8.4.1 INTEREST RATE RISK
The most evident risk affecting reverse mortgages is interest rate risk. Given the long and uncertain maturity of these loans, one may need to rely on mod- els to simulate future paths for interest rates. The recent economic realities where negative interest rates are present worldwide indicate that a very cau- tious approach must be followed when selecting interest rate models that are calibrated to the market.
A very interesting observation (Pfau, 2016) linked to interest rates is how the line of credit of a reverse mortgage grows. The loan balance typically grows at a rate given by the reference interest rate, say one-month LIBOR, plus a fixed spread reflecting the lender’s profit margin and plus a fixed mortgage insurance premium. The sum rate is called the effective rate and is applied to project the growth of the loan balance. The same rate is also applied to increase the overall principal limit, which for line-of-credit reverse mortgage contracts is equal to the balance of the line-of-credit plus the loan balance and plus set-asides.
0 5 10 15 0.5
0.6 0.7 0.8 0.9 1
Simulate mortality using base geometrical process
Years after the age of 65
Survival probability
0 5 10 15
0 0.02 0.04 0.06 0.08 0.1
Simulate loan exit
Years after the age of 65
Loan exit probability
Figure 8.6. The simulation of borrower’s survival and loan termination probability. The borrower takes the loan when he is 75 years old.
The design arbitrage is that interest and insurance premiums are charged only to the loan balance. The line-of-credit and set-aside accrue under the effective rateas if these rates are also charged to these ledgers.
The line-of-credit decreases only when funds are withdrawn. Voluntary pre- payments will boost the line-of-credit again, and any future evolution is subject to effective rate. Hence, trading a reverse mortgage with a line-of-credit and delaying withdrawing funds from it will have the effect over time to have access later on to a larger balance. The actual level of interest rates, high or low, will of course play a role. Higher rates will imply a higher effective rate and a higher rate of growth.
Tomlison et al. (2016) compared reverse mortgages with a tenure and with a line-of-credit design and concluded that the tenure type is superior in gen- erating sustainable retirement income. However, their interest rate projections were ad hoc and it assumed implicitly that the line-of-credits will be used from the beginning.
On the other hand, from a lender perspective, Cho et al. (2013) advocated using a multi-period cash-flow model incorporating house price risk, interest rate risk and termination delay. They argue that the lump-sum mortgages are
Table 8.1. Longevity expectations based on Immediate Annuities Male and Female Lives.
Expectation of life at birth Expectation of life at age 65
Male Female Male Female
1841 40 42 11 12
1900 49 52 11 12
2000 76 80 16 19
2020 79 83 18 21
Notes: Derived from Continuous Mortality Investigation Research 00 tables.
more profitable and less risky than the tenure reverse mortgages. One possible explanation is that the analytical valuation of a reverse mortgage with tenure payments is far more complex5than a lump-sum mortgage.
Lenders of reverse mortgages use two types of rate. The rateRis the rate charged on the loan. This is the rate at which the loan balance grows. Secondly there is the raterwhich is the discount rate used to calculate the present value of the mortgage loan.
8.4.2 LONGEVITY OR MORTALITY RISK
The sellers of reverse mortgages have considered for a long time that longevity risk is diversifiable. Hence, by pooling a large numbers of loans we could use mortality tables to determine the terminations of loans.
The mortality data are derived from the Continuous Mortality Investigation Research (CMIR) 00 mortality tables, to which Norwich Union is a contributor.
The tables are referred to as Immediate Annuities Male Lives (IML00) and the Immediate Annuities Female Lives (IFL00), adjusted for cohort effects (i.e. where rates of improvement in mortality have been different for people born in different periods historically). The tables show the probability of death during any year for an individual of a particular age who is alive at the start of that year. Actuarial experience suggests that females live longer than males.
In many instances the loan is given to a living couple. The loan will survive as long as one of the couple survives. Hence, there is correlation built-in as couples can take care of each other and survive longer. There is also a selection bias, people taking up reverse mortgages have more money to look after themselves and therefore live longer than their peers.
5 A valuation framework that takes into consideration the mortality risk, interest rate risk, and housing price risk is detailed in Lee et al. (2012).
If τ is the time of death of the homeowner then this time is a ran- dom variable. If Fτ(t)=P(τ ≤t) is the cdf of τ then we can define the survival probability function that the homeowner lives longer than a time pointtas
Sτ(t)=1−Fτ(t)=P(τ >t) (8.19) Actuaries use a concept called the instantaneous death rate defined by the force of mortality
λ(t)= −Sτ(t)
Sτ(t) (8.20)
If the instantaneous rate of mortality is assumed to be constant thenFτ(t)= 1−e−λt and the corresponding probability density function isf(t)=λe−λt. The life expectancy is then calculated as
E(τ)= ∞
0
te−λtdt= 1
λ (8.21)
Thus a constant rate of mortality implies a life expectancy that is independent of the current homeowner’s age. Clearly this is too simplistic.
Another common assumption made about mortality (Brockett, 1991) is that the death of the homeowner is uniformly distributed in the interval [0,d].
Then, ifxis the current age of the borrower, conditional thatτ >xthe cdf of τ isFτ(t)= W−xt with the density isf(t)= W−x1 . The life expectancy in this case is equal toW/(W−x).
8.4.3 MORBIDITY RISK
Morbidity is defined as the movement of people into long-term care. This is defined as the inability to carry out at least two activities of daily living (ADLs).
The ADLs test the borrower ability to care for themselves in their own home and include the capacity to feed, clothe and wash themselves, among others.
There is very little data available on the movement of people into long-term care as a result of their inability to perform ADLs and making it difficult to accurately predict the rate of morbidity which will affect the timing of the underlying cash flows entering the transaction.
The people who have contracted a reverse mortgage have a greater incentive to remain in their property. Future governmental policies may benefit the owners. The actuarial market practice in the UK calculates morbidity as a percentage of the mortality rate.
Table 8.2. The adjustment factors for deriving the morbidity rates.
Age Males(%) Females(%)
≤70 2 3
(70, 80] 4 12
(80, 90] 5 13
(90, 100] 4 8
8.4.4 HOUSE PRICE RISK
The house price risk determines the NNE risk which is managed through two channels, by charging a portion of the interest rate risk to cover this potential fall and by insisting on a low LTV. LTVs are in general age-dependent, with lower LTVs for “younger” borrowers and higher LTVs for “older” borrowers, the difference reflecting the expectation of the lender of exit rates. There are lenders who are fine to give larger amounts of cash to borrowers that can prove that they are in poor health.
Although the most common assumption regarding the house price dynam- ics is the geometric Brownian motion. This is assumed for a real-estate index, for which data is available. Thus, for reverse mortgages, basis house price risk is introduced reflecting the difference in the evolution of the house price index and the price of the particular house that is the collateral in a given loan.
Pu et al. (2014) found a way around basis house price risk by using an N-dimensional vector for allNhouses in the loans portfolio. House prices then evolve as a system of GBMs:
dHt(i) =μH(ti)dt+σHt(i)dWt(i) (8.22) Miao and Wang (2007) showed that the total level of volatility for real- estate can be decomposed into a systematic volatility component and a idiosyncratic volatility component. Hence Pu et al. (2014) expand the error term into
dWt(i) =ρidZt+ 1−ρi2dB(i)t (8.23) where {Zt}t≥0 is a GBM that accounts for the systematic component and {B(i)t }t≥0 is a GBM describing the idiosyncratic shock of thei-th house, the two components being independent between them and across houses.
If Ht =N
i=1Ht(i) is the total price of all houses in the portfolio then, exploiting the independence assumption for all processes involved and impos- ing thatρi ≡ρ, we can write
N i=1
dH(i)t =μ N
i=1
Ht(i)dt+σ N
i=1
H(i)t dWt(i) (8.24)
dHt=μHtdt+σHt
ρdZt+ N
i=1
H(ti) Ht
1−ρ2dB(ti)
(8.25)
Denoting byθ =
ρ2+(1−ρ2)N
i=1
H(i)t Ht
2
and defining the Brown- ian process
Wt= 1 θ
ρZt+
N i=1
P(i)t Pt
1−ρ2B(i)t
(8.26)
it follows that, starting fromH0=N
i=1H0(i),
dHt=μHtdt+σ∗HtdWt (8.27) where σ∗=θσ. Sinceθ <1, under this model, the volatility of the entire property portfolio will always be less than the volatility of an individual house, so there is a benefit of portfolio diversification.
If HH(i)t
t = N1 for all house prices thenθ= ρ2+(1−ρ2)N1 and Wt= 1
θ
ρZt+ 1 N
N i=1
1−ρ2B(ti)
(8.28)
Comparing this scenario with the previous heterogeneous scenario it can be observed that the expected house price growth is the same. Thus, under this model, heterogeneity of houses that are collateral in the reverse mortgage loan portfolios only influence the volatility of the future house prices, not their expectation. Moreover, increasing the portfolio size allows us to conclude that limN→∞θ=ρ. Other extreme cases for this model imply that whenρ= ±1 thenθ=1 andσ∗=σ. Whenρ=0, that is for pairwise independent proper- ties, it follows thatθ = N1, the minimum value for a givenN. In this scenario the idiosyncratic risk of the underlying property portfolio can be diversified the most.
8.4.5 PREPAYMENT RISK
Very little is known about the values of the prepayment rate for reverse mort- gages. In the US in the early days of the HECM programme a flat prepayment rate of 0.3 times the mortality rate of the youngest borrower in the family was used. In Korea a prepayment rate of 0.2 times the 2010 mortality rate for females was chosen based on Korean demographic data.