Chinloy and Megbolugbe (1994) describe a continuous-time framework for disentangling the various options and risks embedded in a reverse mortgage.
This framework is reviewed here.
The loan that is originated when the reverse mortgage is issued has the value
L=min(H,λ) (8.1)
whereHis the value of the house andλis the loan limit. The borrower can draw a maximum ofνLwhereνis the LTV. The rateris used for discounting.
The house price grows at the rate h. The limit of the number of payments (as monthly payments) isη. Hence, in the HECM programme, the maximum limit is considered to be an age 100 and the borrower is at least 62 years old.
This implies thatη=456. Thus, the present value of the borrower’s liability is calculated with continuous compounding as
νLe(h−r)η
The borrower will receive a sequence of payments, some of them being possibly zero, at times t=0, 1, 2,. . .,η. The payments may include indexed
adjustments for inflation and lump-sum draws on a line of credit. In order for this formula to work,handrshould be transformed into monthly equivalents.
Using a slight abuse of notation, in order to keep the exposition simple, we are going to denote by the same notation,handr, the equivalent monthly rates.
Denoting byq(t)the loan survival at timet, spanned by the various charac- teristics of the borrowers such as age, sex, correlation (for couples), mortality, morbidity tables and trends, the cash-flow at timet, from a lender perspective, isq(t)A(t). If the inflation index growth rate isithen Chinloy and Megbolugbe (1994) argue that there will be
η
0
q(t)A(t)e−(r−i)tdt
Hence, the liabilities and payments will be matched over the life of the product if
νLe(h−r)η= η
0
q(t)A(t)e−(r−i)tdt (8.2) Simplifying the assumptions to haveA(t)≡Aandq(t)≡qleads to
νLe(h−r)η=qA η
0
e−(r−i)tdt (8.3) This equation can be solved for the annuity paymentAas
A= (r−i) [e−hη(erη−eiη]
νL
q (8.4)
Now we can construct some examples.
Example 8.1. Assume that for a reverse mortgage loan the following values occur: the property with the initial house price estimation at L=500, 000dollars, an LTVν=40%, inflation rate i=3.25%, discount rate r=1%, house price growth rate h=4%, constant exit rate (combining mortality, morbidity, prepay- ment) q=6%, all rates per annum, and usingη=456months as the lifetime of the reverse mortgage that is typically assumed under the HECM programme for a borrower taking the loan at 62 and living until 100 years old. Then, feeding these values into formula(8.4)the value of the annuity A is equal to 173,530 USD.
In order to gauge the sensitivity of the general annuity formula (8.4) to all input factors involved we have conducted an exercise whereby we varied one input factor at the time. The results are illustrated in Figure 8.1. One can see that, under the simplified framework, the value of the annuity decreases in a convex manner with respect to the discount rater, as well as with inflation
−0.02 0 0.02 0.04 0.06 0.08 0.1 0.2
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2x 105
Value of discount rate
Annuity ERM in dollars
Sensitivity to discount rate r
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1 1.5 2 2.5 3 3.5x 105
Value of inflation rate
Annuity ERM in dollars
Sensitivity to inflation rate i
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0
0.5 1 1.5 2 2.5 3 3.5x 109
Value of house price growth rate
Annuity ERM in dollars
Sensitivity to house price rate h
0 0.02 0.04 0.06 0.08 0.1
1 2 3 4 5 6 7 8 9 10 11x 105
Value of average exit rate
Annuity ERM in dollars
Sensitivity to exit rate q
Figure 8.1. The sensitivity of annuity value in an ERM with respect to various important risk drivers, with a constant exit rate over time. The borrower is 62 years old.
ratei. An increase in the flat exit rateqreduces rapidly the value of the annuity but at the same time a small rate of exit that is equivalent to long survivorship of the borrower takes the value of the annuity to a high level, well above the initial house value. The most striking result is the sensitivity with respect to the house price growth rate. If one assumes a house price increase beyond 25%
per annum, as observed in some bull periods in some hot real-estate spots like London, New York, Tokyo, Hong Kong, Singapore, the value of the annuity effectively explodes.
Here we recalculate the annuity rate by considering a time evolving exit rate.
As in Chinloy and Megbolugbe (1994) we consider that
q(t)=b(1−b)t−1 (8.5) wherebis a base exit rate, taken atb=0.1. Thus, the annuity value is calculated from the equation
νLe(h−r)η=A η t=1
b(1−b)t−1e−(r−i)t
=A b 1−b
η t=1
(1−b)te−(r−i)t
=A b 1−b
ωη+1−ω ω−1 whereω=(1−b)e−(r−i). Hence
A= νLe(h−r)η b/(1−b)
ω−1
ωη+1−ω (8.6)
Redoing the sensitivity calculations, under the same initial starting point values, we get the graphs depicted in Figure 8.2. The same conclusions hold as before. One clear difference though is the linearization of the annuity value with respect to variation in inflation and base exit rate, respectively, the decrease and respectively the increase, are almost linear.
One may wonder how the analysis will vary if the borrower is older. Many borrowers of reverse mortgages enter this market when they are 75 years old. This means that instead of 456 months up to a fixed exit maturity of 100 years, one has 300 months up to the same exit time. In Figures 8.3 and 8.4 we illustrate the same calculations as carried out in this section.
One can see similar profiles for all input parameters except discount rate where there is a tendency towards more linear profiles suggesting a loss in convexity.
The crossover option at η is determined by the strike price Aerηr−1. The present value of the strike crossover barrier is
X(t)=e−rtAerη−1
r (8.7)
Then ifM(H,t;θ)denotes the value of the reverse mortgage, withθdenoting model parameters vector, the payoff of the crossover option at timetis
P(t)=max[X(t)−M(H,t;θ), 0] (8.8) The assignment option refers to the possibility of transferring the NNEG risk to an insurer. The assignment option is issued relative to a bondB(r,t;θ) representing the market value of the annuity contract. The fixed dollar annuity has a future valueνL. The lender has an option to sell the loan with maturity
−0.020 0 0.02 0.04 0.06 0.08 0.1 1
2 3 4 5 6 7x 105
Value of discount rate
Annuity ERM in dollars
Sensitivity to discount rate r
0 0.1 0.2 0.3 0.4 0.5
0 1 2 3 4 5 6 7x 105
Value of inflation rate
Annuity ERM in dollars
Sensitivity to inflation rate i
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0
1 2 3 4 5 6 7 8 9 10x 109
Value of house price growth rate
Annuity ERM in dollars
Sensitivity to house price rate h
0 0.02 0.04 0.06 0.08 0.1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
5x 105
Value of base exit rate
Annuity ERM in dollars
Sensitivity to exit rate q
Figure 8.2. The sensitivity of annuity value in an ERM with respect to various important risk drivers, when exit rate evolves geometrically. The borrower is 62 years old.
ηto an insurer, forνL. At the term date the value of the assignment option is given by
S(r,η;θ)=max[νL−B(r,η;θ), 0] (8.9) If the accumulating balance reachesνLat some timetprior to the maturity of the loanη then it is optimal that the lender will exercise the assignment option. In this way crossover risk is transferred from the lender to the insurer.
What makes the assignment option valuable is the adjustable rates used for growing the outstanding balance. This reinsurance scheme operated in the US with HUD as the insurer for HECM lenders.
−0.02 0 0.02 0.04 0.06 0.08 0.1 0.5
1 1.5 2 2.5x 105
Value of discount rate
Annuity ERM in dollars
Sensitivity to discount rate r
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1 1.5 2 2.5 3 3.5x 105
Value of inflation rate
Annuity ERM in dollars
Sensitivity to inflation rate i
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0
2 4 6 8 10 12 14x 105
Value of house price growth rate
Annuity ERM in dollars
Sensitivity to house price rate h
0 0.02 0.04 0.06 0.08 0.1 0
2 4 6 8 10 12 14x 105
Value of average exit rate
Annuity ERM in dollars
Sensitivity to exit rate q
Figure 8.3.The sensitivity of annuity value in an ERM with respect to various important risk drivers, with a constant exit rate over time. The borrower is 75 years old.