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11-Interest Rate Risk

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11-Interest Rate Risk

- 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

- 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

- 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

- Rates increase

- 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

- Short-term deposits

- 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

- 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%

- 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 = IRSA – IRSL
- IRSA = dollar value of interest rate sensitive assets
- IRSL = dollar value of interest rate sensitive liabilities

- From Previous Example
- IRSA (millions) = 5
- IRSL (millions) = 24

- Dollar Gap (millions): 5 – 24 = -19
- Change in profits= Dollar Gap´Di
- From previous example:
- Change in profits (millions)
-19 ´.01 = -0.19

- Change in profits (millions)

- 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 GAPDiDProfits

NegativeIncreaseDecrease

NegativeDecreaseIncrease

PositiveIncreaseIncrease

PositveDecreaseDecrease

Zero EitherZero

- 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

- Time horizon to determine IRS is ambiguous
- Ignores differences in rate sensitivity due to time horizon
- Focus on profits rather than shareholder wealth

- 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)

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.

- 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

- 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

- 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.

- 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

- 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

- 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)

- 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)

- 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)

- 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

- 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.

PV

y

- 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.

- Let DPV = “change in present value”
- The change in PV for any asset or liability is approximately

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

- Duration Approximation

- From before:
- Original PV of liabilities: $75M
- When rate increased 10 bp, PV dropped to 74.93M
- DPV=74.93M -75M = -0.074M

- Duration Approximation

- Change in bank equity using duration approximation:
- Before, the change in equity was -2.20.

- Modified Duration is defined as
where “D” is called “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

- 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:

- Macaulay’s Duration:
- Modified Duration: