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Spreads (DOC)


   Calculating and Interpreting Yield Spreads over Reference
Fin 510 – Topics in Investment Analysis
Spring 2009
John Settle

Basic idea
A spread is how much a yield exceeds the yield on a reference bond. The spread
compensates the buyer for risks in the bond that are not in the reference bond. For
example, the AAA-Treasury spread for equal maturities represents compensation for
credit risk in AAA bonds. It may also represent underpricing (spread too large for risk
assumed) or overpricing (spread too small for risk assumed). See Chapter 9 Section III
for examples.

There is some analogy between spreads and P/Es, in that differences in characteristics of
bonds that are not controlled for get swept into the spreads in the same way that
characteristics of stocks that are not controlled for get swept into the P/E ratios. Both are
used (1) to compare securities, deciding whether differences in spreads (P/Es) are
attributable to differences in risks (risks and growth), or mispricing; (2) to look at trends
over time.

Types of risk:
   1. Interest rate
   2. Credit risk
   3. Liquidity risk
   4. Risk due to embedded options.
A Short Taxonomy of Spreads

          Yield complexity                 Spread Measure               Spread reflects
                                                                       compensation for
                                                                         (over T-bills):
1       Simple yield to maturity     Nominal spread –               Credit risk; option
                                     difference between YTM         risk; liquidity risk
                                     for two bonds of equal
2       Spot rates (Zero rates)      Zero-volatility spread. The    Credit risk; option
                                     (Z-Spread or static spread):   risk; liquidity risk
                                     percentage that, if added to   (Better measure than
                                     each spot and implied          nominal spread
                                     forward rate in the            because it considers
                                     reference, discounts the       the yield curve)
                                     bond to its actual value.
                                     (See Exhibit 9, page 144)
3       Spot and implied forward
        rates (inferred from
        current zero rates)
4       Forecast distribution of     Option-adjusted spread         Credit risk; liquidity
        possible future one-         (OAS)                          risk. (The spread
        period spot rates.                                          process controls for
        Binomial model can be                                       the embedded options
        used.                                                       in the security.)
5       Monte Carlo simulation       Also called Option-            Credit risk; liquidity
        of possible future spot      adjusted spread (see           risk. (The spread
        rates (needed when cash      Chapter 12)                    process controls for
        flows depend on the                                         the embedded options
        entire path of prior spot                                   in the security.)
        rates – for example with
        MBSs, which have
        prepayment patterns that
        depend on the entire prior
        path of interest rates.)

    Explaining the processes
    1. Find YTM of subject and reference securities and compare.
    2. Evaluate the subject security using a series of spot rates, each of which has the
       same spread over the spot rates applied to the reference security. (This is, in
       effect, shifting the yield curve up parallel to the point where it discounts the
       subject security to its actual market value.)
    3. No method here
    4. A lot of iteration here.
           a. Devise or obtain a model of how interest rates evolve over time. For
                example, it could be that r evolves according to a fixed volatility measure:
   a typical assumption is a continuous lognormal distribution, where ln[rT}
   has a distribution with mean equal to ln[S0 + ( – 2/2)T, and variance
   equal to 2T.
b. Use this model to predict up- and down- values of interest rates in each
   step of a binomial tree. (There are other models, but binomial models are
   the simplest. The text illustrates binomial models where each step is one
   year, but more realism can be approximated by shortening these steps.
   Good approximations can be obtained without too many steps. About 100
   per year should be decent.
         i. The root rate is given: it is the one-period spot rate for the
            reference security.
        ii. Predict the up and down rates for the second period (rH and rL)
            using your interest rate model.
      iii. Use these rates to discount the reference two-period security. If
            you don’t arrive at the actual value, adjust rL and keep the same
            spread between rL and rH to determine rH. Discount again. Keep
            up this trial and error until your reference security evaluates
            correctly. Keep these rates.
       iv. Cycle back to ii, but go up one period. (In the third year, you will
            have three rates to predict, and so forth.) For each new period
            added, use Step iii to calibrate the value of the reference security to
            its actual value. Exhibit 6 shows the results of applying this to the
            three-period iteration.
        v. Repeat until you have enough periods to cover the life of the
            security. When you are done, you have an interest rate tree like the
            one depicted in Exhibit 5, page 231. This is the rate tree you will
            use to do all of your discounting.
c. For any security with any series of contingent payments, put in the
   payments that will occur in each node. Start with the latest period in the
   tree and work back.
         i. Exhibit 7 – a straight bond, is verified to have the value previously
            calculated using the forward rate structure. (Notice: the author has
            simplified the whole exposition by supposing that the reference
            security is the on-the-run securities of the issuer of the subject
            security – see the comment in the second paragraph of Section IV:
            “We will use as our benchmark in the rest of this chapter, the
            securities of the issuer whose bond we want to value. Hence, we
            will start with the issuer’s on-the-run yield curve.”
        ii. Exhibit 8 – a callable bond, no call premium: Note how it is
            necessary in each node to decide whether to call the bond at that
      iii. Exhibit 9 – a callable bond with a declining call premium.
       iv. Exhibits 14-22 – other types of bonds with embedded options. The
            only difference is how you calculate the values and cash flows at
            each of the nodes. This follows whatever stipulations are in the
         d. To get the OAS for this security, guess at an OAS and add that to every
             rate in the tree. Discount to see if you get the actual value of the subject
             security. Keep trying until you get the right OAS.
                  i. The callable bond in Exhibit 8, has its OAS found in Exhibit 11, to
                      be 35 bp.
         e. Some other observations in these sections:
                  i. If you change the assumed volatility of interest rates, you can have
                      a strong effect on the value and the OAS. For example, in the case
                      of a callable bond, more volatile interest rates benefit the issuer, so
                      the value of the option rises and the value of the security falls. The
                      OAS falls from 35 bp to 6 in the example on page 236, where
                      assumed rate volatility rises from 10% to 20%. Reason: this is
                      given the $102.218 price. At that price, you are giving more away
                      to the issuer (lower OAS) at a 20% volatility than at a 10%
                 ii. The value of a call option = [the value of an option-free bond] –
                      [the value of a callable bond] (p 234)
                iii. You can calculate effective duration, but to do so, you have to
                      regenerate the entire interest rate tree from scratch after shifting the
                      on-the-run yield curve up and down by y (which is 25 bp in the
                      author’s example in Exhibits 12 and 13. (Notice: he does not say,
                      and I don’t follow, how the interest rates can change the way they
                      do in these examples. For example the year zero spot rate is
                      increased in both Exhibit 12 (positive 25 bp change in rates) and
                      Exhibit 13 (negative 25 bp change in interest rates). It seems to me
                      that the year zero spot rate ought to move up (down) 25 bp, and
                      then the reconstruction start from there.
   5. This gets much more computer intensive, but the narrative in Chapter 12 Section
      IV gives some of the flavor. In addition to the above equipment, you need a
      model about how homeowners decide to prepay their mortgages. Since not all
      contingencies can be covered in a tree of reasonable size, the space is sampled
      with Monte Carlo simulation.

Final notes:
   1. This is an elaborate process that is desired mainly so that analysts can make yield
   2. The binomial model looks a lot like the model that would be used in option
       pricing, but it makes some fundamentally different assumptions. The option
       pricing model operates on the assumption that a perfect hedge can be formed
       using the option and the underlying, for the duration of one period. It is this
       perfect hedge that fixes the option price in terms of the underlying. This process
       doesn’t take place in determining values or OASs of bonds with embedded
       options. In this respect, it doesn’t actually consider the riskless hedge aspects of
       the options in finding their values. That is, these bonds are not valued using
       option valuation models, and may therefore not precisely reflect those option

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