# An insurance plan to guarantee reverse mortgages: comment.

An Insurance Plan to Guarantee Reverse Mortgages: CommentIn Weinrobe's (1988) article arrangements of reverse mortgage plans (RM) are discussed and analyzed, particularly with respect to the deficit risk of the lender. This comment simplifies the situation by translating the RM concepts into traditional life insurance terms and using actuarial calcultions, making the RM commercially accessible for the life insurance industry. Further, the RM possesses interest immunization properties in conjunction with a usual life insurance portfolio."

In its simplest form, an RM is an annuity paid to an individual by a financial institution with the corresponding lump-sum value paid to the institution at the end of the annuity period. The lump-sum payment is secured by a mortgage on residential property owned by the insured. The reversion, in comparison to an annuity investment, applies to the date the lump-sum payment is due, at the beginning of the annuity period in the case of the annuity investment. To extend these concepts to payments depending on the annuitant's being alive, the annuity investment must be translated to an immediate life annuity, purchased by a lump-sum payment at the beginning of the life annuity. The reversion makes the lump sum payment due at the end of the life annuity period, i.e., death.

The size of the payment for a given size of lump sum is here calculated as an expected value by (numerical) integration instead of the corresponding simulation calculation in the Weinrobe's article.

The value of a single life annuity of $1 yearly, paid continuously, is

[Mathematical Expression Omitted]

The value of a payment of $1 at death is

[Mathematical Expression Omitted]

where

i = annual interest rate [mu]x = force of mortality for an x-year old person

[Mathematical Expression Omitted] = corresponding probability for an x-year-old person to survive a further t years.

The reverse life annuity for an insured aged [xi] years at agreement is defined so that the insured receives [$P.sub.[xi]] yearly (continuously, or approximately equal to 1/12 monthly) from the agreement date to death, and the insurer receives a lump sum of [$S.sub.[xi]] at death. The connection between [p.sub.[xi]] and [s.sub.[xi]] is

[Mathematical Expression Omitted]

For example: Let i = 10 percent p.a. Consider females only with mortality [[mu].sub.x] = 0.0005 + [10.sup.5.728 + 0.038x - 10 1] and let the insured sum be $80,000. For an agreement age of [xi] = 70 years, the values (1,2) are calculated as

[Mathematical Expression Omitted]

and to the yearly payment for an insured sum of [S.sub.70] = $80,000 by rearrangement is (3) becomes $3,779. For the same insured sum, the size of the yearly payment for the different ages at the agreement date is:

If the insured vacates the property before death, a surrender debt would have to be paid by the insured. The surrender debt n years after the agreement date can be calculated as:

[Mathematical Expression Omitted]

but this is only acceptable for the insurer if the health of the insured has not deteriorated (in comparison with that of the age group in general), i.e., if the actual mortality does not exceed the assumed mortality [[mu].sub.[xi]] + n. If the insured is close to death, a more true surrender debt will be close to [S.sub.[xi]].

Instead of surrender termination, the insurance agreement is continued after the insured has vacated the property. The property is sold and the insurer retains the whole sum [S.sub.[xi]]., which originally was due at death. As compensation (also as a partial rent compensation), the insured is paid interest on [S.sub.[xi]] of [i-S.sub.[xi]] until death, in addition to the life annuity [p.sub.[xi]].

Reverse mortgage agreements are net assets for a net insurer, in contrast to usual premium paid insurances, which are liabilities. The payment profiles for these two types are of the same structure, but with opposite signs. Hence the interest risk of the insurer can be eliminated by cash-flow matching. As a consequence, the insurer can guarantee an unchanged future interest rate, fixed at agreement date, for a balanced protfolio of RMs and premium paid insurances. The balance is achieved by calculating expected cash flows and matching these or by a more general duration immunization.

(1) Mortality used in the Danish G82 valuation basis.

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Author: | Schapira, Steffen |
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Publication: | Journal of Risk and Insurance |

Date: | Dec 1, 1990 |

Words: | 726 |

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