FixedIncome

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Fixed-Income Portfolio Management

Mila Getmansky Sherman
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Interest Rate Sensitivity

• Generally, when interest rates increase (yields
  increase), bond prices decrease
  
• Prices of long-term bonds tend to be more
  sensitive to interest rate changes than prices of
  short-term bonds
  
• As maturity increases, price sensitivity to yield
  changes increases at a decreasing rate.
  

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Interest Rate Sensitivity

• An increase in a bonds YTM results in a smaller
  price decline than the price gain associated with
  a decrease of equal magnitude in yield.
  
• Interest rate risk is inversely related to the
  bonds coupon rate. Prices of high-coupon bonds
  are less sensitive to changes in interest rates
  than prices of low-coupon bonds.
• Bond prices are more sensitive to changes in
  yield when the bond is selling at a lower initial
  yield to maturity.

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Sensitivity of Price to YTM
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Maturity

• The concept of maturity is ambiguous. For
  example, we can have Bond A with maturity 20
  years that pays coupons every year, and a Bond B
  with maturity 20 years that is a zero-coupon
  bond. Even though maturity is the same, bond B
  is more sensitive to interest rate changes than
  bond A. How do we explain it?

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Maturity

• In effect, bond A has several payments with their
  own maturity dates and the zero-coupon bond has
  one payment at maturity, so it has a single
  maturity date. What is the correct way of
  measuring a maturity of a bond?

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Duration

• Duration of a bond is computed as the weighted
  average of the times to each coupon or principal
  payment made by the bond. Weights are
  proportional to the payment.

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Duration Formula

• Therefore, the weight, denoted wt, associated
  with the cash flow made at time t (CFt) would be
  
• where y is the bonds YTM. CF(t) is a cash flow
  at time t. D is duration.

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Duration Calculation

• Question 1.
• Calculate Duration of a Bond A which pays 8
  coupon seminannually, maturity is 2 years, and
  YTM10 (semiannual yield 5). Face Value
  1,000.
• Question 2.
• Calculate Duration of a Bond B which pays 0
  coupon, maturity is 2 years, and YTM10. Face
  Value 1,000.
  

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Duration Calculation

• Answers
• Question 1 Duration 1.8853 years
• Question 2 Duration 2 years

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Modified Duration

• The percentage change in bond price is just the
  product of modified duration and the change in
  the bonds YTM
  
• Where modified duration, DD/(1y)

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Price Sensitivity

• Task Compare Bond A with duration of 1.8853
  years to the sensitivity of a zero-coupon bond C
  with maturity and duration of 1.8853 years.
  
• Both should have equal price sensitivity if
  duration is a useful measure of interest rate
  exposure.

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Price Sensitivity Solution

• The coupon bond A sells for 964.5405 at the
  initial seminannual interest rate of 5. If the
  bonds seminannual yield increases by 1 basis
  point (0.01) to 5.01, its price will fall to
  964.1942, a percentage decline of 0.0359.

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Price Sensitivity Solution

• The zero-coupon bond has a maturity of
  1.885323.7706 half-year periods. Because we
  use a half-year interest rate of 5, we also need
  to define duration in terms of a number of
  half-year periods to maintain consistency of
  units. If interest rate is 5, the bond
  initially sells at 831.9623 (1,000/1.053.7706).
  Its price falls to 831.6636 (1,000/1.05013.77
  06) when the interest rate increases, for an
  identical 0.0359 capital loss.

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Price Sensitivity Conclusion

• Equal-duration assets are in fact equally
  sensitive to interest rate movements.
  
• Note, if duration is the same for two bonds with
  the same YTM, then modified duration for the two
  bonds is the same.

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Reality

• Duration rule for the impace of interest rates on
  bond prices is only an approximation.

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When to Use Duration

• For small changes in the bonds YTM, the duration
  rule is quite accurate. However, for larger
  changes in YTM, duration become progressively
  less accurate.

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Convexity

• The duration approximation always understates the
  value of the bond it underestimates the increase
  in bond price when the yield falls, and it
  overestimates the decline in price when the yield
  rises.
• This is due to the curvature (convexity) of the
  true price-yield relationship

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Convexity

• Convexity allows us to improve the duration
  approximation for bond price changes
  

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Convexity Calculation

• The formula for the convexity of a bond with a
  maturity of n years, making annual coupon
  payments, is
  

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Convexity Example

• Consider a bond with a 30-year maturity, 8
  coupon that sells at an initial YTM of 8.
  
• Because coupon rate YTM, PriceFace Value
  1,000. The modified duration of the bond
  (D)11.26 years, and its convexity is 212.4.

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Convexity Example, Cont.

• If the bonds yield increases from 8 to 10, the
  bond price will fall to 811.46, a decline of
  18.85.
  
• The duration rule would predict a price decline
  of -11.260.02-22.52
  
• The duration-with-convexity formula predicts
  -11.260.020.5212.40.022-18.27, which is
  more accurate.
  
• Note, for small changes in YTM, both formulas are
  accurate.