Fixed Income Securities

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Fixed Income Securities and their Derivatives
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Bond Price Volatility
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Price/Yield Relationship

• Recall that for option-free bonds the
  relationship between price and yield to maturity
  is given by

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The Price/Yield Relationship
Inverse relationship (downward sloping)
Nonlinear convex
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Bond Price Volatility

• Describes how much a bonds price changes(usually
  in percentage terms) when yield to maturity
  changes.
• Bond price volatility varies from bond to bond

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Bond Price Volatility

• For small changes in yield, the percentage price
  change up or down occasioned by a decrease or
  increase in yield is about the same in absolute
  value.

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Bond Price Volatility

• For bigger changes in yield, a decrease in yield
  causes a bigger (absolute) change in price than
  does an increase in yield.

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Bond Price Volatility

• Term effect
• For given coupon and initial yield to maturity,
  bond price volatility is greater the greater is
  the bonds term to maturity.

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Bond Price Volatility

• Coupon effect
• For given term and initial yield to maturity,
  bond price volatility is greater the lower is the
  bonds coupon.

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Bond Price Volatility

• Yield to maturity effect
• For given coupon and term to maturity, bond price
  volatility is greater the lower the yield to
  maturity.

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Measures of Price Volatility

• Price value of a basis point (P01)
• The price value of a basis point is the
  (absolute) dollar price change caused by a 1
  basis point change in yield to maturity

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Measures of Price Volatility

• Yield value of 1/32
• The yield value of 1/32 is the (absolute) yield
  change caused by a 1/32 change in bonds (clean)
  price

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Measures of Price Volatility

• The price value of an 01 and the yield value of a
  32nd are related approximately by

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Measures of Price Volatility

• The price value of an 01 is a discrete measure of
  the change in price associated with a change in
  yield

Here the change in y is a discrete (.01) change.
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Measures of Price Volatility

• The slope of the price/yield curve at any point
  is a continuous measure of the change in price
  associated with a change in yield

Here the change in y is an infinitesimal change.
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Measures of Price Volatility

• The proportionate price change associated with an
  infinitesimal change in yield is

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Measures of Price Volatility

• The negative of this proportionate change is
  called the modified duration of a bond

The minus sign is needed to make modified
duration a positive number. Can you see why?
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Measures of Price Volatility

• For semiannual pay bonds

In this case, duration is measured in half-years.
To get modified duration in years, divide this
figure by two.
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Measures of Price Volatility

• Macaulay duration, or just plain duration, is
  (for semiannual pay bonds)

To get duration in years, divide D by two. Or
you can multiply modified duration measured in
years by (1y).
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Measures of Price Volatility

• Market participants often refer to dollar
  duration, which is defined as modified duration
  times a bonds price.
• Dollar duration is useful for approximating the
  price change associated with a given change in
  yield

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Calculating Duration

• Use MSExcel functions DURATION and MDURATION
• Note mistake in Excels description
• Use formula for modified duration

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Properties of Duration

• The duration of a zero coupon bond is equal to
  the maturity of the bond.
• For bonds having the same maturity and priced at
  the same yield, duration varies inversely with
  coupon.
• Even though it is dimensioned in time units,
  duration is not a measure of time. It is a
  measure of bond price volatility.

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Duration

• Duration is useful for estimating how much a
  bonds price will change if yield changes by a
  small amount.
• However, if you just use duration you will
• Underestimate the increase in price when yield
  decreases, and
• Overestimate the decrease in price when yield
  increases.
• Reason convexity

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Convexity
Duration is not constant It increases as yield
to maturity decreases
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Convexity

• Formally, convexity refers to the second order
  term in the Taylor series expansion of the price
  equation

This term is called dollar convexity
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Properties of Convexity

• For a given bond, convexity increases as yield to
  maturity decreases.
• Comparing two bonds with the same yield to
  maturity and term, the one with the lower coupon
  has higher convexity.
• Comparing two bonds with the same yield to
  maturity and modified duration, the one with the
  lower coupon has the lower convexity.

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Applications

• Immunization example
• Problem set

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Review

• Key properties of the price/yield relationship
• Several measures of price volatility
• Three measures of duration
• Properties of duration
• Two measures of convexity
• Properties of convexity

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Next
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Term Structure