLs:
∆𝑁𝐼𝐼𝑖= (𝐺𝐴𝑃𝑖)∆𝑅𝑖 = 𝑅𝑆𝐴𝑖 − 𝑅𝑆𝐿𝑖 ∆𝑅𝑖
– ∆𝑁𝐼𝐼𝑖 is the change in net interest income in maturity bucket 𝑖.
– ∆𝑅𝑖 is the change in the level of interest rates impacting assets and
liabilities in the 𝑖𝑡ℎ bucket.
– 𝐺𝐴𝑃𝑖 is called the repricing gap in maturity bucket 𝑖.
A negative repricing gap (𝑅𝑆𝐴𝑖 < 𝑅𝑆𝐿𝑖) exposes the bank to
refinancing risk in that a rise in interest rates would lower the FI’s
NII.
A positive repricing gap (𝑅𝑆𝐴𝑖 > 𝑅𝑆𝐿𝑖) exposes the bank to
reinvestment risk in that a drop in interest rates would lower the
FI’s NII.
9
Repricing Model
FI can restructure assets and liabilities, on- or off- the balance
sheet, to benefit from projected interest rate changes
– When projecting an increase in interest rates, maintaining a
positive repricing gap will increase net interest income.
– When projecting a decrease in interest rates, maintaining a
negative repricing gap will increase net interest income.
10
Cumulative Gap
If a bank keeps track of repricing gaps for several consecutive
repricing intervals, the repricing gap over a broader repricing
interval can be calculated by summing over the repricing gaps
over the narrower intervals contained by the broader interval. The
sum of the repricing gaps is called the cumulative gap.
11
Applying the Repricing Model
The first step in applying the repricing model involves identifying
assets and liabilities that will be repriced over a certain time
horizon (or maturity bucket).
Example: Suppose interest rate is expected to rise by 1% over
the next 3 months, what is the expected annualized change in
the bank’s net interest income over the next 3 months?
– 𝐺𝐴𝑃 = −$20 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
– ∆𝑁𝐼𝐼 = −$20 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 1% = −$200,000
With interest rate quoted on per annum basis, the repricing
model calculates the annualized change in an FI’s net interest
income.
The formula assumes that both RSA and RSL are repriced at the
beginning of the repricing interval.
12
Identify RSAs and RSLs over an One-Year Interval
13
Identify RSAs and RSLs over an One-Year Interval (cont.)
Rate Sensitive Assets:
– Short-term consumer loans ($50).
• If repriced at year-end, would just make one-year cutoff.
– Three-month T-bills repriced on maturity every 3 months ($30)
– Six-month T-notes repriced on maturity every 6 months ($30)
– 30-year floating-rate mortgages repriced (rate reset) every 9
months ($40)
Rate Sensitive Liabilities
– Three-month CDs ($40)
– Three-month bankers acceptances ($20)
– Six-month commercial papers ($60)
– One-year time deposits ($20)
14
Identify RSAs and RSLs over an One-Year Interval (cont.)
Demand deposits and passbook savings accounts are generally
considered to be rate-insensitive (act as core deposits).
– The explicit interest rate on demand deposits is zero by regulation.
Further, although explicit interest is paid on transaction accounts
such as NOW accounts, the rates paid by FI do not fluctuate
directly with changes in the general level of interest rates.
However, there are arguments for their inclusion as rate-sensitive
liabilities.
– When interest rates rise, individuals may draw down their demand
deposits or savings account and move the money to alternative
instruments paying a higher interests, forcing the bank to replace
them with more expensive fund substitutions.
15
Alternative Expressions of Repricing Gap
Often FIs express interest rate sensitivity as a percentage of
assets, 𝐶𝐺𝐴𝑃
𝐴𝑠𝑠𝑒𝑡𝑠.
Expressing the repricing gap this way is useful since It tells us
– (1) the direction of the interest rate exposure
– (2) the scale of that exposure.
Alternatively, FIs calculate a gap ratio defined as rate-sensitive
assets divided by rate-sensitive liabilities, 𝑅𝑆𝐴
𝑅𝑆𝐿.
– A gap ratio below 1 indicates that the FI is exposed to a refinancing
risk.
– A gap ratio greater than 1 indicates that the FI is exposed to a
reinvestment risk.
16
U.S. Three-Month CD Rates vs. Prime Rates (1990-2012)
17
Unequal Changes in Rates on RSAs and RSLs
If changes in rates on RSAs and RSLs are not equal, the NII
effect should be calculated as
∆𝑁𝐼𝐼 = 𝑅𝑆𝐴 × ∆𝑅𝑟𝑠𝑎 − 𝑅𝑆𝐿 × ∆𝑅𝑟𝑠𝑙
This can be decomposed into a CGAP effect and a Spread effect
as follows:
∆𝑁𝐼𝐼 = 𝑅𝑆𝐴 − 𝑅𝑆𝐿 × ∆𝑅𝑟𝑠𝑎 + 𝑅𝑆𝐿 × (∆𝑅𝑟𝑠𝑎−∆𝑅𝑟𝑠𝑙)
When changes in interest rates on RSAs and RSLs are unequal,
there is a spread effect in addition to the GAP effect.
18
CGAP effect Spread effect
Unequal Changes in Rates on RSAs and RSLs
The spread effect is such that change in spread (∆𝑅𝑟𝑠𝑎−∆𝑅𝑟𝑠𝑙) is
always positively related to the change in net interest income
∆𝑁𝐼𝐼.
When the CGAP effect and the Spread effect work in opposite
directions, the change in net interest income cannot be predicted
without knowing the size of CGAP and the Change in Spread.
19
Weaknesses of the Repricing Model
It ignores market value effects of interest rate changes.
– The present values of cash flows on assets and liabilities change
in addition to the immediate interest received and paid on them,
as interest rates change.
– The repricing model ignores the market value effect – implicitly
assuming a book value accounting approach.
– As such, the repricing gap is only a partial measure of the true
interest rate risk exposure of an FI.
It ignores information regarding the distribution of assets and
liabilities within each maturity bucket.
– Assets and liabilities may be repriced at different time within the
bucket.
– The shorter the range over which bucket gaps are calculated, the
smaller this problem is.
20
The Overaggregation Problem
21
Weaknesses of the Repricing Model
It ignores runoff cash flows.
In the simple repricing model we’ve discussed so far, rate
sensitive assets and liabilities are defined by the original
maturities of these instruments. In reality, FIs continuously
originates and retires assets as it creates new liabilities. For
example, today, some 30-year original maturity mortgages may
have only 1 year left before they mature.
In addition, even if an asset or liability is rate insensitive, virtually
all assets and liabilities pay some principals and/or interest back
to the FI in any given year. As a result, the FI receives a runoff
cash flow from its rate-insensitive portfolio that can be reinvested
at current market rates.
22
Weaknesses of the Repricing Model
The FI manager can deal easily with this in the repricing model by
identifying for each asset or liability item the estimated dollar cash
flow that will run off within the next repricing interval and adding
these amounts to the value of rate sensitive assets and liabilities.
It ignores cash flows from Off-Balance-Sheet Activities
– The RSAs and RSLs used in repricing model generally include only
the assets and liabilities listed on the balance sheet. Changes in
interest rates will affect the cash flows on many off-balance-sheet
instruments as well.
– For example, an FI might have hedged its interest rate risk with an
interest futures contract. The mark-to-market process could
produce a daily cash flow for the FI that may offset any on-
balance-sheet gap exposure.
23
Market Value Effects of Changes in Interest Rates
Change in interest rate leads to changes in market value of
assets and liabilities and thus the net worth of a FI.
Net worth is the difference between the market value of assets
and liabilities.
– This is different from book values of assets and liabilities used in
accounting.
Example: Assume a bank’s asset consists of $100 million face
value 5-year U.S treasury zero-coupon bonds and liability
consists of $75 million 1-year CDs. The yield on the U.S treasury
is 6.65% and that on the 1-year CD is 5.75%. If all yields increase
by 100 basis points, what happens to the market value of equity?
24
Market Value Effects of Changes in Interest Rates
Before the rise in yields
After the rise in yields
25
Assets Liabilities and Equity
Government Bond $72.48 CD $70.92
Equity $1.56
Total $72.48 Total $72.48
Assets Liabilities and Equity
Government Bond $69.17 CD $70.26
Equity -$1.09
Total $69.17 Total $69.17
Market Value Effects of Changes in Interest Rates
In the example above, we see that a moderate increase in
interest rates results in insolvency of the bank.
– The reason is the mismatch of maturities of assets and liabilities.
When interest rates rise, the value of longer-maturity assets fall by
more than the value of shorter-maturity liabilities.
The maturity model measures the effect of interest changes on an
FI’s net worth using the weighted average maturity gap between
assets and liabilities: 𝑀𝐴 −𝑀𝐿.
However, 𝑀𝐴 −𝑀𝐿 = 0 does not immunize the FI from interest
rate risk because
– The timing of intermittent cash flows can still be different.
– Even the timings are the same, the amounts of assets and
liabilities are different due to leverage.
26
Duration Model
The appropriate measure of the market value effect of interest
rate changes is duration.
We now derive the duration measure from first principles.
As a first order approximation, we can calculate the change in
bond price, ∆𝑃, for a change in interest rate, ∆𝑅, by:
∆𝑃 ≈𝑑𝑃
𝑑𝑅∆𝑅 (1)
The price of a bond can be written as
𝑃 = 𝐶𝑡
(1 + 𝑅)𝑡
𝑁
𝑡=1
Hence, 𝑑𝑃
𝑑𝑅= −
𝐶𝑡
(1+𝑅)𝑡+1× 𝑡𝑁
𝑡=1 (2)
27
A Graphic Illustration of the Derivation of Duration
28
𝑇𝑟𝑢𝑒 ∆𝑃
𝑑𝑃
𝑑𝑅∆𝑅
∆𝑅
𝑅
𝐸𝑟𝑟𝑜𝑟
𝑃
𝑅
𝑆𝑙𝑜𝑝𝑒 =𝑑𝑃
𝑑𝑅
Duration Model
Plug the expression for 𝑑𝑃
𝑑𝑅 into equation (1), we get
∆𝑃 = − 𝐶𝑡
1+𝑅 𝑡+1× 𝑡 × ∆𝑅𝑁
𝑡=1 = − 𝐶𝑡
(1+𝑅)𝑡× 𝑡 ×
∆𝑅
1+𝑅𝑁𝑡=1 (3)
where 𝐶𝑡
(1+𝑅)𝑡= 𝑃𝑉𝑡 is simply the present value of the cash flow at
time 𝑡.
Divide both sides of equation (3) by the bond price 𝑃, we have
∆𝑃
𝑃= −
𝑃𝑉𝑡
𝑃𝑁𝑡=1 × 𝑡 ×
∆𝑅
1+𝑅
We denote the term 𝑃𝑉𝑡
𝑃𝑁𝑡=1 × 𝑡 by 𝐷, then
∆𝑃
𝑃= −𝐷 ×
∆𝑅
1+𝑅 (4)
𝐷 𝑖𝑠 𝑤ℎ𝑎𝑡 𝑤𝑒 𝑐𝑎𝑙𝑙 𝑡ℎ𝑒 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛.
29
Duration
Equation (4) defines the economic meaning of Duration.
Duration is the proportional constant that relates percentage
change in 𝑃 to a small change in 𝑅 in the equation above.
Duration is calculated as
D = 𝑃𝑉𝑡
𝑃𝑁𝑡=1 × 𝑡 where 𝑃𝑉𝑡 =
𝐶𝑡
(1+𝑅)𝑡
Hence, duration equals to the weighted-average time to maturity
on an instrument using the relative present value of the cash
flows as weights.
The unit of duration is ‘time’. The ‘weights’ are ratios thus ‘pure
numbers’.
30
Duration Model
What is the duration of a 5-year zero-coupon bond?
Is the duration of coupon-bearing bond greater than or less than
its maturity?
A console bond pays a fixed coupon each year and it never
matures. However, its duration is not infinity
– Price of a console bond 𝑃 = 𝐶
(1+𝑅)𝑡=𝐶
𝑅∞𝑡=1
– 𝐷𝑐 = 1 +1
𝑅
If we know the duration of a debt instrument, we can calculate
change in 𝑃 for a small change in 𝑅 by:
∆𝑃 = −𝐷 × 𝑃 ×∆𝑅
1+𝑅
31