life

Hence, if ar is greater than ax+1, ax+2,......i.e., if the value of an annuity on a's life is greater than the value of an annuity on any older than x,

da

then

a(1+a) > v Tvi

is a positive quantity: in other words, as v increases so

∞(=

1+

11) increas

increases, i diminishes; therefore the

also does V. But as v

lower the rate of interest the greater is the value of a policy of one year's standing on a's life.

increases in

Similarly, if ar+1 is greater than αx+2, αx+з........ value as the rate of interest diminishes; and generally, if the values of annuities on the lives x, x+1, x+2,. the lives x, x+1, x+2,...... form a continually decreasing series, the value of a policy of one year's standing taken out at any age from a upwards is greater the less the rate of interest. Whence it follows from (1) that V increases as the rate of interest decreases.

n

The last line on p. 4 of this volume is misplaced and should be

carried over so as to be the last line on p. 8.

JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

On Reversionary Life Interests as Securities for Loans. By T. B. SPRAGUE, M.A., Vice-President of the Institute of Actuaries. [Read before the Institute, 25 November 1872.]

Of all the securities proposed to insurance companies, reversionary life interests are the most troublesome to deal with in the ordinary way of mortgage; and the objections to so dealing with them have been considered by many actuaries so serious, that they have laid down the rule that the only safe way of making an advance on the security of a reversionary life interest is by way of reversionary charge. The transaction, they think, should be in the nature of a sale rather than a mortgage, a portion of the reversionary life interest being sold in consideration of the present advance.

When a borrower applies for a loan on a reversionary life interest, he generally expects to be called on to effect an insurance of about the same amount as would be required to secure a loan on an immediate life interest; and it is difficult to make him understand why a greatly larger insurance is necessary to protect the lender. The reason becomes obvious enough when we consider what remedy a lender has in the event of the borrower failing to pay his interest and premiums. In that case, there appear to be three possible courses open to the lender. First, he may, by agreement with the borrower, allow the premiums and

VOL. XVII.

upon it at a fixed rate and also the premiums on policies sufficient to secure the charge. Each of these methods has its own advantages for the company. In both, the company must, at the time of making the advance, effect policies for the whole term of the borrower's life of such an amount as is considered sufficient to secure the transaction. In the latter case, where the office is entitled to demand payment of the charge on the death of the life tenant, and of course has the usual power of sale, which it can exercise whatever may be the state of health of the borrower, justice requires that the policies shall become the property of the borrower, from the date when he becomes liable to pay the charge. They will then be treated as mortgaged to the company to secure the amount of the charge and interest due to it; and any excess of insurance beyond that amount will belong to the borrower's estate. It seems open to question whether the policies should not be considered in this light from the very outset. The office will naturally effect policies without profits, because the cost of such is less than that of participating policies. In the former case, it will be more advantageous to the company to effect policies with profits, for the reason that in fairly successful offices, the bonuses declared on a policy greatly exceed in value the difference between the participating and non-participating premiums. Then, if the borrower does not exercise his option to pay off the charge, the company has the benefit of the bonuses accruing from time to time on the policies, which may easily, if the offices have been well selected, increase the rate of interest realized by one per-cent per annum. If the borrower redeems the annuity, his right to the policies is matter of arrangement, and it seems fair that the bare policies should be assigned to him, the bonuses being retained by the company. It is clearly unjust that the policies should belong absolutely to the company; but as the company is under no obligation to effect participating policies, it is reasonable that if it pays the higher premium required for them, the bonuses thus gained should be, in any event, its property. When the terms of the advance are arranged in the latter of the above methods, the company, in addition to the right of requiring payment of the charge, has the advantage of being subject to a smaller deduction for income tax; for this is calculated only on the interest, whereas, in the case where the company receives an annuity, the tax is calculated, not on the portion of it which represents the interest on the cost, but on the full amount. As I mentioned in my paper above referred to, when the advance is made in the former of the above

methods, it seems the best course in practice for the company to effect at the outset policies considerably less than the amount required fully to protect the security, trusting that the bonus additions to the policies will increase the sum assured to a sufficient extent by the time the annuity becomes payable.

In the case where the amount borrowed is very small in comparison with the value of the reversionary life interest, so that the reversionary charge is either less or but little more than a year's income of the property charged, it is sometimes thought sufficient, instead of effecting a policy for the whole term of the reversioner's life, to effect one on his life against that of the tenant for life and for, say, 3 years longer. In this case it is of course stipulated that the borrower shall pay a fixed sum on coming into possession; and it is anticipated that if he should fail to do so, or to make some other satisfactory arrangement, within, say, a year after the death of the life tenant, the company would have no difficulty in going into receipt of the rents or other income, and obtaining the full amount of the charge with interest, premiums, and costs, before the expiration of the insurance.

The formula for the amount of the reversionary charge in this case is precisely the same as that by which the value of a contingent reversion is found. The value of 1 to be paid by the borrower aged x, on the death of the life tenant aged y, if he is then alive, is

1−(P+d) (1+αxy)*;

P, in the case under consideration, denoting the annual premium for an insurance on the life of x against y and for 3 years longer. This assumes that the amount of the insurance effected is the same as that of the reversionary charge. It is true that in practice the reversionary charge will bear interest from the day of the death of the tenant for life, and that the lender may also have to pay the premium on the policy, and law costs; but as by supposition the income of the mortgaged property is large in comparison with the loan, and if the borrower should die there will be always current income available towards paying the charge, it seems unnecessary to require a larger insurance than for the amount of the loan. It is of course provided, that until the borrower pays the amount of the charge and interest, he shall pay the premiums that fall due on the policy, those which fall due before the death of the life tenant being paid

*If a single premium is payable for whole world license or for an insurance against the birth of issue, this must of course be subtracted from the above formula.

by the lender. If now the policy is drawn in the ordinary way, it may happen that the life tenant dies the day after the lender has paid a premium, so that the borrower would have no premium to pay for a year; and, on the other hand, the life tenant might die the day before the premium fell due, in which case the borrower would immediately have to pay a premium. To avoid this inequality, it seems better to provide that the premiums falling due before the life tenant dies shall be commuted by a single payment, and that the first premium to be paid by the borrower shall fall at a fixed time, say 3 or 6 months after the death of the life tenant. Taking the interval as 6 months, the policy would not be on the life of x against y and 3 years longer, but on the life of x against y and 2 or 3 years longer. It seems, in fact, only fair to the borrower to give him a fixed time, from the day of his coming into possession, in which he can pay the charge and interest without being called on for any additional payment on account premiums. Assuming the reversioner to have come into possession, he may prefer, if in good health, instead of paying the charge, to borrow the amount on mortgage of his life interest in possession, effecting a policy for the whole term of his life; or he may prefer to pay the charge out of income. In the event of his choosing the latter alternative, it is only fair that as his debt is reduced, the insurance to secure it should be also reduced; but this is a matter of detail which it would be quite unnecessary to insert in the deed.

of

Another method which has been suggested for the purpose of reducing the cost of the transaction to the borrower is to effect a policy on his life, subject to a low premium during the life of the life tenant, and a larger premium afterwards. It is clear that the premium to be paid during the joint lives must be not less than would be required for the insurance of a against y; and assuming arbitrarily a premium somewhat larger than this to be payable during the joint lives, it is a simple matter to calculate what premium should be payable after the death of y in order that the present value of the total premiums may be equal to the ordinary whole life premium on x. But in practice this arrangement is not applicable; for under it the borrower has an option for which the company has no equivalent. If the life tenant should die soon, x, if in good health, would be able to effect an insurance for the whole term of his life at a rate considerably lower than the agreed rate, and there would be nothing to prevent him going to another company and borrowing from them in the