8.3 Equity Release Programmes Around the World
8.3.2 THE KOREAN GOVERNMENT SPONSORED REVERSE
The minimum age for a borrower should be 60 in Korea for the Korean government-insured reverse mortgage programme. Ma and Deng (2013) describe the model used by the Korean government sponsored reverse
4 If only one spouse is party in the contract as a borrower, upon the death or move into care of that spouse, the reverse mortgage became due. New rules implemented in 2015 allows the spouse who is not listed in the contract to stay in the house as long as it is her/his primary residence.
mortgage. The break-even level of monthly payments can be calculated by equating the present value of the mortgage insurance premium (PVMIP) with the expected loss (PVEL).
To see how this works, we need some notation. Here we denote byp(x,t) the probability that a borrower at agexwill survive atx+t, by d∗(x,t)the probability that a loan will survive to the agex+tbut not to the agex+t+1, P0 is the upfront mortgage insurance premium. T(x) is the remainder of the payment period until loan termination when the loan starts at the agex of the borrower,μis the rate of mortgage insurance premium, OLBt is the outstanding loan balance at timet,pmt is the monthly payment, Ht is the expected house value at time t, h is the growth rate of housing price, r is the expected value of the actual interest rate of the loan during its period or the discount rate.
Then
PVMIP=P0+
t=T(x)
t=1
(OLBt+pmt)×μ×p(x,t)
(1+r)t−1 (8.10)
PVEL=
t=T(x)
t=1
max[(OLBt−Ht), 0] ×d∗(x,t) (1+r)t
(8.11) Requiring that PVMIP = PVEL allows the calculation to be done by a search method of the breakeven level ofpmt.
The evolution of loan balance is given by the equation
OLBt=(OLBt−1+pmt)(1+μ)(1+r) (8.12) and the house price is supposed to evolve according to
Ht=H0×(1+h)t (8.13)
The Korean government sponsored programme uses a fixed interest rate, the borrowers receiving payments similar to an annuity until the loan is ter- minated. The expected interest rateμis used for calculating the outstanding loan balance in each period and the PVMIP and PVEL. As already commented earlier in the chapter there is one rate used to accumulate the balance and another to discount. Commercial lenders in this programme will charge the certificate of deposit (CD) rate plus 1.1%. Hence, the insurer faces interest rate risk.
When payments are like an annuity fixed scheme, using (8.10) and (8.11) and asking thatPVMIP=PVELallows the calculation of thepmtin closed-form
pmt = H0×LTV−P0
T(x)−1
t=0
1 (1+r)(1+μ)
t (8.14)
The Korean programme also allows agraduate monthly scheme, with pay- ments indexed to the mean value of the growth rate of consumer prices. What is of interest in this situation is the base amount of graduate monthly payment pm0from which payments will be indexed and paid in the future. Once again, by imposing thatPVMIP=PVELone can derive
pm0 = H0×LTV−P0 T(x)−1
t=0
1+i (1+r)(1+μ)
t (8.15)
whereidenotes the mean value of the growth rate of consumer prices.
The parameter values used in the Korean programme were reported in Lew and Ma (2012). The housing price growth rate was assumed to be 3.5% per annum, reflecting the average house price growth rate in Korea between 1986 and 2006. The average value of the 10-year government bond rates was 5.12%
between 2002 and 2007 so the expected interest rate was calculated as 7.12%
after adding 2% lender’s margin. Those values were adjusted in Feb 2012 to be 3.3.% for house prices and 6.33% for the expected interest rate. The monthly mortgage insurance premium is 0.5% of the loan’s outstanding balance.
Example 8.2. Suppose that the house price growth rate is h=3.5%, the lenders margin rate isμ=2%which is added to the discount rate r=5.12%. The LTV is taken as 40% of a house that costs initially 500,000. In Korea the insurance premium is taken as 2% of the initial value of the house. The inflation rate is estimated as i=3%. Then for a borrower that is 62 years old and may live up to 100 the fixed monthly coupon that can be paid in the reverse mortgage is equal to pmt =1202.8while if a graduate scheme is used pm0=822.2121.
Example 8.3. Under the same conditions as in Example 8.2. but considering a borrower that is 75 years old and may live up to 100 the fixed monthly coupon that can be paid in the reverse mortgage is equal to pmt=1350.5while if a graduate scheme is used pm0=1011.9.
A very important quantity for reverse mortgages is thetotal annual loan cost rate(TALCR). This is an average annual rate at a specific montht=nthat absorbs all costs in a reverse mortgage, assuming that the borrowers will reach the age of 100 and ignoring survival probabilities. This rate is used as a yardstick to compare various reverse mortgage products.
For the Korean reverse mortgage programme the TALCR can be calculated for each payment type. For constant monthly payment the equation giving the TALCR is
Min(OLBn,Hn)=pmt t=n t=0
(1+TALCR)n−t (8.16)
or in a more concise form
(1+TALCR)n+1−1
TALCR = Min(OLBn,Hn) pmt while for graduate monthly payment the equation is
Min(OLBn,Hn)=pm0
t=n
t=0
[(1+i)(1+TALCR)]n−t (8.17)
or in a more concise form
[(1+i)(1+TALCR)]n+1−1
(1+i)(1+TALCR)−1 = Min(OLBn,Hn) pm0
Both equations (8.16) and (8.17) must be solved by searching methods given their high nonlinearity. One can easily prove that both equations have unique solutions.
There is a clear link between the mortality survival probabilitiesp(x,t)and the termination probabilitiesd∗(x,t)given by
d∗(x,t)=p(x,t)×d(x,t)=p(x,t)−p(x,t+1) (8.18) where d(x,t) is a modified mortality rate that takes into consideration the prepayment rate at agex+t. For the Korean programmed(x,t)=1.2q(x,t) whereq(x,t)is the 2010 mortality rate for females at agex+t.
In Figure 8.5 we illustrate the simulation of the borrower’s survival up to a maximum of 100 years and the derived calculation for the probability of loan exit. These calculations are done based on (8.18) and assuming that the base mortality curveq(x,t)evolves geometrically from an initial value of 0.04 and using the recursive calculations
q(x,t+1)=q(x,t)(1−q(x,t))(t−1).
In Figure 8.6 we present the same calculations but starting fromq(75, 1)= 0.07 or 7%. Note that all calculations are presented on an annual basis since this is common to actuarial mortality calculations. However, for reverse mortgage valuations a monthly valuation grid is needed.
00
5 10 15 20 25 30 35
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Simulate mortality using base geometrical process
Years after the age of 65
Survival probability
0 5 10 15 20 25 30 35
0 0.01 0.02 0.03 0.04 0.05
Simulate loan exit
Years after the age of 65
Loan exit probability
Figure 8.5.The simulation of borrower’s survival and loan termination probability. The bor- rower takes the loan when he is 65 years old.