Annuities and amortising securities
19.3 Mortgage-backed securities
19.1 INTRODUCTION
An amortising security is superficially similar to a bond, in that it is issued by a borrower who intends to repay the funds to the lender via scheduled payments. Unlike a bond, however, the principal of the loan is repaid over the lifetime of the loan rather than as a single bullet payment at the bond’s maturity.
The best-known amortising security is a mortgage on a house. Typically, the mortgage is structured so that the lender makes equal payments over the lifetime of the loan. These payments are made up of repayments of principal and interest. At the outset, the payments go almost entirely to paying off interest, but as the principal reduces the proportions change until near the end of the security’s lifetime the payments are almost entirely directed to principal reduction.
The cash flow structure of an amortising bond is therefore very different to that of a vanilla bond with a similar maturity:
- The modified duration of the amortising bond is much shorter than that of the bond, since its cash flows are uniformly distributed over all payment dates, rather than being concentrated around the bond’s maturity date. The interest rate sensitivity of an amortising bond’s price is therefore lower than that of a bond with the same maturity date.
- The credit rating of an amortising bond tends to be higher than that of a vanilla bond, since repayments of principal are made throughout the bond’s lifetime rather than at a date in the far future, thereby reducing the risk to the lender.
The expression for an amortising bond’s payments is relatively straightforward:
where
- A is the periodic payment amount;
- P is the principal;
- i is the periodic interest rate;
- n is the total number of payments over the lifetime of the loan.
Here the periodic interest rate refers to the amount per interval. For instance, if the bond pays 6% per year but payments are due monthly, then the relevant interest rate is 6%/12 = 0.5%.
The total return of amortising securities is complicated slightly by the paydown of principal over the security’s lifetime. See Chapter 12 for a discussion of this topic.
19.2 PREPAYMENTS
A common feature of many such loans is the ability to make early payments towards the principal of the loan. If even a modest amount can be added to the principal payment in the first few years of the loan, its lifetime can be drastically reduced.
The ability to make prepayments, or to repay the debt earlier, is a valuable feature for the borrower, since it allows more flexibility in how the debt is treated and the ability to refinance at lower rates should interest rates drop during the term of the loan. The ability to make prepayments is equivalent to the borrower holding an embedded call option.
Securities that are based on such loans are more complex than vanilla bonds or callable bonds, because of the presence of both variable cash-flows and an embedded option.
19.3 MORTGAGE-BACKED SECURITIES
A mortgage-backed security (or MBS) is a legal entity that pools together a group of mortgages into a single tradeable entity. The cash flows from each individual mortgage are aggregated by the administrating body into regular coupons, and any prepayments are returned to the MBS holder.
Many of the same terms that are used to describe other interest rate securities can also be used for MBS, as they have maturity dates, coupons, interest rate and credit risk, and yield. The major points to note are as follows:
19.3.1 Yield and risk
The ability to make early repayment on the principal means that the purchaser carries extra risk.
For instance, suppose an MBS is issued at 10%. A naïve purchaser of the MBS may expect to receive around 10% a year on their principal.
However, if interest rates fall to 2% there will be a flood of refinancing by mortgage holders. The purchaser of the bond will receive most or all of the invested capital back much earlier than expected, and then find that it can only be invested at 2% – a drastic drop in return.
To compensate the investor for this reinvestment risk, MBS pay a higher yield than bonds with similar credit ratings. There are various ways to put a price on reinvestment risk, including treating the MBS as a mixture of a bond and an option, or (more accurately) by considering various pathways over which interest rates can develop and their effect on the price of the security. This topic is covered further in Chapter 21.
19.3.2 Convexity
The presence of an option in an MBS often leads such securities to have negative convexity. The reason is that if interest rates fall, there will be more incentive for the mortgage-holders to refinance. The holder of an MBS will be paid off faster than expected, and the price of the MBS will not rise as quickly as the price of a bond without the embedded option.
Whether this is a problem depends on the investor’s strategy and view of interest rates. The effects of negative convexity are offset to some degree by the higher yields paid by an MBS.
19.3.3 Interest rate sensitivity
Unlike a vanilla bond, where the main cash flows are concentrated at maturity, the cash flows of an MBS are distributed over a wide range of maturities. The dependence of an MBS’s price on interest rates is therefore much more complex than a vanilla bond, and this can make use of simple interest rate risk measures, such as modified duration, misleading.
For portfolios where MBS represent only a small proportion of the interest rate risk, this may not matter. For portfolios that have a large exposure to such securities, it may be preferable to use interest rate measures such as key rate durations to analyse the sensitivity of the portfolio to curve movements at individual maturities.
19.3.4 Prepayment models
While it is seldom possible to make any firm predictions about future prepayment streams for an individual mortgage stream, one can make use of external research to model prepayments for pools of mortgages.
One of the most widely used models is that published by the Public Securities Association (PSA). The main features of the model, which is based on extensive observations of real borrowers, are that prepayment rates start at zero when the mortgage is first issued, then rise for the first 30 months of the mortgage, and are constant thereafter.
This incorporates the views that during the first few years of a mortgage, borrowers:
- are less likely to move to a different home;
- are less likely to refinance;
- cannot afford to make additional payments.
The standard PSA model assumes that repayments rise linearly over the first 30 months to a maximum annualised prepayment level of 6% and stay constant thereafter. This 6% level is known as 100% PSA. If repayment rates rise to 9%, the corresponding PSA rate is 9%/6% × 100% = 150% PSA.
The presence of prepayments can drastically affect the cash flow patterns of an MBS; see, for instance, Fabozzi (2001).
Any change in the repayment rate will therefore affect the price, and hence the return, of the MBS, and it is possible in principle to assign a return to changes in this factor. An amortising bond can be seen as a special case of an MBS for which the repayment rate is zero. Some managers make heavy use of prepayment models to price and manage their MBS holdings. The rate at which the MBS will be paid down is forecast using a wide range of inputs.
Attribution on more complex prepayment models will require custom software that can, at the least, allow the user to input prepayment rates, or allow the model to be integrated into the code. Such models are highly specialised and fall outside the scope of this book.