Lecture 5rssrtrs

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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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Identify RSAs and RSLs over an One-Year Interval

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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)

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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.

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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.

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U.S. Three-Month CD Rates vs. Prime Rates (1990-2012)

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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.

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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.

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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.

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The Overaggregation Problem

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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.

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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.

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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?

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Market Value Effects of Changes in Interest Rates

Before the rise in yields

After the rise in yields

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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.

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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)

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A Graphic Illustration of the Derivation of Duration

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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.

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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’.

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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

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