Topic 14: Valuation of Mortgage-Related Securities
(Suggested textbook reading: Chapter 11)
Recall that the value of any financial asset can be represented as
(an equation in which VAis unknown but k and the various CFs are known).
Furthermore, we can compute the yield on any asset as
(the same equation, but with V A and the various CFs known while k is unknown).
If a security is not callable, then using these formulas is easy (as long as it is easy to project the cash flows, the situation that tends to hold for bonds but not for stocks).
Consider a traditional corporate bond with a face value of $1,000 , a 10% coupon rate
(for .10 x $1,000 = $100 in annual interest, broken into two semiannual payments of $50 each), and 10 years (= 20 half-years) until maturity. If the required return (“market rate”) is 10% in APR terms [for a semiannual discount rate of .10/2 = .05 and a yield to maturity of (1.05)2 – 1 = 10.25%], then its value is:
= $623.11 + $376.89 = $1,000 .
However, its value is:
= $573.50 + $311.81= $ 885.31
if the required return (“market rate”) is 12% in APR terms [for a semiannual discount rate of .12/2 = .06 and a yield to maturity of (1.06)2 – 1 = 12.36%].
Finally, its value is:
= $679.52 + $456.39 = $1,135.91
if the required return (“market rate”) is 8% in APR terms [for a semiannual discount rate of .08/2 = .04 and a yield to maturity of (1.04)2 – 1 = 8.16%].
But if the values were given and the appropriate discount rates were not, we would solve for the rates, and would find that the yield to maturity is 12.36% if the investor pays $885.31 but only 8.16% if the investor pays $1,135.91.
What if the bond were callable (we might assume that five years remain on an original 10-year call protection period) at a premium of $100 above the face value? Then we would be trying to find the yield to call, and if the price were $885.31 we would solve for k in the equation
= $885.31.
After solving for k (the semiannual rate of return represented by the cash flows), we would take (1 + k)2 – 1 to compute the yield to call. In this example, the yield to call would not be 12.36%. Rather, k would be 7.3723% (found through trial and error), so the yield to call would be (1.073723)2 – 1 = 15.2881%. Let’s check this result:
= $50 (6.904284725) + $1,100 (.490995417) = $345.21+ $540.10 = $885.31.
On the other hand, if the price were $1,135.91 we would find the yield to call by solving for k in the equation
= $1,135.91.
In this example, the yield to call would be 8.2803%. Let’s check this result:
= $50 (8.0546017) + $1,100 (.6665274) = $402.73 + $733.18 = $1,135.91.
It should be clear that, to compute values (or yields), we must take into account both the amounts and the timing of cash flows. Early repayment (i.e., prepayment) can have a big impact on both.
What might cause early repayment on a home loan?
- Default (with payment subsequently made through resale, or by FHA/VA/PMI/GNMA)
- Changes in employment
- Fire or other peril that destroys improvements
- Changes in interest rates that cause home owners to
- Refinance
- Choose home equity as the best investment alternative
- Move voluntarily (finding the “right time” to move to bigger house)
Note that there is some degree of early repayment even if interest rates rise. One study found that, even if the coupon rate on a pool of loans is equal to the market rate, the prepayment rate is 1.5% per year.
It has also been estimated that :
- If the market rate falls by 200 basis pts., the prepayment rate rises to 8.4% per year.
- If the market rate falls by 400 basis pts., the prepayment rate rises to 30% per year.
- If the market rate rises by 200 basis pts., the prepayment rate falls to 1% per year.
How can analysts try to explain and anticipate prepayments? 5 methods have been used:
1. Average maturity – just treat all prepayment as coming at the end of the historical average life of loans (say, 12 years).
What’s good: it’s easy.
What’s bad: does not take into account changes in the factors that cause early repayment.
2. Constant prepayment – assume that there is a constant rate of prepayment, such as 1% per month.
What’s good: it’s easy.
What’s bad: does not take into account changes in the factors that cause early repayment. Also tends to understate prepayment in early years, and to overstate it in later years. (Empirical evidence shows, for example, that default tends to occur early or not at all.)
3. Based on experience – FHA experience is a good base to use, because of the large size of the market. There are “mortality tables” that show the number of FHA loans still outstanding at any date.
What’s bad: it makes use of historical information in predicting future events, even though the underlying relationships may have changed.
What’s good: it reflects actual market experience. It also allows for adjustments that reflect the pool’s own characteristics by assuming prepayment at a rate of, say, 200% of FHA experience.
4. The Securities Industry and Financial Markets Association model – a simplified version of the FHA approach, widely used by mortgage market analysts. It’s actually a hybrid of the constant percentage and FHA methods. This approach became well known in the mortgage lending industry as the “PSA” model, because one of the earlier groups that was merged to form the SIFSA organization was called thePublic Securities Association. To keep the well-known PSA name rather than calling it the SIFSA model, the organization now lists PSA as standing for “Prepayment Speed Assumptions.”
The basic PSA assumption is .2% in month 1, .4% in month 2, .6% in month 3, and so forth, up to 6% in months 30 and up. We refer to this pattern as “100% PSA.” (A good working estimate is 2% in year 1, 4% in year 2, and 6% in years 3 and up.)
A “200% PSA” pool of loans is one with characteristics such that it is expected to be prepaid .4% in month 1, .8% in month 2. etc. A “50% PSA” pool would be expected to be prepaid half as quickly as the typical case.
Issuers of mortgage-backed securities typically offer yield quotes based on several possible prepayment assumptions, e.g. 75%, 100%, ad 150% PSA.
5. Econometric models
Now let’s look more closely at the effect of prepayments on cash flows.
Consider a callable bond with a 25-year original life, and 20 years remaining. As in the case above, he bond was issued to be callable after 10 years (so it is callable in 5 more years), and it has a 10% coupon rate. The bond sells for $1,197.93. What is its yield to maturity?
= $1,197.93.
Solve with trial and error for k; the answer is that 4% is the semiannual return indicated by the cash flows, such that the yield to maturity is (1.04)2 – 1 = 8.16%.. [Though you could ask why the price would rise above $1,100 at all if prepayment, i.e. a call, could take place.]
What if the bond is called (early repayment)? Then
= $1,197.93
[solve with trial and error to find k, and take (1 + k)2 – 1 to compute the yield to call]. If the market’s semiannual required rate of return were:
4%, then the indicated value would be$405.54 + $743.12 = $1,148.66
3.5%, then the indicated value would be$415.83 + $779.81 = $1,195.64
3%, then the indicated value would be$426.51 + $818.50 = $1,245.01
Because our target value is $1,197.93, our answer turns out to be just under 3.5%:k = .03476, for a yield to call of (1.03476)2 – 1 = 7.0728%:
= $1,197.93.
What happens with pass-throughs? Prepayments cause the total cash flows to be lower than they would otherwise be. They also cause interest to constitute a smaller percentage of the total cash flows received. Is that good or bad?
Consider pass-throughs that sell at discount vs. those that sell at premiums.
On a pure discount security of any type (or on a principal-only strip also), early repayment would increase the yield.
Think of a zero-coupon bond with a two-year life, that sells for $826.44. If the $1,000 face amount is received at the end of the second year, the yield to maturity is 10%. But if the $1,000 is received after 1 year, the yield is 21%. (Note that $1,000/$826.44 = 1.21.)
In the same manner, prepayment increases the yield on a pass-through that sells at a discount price. But prepayment reduces the yield on a pass-through selling at a premium price. (Note that if the instrument sells at a premium, the coupon rate is greater than the market rate, so you don’t want prepayments that have to be reinvested at a lower rate.)
So the two (pass-throughs sold at discounts and premiums) can work well in a portfolio together.
Actually, there are two impacts on a pass-through when interest rates change:
1) the expected cash flows are discounted at a new rate (so there is downward impact on value as rates rise because of higher discount rate); and
2) the cash flows themselves change because of the prepayments (total cash flows rise as market interest rates rise because the principal balance stays higher longer, and more interest therefore is paid) [the timing of the receipt of principal changes as well]
So the impact of a rate change on price is less than for a standard type of bond, because the duration is reduced.
Recall that we measure the impact of a rate change on price by computing the duration.
The duration formula is P/P = -D(r/1+r). Therefore,
D = (-P/P) x (1+r/r).
We can not measure duration directly for instruments that have uncertain payments. But we can compute an “implied” or “effective” duration by looking at how the prices of particular instruments have changed with changing rates. Consider this example (from text):
Prices are determined based on assumed prepayments, which in this case are assumed to follow a 200% PSA pattern. The coupon rate on the pool is 11%. If the market rate is 10%, then the price is 104.25 (e.g., the buyer pays $1,042.50 for the right to collect
$1,000 of principal and the accompanying interest).
If the market rate rose to 10.25% and payments were not affected, we are told that the price would fall to 103.16. So
D = [(104.25-103.16)/104.25] x (1.1025/.0025) = .010456 x 441 = 4.61 yrs.
But if the rise in rates caused prepayment to fall to 180% PSA, then the rise in rates would cause the price to fall (according to the information given) only to 103.46.
D = [(104.25-103.46)/104.25] x (1.1025/.0025) = .007578 x 441 = 3.34 yrs.
Prepayment is seen to cause duration to fall. But what if market rates were to fall? First, if the market rate fell by 25 basis points to 9.75% and prepayments were not affected, then we are told that the price would rise to 105.25.
D = [(104.25-105.25)/104.25] x (1.0975/.0025) = .009592 x 439 = 4.21 yrs.
But if the drop in rates caused prepayment to rise to 230% PSA, then the price would rise (according to the information given) only to 104.80.
D = [(104.25-104.80)/104.25] x (1.0975/.0025) = .005276 x 441 = 2.32 yrs.
Again, prepayment causes the effective duration to fall.
So when rates fall, callability (prepayment) causes values of pass-throughs not to rise as much as they otherwise would. In fact, an acceleration in prepayment (relative to what had initially been expected) drives the prices of pass-throughs down. Note that in 1989 FHA offered a “streamlined refinance plan,” in which it allowed people with 15% rate loans refinance to 10% with no out-of-pocket refinancing costs. GNMA investors lost millions of dollars.
We've looked at pass-throughs in some detail, because these instruments behave like individual mortgage loans. What about some other types of mortgage-backed securities?
- The subordinated portion of a “senior/sub” – these are not GNMAs, so a default/foreclosure results in a loss of money to the subordinated security holder (typically the issuer). That party receives a premium for bearing this risk (at the expense of the senior security holders, who get lower returns because they face less risk). But if defaults are greater than had been expected, the added return to the holder of the subordinated position is wiped out.
- The equity position on a mortgage-backed bond (overcollateralization)
The bond holders want this residual to be large, but the issuer wants it to be small (small equity position results in a higher return on equity)
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FIL 360/Trefzger