Fixed-Income Securities. Solutions 8

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1 FIN 472 Professor Robert Hauswald Fixed-Income Securities Kogod School of Business, AU Solutions 8 1. Determine the one-year and two-year forward swap rates, which reset on 9/13/1999, and start paying in quarterly cycles from 12/13/1999. Assume that the spot LIBOR from the settlement date to 9/13/1999 is 5.500%. In the worksheet below, we estimate the zero prices based on the information provided. Since swaps reset dates precede payment dates by one quarter, we shift down the Eurodollar futures prices by one quarter in the worksheet below to reflect it. Using ED futures, we compute implied LIBOR and based on this data we estimate the forward prices. These forward prices form the basis for computing zero prices.

2 FORWARD SWAP RATE (ONE-YEAR SWAP) PRICING Settlement 8/16/1999 Start Date 9/13/1999 Notional Principal 1 Million End Date 9/17/2000 Fixed Paid 30/360 semi-annual % Spot LIBOR % Floating 3-months quarterly actual/360 Forwards zero prices Floating (Floating) PV(Floating) Fixed PV(Fixed) 8/16/ /360 Actual 9/13/ /13/ % 14, /13/ % 15, ,235 29, /12/ % 15, /11/ % 15, ,235 28, TOTAL ,613 Floating leg pays on 12/13, 3/13, 6/12 & Fixed pays on 3/13 & 9/11 9/11 FORWARD SWAP RATE (TWO-YEAR SWAP) PRICING Settlement 8/16/1999 Start Date 9/13/1999 Notional Principal 1 Million End Date 9/17/2001 Fixed Paid 30/360 semi-annual % Spot LIBOR % Floating 3-months quarterly actual/360 Forwards zero prices Floating (Floating) PV(Floating) Fixed PV(Fixed) 8/16/ Actual 9/13/ /13/ % 14, /13/ % 15, ,987 30, /12/ % 15, /11/ % 15, ,987 30, /11/ % 16, /12/ % 16, ,987 29, /11/ % 16, /10/ % 17, ,987 28, TOTAL 118, , What is the relationship between options on Eurodollar futures contracts, caps, floors, and collars? It is important to note that these options settle by cash at maturity, and they expire on the same day as the underlying futures contract. Let us examine the call with a strike price of At maturity, this call will pay an amount equal to Max[ 0,H 94.25]. We denote by H the Eurodollar futures price at maturity. We know that H = 100 LIBOR from our discussions earlier. Using this, we get the payoff of the call option on Eurodollar futures to be Max[ 0,5.75 LIBOR]. This is also the payoff of a put option on LIBOR with a strike rate of 5.75%. From Table 10, we find that this call costs 24 basis points. Each basis point costs \$25 so the value of this option is = \$600.

3 In a similar way, the put with a strike price of will pay at maturity an amount equal to Max[ 0,94.75 H]. Again, we denote by H the Eurodollar futures price at maturity. Since at maturity, the Eurodollar futures price settles to LIBOR by the condition H = 100 LIBOR, we can rewrite the payoff of the put at maturity as Max[ 0,LIBOR 5.25]. This is the payoff of a call option on LIBOR with a strike rate of 5.25%. 3. Prepayment factors. (a) Prepayments of mortgages are driven by a number of factors, each of which merits further elaboration. 1. Refinancing Incentive: The refinancing incentive is perhaps the most important reason for prepayments. If the market rates for mortgage loans drop significantly below the rate that a borrower is paying, then the borrower has a very strong reason to prepay as long as the borrower is able to qualify for a new loan. This incentive implies that the prepayments should accelerate in periods of falling interest rates, especially when the belief in the market is that the rates have bottomed out. 2. Seasonality Factor: Families typically do not move during the school year. This implies that, other things remaining the same, families will move during the period starting from the middle of June through the first week of September. This should result in increased prepayments during this part of the year. This can be thought of as the seasonality factor, or the school year factor. 3. Age of the Mortgage: During the early part of the mortgage loan, interest payments far exceed the principal component. This, in part, implies that the interest savings associated with refinancing are greater during the earlier part of the mortgage loan. Prepayments are typically greater during the early life of the loan and tend to stabilize afterwards. Indeed, the prepayments appear to be higher when the life of the loan ranges from two to eight years. This may be due to factors such as increasing family size, job switches, and the like. In a similar manner, when the mortgage is more than 25 years old, there may be an incentive to pay it off to secure the property s title. The rate of prepayments appears to slow down for loans in the 10- to 25-year range. 4. Family Circumstances: A number of family circumstances dictate the prepayment decision of a household. These factors include marital status (divorce decisions often might lead to prepayments), job switching, and so on. Sometimes, the inability of a household to make the monthly payment (due to job loss or disability) might lead to a default. Under some circumstances, this can precipitate a prepayment. For instance, there are two forms of mortgage insurance. In one form, the lender initiates the insurance and the policy typically guarantees that the insurance company will pay some percentage or all of the loan if the homeowner defaults. In the other form, initiated by the homeowner, the insurance company will pay off the loan obligations if the insured person dies. If the loan taken by the household is assumable, when the family moves, the next family that moves into the home can assume the mortgage. If the loan is not assumable, it has to be paid in full. 5. Housing Prices: The price of the home is yet another factor in the prepayment decision of the household. The housing price affects the LTV ratio, which in turn affects the ability of the household to qualify for refinancing. If the house price increases, then the LTV decreases. This enhances the homeowner s ability to refinance if the going interest rates and family circumstanc-

5 (b) As interest rates drop, the prepayments start to increase and the value of a GNMA tends to approach its par amount. For a noncallable Treasury, as the interest rates drop the market value increases. Thus, the Treasuries will tend to outperform the GNMAs under those conditions. Of course, refinancing rates do not necessarily move parallel with the Treasury yields. Premium GNMAs will have the greatest prepayment risk, although the premium burnout effect should be kept in mind. The discount and par mortgage-backed securities will also show greater prepayments. 6. Real Estate Mortgage Investment Conduits, or "REMICs," (sometimes also called Collateralized mortgage obligations) are a type of special purpose vehicle used for the pooling of mortgage loans and issuance of mortgage-backed securities. They were introduced in 1987 [1] and are defined under the United States Internal Revenue Code (Tax Reform Act of 1986), and are the typical vehicle of choice for the securitization of residential mortgages in the US. The REMIC must be heavily over-collateralized. The purpose of this over- collateralizing is to create an insurance cushion that helps to offset any cash flow shortages that may result due to a fall in reinvestment income from the underlying monthly cash flows. Trustees ensure that the remaining collateral is large enough at all times for all tranches to get their promised cash flows. Most of the REMICs are rated AAA by the usual rating agencies. To provide AAA rating, these agencies require that the present value of zero-prepayment cash flows from the collateral at a discount rate equal to the maximum coupon of the bond determines the maximum amount of bonds that may be issued. 7. CDO Value Determinants. Three factors determine the value of different CDO securities: a) the probability of default, b) loss conditional on default, and c) the correlation of defaults. The first two factors influence CDO values in the usual manner: the greater the probability of default, and the greater is the loss given default, the worse off will be the CDO investors. The third factor, the correlation of defaults, also is an important consideration. The higher the correlation of defaults, the lower will be the overall value of CDO, although some tranches could do better at the expense of other tranches. 8. Correlation, Default and CDO Value. Correlation plays an important role in the value of equity tranche. When there is a high correlation, two possibilities present themselves: all the underlying names can default together or they all survive together. For the equity tranche holders, the attachment point is reached even if just a few names default together. As a result the fact that many names default together has no incremental adverse effect on their values. On the other hand, the scenario in which many names survive together increases the expected life of equity tranche and hence its value.

6 9. Prepayment risk and RMBS Analysis. This exhibit presents estimated cash flows for a 7.5%, \$1 million mortgage pool under various prepayment assumptions. See attached XLS sheet. 10. CDO Risks and Tranching. An interest only (IO) strip may be carved from collateral securities to receive just the interest portion of a payment. Once an underlying debt is paid off, that debt's future stream of interest is terminated. Therefore, IO securities are highly sensitive to prepayments and/or interest rates and bear more risk. These securities usually have a negative effective duration. IOs have investor demand due to their negative duration acting as a hedge against conventional securities in a portfolio, their generally positive carry (net cashflow), and their implicit leverage (low dollar price versus potential price action). The various hedges available require an overview of the various tranching techniques. IO/PO pair. The simplest coupon tranching is to allocate the coupon stream to an IO, and the principal stream to a PO. This is generally only done on the whole collateral without any prepayment tranching, and generates strip IOs and strip POs. In particular FNMA and FHLMC both have extensive strip IO/PO programs (aka Trusts IO/PO or SMBS) which generate very large, liquid strip IO/PO deals at regular intervals. IO/discount fixed rate pair. A fixed rate CMO tranche can be further restructured in to an Interest Only (IO) tranche and a discount coupon fixed rate tranche. An IO pays a coupon only based on a notional principal, it receives no principal payments from amortization or prepayments. Notional principal does not have any cash flows but shadows the principal changes of the original tranche, and it is this principal off which the coupon is calculated. For example a \$100mm PAC tranche off 6% collateral with a 6% coupon ('6 off 6' or '6-squared') can be cut into a \$100mm PAC tranche with a 5% coupon (and hence a lower dollar price) called a '5 off 6', and a PAC IO tranche with a notional principal of \$ mm and paying a 6% coupon. Note the resulting notional principle of the IO is less than the original principal. Using the example, the IO is created by taking 1% of coupon off the 6% original coupon gives an IO of 1% coupon off \$100mm notional principal, but this is by convention 'normalized' to a 6% coupon (as the collateral was originally 6% coupon) by reducing the notional principal to \$ mm (\$100mm / 6). PO/premium fixed rate pair. Similarly if a fixed rate CMO tranche coupon is desired to be increased, then principal can be removed to form a Principal Only (PO) class and a premium fixed rate tranche. A PO pays no coupon, but receives principal payments from amortization and prepayments. For example a \$100mm sequential (SEQ) tranche off 6% collateral with a 6% coupon ('6 off 6') can be cut into an \$ mm SEQ tranche with a 6.5% coupon (and hence a higher dollar price) called a '6.5 off 6', and a SEQ PO tranche with a principal of \$ mm and paying a no coupon. The principal of the premium SEQ is calculated as (6 / 6.5) * \$100mm, the principal of the PO is calculated as balance from \$100mm. Floater/inverse floater pair. By now it should be clear that coupon tranching always results in pairs of securities with opposite risk attributes. The construction of CMO Floaters is the most

7 effective means of getting additional market liquidity for CMOs. CMO floaters have a coupon that moves in line with a given index (usually 1 month LIBOR) plus a spread, and is thus seen as a relatively safe investment even though the term of the security may change. One feature of CMO floaters that is somewhat unusual is that they have a coupon cap, usually set well out of the money (e.g. 8% when LIBOR is 5%) In creating a CMO floater, a CMO Inverse is generated. The CMO inverse is a more complicated instrument to hedge and analyse, and is usually sold to sophisticated investors. The construction of a floater/inverse can be seen in two stages. The first stage is to synthetically raise the effective coupon to the target floater cap, in the same way as done for the PO/Premium fixed rate pair. As an example using \$100mm 6% collateral, targeting an 8% cap, we generate \$25mm of PO and \$75mm of '8 off 6'. The next stage is to cut up the premium coupon into a floater and inverse coupon, where the floater is a linear function of the index, with unit slope and a given offset or spread. In the example, the 8% coupon of the '8 off 6' is cut into a floater coupon of: 1 * LIBOR % (indicating a 0.40%, or 40bps, spread in this example) The inverse formula is simply the difference of the original premium fixed rate coupon less the floater formula. In the example: 8% - (1 * LIBOR %) = 7.60% - 1 * LIBOR The floater coupon is allocated to the premium fixed rate tranche principal, in the example the \$75mm '8 off 6', giving the floater tranche of '\$75mm 8% cap + 40bps LIBOR SEQ floater'. The floater will pay LIBOR % each month on an original balance of \$75mm, subject to a coupon cap of 8%. The inverse coupon is to be allocated to the PO principal, but has been generated of the notional principal of the premium fixed rate tranche (in the example the PO principal is \$25mm but the inverse coupon is notionalized off \$75mm). Therefore the inverse coupon is 're-notionalized' to the smaller principal amount, in the example this is done by multiplying the coupon by (\$75mm /\$25mm) = 3. Therefore the resulting coupon is: 3 * (7.60% - 1 * LIBOR) = 22.8% - 3 * LIBOR In the example the inverse generated is a '\$25mm 3 times levered 7.6 strike LIBOR SEQ inverse'. Other structures include Inverse IOs, TTIBs, Digital TTIBs/Superfloaters, and 'mountain' bonds. A special class of IO/POs generated in non-agency deals are WAC IOs and WAC POs, which are used to build a fixed pass through rate on a deal. Credit tranching is very different from coupon tranching. It is the most common form of credit protection and means that any credit losses will be absorbed by the most junior class of bond-

8 holders until the principal value of their investment reaches zero. If this occurs, the next class of bonds absorb credit losses, and so forth, until finally the senior bonds begin to experience losses. More frequently, a deal is embedded with certain triggers related to quantities of delinquencies or defaults in the loans backing the mortgage pool. If a balance of delinquent loans reaches a certain threshold, interest and principal that would be used to pay junior bondholders is instead directed to pay off the principal balance of senior bondholders, shortening the life of the senior bonds. Typical mechanisms to carve out different credit exposures are: Overcollateralization: in CMOs backed by loans of lower credit quality, such as subprime mortgage loans, the issuer will sell a quantity of bonds whose principal value is less than the value of the underlying pool of mortgages. Because of the excess collateral, investors in the CMO will not experience losses until defaults on the underlying loans reach a certain level. If the "overcollateralization" turns into "undercollateralization" (the assumptions of the default rate were inadequate), then the CMO defaults. CMOs have contributed to the subprime mortgage crisis. Excess spread: another way to enhance credit protection is to issue bonds that pay a lower interest rate than the underlying mortgages. For example, if the weighted average interest rate of the mortgage pool is 7%, the CMO issuer could choose to issue bonds that pay a 5% coupon. The additional interest, referred to as "excess spread", is placed into a "spread account" until some or all of the bonds in the deal mature. If some of the mortgage loans go delinquent or default, funds from the excess spread account can be used to pay the bondholders. Excess spread is a very effective mechanism for protecting bondholders from defaults that occur late in the life of the deal because by that time the funds in the excess spread account will be sufficient to cover almost any losses. The third risk category is obviously prepayment risk. Once again, tranching allows us to cater to almost any investor taste or preference. Prepayment risk is the risk that the term of the security will vary according to differing rates of repayment of principal by borrowers (repayments from refinancings, sales, curtailments, or foreclosures). If principal is prepaid faster than expected (for example, if mortgage rates fall and borrowers refinance), then the overall term of the mortgage collateral will shorten, and the principal returned at par will cause a loss for premium priced collateral. This prepayment risk cannot be removed, but can be reallocated between CMO tranches so that some tranches have some protection against this risk, whereas other tranches will absorb more of this risk. To facilitate this allocation of prepayment risk, CMOs are structured such that prepayments are allocated between bonds using a fixed set of rules. The most common schemes for prepayment tranching are described below. Sequential tranching (or by time). All of the available principal payments go to the first sequential tranche, until its balance is decremented to zero, then to the second, and so on. There are several reasons that this type of tranching would be done: The tranches could be expected to mature at very different times and therefore would have different Yields that correspond to different points on the Yield Curve.

9 The underlying mortgages could have a great deal of uncertainty as to when the principal will actually be received since home owners have the option to make their scheduled payments or to pay their loan off early at any time. The sequential tranches each have much less uncertainty. Parallel tranching simply means tranches that pay down pro rata. The coupons on the tranches would be set so that in aggregate the tranches pay the same amount of interest as the underlying mortgages. The tranches could be either fixed rate or floating rate. If they have floating coupons, they would have a formula that make their total interest equal to the collateral interest. For example, with collateral that pays a coupon of 8%, you could have two tranches that each have half of the principal, one being a floater that pays LIBOR with a cap of 16%, the other being an inverse floater that pays a coupon of 16% minus LIBOR. A special case of parallel tranching is the IO/PO split explained above. IO and PO refer to Interest Only and Principal Only. In this case, one tranche would have a coupon of zero (meaning that it would get no interest at all) and the other would get all of the interest. These bonds could be used to speculate on prepayments. A principal only bond would be sold at a deep discount (a much lower price than the underlying mortgage) and would rise in price rapidly if many of the underlying mortgages were prepaid. The interest only bond would be very profitable if few of the mortgages prepaid, but could get very little money if many mortgages prepaid. A Z bond is a tranche which supports other tranches by not receiving an interest payment. The interest payment that would have accrued to the Z tranche is used to pay off the principal of other bonds, and the principal of the Z tranche increases. The Z tranche starts receiving interest and principal payments only after the other tranches in the CMO have been fully paid. This type of tranche is often used to customize sequential tranches, or VADM tranches. Schedule bonds (also called PAC or TAC bonds) This type of tranching has a bond (often called a PAC or TAC bond) which has even less uncertainty than a sequential bond by receiving prepayments according to a defined schedule. The schedule is maintained by using support bonds (also called companion bonds) that absorb the excess prepayments. Planned Amortization Class (PAC) bonds have a principal payment rate determined by two different prepayment rates, which together form a band (also called a collar). Early in the life of the CMO, the prepayment at the lower PSA will yield a lower prepayment. Later in the life, the principal in the higher PSA will have declined enough that it will yield a lower prepayment. The PAC tranche will receive whichever rate is lower, so it will change prepayment at one PSA for the first part of its life, then switch to the other rate. The ability to stay on this schedule is maintained by a support bond, which absorbs excess prepayments, and will receive less prepayments to prevent extension of average life. However, the PAC is only protected from extension to the amount that prepayments are made on the underlying MBSs. When the principal of that bond is exhausted, the CMO is referred to as a "busted PAC", or "busted collar".

10 Target Amortization Class (TAC) bonds are similar to PAC bonds, but they do not provide protection against extension of average life. The schedule of principal payments is created by using just a single PSA. Very accurately defined maturity (VADM) bonds Very accurately defined maturity (VADM) bonds are similar to PAC bonds in that they protect against both extension and contraction risk, but their payments are supported in a different way. Instead of a support bond, they are supported by accretion of a Z bond. Because of this, a VADM tranche will receive the scheduled prepayments even if no prepayments are made on the underlying. Non-accelerating senior (NAS) NAS bonds are designed to protect investors from volatility and negative convexity resulting from prepayments. NAS tranches of bonds are fully protected from prepayments for a specified period, after which time prepayments are allocated to the tranche using a specified step down formula. For example, an NAS bond might be protected from prepayments for five years, and then would receive 10% of the prepayments for the first month, then 20%, and so on. Recently, issuers have added features to accelerate the proportion of prepayments flowing to the NAS class of bond in order to create shorter bonds and reduce extension risk. NAS tranches are usually found in deals that also contain short sequentials, Z-bonds, and credit subordination. A NAS tranche receives principal payments according to a schedule which shows for a given month the share of pro rata principal that must be distributed to the NAS tranche. NASquential NASquentials were introduced in mid 2005 and represented an innovative structural twist, combining the standard NAS (Non-Accelerated Senior) and Sequential structures. Similar to a sequential structure, the NASquentials are tranched sequentially, however, each tranche has a NAS-like hard lockout date associated with it. Unlike with a NAS, no shifting interest mechanism is employed after the initial lockout date. The resulting bonds offer superior stability versus regular sequentials, and yield pickup versus PACs. The support-like cashflows falling out on the other side of NASquentials are sometimes referred to as RUSquentials (Relatively Unstable Sequentials).

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