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Duration – Measures of Interest Rate Sensitivity

Macaulay Duration – Percentage change in Price for a percentage change in Yield.

(Average life of PV of Cash Flows)

Where,

( in Years, yield BEY)

Modified Duration – Percentage change in Price for a 100 basis point change in Yield

Effective Modified Duration – Percentage change in Price for a 100 basis point change in

Yield, taking optionality of cash flows into account.

Spot rates can be calculated from Par rates and vice versa. Forward Rates can be calculated from Spot Rates and vice versa.

1. Calculating Spot Rates from Par Rates

• Example Treasury Par Curve as of 2/13/2004
• Period        Year        Par               Spot
• 1                       .5            .978            .978
• 2                      1.0           1.203        1.204
• 3                      1.5            1.435       1.437
• 4                        2.0         1.667       1.672
• Spot(1)=Par(1)=.978
• Spot(2) x=1.203678
• Spot(3) x=1.436854
• Spot(4)
• x=1.671858

2. Calculating Forward Rates From Spot Rates

• Period         Year         Par        Spot          Forward
• 1                       .5             .978          .978              .978
• 2                    1.0              1.203     1.204          1.430
• 3                    1.5             1.435        1.437        1.914
• 4                    2.0            1.667         1.672        2.381
• Forward(1) = Spot(1) = .978
• Forward(2) x = 1.43
• Forward(3) x = 1.914
• Forward(4)
• x = 2.381
• 
• 
• t = # of half year compounding periods

3. Calculating Spot Rates from Forward Rates

• 
• Period                Year           Forward              Spot
• 1                              .5                   .978                    . 978
• 2                           1.0                 1.430                    1.204
• 3                            1.5                1.914
• 4                            2.0               2.381
• Spot(1) = Forward(1) = .978
• Spot(2) = 200 x [(1+.978/200) x (1+1.430/200)^(1/2) – 1} = 1.2048 (rounding error)

4. Calculating Par Rates From Spot Rates

• Period Year Forward Spot Par
• Period                Year           Forward              Spot
• 1                              .5                   .978                    . 978
• 2                           1.0                 1.430                    1.204
• 3                            1.5                1.914
• 4                            2.0               2.381
• Par(1)=Spot(1)=.978
• Par(2): x =.6016611
• Par(2)= 2*.60166= 1.203
• Par(3):

5. Calculating Forward Spot Curve From Forward Rates

1-Year Forward Spot Rates

• Period                Year           Forward              Spot
• 1                              .5                   .978                    . 978
• 2                           1.0                 1.430                    1.204
• Spot(2) = 200 x [(1+1.914/200) x (1+2.381/200)]^(.5) -1] = 2.147

6. Calculating Forward Par Curve From Forward Spot Curve

• Period             Year               Forward             Spot             Par
• 1                          .5                    1.914             1.914          1.914
• 2                       1.0                   2.381             2.147           2.145
• Par(1)=Spot(1) = 1.914
• Par(2): x=1.073, Par(2)=2.146

Prepayment Modeling

Sources of Prepayments

• ¨ Refinancing – Sensitive to interest rates
• ¨ Home Turnover
• ¨ Defaults
• ¨ Partial Prepayments (curtailments)

Likelihood to refinance depends on

• 1. Interest rate of the loan
• 2. Age of the loan
• 3. Current Interest Rates
• 4. History of Interest Rates
• 5. Loan to value ratio
• 6. Loan size
• 7. Seasonality
• 8. Local economic conditions

Since historical data is not available for all possible combinations of above prepayment models look at the ratio (or spread between) loan interest rate and current interest rates. The higher the ratio or higher the spread the greater likelihood of prepayment. Relationship is an S curve. Combines the effects of Relative Interest Rates, Loan Age, and Burnout

Effective Risk Measures

Effective Modified Duration – Percentage change in Price for a 100 basis point change in

Yield, taking optionality of cash flows into account.

Effective Risk Measures – Take Into Account Optionality – Based on Dynamic Cash Flows (Simulation of Multiple Interest Rate Paths)

Volatility Measurement – Based on relative (ratio) yield movements. Assumes a log normal distribution of relative interest rate moves.

1. Effective Duration – Percentage change in full price for a 100 basis point parallel shift in the par yield curve.

2. Effective DV01 – Absolute Change in price for a 1 basis point parallel shift in the par yield curve.

3. Effective Convexity - measures the degree to which the price/parallel-shift curve of a security differs from the (+50, -50) tangent at the current yield curve. Positive Convexity implies P/Y curve is above tangent. A measure of the sensitivity of the Effective Duration of a security to a parallel shift of the Yield Curve so as to measure the sensitivity of price to “large” rate moves.

4. Effective Spread Duration – Percentage change in full price for a 100 basis point change in Option Adjusted Spread

5. Effective Partial (Key Rate) Durations - A measure of the sensitivity (percent change) of the full price of a security to a move in a single “key rate” point of the Yield Curve. Percent change in full price for a 100 basis point change in the corresponding point in the yield curve. Utilized to measure a security’s sensitivity to reshaping of the Yield Curve.

6. Volatility Duration – A measure of the sensitivity (percent change) of the full price of a security to a 1% change in Volatility.

Total Return

Nominal Return – Rate of Return on a security assuming it was purchased on a certain begin (settlement) date and sold on a certain horizon date. The return calculation takes into account the settlement full price, the horizon full price, intermediate cash flows from the security (coupon plus any principal payments) plus reinvestment of any intermediate cash flow payment to the horizon date.

Mortgage P/Y Measures and their Dependencies

• ¨ Nominal Measures
  • o Yield – Price, Prepayment Assumption (CPR, PSA, Model)
  • o Spread – Yield, Yield Curve (Treasury On The Run, Tsy Off The Run, Swap)
  • o Nominal Duration – Price, Prepayment Assumption
• ¨ Effective Measures (OAS, Eff Dur, Eff Convexity) – Price, Prepay Model Settings, Yield Curve, Volatility Assumption 1-factor, 2-factor (term structure of volatilities)

Scenario Analysis Assumptions

• ¨ Yield Curve Shifts
  • o Parallel Shifts
  • o Reshaping Scenarios
    • § Arbitrary Shifts
    • § Principal Component Scenarios
• ¨ Horizon Pricing Method
  • o Horizon Price
  • o Horizon Yield
  • o Horizon Nominal Spread
    • § Absolute Spread
    • § Spread Change
  • o Horizon OAS
    • § Absolute OAS
    • § OAS Change
      • · Positive/negative OAS Changes by sector/security
      • · Constant OAS (Most Common Assumption)
• ¨ Horizon Length
  • o Immediate – used for hedging etc.
  • o Positive period (usually 1month – 1 Year) Used for measuring yield vs risk trades.
• ¨ Prepayment Assumptions
  • o PSA Rate
  • o CPR Rate
  • o Vector of Monthly CPR Rates
  • o Prepayment Model (Most Common Assumption)
• ¨ Reinvestment Rate (insignificant for short horizon lengths, very significant for long horizon lengths)

Tracking a Mortgage Index

Constraints

• Effective Duration
• Partial Durations
• Issuer Diversification
• Coupon Diversification
• Principal Components Returns

Objective

· Maximize OAS

· Maximize ROR

· Match OAS as closely as possible to Index OAS

· Match ROR as closely as possible to Index ROR

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