Chapter 2.
Interest Rate Risk (Part 2)
Content
• Duration
• Duration model
• Manage single security
• Manage balance sheet
Problem
• Besides its effect on NII, interest
rate movement also impacts
market values of an institution’s
assets, liabilities, and thus equity
• Example: as discount rate ↑,
debt instrument’s price ↓
Duration
• Objective: measure change to bank’s equity given a change in market
interest rates
• For each asset/liability, we can plug new R in the formula to get new
MV , then get ∆MV → not very convenient for modeling
• A simpler tool: approximate ∆MV using duration
Duration
• Duration is the weighted average time to maturity on an investment
• Takes into account the timing of cash flow arrivals
Duration
• What is the duration for a zero-coupon bond?
Duration and maturity
• Duration increases with maturity but
at a decreasing rate
Duration and interest rate
• Duration decreases with interest rate
Duration and coupon
• Duration decreases with coupon interest
Duration model
• Economic meaning of duration: duration measures the
elasticity of security’s price to a small change in interest rate
(yield to maturity)
Economic meaning of duration
• Rearrange: duration is the percentage change in security
price given a 1% change in interest rate.
Modified Duration
Dollar Duration
• Call Dollar Duration
• This is the dollar value change in security price given 1%
change in yield
Interest rate risk management
• Use duration to manage interest rate risk:
• Manage interest rate risk of a single security
• Manage interest rate risk of the whole balance sheet
Duration of
First National
Bank's Assets
and Liabilities
Manage single security
• Objective: earn a certain return on debt security regardless of interest
rate movements during the investment period (e.g., 3 years)
• Simplest solution: buy and hold a zero-coupon bond with 3-year
maturity
• Duration is also three years
• No intervening cash flow generated → not subject to reinvestment risk
Manage single security
• If no zero-coupon bond is available → buy a coupon bond
• Interest rate can suddenly change right after investor buys the bond
• To immunize interest rate risk, buy coupon bond with duration ≈ 3
years
Manage balance sheet
• Interest rates change → Market values of assets and liabilities change
• Use duration to evaluate the overall interest rate exposure
Manage balance sheet
Where k = L/A is a measure of the bank’s leverage
Duration gap analysis
Example of the First National Bank
• For each asset item, calculate its weighted duration =
duration_of_item*(amount_of_asset_item/total_assets)
• Example: securities less than a year: 5*0.4/100=0.02
• Do this for all assets
• Add up all obtained values: Average duration of assets = 2.70
• Do the same with liabilities: note that total liabilities exclude capital ($95million).
• Average duration of liabilities = 1.03
Consider a change in rates from 10% to 15%
• ∆Asset Value = −2.7 × .05/(1 + .10) × $100m = −$12.3m
• ∆Liability Value = −1.03 × .05/(1 + .10) × $95m = −$4.5m
• Net Worth: ∆NW = ∆Assets – ∆Liabilities
∆NW = −$12.3m − (−$4.5m) = −$7.8m
Duration gap analysis
DURgap = DURa − [L/A × DURl]
%∆NW=∆NW/A= −DURgap × ∆i/(1 + i)
Apply to the example:
• DURgap = DURa − [L/A × DURl] = 2.7 − [(95/100) × 1.03] = 1.72
• %∆NW= −DURgap × ∆i/(1 + i) = −1.72 × .05/(1 + .10) = −.078, or −7.8%
Example: Duration Gap Analysis
Duration of the Friendly Finance Company’s Assets and Liabilities
Factors that change the market value of a
bank’s equity or net worth
• The leverage adjusted duration gap = [DA − kDL]
• measured in years and reflects the degree of duration mismatch in a bank’s
balance sheet. S
• the larger this gap is in absolute terms, the more exposed the bank is to
interest rate shocks.
• The bank size:
• The larger the scale, the larger the dollar size of the potential net worth
exposure from any given interest rate shock.
• The size of the interest rate shock = ∆R/(1 + R).
• The larger the shock, the greater the bank’s exposure.
Immunize interest rate risks
• Make adjusted duration gap DA - k DL ≈ 0
• A bank typically has positive duration gap DA - k DL > 0, so for gap = 0
• Reduce DA
• Increase DL
• Change k
• A combination of the change above
Alternative objective
• Banks have to maintain a minimum capital ratio E/A
• May prefer to immunize interest rate risk to E/A: ∆(E/A) = 0
• Immunize: DA ≈ DL
Considerations
• Expensive to change DA and DL
• Security duration changes over time → Immunization is a continuous
process
• Large interest rate change makes approximation using duration less
accurate
• Duration model estimates a linear change in security price
• But price-yield relationship is convex, not linear
Convexity of bond price-yield relationship
Convexity Gap Model
Where CG = MCA – k x MCL
Problem sets
• Chapter 9: 4, 7, 11, 13, 17, 19, 20, 21, 23, 24, 25, 31