The key ingredients for implementing a simulation model for pricing MBS securities are
1. aninterest rate model 2. aprepayment model
3. if there is a liquid market, an OAS model may be used for tuning the market risk factors
The interest rate model must produce the short-term rate and the long-term rate, the latter sometimes being an input into the prepayment models.
A Monte Carlo simulation will simulate interest rate paths and prepay- ment paths, then combine them to cover a wide set of market scenarios and ultimately calculate the price of the MBS along each dual path by simple discount cash-flow calculus. The price of the MBS is then the average of all those prices.
4.5.1 PRICING MBS FRAMEWORK
The general discounting cash-flow techniques are based on the formula MBS0=E
df(t)c(t)
(4.1) wheredf(t)s the discounting factor for tenortandc(t)is the cash-flow realized at timet.
The discount factor is determined from the short-term interest rate process df(t)=df[0, 1]df[1, 2]. . .df[t−1,t] (4.2)
=
t−1
k=0
exp(−skt)=exp[−
t−1
k=0
skt] (4.3)
wheredf[k−1,k]is the discount factor calculated at time k−1 for time k,r(k) is the short-term rate used to generate df[k,k+1], observed at the end of period k, andtis the time step used in simulation.
The cashflows are given by standard calculations with mortgage payments.
4.5.2 BOND-EQUIVALENT MBS YIELD AND THE OAS ADJUSTMENT RATE
The cash-flows on MBS are typically monthly. Some investors compare the yield on MBS with the yield on a treasury coupon. Ifri= the monthly interest rate that will equate the present value of the projected cash-flows for the MBS security to the market price of that security plus accrued interest. Then the MBS bond-equivalent yield is 2[(1+ri)6−1].
There are two problems with this measure. Firstly, there is reinvestment risk:
cash-flows must be reinvested at the rateri on the market. Secondly there is interest rate maturity risk because the bond must be hold to maturity. The difference between the cash-flow yield and the treasury yield is anominalyield.
This spread will include prepayment risk.
The prepayment model that market participants are using is very likely to be different from bank to bank. This means that in aliquidmarket, the market price of an MBS may be different from the internal model price. In order to apply a more marked-to-market approach an additional spreadrOAS, the OAS spread, was applied to the interest rates used to discount the cash-flows, such that the market price and model price coincide.
In order to simulate the one-month future interest rates, ifft(j)is the one- month future interest rate for monthton pathj, interest rate paths are collected in the matrix illustrated in Table 4.1. These futures rates are then assumed to be the future realized rates in a Monte Carlo risk management exercise. The same thing is now done for simulating future refinancing mortgage rates. Ifrt(j)is the one-month future interest rate for monthton pathj, then the refinancing
Table 4.1. Pathway Monte Carlo simulation for future interest rates.
Path
Month 1 2 ã ã ã j ã ã ã n
1 f1(1) f1(2) ã ã ã f1(j) ã ã ã f1(n)
2 f2(1) f2(2) ã ã ã f2(j) ã ã ã f2(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
t ft(1) ft(2) ã ã ã ft(j) ã ã ã ft(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
M−1 fM−1(1) fM−1(2) ã ã ã fM−1(j) ã ã ã fM−1(n)
M fM(1) fM(2) ã ã ã fM(j) ã ã ã fM(n)
Table 4.2. Pathway Monte Carlo simulation for future mortgage rates.
Path
Month 1 2 ã ã ã j ã ã ã n
1 r1(1) r1(2) ã ã ã r1(j) ã ã ã r1(n)
2 r2(1) r2(2) ã ã ã r2(j) ã ã ã r2(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
t rt(1) rt(2) ã ã ã rt(j) ã ã ã rt(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
M−1 rM−1(1) rM−1(2) ã ã ã rM−1(j) ã ã ã rM−1(n)
M rM(1) rM(2) ã ã ã rM(j) ã ã ã rM(n)
Table 4.3. Pathway Monte Carlo simulation for future pool cash-flows.
Path
Month 1 2 ã ã ã j ã ã ã n
1 C1(1) C1(2) ã ã ã C1(j) ã ã ã C1(n)
2 C2(1) C2(2) ã ã ã C2(j) ã ã ã C2(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
t Ct(1) Ct(2) ã ã ã Ct(j) ã ã ã Ct(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
M−1 CM−1(1) CM−1(2) ã ã ã CM−1(j) ã ã ã CM−1(n)
M CM(1) CM(2) ã ã ã CM(j) ã ã ã CM(n)
rates will correspond pathwise to the simulated monthly future interest rates.
This process is depicted in Table 4.2. The refinancing rate can be taken as a fixed spread over a proxy interest rate such as the 10-year rate or as a linear combination between the 2-year and 10-year government spot rates.
Combining the above two facilitates the simulation of future cash-flows for the pool. Denoting byCt(j)the loan pool cash flow for monthton pathj, the future monthly cash-flows generated by the mortgages in the pool are collected in the matrix described in Table 4.3.
The cash-flows are obtained from the schedule of payments and also apply- ing prepayments and defaults from internal models.
Now it becomes straightforward to slice the cashflows according to all tranche specifications. IfNCt(j)is the cash flow arriving to a given tranche (note) for monthton path jthen Table 4.4 illustrates the cash-flows along each simulated path. Note that the actual number of future monthsMmay be different for different notes due to different prepayment speeds.
From any term structure of future rates one can derive the spot rates to any month maturity. The spot rate for maturity given by monthton thej-th path is
st(j)=
[1+f1(j)][1+f2(n)] ã ã ã [1+ft(j)]1/t
−1
Table 4.4. Pathway Monte Carlo simulation for future tranche cash-flows.
Path
Month 1 2 ã ã ã j ã ã ã n
1 NC1(1) NC1(2) ã ã ã NC1(j) ã ã ã NC1(n)
2 NC2(1) NC2(2) ã ã ã NC2(j) ã ã ã NC2(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
t NCt(1) NCt(2) ã ã ã NCt(j) ã ã ã NCt(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
M−1 NCM−1(1) NCM−1(2) ã ã ã NCM−1(j) ã ã ã NCM−1(n)
M NCM(1) NCM(2) ã ã ã NCM(j) ã ã ã NCM(n)
Table 4.5. Pathway Monte Carlo simulation for spot rates on a monthly grid.
Path
Month 1 2 ã ã ã j ã ã ã n
1 s1(1) s1(2) ã ã ã s1(j) ã ã ã s1(n)
2 s2(1) s2(2) ã ã ã s2(j) ã ã ã s2(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
t st(1) st(2) ã ã ã st(j) ã ã ã st(n)
ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã
M−1 sM−1(1) sM−1(2) ã ã ã sM−1(j) ã ã ã sM−1(n)
M sM(1) sM(2) ã ã ã sM(j) ã ã ã sM(n)
This allows the calculation of the present values of all cash-flows PV(NCt(j))= NCt(j)
[1+st(j)+rOAS] (4.4) PV[Path(j)] =PV[NC1(j)+PV[NC2(j)] +. . .+PV[NCM(j)]] (4.5) The theoretical value under OAS spread discounting framework is then calcu- lated as with any Monte Carlo exercise by averaging all present values across all simulated paths
PV(Path(1))+PV(Path(2))+. . .+PV(Path(n)) n
In order to be able to calculate the present values of future cash-flows one needs to be able to simulate paths of monthly interest spot rates that are the discounting rates. These are collected in a similar matrix or table as shown in 4.5 wherest(j)is the spot rate up to monthton pathj.
With all the above calculated, the OAS risk-adjusted discount factors are calculated using the formula
df(t)=
t−1
k=0
exp(−(sk+rOAS)t)=exp[−
t−1
k=0
(sk+rOAS)t] (4.6)
The OAS spread is calculated by trial and error until MBS0=E
df(t)c(t)
=market price+accrued interest (4.7)