11-Interest Rate Risk

Review Interest Rates are determined by supply and

demand, are moving all the time, and can be difficult to forecast.

The yield curve is generally upward sloping

Interest Rate Risk: The uncertainty surrounding future interest rates. Unforeseen parallel shifts in the yield curve Unforeseen changes in the slope of the yield curve

Our Focus

Where We are Going Dollar Gap

Method to understand the impact of interest rate risk on bank profits

Simple, and requires some ad-hoc assumptions Not discussed in Bodie-Kane-Marcus

Duration Method to understand the impact of interest rate

risk on the value of bank shareholder equity More elegant and mathematically intense The focus of the reading in Bodie-Kane-Marcus Used extensively as well by bond traders

Interest Rate Risk Banks assets

Generally long-term, fixed rate Bank liabilities

Generally short term, variable rate

Impact on profits: Rates increase

Interest received stays fixed Interest paid increases Profits decrease

Interest-Rate Sensitive An asset or liability whose rate is reset within

some “short period of time”e.g. 0-30 days, 1-year, etc.

Interest rate sensitive assets: Short-term bond rolled over into other short-term

bonds Variable rate loans

Interest rate sensitive liabilities: Short-term deposits

Interest-Rate Risk Example

Assets: 50 billion 5B IRS; rate = 8% per year

Liabilities: 40 billion 24B IRS; rate = 5% per year

Profits ($billion) 50(.08)-40*(.05) = 2

Parallel Shift in Yield Curve Suppose all rates increase by 1%

Assets IRS (5B): rate = 9% Not IRS (45B): rate = 8%

Liabilities IRS (24B): rate = 6% Not IRS (16B): rate = 5%

Profits After Rate Increase Interest Expenses ($billion)

16(.05)+24(.06)=2.24

Interest revenues ($billion) 5*(.09)+45*(.08)=4.05

Profits Before: 2 billion After:4.05-2.24=1.81 billion Decrease: 0.19 billion or 9.5% drop in profits

(1.81/2)-1=.095

Gap Analysis

Gap = IRSA – IRSL

IRSA = dollar value of interest rate sensitive assets

IRSL = dollar value of interest rate sensitive liabilities

Gap From Previous Example

IRSA (millions) = 5 IRSL (millions) = 24

Dollar Gap (millions): 5 – 24 = -19

Change in profits= Dollar Gap i From previous example:

Change in profits (millions) -19 .01 = -0.19

Gap Analysis

If the horizon is long enough, virtually all assets are IRS

If the horizon is short enough, virtually all assets become non-IRS

No standard horizon

Dollar Gap Summary

Dollar GAP i ProfitsNegative Increase DecreaseNegative Decrease IncreasePositive Increase IncreasePositve Decrease DecreaseZero Either Zero

What is the right GAP? One of the most difficult questions bank

managers face

Defensive Management Reduce volatility of net interest income Make Gap as close to zero as possible

Aggressive Management Forecast future interest rate movements If forecast is positive, make Gap positive If forecast is negative, make Gap negative

Problems with Gap

Time horizon to determine IRS is ambiguous

Ignores differences in rate sensitivity due to time horizon

Focus on profits rather than shareholder wealth

Building a Bank

Suppose you are in the process of creating a bank portfolio.

Shareholder equity: $25 million You’ve raised $75M in deposits (liabilities) You’ve purchased $100M in 30-yr annual coupon

bonds (assets)

Bank Equity and Interest Rates

Assets 30-yr bonds

– FV: $100M– Coupon rate: 1.8%– YTM=1.8%

Liabilities Deposits

– $75M– Paying 1% per year

Annual profits: $1.8M - $0.75M = $1.05M

Rate on bonds is fixed – no matter how rates change.Rate on deposits resets every year.

Gap Analysis

IRSA – IRSL = 0 – 75M = -75M Assume rates increase by 10 basis points We must now pay 1.1% on deposits

Change in profits: -75M(.001) = -75,000– Profits down 7% = 75K/1.05M

Gap Analysis

One Solution: To protect profits from interest rate increases, sell your holdings in the long term bonds and buy shorter term bonds

But since yield curve is usually upward sloping (liquidity risk-premiums), shorter term bonds will usually earn lower yields.

Result: Lower profits

Manager’s Objective

Managers should probably not be concerned about protecting profits.

Instead, should be concerned about protecting value of shareholder equity: the value shareholders would get if they sold their shares.

Market Value of Bank Assets

Before Rates Increase: Assets =$100M After rate increase?

– Bank is earning 1.8% on 30-yr bonds– Other similar 30-year bonds are paying a YTM of 1.9%- Market value of bank assets:

- N=30, YTM=1.9%, PMT=1.8M, FV=100M- Value = $97.73M

Market Value of Bank Liabilities

Liability of $75.75M due in one year– Principal and interest– Depositors will “redeposit” principal with you at new rate in

1-year

Market Value of liabilities = amount I would have to put away now at current rates to pay off liability in one year = present value

Before rates increase: 75.75/1.01=$75M After rates increase: 75.75/1.011=$74.93M

Market Value of Equity

PV(assets) – PV(liabilities)

One way to think of it: – Assume 1 individual were to purchase the bank– After purchasing the bank she plans to liquidate– When she sells assets, she will get PV(assets)– But of these assets, she will have to set aside some

cash to pay off liabilities due in 1 year, PV(liabilities)

Market Value of Equity

One way to think of it (continued)

– When considering a purchase price, she shouldn’t pay more than PV(assets)-PV(liabilities)

– But current shareholders also have the option to liquidate rather than sell the bank

– Current shareholders shouldn’t take anything less than PV(assets) - PV(liabilities)

Interest Rates and Bank Equity

Before rates increase: equity=$25M After rates increase: 97.73-74.93=$22.8M

Change in equity: 22.8M-25M = -2.2M

A 10 basis point increase in rates leads to a drop in equity of 8.8% (22.8/25-1=-.088)

Solutions

Sell long term bonds and buy short-term bonds– Problem: Many assets of banks are non-tradable loans

(fixed term) – more on this later.– Some bank loans are tradeable: securitized mortgages– How much do we want to hold in long vs. short bonds?

Refuse to grant long-term fixed rate loans– Problem: No clients – no “loan generation fees”– Bank wants to act as loan broker

Solutions

What should be our position in long versus short-term bonds?

How much interest rate risk do we want? Longer term bonds earn higher yields, but the

PVs of such bonds are very sensitive to interest rate changes.

We need a simple way to measure the sensitivity of PV to interest rate changes.

$0.00

$5,000,000.00

$10,000,000.00

$15,000,000.00

$20,000,000.00

$25,000,000.00

$30,000,000.00

0 0.02 0.04 0.06 0.08 0.1 0.12

PV and Yields

y

PV

Modified Duration

Duration: a measure of the sensitivity of PV to changes in interest rates: larger the duration, the more sensitive

Bank managers choose bank portfolio to target the duration of bank equity.

yPVDPV

PVDy

PV

*

*

rather or

Duration and Change in PV

Let PV = “change in present value”

The change in PV for any asset or liability is approximately

curve) yield of shift (parallel yields in change

Duration" Modified"

y

D

yPVDPV*

*

Example

From before:– Original PV of 30-year bonds: $100M– When YTM increased 10 bp, PV dropped to 97.73 PV=97.73M -100M = -2.27M

Duration Approximation

MM-ΔPV

DD

y

MPV

30.2100001.02.23

02.23

001.0

100

**

yet) find tohow know tdon' (you

Example

From before:– Original PV of liabilities: $75M– When rate increased 10 bp, PV dropped to

74.93M PV=74.93M -75M = -0.074M

Duration Approximation

MM-ΔPV

DD

y

MPV

074.075001.99.0

99.0

001.0

75

**

yet) find tohow know tdon' (you

Example

Change in bank equity using duration approximation:

Before, the change in equity was -2.20.

MMM

LAE

23.2)07.0(30.2

Modified Duration

Modified Duration is defined as

where “D” is called “Macaulay’s Duration”

y

DD

1*

Macaulay’s Duration

Let t be the time each cash flow is received (paid)

Then duration is simply a weighted sum of t

The weights are defined as

T

ttwtD

1

Price Bond

)1/( tt

tyCF

w

Example

Annual coupon paying bond– matures in 2 years, par=1000, – coupon rate =10%, YTM=10%

Price=$1000 Time when cash is received:

– t1=1 ($100 is received), t2=2 ($1100 is received) weights:

2

1 2

100 /(1.10) 1100 /(1.10)0.0909, 0.9091

1000 1000w w

Example

Macaulay’s Duration:

Modified Duration:

91.129091.010909.0 D

74.110.1

91.1* D