Duration and Yield Changes

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Duration and Yield Changes

• Duration provides a linear approximation of the
  price change associated with a change in yield.
• The duration of an asset will change depending
  upon the original yield used in its calculation.
  
• As the yield decreases, the price change
  associated with a change in yield increases.
• Likewise duration will increase as the yield of
  an option free bond decreases. This is
  illustrated as a steeper line approximately
  tangent to the price yield relationship.

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Convexity

• The approximation was further from the actual
  price change the larger the yield change due to
  the shape of the price yield relationship.
• It is useful to attempt to measure the error
  present in the linear duration approximation of
  the convex price yield relationship.

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Approximating the price change

• Duration provides a linear approximation of the
  price change, a better approximation would be to
  use a Taylor series expansion of the
  relationship.

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Modified Duration

• Previously we showed that modified duration can
  be estimated by
• Modified duration can be interpreted as the
  approximate percentage change in price for a 100
  basis point change in yield

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Dollar Duration

• The dollar duration represents the dollar change
  in price for a change in yield, it can be found
  by multiplying modified duration (which is an
  approximation for the change in price) by the
  original price.

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Measuring Convexity

• The first term in the Taylor expansion is then
  simply the dollar duration
• The second term includes the second derivative of
  the price equation, which is referred to as the
  dollar convexity measure.

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Convexity Mathematics
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Dollar Convexity Measure

• The second derivative of the price equation is
  referred to as the dollar convexity measure
• The Convexity Measure and is simply the dollar
  convexity measure divided by price

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Approximating the change in price

• Previously we showed that we could approximate a
  price change using a Taylor expansion.
• Dividing both sides by Price produces an
  approximation of the percentage change in price

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The percentage change in price

• Starting with the Taylor expansion divided by
  price And substituting from our previous results
• We can approximate the price change as

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Duration Example

• 10 30 year coupon bond, current rates 12, semi
  annual payments

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Example continued

• Since the bond makes semi annual coupon payments,
  the duration of 17.389455 periods must be divided
  by 2 to find the number of years.
• 17.389455 / 2 8.6947277 years

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Approximating the percentage price change

• The Taylor series combined two components, the
  duration estimation and the convexity measure.
• The first term in the approximation is equal to
  the modified duration multiplied by the change in
  yield.
• Lets assume a 200 basis point change in yield
• Modified duration would equal
• -8.6947/1.06 -8.2025733

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Approximating the change in yield continued

• The percentage change in price from duration for
  a 200 BP increase in yield would then be
• -8.2025733(.02) -.164
• The convexity measure is 124.56 and the
  percentage change in price from convexity is then
  
• (0.5)124.56(.02)2 .0249

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Approximating change in Price continued

• The approximate percentage change in price
  associated with a 200 Bp increase in yield is
  then the sum of the duration and convexity
  approximations
• Approximate change -.164 .0249 -.13914
• The actual price would be 719.2164 which would
  be a percentage change of
• (719.2164-838.3857)/838.3857 -.14214

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Approximating the change in yield continued

• The percentage change in price from duration for
  a 200 BP increase in yield would then be
• -8.2025733(-.02) .164
• The convexity measure is 124.56 and the
  percentage change in price from convexity is then
  
• (0.5)124.56(.02)2 .0249

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Approximating change in Price continued

• The approximate percentage change in price
  associated with a 200 Bp decrease in yield is
  then the sum of the duration and convexity
  approximations
• Approximate change .164 .0249 18.896
• The actual price would be 1,000 which would be a
  percentage change of
• (1,000-838.3857)/838.3857 .1927

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Convexity Adjustment

• In both cases the approximation using convexity
  is much closer to the actual price change than
  the approximation using only duration.

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Convexity Intuition

• Unlike the interpretation of modified duration,
  there is not a straightforward intuitive
  explanation of convexity.
• Be careful, measures of convexity are often
  referred to as just convexity which is incorrect,
  convexity is the shape of the price yield
  relationship.

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Value of Convexity

• In comparing two bonds with the same duration,
  the same yield, and selling a the same price the
  one that is more convex will have a higher price
  if yield changes.
• This implies that the capital loss associated
  with an increasing yield will be less for the
  more convex bond and the capital gain associated
  with a decrease in yield will be greater.
• Generally the greater convexity, the lower the
  yield since the price risk is less.

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Properties of Convexity on option free bonds

• As the required return increases the convexity
  decreases (positive convexity). This implies
  that the change in the change in price is always
  in the favor of the bond holder.
• For a given yield and maturity, the lower the
  coupon the greater the convexity.
• For a given yield and modified duration, the
  lower the coupon the lower the convexity.

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Approximating Convexity

• The procedure used so far is lengthy and can be
  approximated similar to the approximation of
  duration.

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Approximate Convexity Measure

• Let P0 be the original price and P- be the price
  following a small decrease in yield. and P be
  the price following a small increase in yield.
  The convexity measure can be approximated by

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Empirical example

• Using our numbers from before, a 200 Bp increase
  in yield caused the price of the 30 year 10
  coupon bond to be 719.2164, a 200 Bp decrease in
  yield moved the price to 1,000. Convexity would
  then be
• Our earlier estimation was 124.56.

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Convexity and software

• The calculation of convexity varies when using
  software.
• Often the convexity measure is already divided by
  2. In our example before we divided it by 2 when
  calculating the price change.
• Other times it is scaled to account for the par
  value.
• The key is knowing how it is calculated when
  making the adjustment to the duration estimation.
  

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Effective Convexity

• Similar to the duration estimate, the original
  measure of convexity assumes that the cash flows
  do not change when the yield changes.
• Effective convexity includes an adjustment from
  the change in cash flows associated change in
  interest rates. For bonds with embedded options
  the difference between effective convexity and
  out measure of convexity can be large.

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Bonds with call and put options

• So far we have assumed that the bond is option
  free.
• If the bond has embedded options it can change
  the shape of the price-yield relationship.

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Call Options

• With a call option, as yield declines it is more
  likely that the bond will be called.
• The value of the bond if called is less than than
  if it isnt.
• As the yield declines the bond may exhibit
  negative convexity the increase in price
  associated with successive decreases in yield
  will be less (not greater as in the case of
  positive convexity).

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Put options

• With a put option, as the yield increases the
  likelihood of the option being exercised
  increases.
• Since this adds value to the bond, the price
  yield relationship will become more convex as
  yield increases compared to an option free bond
  with the same characteristics.

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Interest Rates

• So far we have assumed that interest rates behave
  the same in relation to all bonds.
• However because of the term structure of interest
  rates, and other factors impacting the risk
  premium, the interest rate volatility of bonds
  differs
• To understand how this impacts the bond valuation
  process and price yield relationship we need to
  expand our understanding of interest rates.

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Theoretical Spot Rate

• The theoretical spot rate is the rate that
  represents the return earned on a zero coupon
  instrument.
• In other words it attempts to eliminate many of
  the other sources of risk other than maturity.
• The most common approach it to attempt to also
  eliminate the impact of default risk, so we will
  want to construct a spot rate curve for US
  Treasuries.

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Theoretical Spot Rate Curves

• Two main issues
• Given a series of Treasury securities, how do you
  construct the yield curve?
• Linear Extrapolation
• Bootstrapping
• Other
• What Treasuries should be used to construct it?
• On the run Treasuries
• On the run Treasuries and selected off the run
  Treasuries
• All Treasury Coupon Securities and Bills
• Treasury Coupon Strips

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Observed Yields

• For on-the-run treasury securities you can
  observe the current yield.
• For the coupon bearing bonds the yield used
  reflects the yield that would make it trade at
  par. The resulting on the run curve is the par
  coupon curve.
• However, you may have missing maturities for the
  on the run issues. Then you will need to
  estimate the missing maturities.

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Example

• Maturity Yield
• 1 mo 1.7
• 3 mo 1.69
• 6 mo 1.67
• 1 yr 1.74
• 5 yr 3.22
• 10 yr 4.14
• 20 yr 5.06

• The yield for each of the semiannual periods
  between 1 yr and 5 yr would be found from
  extrapolation.

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• 5 yr yield 3.22 1 yr yield 1.74
• 8 semi annual periods

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Bootstrapping

• To avoid the missing maturities it is possible to
  estimate the zero spot rate from the current
  yields, and prices using bootstrapping.
• Bootstrapping successively calculates the next
  zero coupon from those already calculated.

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Treasury Bills vs. Notes and Bonds

• Treasury bills are issued for maturities of one
  year or less. They are pure discount instruments
  (there is no coupon payment).
• Everything over two years is issued as a coupon
  bond.

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Bootstrapping example

• Assume we have the following on the run treasury
  bills and bonds
• Assume that all coupon bearing bonds (greater
  than 1 year) are selling at par (constructing a
  par value yield curve)
• Maturity YTM Maturity YTM
• 0.5 4 2.5 5.0
• 1.0 4.2 3.0 5.2
• 1.5 4.45 3.5 5.4
• 2.0 4.75 4.0 5.55

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Bootstrapping continued

• Since the 6 month and one year bills are zero
  coupon instruments we will use them to estimate
  the zero coupon 1.5 year rate.
• The 1.5 year note would make a semiannual coupon
  payment of 100(.0445)/22.225
• Therefore the cash flows from the bond would be
• t0.52.225 t12.225 t1.5102.225

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Bootstrapping continued

• A package of stripped securities should sell for
  the same price (100 par value) as the 1.5 year
  bond to eliminate arbitrage.
• The correct semi annual interest rates to use
  come from the annualized zero coupon bonds
• r0.5 4/2 2 r1.0 4.2/2 2.1

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Bootstrapping continued

• r0.5 4/2 2 r1.0 4.2/2 2.1
• The price of the package of zero coupons should
  equal the price of the theoretical 1.5 year zero
  coupon

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Bootstrapping continued
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Bootstrapping continued

• The semi annual rate is therefore 2.2293 and the
  annual yield would be 4.4586
• Similarly the 2 year yield could be found
• the coupon is 4.75 implying coupon payments of
  2.375 and cash flows of
• t0.52.375 t1.02.375 t1.52.375
  t2.0102.375

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Bootstrapping continued
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Bootstrapping continued

• What is the 2.5 year par value zero coupon rate?
  The coupon is 5

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Adding off the run securities

• You can fill in the larger gaps in maturity by
  adding selected off the run securities especially
  at the higher end of the maturity range.
• After extrapolating the missing maturities use
  the bootstrapping method to calculate the
  hypothetical zero coupon spot curve.
• You can also include all securities and use an
  exponential spline methodology

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Treasury Strips

• Another possibility is to use stripped treasury
  securities as an observed estimate of the zero
  coupon spot curve.
• Problems
• Liquidity of strips is not as great as treasury
  coupon market
• Tax treatment of strips differs from coupons
• Foreign investors tax treatment

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Using the spot yield curve

• Arbitrage should force the price of the treasury
  to be equal to the total of the cash flows
  discounted at the zero spot curve.
• If it does not there is an opportunity for a risk
  free profit.

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Arbitrage Example

• Assume you have a 9.4 coupon 3 year treasury
  notes selling at par. You have purchased a total
  of 10,000 par value of the note.
• Current Value
• 10,000 (since it sells at par)
• Coupon Payments
• 10,000(.094)/2 470 each 6 months

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• The current term structure is
• 6 mos 7.8 1yr 8 1.5 yr 8.4
• 2 yr 8.8 2.5 yr 9.2 3 yr 9.4
• The PV of the stripped bond is

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Arbitrage continued

• You could buy the treasury in the market then
  sell the stripped coupons for a greater amount.
• The arbitrage transaction could make 21.48 per
  10,000 if exploited.
• Because of this if the price is not equal to the
  price of the stripped security using the zero
  coupon curve, the price should move toward the
  theoretical price.
• The theoretical price is termed the Arbitrage
  free valuation or arbitrage free price.

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Forward Rates

• Using the theoretical spot curve it is possible
  to determine a measure of the markets expected
  future short term rate.
• Assume you are choosing between buying a 6month
  zero coupon bond and then reinvesting the money
  in another 6 month zero coupon bond OR buying a
  one year zero coupon bond.
• Today you know the rates on the 6 month and 1
  year bonds, but you are uncertain about the
  future six month rate.

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Forward Rates

• The forward rate is the rate on the future six
  month bond that would make you indifferent
  between the two options.
• Let z1 the 6 month zero coupon rate
• z2 the 1 year zero coupon rate (semiannual)
• f the rate forward rate from 6 mos to 1 year.

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Returns

• Return on investing twice for six months
• (1z1)(1f)
• Return on the one year bond
• (1z2)2
• If you are indifferent between the two, they must
  provide the same return
• (1z1)(1f) (1z2)2
• or
• f ((1z2)2/(1z1))-1

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Forward Rates

• Forward rates do not generally do a good job of
  actually predicting the future rate, but they do
  allow the investor to hedge
• If their expectation of the future rate is less
  than the forward rate they are better off
  investing for the entire year and lock in the 6
  month forward rate over the last 6 months now.