Results and Analysis

As shown in section 2.3.1, the delta hedging strategy is perfect in a frictionless market where securities can be traded continuously without transaction cost. This means with an initial premium equal to the time-zero policy value being charged at loan origination and the corresponding hedging portfolio being constructed and continuously rebalanced over time, the insurer’s net position in the policy (i.e. the value of the hedging portfolio) is precisely zero at the end of loan term after all claims have been paid.

16_{This usually happens between year 6 and 8. We assume in this case the policy value becomes zero}

and no more hedging is needed. In addition, the calculation of ˆ∆d

t(FtT) is difficult and inaccurate when

Vd

Frequency Mean(SE) Std Min 10% Median 90% Max

weekly 18(35) 1111 -3835 -1372 62 1381 4082

biweekly 80(47) 1499 -6760 -1833 141 1870 5919

monthly 108(63) 1989 -8569 -2358 174 2492 7798

Table 2.2: Statistics of HEs based on 1000 independent scenarios. U0 = 380,000, V0d ≈

5551(1.46% of U0). SE stands for standard error.

However in practice, as well as in our implementation, continuous trading is not feasible and policy values and deltas are approximated numerically. Consequently the value of the hedging portfolio at the end of loan term deviates from zero. Discount this value to the time of loan origination using risk-free rate and call it hedging error (HE). Since HE depends on security prices, we can obtain the distribution and statistics of HE using the 1000 security price paths we have simulated. We consider different rebalancing frequencies, modifications of hedging strategies, alternative parameter values and security price models.

Rebalancing Frequency

The policy value at origination as a percentage of loan sizeU0is 1.46%, with 95% confidence

interval of (1.4553%,1.4660%).17 For each scenario of security prices, we consider monthly, biweekly and weekly rebalancing. The statistics of HE are shown in Table 2.2.

The mean HEs are all positive but tend to zero as frequency increases. The results are expected and confirm the validity of delta hedging theory. In the case of weekly rebalancing, the p-value for the null hypothesis “mean HE is zero” is 0.62, indicating that zero mean HE cannot be rejected. The large absolute values of minimum and maximum imply ineffectiveness of discrete rebalancing in some scenarios.

To visualize how the hedging portfolio tracks policy value, we pick one scenario and simultaneously plot portfolio and policy values in the first 8 years of the mortgage term. Figure2.4shows that weekly rebalancing performs much better than biweekly and monthly rebalancing and generates a small positive HE. It is also clearly demonstrated in this figure that policy values move in opposite direction as security prices. Unless otherwise specified, we use weekly rebalancing for the rest of this section.

17_{We do not use the same initial value and delta for all scenarios but estimate them independently. The}

Figure 2.4: A scenario where weekly rebalancing provides a good hedge.

Hedging Period and Proportion

We now consider some modifications of delta hedging, including hedging only in the first 3 years, hedging 50% of the policies, and no hedge. The results are shown in table 2.3.

We can make the following observations: 1) Full Hedge produces the mean HE closest to zero, and HEs are about symmetric around zero. HEs from other strategies are more skewed with mean deviated significantly from zero. 2) All modified delta hedging strategies produce positive mean HE but have a heavy lower tail. This implies the insurers benefits in most cases but faces greater risk of extreme loss. These strategies are vulnerable in a

Hedging strategy Mean(SE) Std 10% Median 90% Skewness

full hedge 18(35) 1111 -1372 62 1381 -0.09

first-3-year 330(113) 3564 -1636 378 3345 -2.54

50% of policies 795(138) 4367 -3923 2412 3278 -3.40

no hedge 1572(272) 8608 -7523 5100 5791 -3.48

Table 2.3: Statistics of HEs based on 1000 independent scenarios. U0 = 380,000, V0d ≈

Figure 2.5: A scenario where modified hedging strategies fail.

distressing market where protection is most needed. Figure 2.5 shows such a scenario. 3) HEs from no hedge are positive in 804 scenarios, 307 among which produce HEs equal to the initial policy value, indicating zero claim. One may conclude that under this particular model no hedge generates more profit than full hedge in most scenarios (806 out of 1000), but this does not necessarily imply no hedge is a better strategy than full hedge from a risk management perspective. Section 2.3.4.4 discusses this issue in more detail.

Test for Robustness

We consider alternative values for contract rate rc, initial LTV R0, volatility of security

priceσ and sensitivity of default to current LTVβ1 to test the robustness of delta hedging

under discrete rebalancing. One parameter is changed in each case. Table 2.4 indicates that initial policy value is sensitive and increasing in all four parameters. None of the HE means is significantly different from zero at a 95% level based on a student t test.

Full Hedge vs. No Hedge

The goal of hedging is to mitigate house price risk. The non-zero HE of full hedge is a result of discrete trading; it can be affected by house price fluctuations but the dependency is different from and weaker than the dependency of HE from no hedge on house prices. In the benchmark, we have a bullish market where property value increases in most of the

Vd 0(% of U0) HR Mean(SE)[% of V0d] HR Std[% ofV0d] Benchmark 5551 (1.46) 18(35)[0.32] 1111[20.00] rc= 0.1 7204 (1.90) 40(39)[0.55] 1218[16.91] R0 = 0.85 2528 (0.74) 6(23)[0.22] 725[28.69] σ= 0.15 2060 (0.54) 16(18)[0.75] 581[28.19] β1 = 3.5 7974 (2.10) -65(50)[-0.82] 1579[19.81]

Table 2.4: U0 = 380,000, the last two columns are mean and standard deviation of HEs.

Benchmark: rc= 0.06, R0 = 0.95, σ= 0.20, β1 = 3.0

V₀d Full hedge No hedge out of

Mean(SE) VaR CTE Mean(SE) VaR CTE 1000 Benchmark 5551 18(35) -1372 -2009 1572(272) -7523 -20197 194 µ= 0.03 5530 -18(37) -1416 -2165 -1965(429) -19062 -37131 310 µ= 0 5531 -33(41) -1558 -2436 -6625(540) -32441 -47623 458 σ = 0.30 16316 -4(63) -2549 -3657 2685(634) -25637 -46282 287 σ = 0.40 29849 -67(97) -3968 -5739 3779(957) -43258 -64209 340 RSLN 1 1624 -324(72) -3017 -5895 -900(220) -6088 -18158 232 RSLN 1877 -433(80) -3627 -6712 -1635(278) -9446 -23823 271 RSLN 2 2770 -487(97) -4788 -7401 -3395(402) -16420 -36193 386

Table 2.5: Both VaR and CTE are at 90% level. The last column is the number of scenarios where full hedge has a larger HE than no hedge. Monthly rebalancing for RSLN model. Benchmark: µ= 0.064, σ= 0.20.

world and no hedge tends to outperform full hedge. To compare the two strategies in less optimistic markets, we change the model parametersµand σand calculate the mean, 90% value at risk (VaR) and conditional tail expectation (CTE) of HEs, as well as the number of scenarios where full hedge generates higher HE than no hedge.

The results are shown in the first five rows of Table 2.5. Full hedge becomes more advantageous when µ is reduced or σ is increased, implied by the relative small absolute values of VaR and CTE. For no hedge, an increase in volatility increases mean HE but also magnifies tail risk. VaR and CTE from no hedge are ten times of those from full hedge.

In addition to geometric Brownian motion, we consider a regime switching lognor- mal (RSLN) model for security prices. A two regime RSLN model consists of regime 1, representing a thriving environment with higher drift and lower volatility and regime 2, representing an opposite depressing environment. Prices follow geometric Brownian mo- tion in each regime with its specific parameters. To simplify simulation, we assume the

regime switching process is observable.

The market under a RSLN model is incomplete (see Hardy[45]) because the regime switching process is not replicable. It follows that there does not exist a perfect hedging strategy.18 _{For the purpose of demonstration, we keep using delta hedging. Policy value}

is calculated by equations (2.14) & (2.13) and delta is approximated using (2.21). Due to market incompleteness, there are many choices of risk neutral measures. We choose a simple _Q such that the drifts in both regimes equal risk-free rate. Note that _Q is used for policy pricing and delta calculation, and _P is used for simulating scenarios of security prices. Figure 2.6 shows our RSLN model with assumed parameter values. To save computational cost, we assume monthly rebalancing and regimes can only be switched at the end of a month. Transition probabilities at month ends are denoted byp12 and p21

and are independent of stock prices.

Figure 2.6: Two regime RSLN.

The last three rows of Table 2.5 show the statistics of HEs when security price follows RSLN model starting from regime 1, the stationary distribution of regimes and regime 2. Mean HE deviates significantly from zero, suggesting the imperfectness of delta hedging in these cases. Although no hedge is more likely to generate a higher return than full hedge, its average return is lower and it faces severer losses in extreme scenarios.

18_{There are hedging strategies dealing with incomplete market, for example mean variance hedging and}