Bond Prices and the Importance of Duration

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Bond Prices and the Importance of Duration

• Business 5039

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K. Hartviksen
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Bonds

• Please review and then study the chapter on bond
  valuation.
• Challenge the chapter problems.
• You must be able to perform calculations such as
• Current yield
• Yield to maturity
• Realized compound yield
• Input forward rates

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Malkiels Theorums of Bond Price Behaviour

• Be sure to understanding Malkiels bond theorums
• Inverse relationship between yields and prices
• Price volatility increases with time remaining to
  maturity
• Higher coupon rates lower interest rate risk
• The relative importance of theorum 2 diminishes
  as maturity increases
• Capital gains from a fall in interest rates
  exceeds the capital loss from an equivalent
  interest rate increase.

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Diversifying a Bond Portfolio

• Bond diversification
• There is no such thing as systematic and
  unsystematic risk in bond portfolios
• The two types of relevant risk in bond portfolios
  are
• Default risk - this topic has a great deal of
  relevance to the bond portfolio manager, if you
  can select bonds whose issuers will experience an
  improved financial health and a higher bond
  rating, required returns will fall and prices
  will rise
• Interest rate risk again, bond portfolio
  managers may seek to improve returns on their
  portfolio by making bets in accordance with their
  yield curve forecasts

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Valuing Bonds

• Bonds and Bond Valuation
• Bond Features and Prices
• Bond Values and Yields
• Interest Rate Risk
• Finding the Yield to Maturity
• Bond Price Reporting
• Interest Rates
• Short-term interest rates Risk free rate - yields
  on treasury bills
• Determinants of short-term rates - Fisher Effect
  (Inflation, nominal and real rates of return)
• Term Structure of Interest Rates (Expectations,
  Liquidity Preference, Segmentation Hypotheses)

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Bond Features

• Fixed coupon rate expressed as a of the par or
  face value
• face value 1,000
• known term to maturity
• required rate of return is the rate the market
  demands on such an investment YTM
• coupon payments are usually made semi-annually

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Bond Concepts

• Note the difference between Canada Bonds and
  Canada Savings Bonds (CSBs)
• CSBs are not negotiable.if you want to liquidate
  such an investment you redeem them through a
  financial institution like a Bank.

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Bond Terminology

• Par value face value
• coupon rate
• term to maturity
• zero-coupon bonds
• call provision
• convertible bonds
• retractable and extendible bonds
• floating-rate bonds

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Quoted Bond Prices

• Quoted bond prices do not include the accrued
  interest that accrues between coupon payment
  dates.

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Bond Quality

• determinants of bond safety
• coverage ratios
• leverage ratio
• liquidity ratio
• profitability ratio
• cashflow to debt ratio
• bond ratings focus on the foregoing factors

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Bond Indentures

• Contract between the issuer and bondholder
• protective covenants
• sinking funds (two systems)
• subordination of further debt
• dividend restrictions
• collateral

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Bond Pricing

• Present value of all expected future cashflows
• Yield to maturity
• ex ante calculation
• underlying assumptions
• Yield to call
• Realized Compound Yield (ex post) versus Yield to
  Maturity (ex ante)
• Yield to Maturity versus Holding Period Return
• current yield

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After-Tax Returns

• OID - original issue discount - example
  zero-coupon bonds
• OIDs result in an implicit interest payment to
  the holder of the security.
• Revenue Canada requires tax on imputed interest
  each year.

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Strip Bonds

• A derivative security
• a product created by an investment dealer
  decomposing a government bond and selling
  individual claims to the different parts of the
  bond to different investors

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Stripped Bond

• Is a claim on the face value of a coupon-bearing
  bond.
• The types of bonds that are stripped are often
• government of Canada
• provincial bonds
• Ontario Hydro, Hydro Quebec

K. Hartviksen
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Stripped Bonds

• Why are they called a derivative security?
• Because the original or underlying security was a
  normal bond that offered a stream of coupons
  (strip) and a face value due at maturity.
• Investment dealers buy up the original bonds,
  then sell claims to the annuity and face value
  separately.

New Market Price
New Market Price
Market Price
Stripped bond
K. Hartviksen
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What is the appeal of a Stripped Bond for
investors?

• You avoid the problems associated with
  reinvestment of the annual coupons.
• A yield-to-maturity (YTM) calculation assumes
  that all coupon interest that will be received is
  reinvested at the ex ante YTM.
• Because there are no intermediate cash flows
  (coupon interest) involved in a stripped bondthe
  ex ante YTM must equal the ex post YTM. This
  allows the investor to lock in a rate of return
  on their stripped bond investment for the term
  remaining to maturity (as much as 30 years!

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Stripped bond example

• What price would you pay for a stripped bond if
  it has 30 years to maturity, 20,000 face value
  and offers a 12 yield-to-maturity?
• P0 20,000(PVIFn30, k 12)
• 20,000 (.0334)
• 668.00
• Many people who are planning for retirement use
  such securities to lock in a given rate of return
  in their RRSPs.
• Parents can use them to save for their childs
  education through an RESP.

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Disadvantages of Stripped bonds

• You lock in a given rate of returnif interest
  rates rise, the market value of the investment
  will drop very rapidly.
• As a derivative product, the investment is
  illiquid (ie. No secondary market for this
  exists). Your ability to liquidate your
  investment will depend on the underwriters
  willingness to reverse the transaction. Hence
  this is not a good vehicle to use to make
  speculative returns when trying to take advantage
  of interest rate forecasts.
• Held outside of an RRSP or RESP, the imputed
  interest earned each year is subject to tax
  despite the fact that you did not receive a cash
  return.

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Interest Rate Price Sensitivity

• Because there are no coupon payments, stripped
  bonds have a duration equal to their term to
  maturity.
• Because of the distant cashflow involved, the
  current price (present value of that distant cash
  flow) is highly sensitive to changes in interest
  rates.

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Price Sensitivity Example

• Take our previous example
• P0 20,000(PVIFn30, k 12)
• 20,000 (.0334)
• 668.00
• Assume now that interest rates fall by 16.7 from
  12 to 10. What is the percentage change in
  price of the bond?
• P0 20,000(PVIFn30, k 10)
• 20,000 (.0573)
• 1,146.00
• Percentage change in price (1,146 - 668) /
  668
• 71.6

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Price Elasticity of Stripped Bonds
30 year stripped bond price given different YTM.
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Bond Value

• Value C(PVIFAn,r) 1,000(PVIFn,r)
• involves an annuity stream of payments plus the
  return of the principal on the maturity date

Bond Price discounted value of all future cash
flows
1 2 3 4 5 6 M
60 60 60 60 60 60 60
1,000
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Yield to Maturity

• An ex ante calculation
• the discount rate that equates the market price
  of the bond with the discounted value of all
  future cash flows.
• Is the investors required return
• changes in response to
• changes in the general level of interest rates in
  the economy
• changes in the risk of the issuing firm
• changes in the risk of the bond itself

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Bond Price Behaviour

• Bond prices are affected by changes in interest
  rates.
• Bond prices are inversely related to interest
  rates
• Longer term bond prices are more sensitive to a
  given interest rate change
• low coupon rate bond prices are more sensitive to
  a given interest rate change

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Term Structure of Interest Rates

• Liquidity preference theory
• Expectations hypothesis
• Segmentation theory

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Liquidity Preference Theory

• Investors require a premium for tying up their
  investment in bonds over a longer period of time.

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Expectations Theory

• The current spot rate is the geometric average of
  the forward rates expected to prevail over the
  life of the investment.
• r spot rate
• f forward rate
• (1 r2)2 (1 r1)(1 f2)
• f2 (1 r2)2 / (1 r1)

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Segmentation Theory

• There may be separate supply/demand conditions
  present at the short, intermediate, or long-term
  part of the market.
• These conditions can cause the yield curve to be
  disjoint

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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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Prices over time
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Bond Pricing Theorums

• Business 3079

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Theorums about Bond Prices

• 1. Bond prices move inversely to bond yields.
• 2. For any given difference between the coupon
  rate and the yield to maturity, the accompanying
  price change will be greater, the longer the term
  to maturity (long-term bond prices are more
  sensitive to interest rates changes than
  short-term bond prices).

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Theorums about Bond Prices

• 3. The percentage change described in theorum 2
  increases at a diminishing rate as n increases.
• 4.For any given maturity, a decrease in yields
  causes a capital gain which is larger than the
  capital loss resulting from an equal increase in
  yields.

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Theorums about Bond Prices

• 5. The higher the coupon rate on a bond, the
  smaller will be the percentage change for any
  given change in yields.

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Theorum Implications

• 1. It is best to buy into the bond market at the
  peak of an interest rate cycle.
• Because
• as interest rates fall, bond prices will rise
  and the investor will receive capital gain (this
  is important for investors with investment time
  horizons that are shorter than the term remaining
  to maturity of the bond.)

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Theorum Implications

• 1. It is best to buy into the bond market at the
  peak of an interest rate cycle.
• Because
• the bond will be priced to offer a high yield to
  maturity. If your investment time horizon
  matches the term to maturity for the bond, then
  holding the bond till it matures should offer you
  a high rate of return. (If it is a high coupon
  bond, though, you will have to reinvest those
  coupons when received a the going rate of
  interest. If it is a stripped bond, then there
  would be no interest rate risk and the ex ante
  yield to maturity will equal the ex post yield.

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Theorum Implications

• 1. When you expect a rise in interest rates,
  sell short/leave the market/move to bonds with
  fewer years to maturity.
• Because if rates rise, then you will experience
  capital losses on the bond. Of course, paper
  capital losses may not be particularly relevant
  if your investment time horizon equals the term
  to maturity because, as the maturity date
  approaches, the bond price will approach its par
  value regardless of prevailing interest rates.

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Theorum Implications

• 2. If interest rates go up your capital losses
  will be smaller if you are in the short end of
  the market.
• So your choice of investing in bonds with short
  or long-terms to maturity should be influenced by
  your expectations for changes in interest rates.
  If you think rates will rise (and bond prices
  fall) invest short term. If you think rates will
  fall (and bond prices rise) then invest in
  long-term bonds. (The foregoing assumes that you
  are not interested in purely immunizing your
  position.

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Theorum Implications

• 2. If you are at the peak of the short-term
  interest rate cycle, buying into the long end of
  the market will bring you the greatest returns.

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Theorum Implications

• 3 It is not necessary to buy the longest term to
  get large price fluctuations.

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Theorum Implications

• 4 For a given change in interest rates an
  investor will receive a greater capital gain when
  rates fall and he/she is in a long position, than
  if he/she is short and interest rates rise.

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Theorum Implications

• 5 Bonds with low coupon rates have more price
  volatility (bond price elasticity) than bonds
  with high coupon rates, other things being equal.
• It follows, that stripped bonds have the
  greatest interest rate elasticity.

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How a change in interest rates affects market
prices for bonds of varying lengths of maturity.
1,055.35
10 yield-to-maturity
Years to maturity
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Bond Price Elasticity

• Business 3079

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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
  keep 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.