Amortizing bonds

Amortizing bonds[edit]

An amortizing security is 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 lump payment at the bond's maturity.

The most common form of an amortizing security is a mortgage on a house. Typically, this is structured so that the lender makes equal payments over the lifetime of the the loan. These payments are made up of repayments of principal and interest. At the outset, the payments are almost entirely 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 amortizing bond is therefore very different to that of a vanilla bond with a similar maturity. In particular,

• The duration of the amortizing bond is much shorter that 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 amortizing bond is therefore lower than that of a bond with the same maturity date.
• The credit rating of an amortizing 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.

The mathematics underling the calculation of an amortizing bond's payments is relatively straightforward:

<math>A = P\frac{i(1 + i)^n}{(1 + i)^n - 1} = P\left(i + \frac{i} {(1 + i)^n - 1}\right)</math>

where:

• A = periodic payment amount
• P = principal
• i = periodic interest rate
• n = total number of payments

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%.

Figure 1

Figure 1 shows principal and interest payments for a loan of $100,000 over 30 years with monthly payments, paying 10% annually. Note that all payments are equal at $878, although the relative amounts of principal and interest change drastically over the lifetime of the loan.

Prepayments[edit]

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.

MBS are therefore more complex than vanilla bonds or callable bonds, due to the presence of both variable cash-flows and an embedded option.

Mortgage-backed securities[edit]

A mortgage-backed security is a legal entity that pools together a group of mortgages into a single tradeable entity. The cash flows from each indvidual mortgage are aggregated by the administrating body into regular coupons, and any prepayments are returned to the MBS holder.

MBS may be described using many of the same terms as other interest rate securities; they have maturity dates, coupons, interest rate and credit risk, and yield. The major points to note are as follows:

Yield and risk[edit]

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 naive purchaser of the MBS may expect to recieve around 10% a year on their principal. However, if interest rates fall to 2% there will be a flood of refinancing. 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.

For this reason, MBS pay a higher yield than do bonds of similar characteristics to compensate the investor for this reinvestment risk. 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 MBS.

Convexity[edit]

Positive convexity[edit]

Unlike a vanilla bond, for which convexity generates positive returns, most MBS show negative convexity.

Consider a long-dated vanilla bond. If interest rates fall, the price of the bond will rise; and if rates fall, the price of the bond will fall.

However, this relationship is not always linear. If the bond has a desirable property called positive convexity, then the bond's price will rise faster than expected if rates fall, and slower than expected if rates fall. If all else is equal, therefore, the holder of a bond with high convexity will beat the market whether rates rise or fall. Convexity increases roughly as the square of the security's maturity, so longer-dated bonds tend to have much higher convexities than do money-market instruments.

Convexity is such an attractive property that it can lead a bond's price to be bid up in the marketplace. As a result, the bond's yield will fall. So a high-convexity strategy is best suited to cases where the investor belives there will be market movements, but does not know in which direction they will occur. If nothing moves, the lower yield will drive down the overall return of the managed portfolio.

Negative convexity[edit]

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 embdedded option.

Whether this is a problem or not 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.

Interest rate sensitivity[edit]

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 (KRDs) to analyse the sensitivity of the portfolio to movements at individual maturities. FIA allows both options, depending on the user's requirements.

Prepayment models[edit]

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, or 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, and
• cannot afford to make additional payments.

The standard PSA model assumes that repayments rise linearly over the first 30 months to a maximum annualized 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. Figures 2, 3 and 4 show the cash flows for the same bond as in Figure 1, but with 50%, 100% and 150% PSA repayments:

Figure 2

Figure 3

Figure 4

Differences between amortizing bonds and MBS[edit]

An amortizing bond can be seen as a special case of an MBS for which the repayment rate is zero. We adopt this approach in FIA for simplicity.

Differences between sinking securities and vanilla bonds[edit]

A sinking security requires two additional pieces of extra information above that needed by a vanilla bond for attribution purposes.

• In addition to the maturity date of the security, FIA requires the tenor (or lifetime) of the MBS as we need to know how much of the bond's lifetime has passed in order to estimate its cashflows. This is calculated by using the issue date of the security, as well as its maturity date.
• Repayment rate: an estimate of the PSA rate.

These two factors allow the calculation of a graph similar to those shown above, and the correct point to be taken to forecast the security's remaining cash flows.

Repayment factor[edit]

The repayment factor is a time-varying measure of how much of the initial face value of the sinking security has been repaid.

In FIA, this quantity is assumed to have been factored into the security exposures supplied in the returns file.

Security code[edit]

Both amortizing bonds and MBS are represented by the same security type SINKER.

Calculation of returns[edit]

Sinking securities are priced as the sum of the various discounted cash flows generated by the security. Cash flows are calculated using the same algorithm for the illustrations shown above, and the value of these cash flows is then calculated using the appropriate zero coupon yield curve.

Security file setup[edit]

A sinking security is set up as follows:

Returns file setup[edit]

An MBS or amortizing bond requires the following information in the returns file:

In addition, information on the bond's yield to maturity, modified duration and convexity can also be supplied. If provided, they will be used in all subsequent attribution calculations. If not supplied, FIA will calculate its own values for these quantities using the supplied security parameters and market data.

Mortgage-backed securities

Cash flow pattern

A mortgage-backed security can be seen as a special case of an amortizing bond.

Unlike a standard bond, that repays all of the outstanding amount owing to the purchaser at maturity, an amortizing bond pays back its principal over the bond's lifetime. A commonly used arrangement is for equal payments to be made at each payment date.

A mortgage-backed (or asset-backed) security differs from an amortizing bond in that the issuer can make extra payments to the bond-holder during the lifetime of the security.

The PSA prepayment model

[Characteristics of the model]

[Effects of an amortizing schedule]

The cash flows of an amortizing bond are completely different to those of a vanilla 'bullet' bond and this changes its risk characteristics accordingly.

One effect is that the bulk of an MBS's cash flows are made much earlier in its lifetime than those of a vanilla bond.

Embedded option - repayment will be faster when interest rates are lower

[Convexity of bond is >0, convexity of MBS is <0]