This is a proportional hazard model proposed by Schwartz and Torous (1989) and given by
SMM(t)≡π(t;V,β)=π0(t;λ,p)exp 4
i=1
βiVi(t)
= λp(λt)p−1 1+(λt)p exp
4
i=1
βiVi(t)]
(3.8) where
1. V1(t)=c−l(t−s),s≥0, withcis the contracted mortgage rate,lis the long-term treasury rate for monthtwith ans-month lag. In generals=3 is used.
2. V2(t)= [c−l(t−s)]3,s≥0 represents an acceleration factor when the refinancing rates are sufficiently lower than the mortgage contract rate.
3. V3(t)=lnBB(t)∗(t) accounts for the burn-out effect whereB(t)is the dollar amount of the pool outstanding at timetandB∗(t)is the pool’s principal that would prevail attin the absence of prepayments but would consider the amortization of the underlying mortgages.
4. V4(t)is +1 iftis May to August and 0 iftis September to April.
Hereπ0(t;λ,p)is the baseline function which explains the CRP when the factor vectorV =0. The baseline value of π0 shows us that the mortgage will suffer a baseline risk at any time of the mortgage. The factor exp(βV) represents the mortgage specified risk.
The explanatory variables in the Schwartz and Torous (1989) model may include the costs of refinancing, borrowers characteristics, property character- istics. By using a proportional hazard model the explanatory variables impact equi-proportional at all mortgage ages.
Focusing only on the covariate variables listed above as in the original paper, one should expectβ1>0 because a larger value ofV1implies a greater incen- tive to prepay;β2>0 since if the proxy of a new mortgage ratelindicates that lower available rates increase further the incentive to refinance;β3>0 because the covariateV3is likely to be negative and higher in absolute value when more mortgages in the pool have been prepaid already, so there will be less mortgage borrowers looking to prepay. Clearly,β4>0 because it is well known that prepayment activities are higher in the spring and summer due to the effects of the school calendar year.
Example 3.4. Calibrating on Ginnie Mae 30-year single-pool family prepayment rates for the period Jan 1978 to Nov 1987 the following values
were reported5in Schwartz and Torous (1989) when the covariates included the seasonality covariate,
λ=0.0149(0.001), p=2.312(0.139), β1=0.380(0.064), (3.9) β2=0.003(0.001), β3=3.577(0.345),β4=0.266(0.329)
Another estimation carried out by leaving out the seasonality covariate gave the estimation results
λ=0.0157(0.002),p=2.350(0.121),β1=0.397(0.043) (3.10) β2=0.004(0.001), β3=3.743(0.447)
0 50 100 150 200 250 300 350 400
1 2 3 4 5 6 7 8 9 10
Months
Spot rate (%)
CBOE Ten Year Interest Rate US Govt
Figure 3.6. Historical 10 Year US Government Interest Rates, monthly between April 1986 and May 2016.
Source of Data: CBOE
5The estimates are maximum-likelihood estimates with jacknifed standard deviation estimates in parentheses.
In order to see how the Schwartz-Torous model would be applied, we take as a proxy for the new mortgage market rate the 10 Year US Government Spot Rate. This is illustrated in Figure 3.6. Then we calculate the baseline hazard rate, scheduled balance, prepayment rates amortizing balance for a hypothetical 200,000 USD 30-year fixed rate mortgage. Issued in April 1986. There are 360 months for which we calculate the SMM rate and their corresponding CPR rates. We must remark that the general trend between 1986 and 2016 was a clear downward trend.
For a mortgage with a contractual fixed mortgage rate ofc=15% the pre- payment calculations are depicted in Figure 3.7. For reference calculations of baseline hazard rate, scheduled balance, prepayment rates and amortizing
0 100 200 300 400
0 1 2 3 4 5 6 7 8x 10−3
Time in months
SMM baseline
Loan rate = 15%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Scheduled Balance
Loan rate = 15%
0 100 200 300 400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time in months
CPR curve
Loan rate = 15%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Realized Balance
Loan rate = 15%
Figure 3.7.Application of Schwartz-Torous for a mortgage with fixed rate 15%.
0 100 200 300 400 0
1 2 3 4 5 6 7 8x 10−3
Time in months
SMM baseline
Loan rate = 8.5%
0 100 200 300 400
0 0.5 1 1.5
2x 105
Time in months
Scheduled Balance
Loan rate = 8.5%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1
Time in months
CPR curve
Loan rate = 8.5%
0 100 200 300 400
0 0.5 1 1.5
2x 105
Time in months
Realized Balance
Loan rate = 8.5%
Figure 3.8. Application of Schwartz-Torous for a mortgage with fixed rate 8.5%.
balance are done for a hypothetical 200,000 USD 30-year fixed rate mortgage, issued in April 1986. For comparison we repeat the prepayment calculations for different contractual mortgage rates c∈ {8.5%, 5%, 1%}. The results are presented in Figures 3.8-3.10, respectively.
It can be observed that while the baseline hazard rate curve is the same in all four scenarios – as it should be by design,– the prepayment curve is also very much the same in all four scenarios. The amortization of the scheduled balance changes from a concave shape at high contractual mortgage rates to an almost linear amortization for the 1% mortgage rate. The realized balance
0 100 200 300 400 0
1 2 3 4 5 6 7 8x 10−3
Time in months
SMM baseline
Loan rate = 15%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Scheduled Balance
Loan rate = 15%
0 100 200 300 400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time in months
CPR curve
Loan rate = 15%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Realized Balance
Loan rate = 15%
Figure 3.9.Application of Schwartz-Torous for a mortgage with fixed rate 5%.
has a similar shape in all four scenarios, amortizing faster in the first part of the mortgage life, in contrast with market evidence showing that borrowers do not refinance early on but rather in line with the refinancing incentive, given that the new rates were going lower over time. The prepayment balance slows down in the middle—this was the period of late 1990s and beginning of 2000s when rates increased for a while, and then finally the prepayment rates increase exponentially towards the end leading to a fast amortization of the balance.
One major advantage of the Schwartz-Torous model is that it can be used for a loan-by-loan analysis and for specialized mortgage portfolios such as buy-to- let or first time buyers that exhibit different behaviour to the other borrowers.
0 100 200 300 400 0
1 2 3 4 5 6 7 8x 10−3
Time in months
SMM baseline
Loan rate = 1%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Scheduled Balance
Loan rate = 1%
0 100 200 300 400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time in months
CPR curve
Loan rate = 1%
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 105
Time in months
Realized Balance
Loan rate = 1%
Figure 3.10. Application of Schwartz-Torous for a mortgage with fixed rate 1%.