which disability occurs on account of the second advance payment is

d N +3 T Dis+1

As there are k possible instalments the value at the end of the year in which disability occurs on account of the prepayment of all possible instalments is

and at the inception of the contract the value of the benefit is

As premiums for the benefit are payable until age y the net annual premium for the benefit is

Limited Payment Life Policy.-At the beginning of the contract the value of this benefit is the same as that under a whole life policy issued at the same age and when x + n = or > y the annual premium for the benefit is also the same as under a whole life policy, because in this case the annual premium for the benefit will cease at age y. Where x + n <y the net annual premium for the benefit is

Endowment Policy.-Under an endowment policy the temporary annuity

Ni Ni x+n

D%

measures the value of the interest from the date when an instalment is paid until the date of death or maturity. Furthermore, the payment of instalments may be terminated either by death

or maturity, so that the value of the benefit at the end of the year in which disability occurs, the insured's age then being z + 1,

Where x + n = or <y the annual premium for the benefit is payable for n years, and the net annual premium for the benefit is

but where x+n>y the annual premium for the benefit ceases at age y and the benefit is not received unless disability occurs before age y and the premium for the benefit is

As the benefit ceases unless disability occurs at least two years prior to maturity, the summation in equation (60) must cease when z = x + n - 3. Accordingly when x + n = y + 1 the summation is -2 instead of 1, the formula remaining otherwise the same.

When a policy provides for an annuity or for the payment of the face amount in instalments during disability, it generally also provides for waiver of premiums during disability. The total premium for the disability benefits will then be the sum of the premium for the waiver benefit and the premium for the

annuity or instalment benefit. In calculating the premium for the waiver benefit the premium to be waived is considered to be the premium for the regular insurance exclusive of the disability benefit, because the premium for the disability benefit is calculated on the assumption that it will cease in event of disability.

The waiver of premium benefit contained in a limited payment life or an endowment policy ceases when the last premium is paid. The benefit considered on pages 41 to 43 ceases unless disability occurs prior to one year before the maturity of an endowment policy; the benefit considered on pages 44 to 46 ceases unless disability occurs prior to two years before the maturity of an endowment policy. Where x + n = or < y premiums for these benefits are calculated on the assumption that they will be received for n years, unless the insured dies or becomes disabled before that time. In other words it is assumed that premiums will be received after the benefit ceases. Although many policies grant the insured the privilege of canceling at any time the disability feature and reducing the premium by the amount of the premium for the disability benefits, yet it is not believed that many policyholders will take advantage of this condition.

RESERVES ON ACTIVE LIVES.

Under disability benefits designated on page 2, numbers 1, 2 and 3, the reserve for the disability benefit may be determined from the general formula:

Present value

of benefit

minus

(

Present value of future net premiums

This reserve is, of course, in addition to the regular reserve for the policy. When two or more disability benefits are combined in the same policy the reserve to be held for those disability benefits will be the total of the reserves under all the disability benefits.

The formulæ now to be given for reserves on active lives do not have any meaning when a duration of the policy is reached which makes it impossible for an active life to become entitled to the disability benefit which is dealt with by the formula. The terminal reserve in such cases should be taken as zero.

The Insurance Departments of New York and of Massachusetts have adopted tables of reserves based upon the tables constructed by Hunter. In constructing these tables for ordinary life policies with benefit limited to disability occurring prior to age y, the reserves on active lives were calculated by the formula

It will be noticed that the premium assumed to be waived is Tz, which is the net premium according to the American Table of Mortality.

In

It was found in many cases that the terminal reserves were negative, but it was not deemed advisable to take account of them in practice. Theoretically when the terminal reserve is negative, one half of a net one-year-term disability premium for the attained age should be held as a mean reserve. practice, however, the use of one half of the net disability premium for the policy is simpler and produces satisfactory results in the aggregate. This method was adopted by the Insurance Departments.

It will be a useful exercise for the student to show that the reserve formula used by the Insurance Departments is equal to

Tx

This is formula (61) on page 49, if P is given the value 2 and y is given the value 60.

WAIVER OF PREMIUM Benefit.

Let P be the annual premium that is to be waived. It may be either the gross or the net premium for the policy without the disability benefits.

Let Pa be the net annual premium for the waiver of premium benefit.

Let n be the number of years premiums are payable under a limited payment life policy or the term of an endowment policy. Let t be the duration for which the reserve is desired.

The reserve for the waiver of premium benefit on an active life at the end of t years is as follows:

Whole Life Policy.

(61)

P(Ni - Dia1⁄4+1) — Pa(No4, — Naa)

Daa
x+t

+

Limited Payment Life or Endowment Policy.-Where x + n < Y

The reserve for the waiver of premium benefit will, of course, disappear when the insured attains age y unless he should be disabled before attaining that age. These reserves will frequently be negative and when this happens the company would hold for valuation purposes one half of the net annual premium for the benefit.

ANNUITY DURING DISABILITY.

Let Pd be the net annual premium for the benefit.

Let k be the annuity, that is the amount payable each year. Let n be the number of years' premiums which are payable under a limited payment life policy, or the term of an endowment policy.

Let t be the duration for which the reserve for disability benefit is desired.

The reserve on an active life after a policy has been t years in force, where the first payment of the annuity is made one year after the receipt of proof of disability, will be as follows:

Limited Payment Life Policy.-When x + n = or > y the premium and the benefit are precisely the same as under a whole life policy, consequently the reserve is the same, but when