Chapter 6_3_ by fanzhongqing

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									       Yield Measures, Spot
     Rates, and Forward Rates
                                  by Frank J. Fabozzi



                         PowerPoint Slides by
              David S. Krause, Ph.D., Marquette University



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            Chapter 6
  Yield Measures, Spot Rates, and
          Forward Rates
• Major learning outcomes:
   – Yield and yield spread measures, which help investors gauge
     relative bond value.
   – Computing the spot rates from on-the-run Treasury issues.

   – Understanding the limitations of the nominal spread measure.
      • Explaining the two measures that overcome the limitation:
          – Zero-volatility spread
          – Option-adjusted spread


   – Computing forward rates.
           Key Learning Outcomes
•   Explain the sources of return from investing in a bond (coupon interest
    payments. Capital gain/loss, and reinvestment income).

•   Compute the traditional yield measures for fixed-rate bonds (current yield,
    yield to maturity, yield to first call, yield to first par call date, yield to
    refunding, yield to put, yield to worst, and cash flow yield).

•   Explain the assumptions underlying traditional yield measures and the
    limitations of the traditional yield measures.

•   Calculate the reinvestment income required to generate the yield computed
    at the time of purchase.

•   Explain the factors affecting reinvestment risk.

•   Calculate the spread for life and discount margin measure for a floating-rate
    security and explain the limitations of both.
           Key Learning Outcomes
•   Calculate the yield on a discount basis for a Treasury bill and explain its
    limitations.

•   Compute using the method of bootstrapping the theoretical Treasury spot
    rate curve given the Treasury yield curve derived from the on-the-run
    Treasury issues.

•   Explain the limitations of the nominal spread.

•   Describe and compute a zero-volatility spread given a spot rate curve.

•   Explain why the zero-volatility spread will diverge from, and is superior to,
    the nominal spread.

•   Explain an option-adjusted spread for a bond with an embedded option and
    explain what is meant by the option cost.
          Key Learning Outcomes
•   Explain why the nominal spread hides the option risk for bonds with
    embedded options.

•   Define a forward rate and compute forward rates from spot rates.

•   Demonstrate the relationship between short-term forward rates and spot
    rates.

•   Explain why valuing a bond using spot rates and forward rates produces the
    same value.

•   Calculate the forward discount factor from forward rates.

•   Calculate the value of a bond given forward rates.
    Sources of Bond Return
• A bond investor receives a return from
  one or more of the following:
  – The coupon interest payments made by the
    issuer;
  – Any capital gain or loss when the bond
    matures or is sold;
  – Income from reinvestment of the interim cash
    flows (interest/and or principal payments)

• Yield calculations should consider all
  three.
     Sources of Bond Return
• Coupon interest – periodic income, usually
  semi-annually payments.

• Capital gain or loss – the profit or loss of market
  value (selling / maturity price minus purchase
  price).

• Reinvestment income is the interest income
  generated by reinvesting coupon interest
  payments and any principal payments from the
  time of receipt to the bond’s maturity.
                    Bond Yields
• The current yield relates the annual dollar coupon interest to the
  market price and fails to recognize any capital gain or loss and
  reinvestment income.

• The yield to maturity is the interest rate that will make the present
  value of the cash flows from a bond equal to the price plus accrued
  interest.

• The market convention to annualize a semiannual yield is to
  double it and the resulting annual yield is referred to as a bond-
  equivalent yield.

• When market participants refer to a yield or return measure as
  computed on a bond-equivalent basis it means that a semiannual
  yield or return is doubled.
                Current Yield
• Current yield is greater (less) than the coupon
  rate, when the bond sells at a discount (premium)

• The current yield is a weak measure because it
  does not measure any cash flows other than the
  coupon payments:
  – No consideration is given to the capital gain or loss
  – Reinvestment income is also not considered
  – Time value of money concepts are not followed
           Yield to Maturity
• YTM is the most popular measure
  – It is the interest rate that will make the present
    value of the bond’s cash flow equal to its
    market price plus accrued income.
  – It is the same as the Internal Rate of Return
    (IRR)
  – Calculations:
     • Trial and error / approximations
     • Calculator
     • Spreadsheet (Excel)
                  YTM Computation
• It’s simply a present value problem (solve for y):

         C         C        C                 C       M
   P                            ...            
      1  y  1  y  1  y 
                       2        3
                                          1  y  1  y 
                                                  N        N




   –   P is the bond price
   –   C is the periodic coupon payment
   –   N is the number of years to maturity
   –   M is the (face value) payment at maturity
   –   y is the “risk-adjusted discount rate” (or yield to
       maturity, or IRR)
           Yield to Maturity
• For bonds that pay interest semi-annually,
  the market convention adopted is to
  annualize the semi-annual yield to maturity
  by doubling it (multiplying by 2). This is the
  annual YTM.
  – This is also referred to as the: bond-equivalent
    yield (BEY)


• For a bond selling at a discount (premium),
  the coupon rate is < (>) the current yield,
  and < (>) the yield to maturity.
             Yield to Maturity
• The yield to maturity takes into account all three sources
  of return but assumes that the coupon payments and any
  principal repayments can be reinvested at an interest rate
  equal to the yield to maturity.

• The yield to maturity will only be realized if the interim
  cash flows can be reinvested at the yield to maturity and
  the bond is held to maturity.

• Reinvestment risk is the risk an investor faces that future
  reinvestment rates will be less than the yield to maturity
  at the time a bond is purchased.
   Yield to Maturity Relationship
• The following relationships between the price
  of a bond, coupon rate, current yield, and yield
  to maturity hold:
  Bond-Equivalent Yield Convention
• The market convention to annualize a
  semiannual yield is to double it and the
  resulting annual yield is referred to as a
  bond-equivalent yield.

• When market participants refer to a yield or
  return measure as computed on a bond-
  equivalent basis it means that a semiannual
  yield or return is doubled.
    Bond-Equivalent Yield Convention
• The idea of simply doubling the semi-annual yield to get the
  YTM should be somewhat troubling to those who
  understand the concept of the time value of money. You
  would think that computing the effective rate would be
  proper.

   – However, it is still done this way by many because “that’s the way it
     always has been done.” This is the market convention, much the
     same as the Dow Jones Industrial Average is used to express how
     the overall stock market is performing.
       • This is okay as long as all bond yields are calculated this way and will
         not lead to arbitrage opportunities.

   – Even though the BEY is not the true effective annual yield, we’ll soon
     see that the use of yields has many problems – the least of which
     involves the doubling of the semi-annual rate to obtain a YTM
     estimate.
            Limitations of YTM
• YTM is better than current yield because it
  considers the coupon, capital gain/loss, and
  reinvestment income. It is also based on the time
  value of money concepts.
  – However, it assumes that the coupon payments from a
    bond can be reinvested at an interest rate equal to the
    yield to maturity.


• This assumption can be highly misleading because
  of reinvestment risk
  – We have seen that a term structure of interest rates
    exists, which can result in different yields at different
    points in time.
   Example of the Limitation of YTM
• Suppose an investor has a 15-year 8%
  semi-annual coupon bond purchased at par
  ($100).
  – The YTM is 8%
    • Translated into total future dollars this is:
       – $100 x (1.04)30 = $324.34
       – Decomposed this is $100 of principal return and $224.34 of
         total dollar return


    • Without reinvestment income, the dollar return would
      be:
       – $120 of coupon income and $0 capital gain (because the
         bond is purchased at par)
      Example of the Limitation of
           YTM (continued)
• The dollar return shortfall is $224.34 - $120 = $104.34
   – This shortfall is made up if the coupon payments are reinvested at
     a yield of 8% (the interest rate at the time the bond was purchased)
   – For this bond, the reinvestment income is 46.5% ($104.34/224.34)
     of the total dollar return needed to produce a yield of 8%


• The investor will only realize the YTM of 8% if:
   – The coupon payments can be reinvested at the YTM of 8% (this is
     reinvestment risk)
   – The bond is held to maturity (if the bond is not held to maturity, the
     investor faces the risk of selling for less than the purchase price
     which is known as interest rate risk)
   – These are large and questionable assumptions
           Factors Affecting
          Reinvestment Risk
• Reinvestment risk is the risk an investor faces that
  future reinvestment rates will be less than the yield
  to maturity at the time a bond is purchased.

• Interest rate risk is the risk that if a bond is not
  held to maturity, an investor may have to sell it for
  less than the purchase price.

• The longer the maturity and the higher the coupon
  rate, the more a bond’s return is dependent on
  reinvestment income to realize the yield to
  maturity at the time of purchase.
            Reinvestment Risk
The two factors affecting reinvestment risk are:
  For a given YTM and a given non-zero coupon rate, the
      longer the maturity, the more the bond’s total dollar
      return depends on reinvestment income to realize the
      YTM at the time of purchase. That is, the greater the
      reinvestment risk.

     The implication is that the YTM for a long-term, high coupon bond
        may have a large amount of the total dollar return as
        reinvestment income (which is more risky).
    Reinvestment Risk (continued)
The two factors affecting reinvestment risk are:

    2. For a coupon bond, for a given YTM and maturity, the
    higher the coupon rate, the more dependent the bond’s
    total dollar return will be on the reinvestment of the
    coupon payments in order to produce the YTM at the time
    of the purchase.

      –   The implication is that bonds selling at a premium will be more
          dependent on reinvestment income than bonds selling at par. This is
          because the reinvestment income has to make up the capital loss
          due to amortizing the price premium when holding the bond to
          maturity.

      –   Conversely, a bond selling at a discount will be less dependent on
          reinvestment income.
    Reinvestment Risk (continued)
•   Exhibit 1 shows an example of three
    bonds with different coupons (and selling
    prices)
    – Based on the years to maturity and the
      coupon rate (selling price) it shows:
      •   Reinvestment risk is greatest for longer maturity
          bonds
      •   Reinvestment risk is greatest for higher coupon
          bonds (those selling at a premium)
Reinvestment Risk
         Semi-Annual versus Annual
           Coupon Paying Bonds
•       Some bonds (especially those issued outside the
        U.S.) pay interest annually rather than semi-
        annually

•       The bond-equivalent yield for an annual paying
        bond is computed as follows:
                    2 x [(1+ yield on annual-pay bond)0.5 -1]

    –     The term in the square brackets involves determining what semi-
          annual yield, when compounded, produces the yield on an
          annual-paying bond.
    –     Doubling this semi-annual yield gives the bond-equivalent yield.
      Semi-Annual versus Annual
    Coupon Paying Bonds Example
•   Suppose the YTM on an annual-pay bond
    is 6%. Then the bond-equivalent yield is:
                2 x [(1+ .06)0.5 -1] = 5.91%
    –   The bond-equivalent yield will always be less than
                the annual-paying bond’s YTM
    Semi-Annual versus Annual
  Coupon Paying Bonds Example
It is also possible to convert the bond-equivalent
      yield to an annual-pay basis using the
      following:
        [(1+ (yield on bond-equivalent basis)/2)2 -1]


   The yield on an annual-pay basis is always greater
      than the yield on a bond-equivalent basis because
      of compounding.
                    Yield to Call
•   The yield to call is the interest rate that will make the
    present value of the expected cash flows to the assumed
    call date equal to the price plus accrued interest.

•   Yield measures for callable bonds include yield to first call,
    yield to next call, yield to first par call, and yield to
    refunding.

•   The yield to call considers all three sources of potential
    return but assumes that all cash flows can be reinvested at
    the yield to call until the assumed call date, the investor will
    hold the bond to the assumed call date, and the issuer will
    call the bond on the assumed call date.
                       Yield to Call
•   For callable bonds, the practice has been to
    calculate a YTM and a yield to call (YTC)

•   The YTC assumes the issuer will call the bond on
    the call date at the specified call price
    –   Investors calculate the yield to first call, yield to next
        call, yield to first par call, and a yield to refunding.
        •   Yield to refunding is used when bonds are currently callable but
            have some restrictions on the source of the funds used to buy
            back the debt when a call is exercised.
                 Yield to Call
•    Calculating the YTC is the same as that for
     any YTM calculation.
    – In the case of yield to first call, the expected
      cash flows are the coupon payments to the
      first call date and the call price.
    – For the yield to first par call, the expected
      cash flows are the coupon payments to the
      first par call date and the par value.

•    Exhibit 2 contains an example of a YTC
     under varying interest rate levels.
Yield to Call
                Yield to Call
•   The YTC considers all three sources of
    potential return from owning a bond;
    however, like YTM it is assumed that all
    cash flows can be reinvested at the YTC
    until the assumed call date.
    – As we have learned previously, this may be an
      inappropriate assumption.

•   Other problems with YTC occur when
    comparisons are made with YTM because
    of the different maturity/call dates.
                   Yield to Put
•   When a bond is putable, the yield to the put date
    is computed.
    –   The yield to put is computed assuming that the issue
        will be put on the first put date.


•   The yield to put is the interest rate that will make
    the present value of the cash flows to the first
    put date equal to the price plus accrued interest.

•   Like other yield measures, the yield to put
    assumes that any interim coupon payments can
    be reinvested at the yield calculated.
          Yield to Worst
•   The yield to worst is the lowest yield
    from among all possible yield to calls,
    yield to puts, and the yield to maturity.
                  Yield to Worst
•   A yield can be computed for every possible
    call and put date, as well as the YTM.
    –   The lowest of all the yields is called the yield to
        worst.


•   The yield to worst has little meaning as a
    measure of potential return because
    –   It does not identify the potential return over some
        investment horizon
    –   It does not recognize that each yield calculation
        has different exposures to reinvestment risk.
                 Cash Flow Yield
•       Mortgage- and asset-backed securities (which are
        secured with pools of loans or receivables) have
        periodic cash flows that include interest and principal.
        These are typically monthly.

•       Because the borrowers can prepay, the timing of the
        principal payments are not known with certainty.

•       It is necessary to make an assumption about the rate at
        which principal prepayments occur. This is referred to
        as:
    –     Prepayment rate, or
    –     Prepayment speed (this is measured in units of PSA).

•       Given an assumed payment schedule, a cash flow yield
        can be computed.
            Cash Flow Yield
•   For mortgage-backed and asset-backed securities, the
    cash flow yield based on some prepayment rate is the
    interest rate that equates the present value of the
    projected principal and interest payments to the price
    plus accrued interest.

•   The cash flow yield assumes that all cash flows
    (principal and interest payments) can be reinvested at
    the calculated yield and that the assumed prepayment
    rate will be realized over the security’s life.

•   For amortizing securities, reinvestment risk is greater
    than for standard coupon nonamortizing securities
    because payments are typically made monthly and
    include principal as well as interest payments.
        Bond-Equivalent Cash Flow Yield
•       The interest rate that will make the present value of the
        projected principal and interest payments equal to the
        market price plus accrued interest is a monthly yield that
        gets annualized as follows:
    •     The semi-annual effective yield is computed from monthly yield
          by compounding it for six months: (1 + monthly yield)6 – 1

    •     Effective semi-annual yield is doubled to get the annual cash flow
          yield on a bond-equivalent basis: 2 x effective semi-annual yield

•       Again, it seems odd to be doubling up the semi-annual
        rate by multiplying by 2, but this is the market convention
        (“that’s the way it has always been done”).

•       Remember, too, that this is a minor issue compared to
        the other problems with using yield as a relative
        measure.
    Limitations of Cash Flow Yield
•       The shortcomings of cash flow yield are:
    –     The projected cash flows are assumed to be reinvested at the
          cash flow yield
    –     The mortgage- and asset-backed securities are assumed to be
          held until the final payoff of all loans (or based on some
          assumed prepayment rate)

•       The reinvestment risk is especially significant since the
        payments are monthly and include principal payments.

•       If the actual prepayment differs significantly from the
        actual principal payment rate, the cash flow yield will not
        be close to the actual yield.
     Spread/Margin Measures for
        Floating Rate Bonds
•   For floating-rate securities, instead of a
    yield measure, margin measures (i.e.,
    spread above the reference rate) are
    computed.

•   Two margin measures commonly used are
    spread for life and discount margin.
         Spread/Margin Measures for
            Floating Rate Bonds
•       The coupon rate for a floating (variable) rate bond
        (floater) changes periodically according to a reference
        rate (i.e. LIBOR).

•       Since the future reference rate is not known, it is not
        possible to determine the cash flows – meaning that YTM
        cannot be calculated.
    –     Margin measures are used instead of YTM for floaters.
    –     Margin is simply some spread above the floater’s reference rate.

•       There are several spread or margin measures:
    –     Spread for life
    –     Discount margin
              Spread for Life
•   When a floater is selling at a premium
    (discount), investors consider the premium
    (discount) as another source (reduction) of
    the return.
    – Spread for life (or simple margin) is a measure
      of potential return that accounts for the
      amortization (accretion) of the premium
      (discount) as well as the constant quoted
      margin over the bond’s remaining life.
                         Spread for Life
•       Spread for life is calculated as follows:
          •      [(100(100 – Price))/Maturity + Quoted Margin] x (100/Price)

    –         Where
          •      Price = market price per $100 of par value
          •      Maturity = number of years to maturity
          •      Quoted Margin = quoted margin in the coupon reset formula
                 measured in basis points

    –         Example: a floater with a quoted margin of 80 basis points is
              selling for 99.3098 and matures in 6 years:
          •      [(100(100 – 99.3098))/6 + 80] x (100/99.3098) = 92.14 bps

•       This is a simple calculation, but has the limitations that it considers
        only the accretion / amortization of the discount/premium over the
        floater’s remaining term to maturity and does not consider the level
        of the coupon rate or the time value of money.
          Discount Margin
•   The discount margin assumes that the
    reference rate will not change over the
    life of the security and that there is no
    cap or floor restriction on the coupon
    rate.

•   The discount margin estimates the
    average margin over the reference rate
    that the investor can expect to earn over
    the life of the security.
                 Discount Margin
•   The discount margin is calculated as follows:
    –   Determine the cash flows assuming the reference
        rate does not change over the life of the security
    –   Select a margin
    –   Discount the cash flows by the current value of the
        reference rate plus the margin
    –   Compare the present value of the cash flows to the
        price plus accrued interest.
        •   If the present value is equal to the price plus accrued
            interest, the discount margin is the same as the selected
            margin
        •   If the present value is not equal, then use a different margin
            to equate the two.
    Discount Margin (continued)
•    Exhibit 3 shows the ‘trail and error’ approach to
     calculating the discount margin for a floating
     rate bond

•    The drawbacks of the discount margin as a
     measure of potential return from investing in a
     floating rate bond are as follows:
    –   The measure assumes that the reference rate will
        not change over the life of the security

    –   If the floating-rate security has a cap or floor, is not
        taken into consideration
Discount Margin
          Yield on Treasury Bills
•   Because T-bills have a maturity of one year or
    less, the standard convention is to compute a
    yield on a discount basis. There are two
    variables:
    –   The settlement price per $1 of maturity value (p)
    –   The number of days to maturity which is calculated as
        the number of days between the settlement date and
        the maturity date (NSM)
•   The yield on a discount basis is compute as:
                  Discount = (1 – p) x (360/NSM)
    Example of Yield on Treasury Bills
•   Settlement date of 8/6/06 with maturity of 1/8/07 and a
    price of 0.9776922. The number of days from settlement
    to maturity is 155.

•   The yield on a discount basis is:
                 5.18% = (1 – 0.9776922) x (360/155)


•   For a given yield on a discount basis, the price of a bill
    (per $1 maturity value) is computed as:
                       p = 1 – (d x (NSM/360))


               p = 1 – (.0518 x (155/360)) = 0.97769722
         Yield on Treasury Bills
•   The quoted yield on a discount basis is not a
    meaningful measure of return because:
    –   The measure is based on a maturity investment value
        rather than on the actual dollar amount
    –   The yield is annualized according to a 360-day year
        rather than a 365-day year, making it difficult to
        compare yield on T-bills with T-notes and bonds which
        pay interest based on the actual number of days in a
        year

•   Market participants recognize these limitations of
    yield on discount basis and make adjustments to
    make the yield on a T-bill comparable to other
    Treasury investments.
         Theoretical Spot Rates
•   The theoretical spot rate is the interest rate that should be used to
    discount a default-free cash flow.

•   Because there are a limited number of on-the-run Treasury
    securities traded in the market, interpolation is required to obtain the
    yield for interim maturities; hence, the yield for most maturities used
    to construct the Treasury yield curve are interpolated yields rather
    than observed yields.

•   Default-free spot rates can be derived from the Treasury yield curve
    by a method called bootstrapping.

•   The basic principle underlying the bootstrapping method is that the
    value of a Treasury coupon security is equal to the value of the
    package of zero-coupon Treasury securities that duplicates the
    coupon bond’s cash flows.
       Theoretical Spot Rates
          “Bootstrapping”
•   In order to value default-free cash flows, the
    theoretical spot rate for Treasury securities must
    be determined. (This was given in Chapter 5 –
    now we calculate it).

•   The default-free theoretical spot rate curve is
    constructed from the observed Treasury yield
    curve.

•   Several techniques are used to create the yield
    curve; however, the most commonly employed
    method is called “bootstrapping”
        Bootstrapping Spot Rates
•   Bootstrapping uses the yield for the on-the-run
    Treasury issues (since there are no credit or
    liquidity risks).
    –   A problem exists because there may be an insufficient
        number of data points for on-the-run issues to construct
        a yield curve.


•   Issuance of Treasury securities
    –   3-month, 6-month, 2-year, 3-year, 5-year, and 10-year
        notes (the 30-year has recently been reissued)
    –   This leaves gaps in the yield curve which can be filled
        in with simple linear interpolation
     Bootstrapping Spot Rates
•   To fill in the gap for each missing one year maturity, it is
    possible to start with the lowest maturity and work up to
    the highest maturity with the following formula:

            (yield at higher maturity – yield at lower maturity)
          Number of years between two observed maturity points


•   The estimated on-the-run yield for all intermediate whole-
    year maturities is found by adding the amount computed
    to the yield at the lower maturity.
      Bootstrapping Spot Rates
Example: 2-year 4.52%, 5-year 4.66%, 10-year 4.80%, 30-year 5.03%

Using the above information, to bootstrap the 3- and 4-year Treasury rates,
     the following interpolation of .0466% was computed as follows:
                               (4.66% – 4.52%)
                                    3 years

               Then the interpolated 3-year rate would be:
                         4.52% + .0466% = 4.567%
                 The interpolated 4-year rate would be:
                        4.567% + .0466% = 4.614%

Therefore, when a yield curve is shown, many of the points are only
     approximations. Exhibits 4 and 5 show an interpolated “bootstrapped”
     Treasury yield curve.

This method produces only a ‘crude approximation’
Bootstrapping Spot Rates
Bootstrapping Spot Rates
         Theoretical Spot Rates
•   The basic principle is that the value of a Treasury
    coupon series should be equal to the value of a
    package of zero-coupon Treasuries that
    duplicates the coupon bond’s cash flows.

•   Using the arbitrage-free method, it is possible to
    compute the approximate yield of bonds (spot
    rates) over any maturity range (including months)
    and going forward in time.
    –   These will be more precise that the linear interpolated
        results from bootstrapping.
    –   Exhibit 6 shows the plot of the theoretical spot rates
        and the par value Treasury yield curve.
Theoretical Spot Rates
   Method of Bootstrapping Spot
   Rates from the Par Yield Curve
1. Begin with the 6-month spot rate.
2. Set the value of the 1-year bond equal to the
   present value of the cash flows with the 1-year
   spot rate divided by 2 as the only unknown.
3. Solve for the 1-year spot rate.
4. Use the 6-month and 1-yar spot rates and
   equate the present value of the cash flows of
   the 1.5 year bond equal to its price, with the
   1.5 year spot rate as the only unknown.
5. Solve for the 1.5 year spot rate.
                   Example
• Consider the yields on coupon Treasury bonds
  trading at par (given in the table).
• YTM for the bonds is expressed as a bond
  equivalent yield (semi-annual YTM).

   Par Yields for Three Semiannual-Pay Bonds

     Maturity      YTM       Coupon       Price
    6 months      5.00%       5.00%      $100.00
      1 year      6.00%       6.00%      $100.00
    18 months     7.00%       7.00%      $100.00
1. Begin with the 6-month spot rate.
• The bond with six months left to maturity has a
  semi-annual discount rate of 5%/2 = 2.5% or 5%
  on a bond equivalent yield basis.
• Since this bond will only make one payment of
  $102.50 in six months, the YTM is the spot rate
  for cash flows to be received six months from
  now.
• The bootstrapping process proceeds from this
  point using the fact that the 6-month annualized
  spot rate is 5%.
2. Set the value of the 1-year bond equal to the
     present value of the cash flows with the 1-
        year spot rate divided by 2 as the only
      unknown. Solve for the 1-year spot rate.
The 1-year bond will make two payments, one in six
  months of $3.0 and one in one year of $103.0,
  and that the appropriate spot rate to discount the
  coupon payment (which comes 6 months from
  now), is written as:
$3.0/(1.025)1 + $103.0/(1+z2/2)2 = $100
where z2 is the annualized 1-year spot rate
      3. Solve for the 1-year spot rate.
      $3.0/(1.025)1 + $103.0/(1+z2/2)2 = $100
     where z2 is the annualized 1-year spot rate.

Solve for z2/2 as: $103.0/(1+z2/2)2 = $100 - $3/1.025
                           = $100 - $2.927 = $97.073
or:
$103.0/$97.073 = (1+z2/2)2

So: sq. root of ($103.0/$97.073) -1 = z2/2 = 3.0076%

  1-yr Spot rate (z2) = 3.0076% times 2 = 6.0152%
4. Use the 6-month and 1-year spot rates and
    equate the present value of the cash flows
    of the 1.5 year bond equal to its price, with
      the 1.5 year spot rate as the unknown.
• Now that we have the 6-month and 1-year spot rates,
  this information can be used to price the 18-month bond.
• Set the bond price equal to the value of the bond’s cash
  flows as:
       $3.5/(1.025)1 + $3.5/(1.030076)2 +
               $103.5/(1+z3/2)3 = $100
   where z3 is the annualized 1.5-year spot rate.
    5. Solve for the 1.5 year spot rate.
 $3.0/(1.025)1 + $3.5/(1.030076)2 + $103.5/(1+z3/2)3 = $100
        where z3 is the annualized 1.5-year spot rate.

Solve for z3/2 as: $103.5/(1+z3/2)2 = $100 - $3.5/1.025 -
   $3.5/(1.030076)2 = $100 - $3.415 - $3.30 =$93.285
or:
$103.5/$93.285 = (1+z3/2)2

So: cube root of ($103.5/$93.285) -1 = z3/2 = 3.5244%

     1.5-yr Spot rate (z3) = 3.5244% times 2 = 7.0488%
    Yield Spreads Relative to the Spot
               Rate Curve
•       The nominal spread is the difference between a
        non-Treasury bond’s yield and the YTM for a
        benchmark Treasury coupon security.
    –     The nominal yield spread measures the compensation for the
          additional credit risk, option risk, and liquidity risk an investor is
          exposed to by investing in a non-Treasury security with the same
          maturity.

•       The problems with the nominal spread measure are:
    –     For both bonds, the yield fails to take into consideration the term
          structure of spot rates
    –     In the case of a call or put bond, expected interest rate volatility
          may alter the cash flows of the non-Treasury bond
              Nominal Spread

•   The nominal spread is the difference
    between the yield for a non-Treasury bond
    and a comparable-maturity Treasury
    coupon security.

•   The nominal spread fails to consider the
    term structure of the spot rates and the fact
    that, for bonds with embedded options,
    future interest rate volatility may alter its
    cash flows.
Nominal Spread
                     Z-Spread

•   The zero-volatility spread or Z-spread is a
    measure of the spread that the investor will
    realize over the entire Treasury spot rate curve if
    the bond is held to maturity, thereby recognizing
    the term structure of interest rates.

•   Unlike the nominal spread, the Z-spread is not a
    spread off one point on the Treasury yield curve
    but is a spread over the entire spot rate curve.
             Zero-Volatility Spread
•       The zero-volatility or Z- spread is a measure of
        the spread the investor would realize over the
        entire Treasury spot rate curve if the bond is held
        to maturity.
    –     It is not the spread off of one point on the Treasury yield curve
          (nominal spread), it is an average over all spot rates.

•       The Z-spread is also called a static spread – and is
        calculated as the spread which will make the present
        value of the cash flows from the non-Treasury bond, when
        discounted at the Treasury spot rate plus the spread,
        equal to the non-Treasury bond’s price.
    –     Trial and error is used to determine the Z-spread.
    Zero-Volatility Spread Example
•   Exhibits 8 and 9 show how trial and error
    is used to compute the Z-spread

•   The Z-spread is measured relative to the
    Treasury spot rate curve and represents a
    spread to compensate for the non-
    Treasury bond’s credit risk, liquidity risk,
    and any embedded option risk.
Zero-Volatility Spread
Zero-Volatility Spread
    Zero-Volatility and Nominal Spread
•    For bullet bonds, unless the yield curve is
     very steep, the nominal spread will not
     differ significantly from the Z-spread; for
     securities where principal is paid over
     time rather than just at maturity there can
     be a significant difference, particularly in a
     steep yield curve environment.
    Z-Spread and the Nominal Spread
•       The Z- and nominal spreads will not differ much for standard coupon-
        paying bullet bonds. They will diverge when:

    –      The slope of the term structure is steep
    –      The principal is paid off before maturity (i.e. mortgage- and asset-back
           bonds)

•       The Z-spread can be calculated to any benchmark spot rate curve.
    –      When used for the same issuer, it is possible to isolate liquidity risk
        Option-Adjusted Spread
•   The option-adjusted spread (OAS) converts the cheapness or richness of a
    bond into a spread over the future possible spot rate curves.

•   An OAS is said to be option adjusted because it allows for future interest rate
    volatility to affect the cash flows.

•   The OAS is a product of a valuation model and, when comparing the OAS of
    dealer firms, it is critical to check on the volatility assumption (and other
    assumptions) employed in the valuation model.

•   The cost of the embedded option is measured as the difference between the
    Z-spread and the OAS.

•   Investors should not rely on the nominal spread for bonds with embedded
    options since it hides how the spread is split between the OAS and the option
    cost.

•   OAS is used as a relative value measure to assist in the selection of bonds
    with embedded options.
           Option-Adjusted Spread
•       The Z-spread, which looks at measuring the spread over a
        spot rate curve, has a problem in that it fails to take future
        interest rate volatility into consideration – which could
        change the cash flows for bonds with embedded options.

•       The option-adjusted spread (OAS) was developed to take
        the dollar difference between the fair valuation and the
        market price and convert it to a yield spread measure.
    –      The OAS is used to reconcile the fair price (value) and the market
           price by finding a return (spread) that will equate the two.
    –      The spread is measured in basis points.
           Option-Adjusted Spread
•       The OAS depends upon the valuation model employed.
    –      OAS models primarily differ in how they forecast interest rate changes.

•       What are the key modeling differences?
    –      Interest rate volatility is a crucial assumption. The higher the interest rate
           volatility, the lower the OAS.
    –      The OAS is a spread over the Treasury spot rate curve or the issuer’s
           benchmark. In the model, the spot rate curve is the result of a series of
           assumptions that allow for changes in interest rates.

•       The spread is referred to as “option adjusted” because the bond’s
        embedded options can change the cash flows and therefore the
        value of the security. Note: the Z-spread ignores how interest rate
        changes can impact cash flows – which is why it is referred to as the
        zero-volatility OAS.
                Option Cost
•   The implied cost of the option embedded in a
    bond can be obtained by calculating the
    difference between the OAS at the assumed
    interest rate or yield volatility and the Z-spread.

•   The Z-spread is the OAS plus the option cost.
    Therefore, the option cost equals the Z-spread
    minus the OAS.

•   Thus, the option cost is the difference between
    the spread that would be earned in a static
    interest rate environment (Z-spread) and the
    spread after adjusting for the option (OAS).
                        Option Cost
•       For callable bonds and mortgage- and asset-backed
        securities, the option cost is positive.
    –     This is because the issuer’s ability to alter the cash flows will
          result in an OAS that is less than the Z-spread.

•       In the case of a putable bond, the OAS is larger than the
        Z-spread because of the investor’s ability to alter the
        cash flows.

•       In general, when the option cost is positive (negative),
        the investor has sold (bought) an option to the issuer or
        borrower.
    –     An investor that relies only on the nominal spread may not be
          adequately compensated for taking on option risk – which is one
          of the strengths of the OAS approach.
Summary of Spread Measures


                                                 Reflects
Spread Measure         Benchmark              compensation for:
                                           Credit risk, option risk,
    Nominal         Treasury yield curve           liquidity risk
                    Treasury spot rate     Credit risk, option risk,
  Zero-volatility          curve                   liquidity risk
                    Treasury spot rate
 Option-Adjusted           curve           Credit risk, liquidity risk
             Forward Rates
•   Besides default-free theoretical spot rate curves
    extrapolated from the Treasury yield curve, it is
    possible to compute forward rates.

•   Since forward rates are extrapolated from the
    default-free theoretical spot rate curve, these
    rates are referred to as implied forward rates.

•   Besides using the Treasury yield curve, it is
    possible to compute forward rates from other
    interest rate curves (i.e. LIBOR).
            Forward Rates
•   Using arbitrage arguments, forward rates
    can be extrapolated from the Treasury
    yield curve or the Treasury spot rate curve.

•   The spot rate for a given period is related
    to the forward rates; specifically, the spot
    rate is a geometric average of the current
    6-month spot rate and the subsequent 6-
    month forward rates.
            Forward Rates
• Notation:       1f1

    when issued         time to maturity

• Definition of forward rate: The implied
  rate of return on a security to be issued
  at some future date.

• Definition of spot rate: The rate of return
  on securities already issued.
  Spot and Forward Rates for Fixed
         Income Securities
• A spot rate is a rate agreed upon today, for a
  loan that is to be made today. (e.g. r1 = 5%
  indicates that the current rate for a one-year
  loan is 5%).

• A forward rate is a rate agreed upon today, for a
  loan that is to be made in the future. (e.g. 2f1 =
  7% indicates that we could contract today to
  borrow money at 7% for one year, starting two
  years from today).
                      Forward Rates
Forward rates of interest are implicit in the term structure of interest rates

                          t=0          1          2         3         4…

                                  r1        1f2

                                       r2             2f3

                                             r3                 3f4



Note the notation: 3f4 means “the forward rate from period 3 to period 4.”

When the beginning subscript is omitted, it is understood that the forward
  rate is for one period only: 3f4 = f4 .
General Formula for Forward Rates
                                  (1  rn )n  (1 rn1 ) n1 (1  fn )1
• One-period forward rates:
                                               imply ing that ...

                                                  (1  rn ) n
                                           fn              n1
                                                                1
                                                (1 rn1 )

• n-period forward rates:   (1  rk n )k  n  (1  rk ) k (1         f k  n) n
                                                                    k

                                              imply ing that...
                                                                        1
                                                  (1  r )k  n  n
                                         fk  n        kn       1
                                     k
                                                   (1  rk ) 
                                                  
                                                              k
                                                                  
         Example: Forward Rates
• What one-year forward rates are implied by the
  following spot rates?
       Maturity Year             Spot Rate (rt)          Forward Rate (ft)

             1                       4.0%                            –

             2                       5.0%                      6.01%

             3                       5.5%                     6.507%

  (1  r2 )2  (1 r )(1 f2 )
                   1                        (1  r3 )3  (1  r2 )2 (1  f3 )
                                                    3            2
         2
   (1.05)  (1.04)(1  f2 )                 (1.055)  (1.05) (1  f3 )
        f2  6.01%                                f3  6.507%
   Implied Forward Rate Example
• Suppose the spot term structure of zero-
  coupon yields is: {r1=0.08, r2=0.10, r3=0.13,
  r4=0.14,…}

• If investors wish to invest $1,000,000 for two
  years. They can choose between:
   – buying a 2-yr. discount bond, and
   – buying a sequence of two 1-yr. bonds, i.e.,
    one now and one in one year from now.
  What Will the Investor Choose?
• The alternative that pays the higher
  cumulative return over the 2-yr time horizon.

• Caveat: The rate of return on the bond
  issued one year from now is uncertain.

• How do we estimate it?
  – With the implied forward rate
Estimating the Implied Forward Rate

  (1+ 1 f1 )(1+r1 ) = (1+r2 )2 
                      (1+r2 )2 (1.1)2
         (1+ 1 f1 ) =          =        = 1.12   12%
                       (1+r1 )   (1.08)


• Underlying assumption: These must be
  equal cumulative returns, with no arbitrage
  possible.
Estimation of Implied Forward Rates
(using the spot term structure from a previous slide)
                                                  3f1


               t= 0        1      2         3           4
                                1f2         2f2



                                      1f3
   (i)    f is given by:
         3 1

         (1+ 3 f1 )(1+r3 )3 = (1+r4 ) 4 
                     (1+r4 ) 4
         (1+ 3 f1 )=           = 1.17 or 17%
                     (1+r3 ) 3
    Estimation of Implied Forward
         Rates (continued)
(ii) 1 f2 is defined by:
           (1+ 1 f2 )2 (1+r1 ) = (1+r3 )3 
                         (1+r3 )3   (1.13)3
           (1 + 1 f2 ) =          =         = 1.1558 or 15.58%
                          (1+r1 )    (1.08)
(iii) 1 f3 is defined by:
        (1+ 1 f3 )3 (1+r1 ) = (1+r4 ) 4 
                         (1+r4 )4 1 3  (1.14) 4 1 3
          (1+ 1 f3 ) = (          ) =(         ) = 1.1605 or 16.05%
                          (1+r1 )       (1.08)
   General Formula for Implied
        Forward Rates
                                         1
                      (1+ri+j )i+j         j
           1+ i fj =           i 
                      (1+ri ) 

• Note that implied fwd rates are internally
  consistent, e.g.,
            (1 1f 2 )2 (1 3f1)  (1 1f 3 )3 
                                                 1
            (1 1f 3 )  (1 1f 2 ) (1 3f1)
                         
                                 2
                                             
                                                  3
    Deriving a 6-Month Forward Rate
To compute a 6-month forward rate, it is necessary to utilize
    a yield curve and the corresponding spot rate curve.

•       The following 2 investments should have the same
        value:
    –     1-year Treasury bill and
    –     2 six-month Treasury bills (one purchased now and the other in
          six months)

•       An investor should be indifferent since they should
        produce the same investment income over the same
        investment horizon.
   Deriving a 6-Month Forward Rate
Although an investor does not know the interest rate of the
     second 6-month T-bill, it is possible to compute it because
     the “forward” rate must such that it equalizes the dollar
     return between the two alternatives.

Exhibit 11 shows the timeline for the two investment
    alternatives:
•   The value of first six-month T-bill is: X(1 + z1)
•   The value of the total investment following the second six-
    month T-bill is: X(1 + z1)(1 + f)
   –   Where z1 is one-half the bond-equivalent yield of the 6-month spot
       rate and f is one-half the forward rate on a 6-month Treasury bill
       available 6 months from now. X is the amount of the investment.
Deriving a 6-Month Forward Rate
    Relationship Between Spot Rates and Short-
                Term Forward Rates
•        The value of alternative investment (a 1-year T-bill) is computed as:
                                        X(1 + z2)2

•        Because the two alternatives should generate identical returns:

                          X(1 + z1)(1 + f) = X(1 + z2)2
•        Solving for f   = [(1 + z2)2 / (1 + z1)] -1

     –      Multiplying f by 2 to get the forward rate on a bond-equivalent yield
            basis.

•        Forward rates can be computed on various combinations of short-
         and longer-term interest rates. Exhibit 12 provides the six-month
         forward rates for the entire yield curve. Exhibit 13 is a graph of the
         forward rate curve.

								
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