hedging ununcertain exposure to interest rate risk. Due to two of aforemen- tioned risks, the initial balance of the mortgage portfolios varies in a non- deterministic manner: while prepayment risk leads to a (stochastic) reduc- tion in the total amount of interest initially expected to be received, eventual defaults lead to the reduction in both expected notional and interest payments.
A number of alternative solutions to this hedging problem have been suggested in Fabozzi et al. (2010) and are exemplified in this section; specifically, we consider below the hedging of a portfolio of mortgages using a structured swap (i.e. balance guaranteed swap), and amortizing swap and swaptions, floors or caps. In our numerical example we shall assume a pool of self amortizing loans, with a total notional of $500,000,000, coupon rate of 6% (payable monthly) and maturity of 30 years (i.e. 360 months).
As explained in Fabozzi et al. (2010) balanced guaranteed swaps and bal- ance guaranteed LIBOR-base rate swaps are two examples of structured swaps used by players in the real-estate market like, for example, the banks who originated the loans to hedge the interest rate risk resultant from the mis- match between the coupons/ interest payments received on the pool of (secu- ritized) mortgages, which are either fixed or given by the standard variable rate (SVR), which in turn is intrinsically linked to the base rate set by the Central Bank, and the LIBOR-based payments that they have to pass onto the investors in the securitized loan. As mentioned above, the notional in the balance guaranteed swap is determined each period by the principal balance of the collateral loans. Figure 4.6 depicts how this balance can vary depending on prepayment rates. More specifically, the different curves depicted represent the potential evolution of the balance based on various assumptions for the conditional prepayment rate (CPR). To (partially) eliminate this uncertain exposure risk given by the stochastic nature of the notional of the balance guaranteed swap one can use a portfolio of a self-amortizing swap, combined with swaptions, floors or caps; we shall discuss each of these three possibilities in turn.
In Figure 4.6 we illustrate various amortizing portfolio balance curves gener- ated under various prepayment assumptions. In practice it is difficult to predict which balance curve will be the realized one. The balances may amortize faster or slower depending on the prepayment and defaults.
Firstly, one would need to determine a region that will contain the realized balance evolution. We set the upper boundary of this region as being the 0%
CPR curve (i.e. no prepayments, notional payments made as scheduled for a self-amortizing loan, which underlines the self-amortizing swap), while the lower boundary is represented by the 30% CPR curve (i.e. assuming annual pre-payments of 30% in excess of scheduled notional payments). Subsequently, under the first hedging scenario (i.e. using swaptions), a self-amortizing swap with notional given by the upper curve (i.e. low CPR; notional denoted by ut) in the above graph is entered into: the seller of the balance guaranteed swap would buy the self-amortizing swap (i.e. pay fixed and receive LIBOR
0 50 100 150 200 250 300 350 400 450 500
0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 248 256 264 272 280 288 296 304 312 320 328 336 344 352 360
Month
Million ($)
Balance
0% CPR 1%CPR 3% CPR 6% CPR 10% CPR 20% CPR 30% CPR
Figure 4.6. Balance Schedule for self-amortizing mortgage loans with different prepay- ment rates (CPRs).
(+ a spread). As the realized notional balance (i.e. ρt, the notional of the balance guaranteed swap) moves away from the low CPR curve (i.e. ut, the notional of the self-amortizing swap), the hedge is adjusted (i.e. the upper curve is lowered closer to, but not crossing, the realized curve) using a series of Bermudan swaptions. The swap underlying the swaptions can be either a plain vanilla swap, or an amortizing swap.
The results presented below use the amortizing version of the swap under- lying the swaptions. More specifically, the way this strategy is implemented is as follows: at each point in time, the difference between the upper and lower balance is divided into a constant number of unitsn; in our numerical example below we shall usen=60 corresponding to a five year horizon. The notional of the amortizing swap underlying the swaptions will now be given byw(ut−lt), withw= 1n, and whereutandltrepresent the time-tbalances under the upper (i.e. low CPR) and lower (i.e. high CPR) curves.
For instance, in our numerical example, u1 =$499,502,247 and l1=
$484,874,059. Hence, the notional of the swap (from the swaption) at time time t2, should the swaption be exercised at timet1would be equal to $243,803. The time-(t+1)payoff for this strategy is hence equal to:
ρtt
c(1)−L(t,τ)−s(1)
+utt
L(t,τ)+s(2)−c(2) +Itw(ut−lt)
c(2)−s(2)−L(t,τ)
(4.21)
where1= 121, that is one month,(c(1),c(2))and(s(1),s(2))are the coupon rates and swap spreads for the balance guaranteed swap and amortizing swap off the upper curve, respectively;L(t,τ)is the time-t LIBOR with maturity ẽĎ, where (i.e., six months in our numerical application). Thus,It=0 if none of the(n=60)Bermudan swaptions are exercised at time-tand it is equal tomif exactlymof the Bermudan swaptions are exercised by time-t. In formula (4.21) the first term is the payoff from the balance guaranteed swap, the second term represents the amortizing swap and the third term is the payoff from a swaption with strikec(2)−s(2).
The second hedging option would again involve the same structured (bal- ance guaranteed) swap as well as the same amortizing swap off the upper curve as before. The swaptions however will now be replaced by a series of floors, with notional equal to the difference between the upper and the lower curves (i.e.
ut−lt) and strike equal to the strike of the swaptions above (i.e.c(2)−s(2)).
The time-(t+1)payoff for this strategy is hence equal to:
ρtt
c(1)−L(t,τ)−s(1)
+utt
L(t,τ)+s(2)−c(2) +(ut−lt)
c(2)−s(2)−L(t,τ)
(4.22) The difference in formula (4.22) compared to (4.21) is the third term, which now is a floorlet with strikec(2)−s(2).
Finally, the third hedging strategy would again include the same balance guaranteed swap as in the previous two cases, but now the amortizing swap will be off the lower curve and the strategy will be completed by a series of caps, with notional equal to the difference between the upper and the lower curves (i.e. the same as the notional of the floors above) and strike equal to the difference between the coupon rate (fixed leg) and the price (swap spread) of the amortizing swap (now of the lower curve). The time-t+1 payoff for this strategy is hence equal to:
rtt
c(1)−L(t,τ)−s(1)
+ltt
L(t,τ)+s(3)−c(3) +(ut−lt)
L(t,τ)−c(3)+s(3)
(4.23) wherec(3) ands(3) are the fixed leg and the price of the amortizing swap of the lower curve. Note that the second term of formula (4.23) is the payoff of an amortizing swap, similar to formula (4.21) and formula (4.22), but the notional and the fixed rates are different. The third term in formula (4.23) can be recognized as the payoff of a caplet with strikec(3)−s(3).
It is worthwhile noting that if all coupon rates for the structured swap and the two amortizing swaps, off the upper and lower curves, respectively, are
constant and equal (i.e. c(1) =c(2)=c(3), and also if the swap spreads for the two amortizing swaps (of the lower and upper curve) are also equal (i.e.
s(2) =s(3)), then the latter two strategies yield the same results. If however, coupon and/or swap rates are different, then the results obtained with these two strategies will differ. In our numerical example below we assume:c(1)= c(2) =c(3) =6%;s(1)=0.16% ands(2) =s(3)=0.20%.
Numerical simulations
The LIBOR can be simulated for any maturity, 1-month, 3-month, 6-month, based on a simulated instantaneous short rate, as explained in Appendix 4.8 and using the closed form expression for the term structure.
For example, in panels c and d of Figure 4.7 below, we plot the simulated evolution of 6-month LIBOR vs. that of 1-month and 3-month LIBOR, respec- tively, using the CIR model, over a period of five years (60 months). We notice that, like in the Vasicek case, the shorter maturity rates are more volatile and also that the longer maturity rate can be both above and below the shorter maturity rate.
We now turn to the implementation of the three hedging strategies described above i.e. using swaptions, floors, or caps based on the realized curve and simulated paths for the 6-month LIBOR, using the Vasicek and CIR models, as described above. Also, we assume monthly payments, a maturity of 30 years (i.e. 360 months), n =60 (i.e. 60 Bermudan amortizing swaptions), as well as all coupon rates (for the structured swap as well as the two amortizing swaps, off the upper and lower curve) constant and equal to 6.
In Figure 4.8 we plot the evolution of the hedge using swaptions, for one of the simulated LIBOR path, using the Vasicek model.
If we calculate the total payoff (ignoring the timing of cash flows and hence time-value of money effects) for the two strategies, first with swaptions then with floors) we can conclude, that for this particular scenario, the hedge with floors (caps) appears to be superior: while the total payoff for the swaptions hedge is equal to $4,177,466, the value obtained for the floors (caps) hedge is
$81,781,850. However, since this staggering difference was obtained for one particular simulated path for the 6-month LIBOR, we repeated the simulation 1,000 times and also varied the assumed model; the average payoffs across all simulations are reported in Table 4.11.