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Spreads Calculating and Interpreting Yield Spreads over Reference Bonds 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 maturity. 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 point. 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 contract. 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% volatility. 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 comparisons. 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 values.
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