Bond Prices and the Importance of Duration

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Bond Prices and the Importance of Duration

• Business 3059

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K. Hartviksen
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Duration

• is a handy tool because it can encapsule interest
  rate exposure in a single number.
• rather than focus on the formula...think of the
  duration calculation as a process...
• semi-annual duration calculations simply call for
  halving the annual coupon payments and
  discounting every 6 months.

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Duration Rules-of-Thumb

• duration of zero-coupon bond (strip bond) the
  term left until maturity.
• duration of a consol bond (ie. a perpetual bond)
  1 (1/R)
• where R required yield to maturity
• duration of an FRN (floating rate note) 1/2
  year

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Other Duration Rules-of-Thumb

• Duration and Maturity
• duration increases with maturity of a
  fixed-income asset, but at a decreasing rate.
• Duration and Yield
• duration decreases as yield increases.
• Duration and Coupon Interest
• the higher the coupon or promised interest
  payment on the security, the lower its duration.

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Economic Meaning of Duration

• duration is a direct measure of the interest
  rate sensitivity or elasticity of an asset or
  liability. (ie. what impact will a change in YTM
  have on the price of the particular fixed-income
  security?)
• interest rate sensitivity is equal to
• dP - D dR/(1R)
• P
• Where P Price of bond
• C Coupon (annual)
• R YTM
• N Number of periods
• F Face value of bond

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Interest Rate Elasticity

• the percent change in the bonds price caused by
  a given change in interest rates (change in YTM)

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Economic Meaning of Duration

• interest rate sensitivity is equal to
• dP - D dR/(1R)
• P
• dP/P change in bond price
• dR/(1R) change in interest rate
• Obviously, the relationship is an inverse
  function of Duration (D)

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Example of Calculation of Interest Rate
Sensitivity

• given
• n 6 years (Eurobond ... annual coupon payments)
• 8 percent coupon
• 8 YTM
• if yields are expected to rise by 10, what
  impact will that have on the price of the bond?
• the first step is to calculate the duration of
  the bond.
• If there were no coupon payments the duration
  would be 6.
• since there are coupon payments the duration must
  be less than 6 years.
• D 4.993 years
• the second step is to calculate the change in
  price for the bond.
• -(4.993)(.1/1.08) - 0.4623 - 46.23

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Immunization

• fully protecting or hedging an FIs equity
  holders against interest rate risk.
• elimination of interest rate risk by matching the
  duration of both assets and liabilities. (not
  their average lives or final maturities).
• when immunized
• the gains or losses on reinvestment income that
  result from an interest rate change are exactly
  offset by losses or gains from the bond proceeds
  on sale of the bond.

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Example of Bond Price

• The Canada 10.25 1 Feb 04 is quoted at 123.95
  yielding 5.27. This means that for a 1,000 par
  value bond, these bonds are trading a premium
  price of 1,239.50
• The figure represents bond prices as of June 17,
  1998.
• This bond has 5 years and 8 months
  (approximately) until maturity 5(8/12) 5.7
  years
• Bond Price 102.50(PVIFAn5.7 ,r5.27)
  1,000 / (1.0527)5.7
• 102.50(PVIFAr5.27, n 5.7) 746.21
• 102.50(4.8156653) 746.21
• 493.61 743.42 1,237.03
• Can you explain why the quoted price might differ
  from your answer?

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K. Hartviksen
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Sensitivity Analysis of Bonds
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K. Hartviksen
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Prices over time
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K. Hartviksen
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Bond Price Elasticity

• Business 3059

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Bond Price Elasticity

• The sensitivity of bond prices (BP) to changes in
  the required rate of return (I) is commonly
  measured by bond price elasticity (BPe), which is
  estimated as

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Example of Elasticity

• If the required rate of return changes from 10
  percent to 8 percent, the bond price of a zero
  coupon bond will rise from 386 to 463. Thus
  the bond price elasticity is

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Example of Elasticity
This implies that for each 1 percent change in
interest rates, bond prices change by 0.997
percent in the opposite direction.
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Bond Price Elasticity and Bond Price Theorums

• The following table demonstrates how bond price
  elasticity measures the effects of a given change
  in interest rates on bonds with different coupon
  rates.
• Zero coupon or stripped bonds have the longest
  durations because there are no intermediate cash
  flows, hence they exhibit the greatest
  elasticity.
• The higher the coupon rate, the lower the
  elasticity all other things being equal.

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Sensitivity of 10-year bonds with different
coupon rates to interest rate changes
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Bond Price Sensitivity and Term to Maturity

• The following chart explores the impact of the
  term to maturity on bond price sensitivity
• clearly, the longer the term to maturity, the
  greater the bond price elasticity.
• When interest rates rise, the bond price will
  rise by a greater percentage, than the fall in
  bond price in response to an equal but opposite
  increase in interest rates.

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Sensitivity of 10-year bonds with different
coupon rates to interest rate changes
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Bond Prices and Term to Maturity
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Duration

• An alternative measure of bond price sensitivity
  is the bonds duration.
• Duration measures the life of the bond on a
  present value basis.
• Duration can also be thought of as the average
  time to receipt of the bonds cashflows.
• The longer the bonds duration, the greater is
  its sensitivity to interest rate changes.

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Duration and Coupon Rates

• A bonds duration is affected by the size of the
  coupon rate offered by the bond.
• The duration of a zero coupon bond is equal to
  the bonds term to maturity. Therefore, the
  longest durations are found in stripped bonds or
  zero coupon bonds. These are bonds with the
  greatest interest rate elasticity.
• The higher the coupon rate, the shorter the
  bonds duration. Hence the greater the coupon
  rate, the shorter the duration, and the lower the
  interest rate elasticity of the bond price.

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Duration

• The numerator of the duration formula represents
  the present value of future payments, weighted by
  the time interval until the payments occur. The
  longer the intervals until payments are made, the
  larger will be the numerator, and the larger will
  be the duration. The denominator represents the
  discounted future cash flows resulting from the
  bond, which is the bonds present value.

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Duration Example

• As an example, the duration of a bond with 1,000
  par value and a 7 percent coupon rate, three
  years remaining to maturity, and a 9 percent
  yield to maturity is

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Duration Example ...

• As an example, the duration of a zero-coupon bond
  with 1,000 par value and three years remaining
  to maturity, and a 9 percent yield to maturity is

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Example of a Duration Calculation
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Duration of a Portfolio

• Bond portfolio mangers commonly attempt to
  immunize their portfolio, or insulate their
  portfolio from the effects of interest rate
  movements.
• For example, a life insurance company knows that
  they need 100 million 30 years from now cover
  actuarially-determined claims against a group of
  life insurance policies just no sold to a group
  of 30 year olds.
• The insurance company has invested the premiums
  into 30-year government bonds. Therefore there
  is no default risk to worry about. The company
  expects that if the realized rate of return on
  this bond portfolio equals the yield-to-maturity
  of the bond portfolio, there wont be a problem
  growing that portfolio to 100 million. The
  problem is, that the coupon interest payments
  must be reinvested and there is a chance that
  rates will fall over the life of the portfolio.

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Duration of a Portfolio ...

• The life insurance company example illustrates a
  key risk in fixed-income portfolio management -
  interest rate risk.
• The portfolio manager, before-the-fact calculates
  the bond portfolios yield-to-maturity. This is
  an ex ante calculation. As such, a naïve
  assumption assumption is made that the coupon
  interest received each year is reinvested at the
  yield-to-maturity for the remaining years until
  the bond matures.
• Over time, however, interest rates will vary and
  reinvestment opportunities will vary from that
  which was forecast.

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Duration of a Portfolio ...

• The insurance company will want to IMMUNIZE their
  portfolio from this reinvestment risk.
• The simplest way to do this is to convert the
  entire bond portfolio to zero-coupon/stripped
  bonds. Then the ex ante yield-to-maturity will
  equal ex post (realized) rate of return. (ie.
  the ex ante YTM is locked in since there are no
  intermediate cashflows the require reinvestment).
• If the bond portfolio manager matches the
  duration of the bond portfolio with the expected
  time when they will require the 100 m, then
  interest rate risk will be eliminated.