CONTINGENT ANNUITIES.

892. Contingent Annuities include Life Annuities, Dowers, Pensions, etc. The value of these depends upon the expectation of life.

893. The Expectation of Life is the average number of years that a person at any particular age may be expected to live.

894. A Table of Life Annuities shows the sum to be paid at different ages to secure a life-annuity of $1 during the remainder of the life of the annuitant. See Appendix.

CASE I.

895. To find the present value of a life-annuity, or dower.

1. What must be paid for an annuity of $500, by a person aged 52, at 6%?

SOLUTION. We find in the table that a person aged 52 must pay, at 6%, $10.208 for an annuity of $1; and for an annuity of $500 he must therefore pay 500 times $10.208, or $5104. Hence the following

OPERATION.

$10.208×500=$5104.

Rule. Find in the table the present value of an annuity of $1 at the given age and rate, and multiply this by the given annuity.

NOTE. To find the present value of a life-estate or Widow's Dower (which is a life-estate in one-third of her husband's real estate), we calculate the yearly interest, at an agreed rate, of the value of the property, and find the present value of this interest in the same manner as that of a life-annuity.

2. Mr. Williams, who is 54 years old, wishes to buy an annuity of $850; what will it cost at 6%? Ans. $8296.85.

3. James Stevens, being pressed by his creditors, makes over to them a life annuity of $1000; what should it be considered worth, if he is 45 years old, and it is paid at the rate of 5% ? Ans. $12,648.

4. William Turner has a life estate of $10,500; what is the property on which it is paid, and what is its present value, his age being 40 years and the rate of interest 7% ? Ans. Property, $150,000; present value, $113,872.50. 5. A gentleman dies, leaving real estate to the amount of

$90,000; what is his widow's dower, and what is its present value, her age being 65 years, interest at 5% ?

Ans. Dower, $1500; present value, $11,647.50.

CASE II.

896. To find how large an annuity can be purchased for a given sum.

1. How large an annuity can be purchased for $2000 by a person aged 38, interest 6% ?

$2000

OPERATION.

12.239=$163.41

SOLUTION. We find from the table that an annuity of $1 for the given age and rate would cost $12.239; hence he could obtain for $2000 an annuity of as many dollars as $12.239 is contained times in $2000, which is $163.41. Hence the following

Rule. Find from the table the present value of an annuity of $1 for the given age and rate, and divide the given amount by it.

2. How large an annuity can be purchased for $850 by a person aged 54, at 4% ? Ans. $73.11.

3. The present value of a widow's dower is estimated at $15,000; her age is 59 years, and the interest 6%; what is her dower, and what the value of her husband's estate? Ans. Dower, $1758.71 a year; estate, $87,935.25. 4. John Gibbons sells a life-estate for $6500, he being 48 years old; what is the life-estate, and what the value of the property, interest 7% ?

Ans. Life-estate, $646.637 a year; estate, $9237.67.

CASE III.

897. To find the present value of the reversion of a given annuity.

1. Find the present value of the reversion of an annuity of $400 a year for 50 years, after the death of a person aged 35 years, at 6%.

SOLUTION.-The present value of an annuity of $1 for 50 years at 6%, as found by the table, is $15.761861; and the present value of an annuity on the life of a person 35 years old at 6%, is $12.573; hence the difference of these two numbers, which is $3.188861, will be what remains of an annuity of $1 after the death of the possessor. Multiplying $3.188861 by 400, we have the reversion of an annuity of $400, which is $1275.54.

OPERATION.

$15.761861 12.573

3.188861 400

$1275.5444

Rule. Find the present value of an annuity of $1 for the whole time, and then find its value during the given life ; the difference of these two sums, multiplied by the given annuity, will be the present value of the reversion.

NOTE. The present value of the reversion of a life-estate or dower is most easily found by deducting the present value of the life-estate or dower from the value of the property.

2. Find the present value of the reversion of an annuity of $750 for 50 years at 5%, after the death of a person aged 62 years. Ans. $7326.69. 3. What is the present value of the reversion of a perpetuity of $1000, after the death of a person aged 56 years, interest 7% ? Ans. $5690.71.

4. Henry Morris inherits an estate of $50,000 from his uncle, but it is burdened with the dower of the widow, 45 years old; what is the present value of his estate, including the reversion of the dower, int. 6% ? Ans. $38,572.

5. Samuel Ellis is heir to an entailed estate of $75,000, after the death of his father, aged 75 years; what is the present value of his reversion, at 4% ? Ans. $59,282.97.

CASE IV.

898. Given, the annuity, its present value, and the rate, to find the age of the annuitant.

1. A, receiving a legacy of $6378.60, buys an annuity of $600 at 6%; what was his age?

SOLUTION. If $6378.60 is the present

OPERATION.

value of an annuity of $600, the present $6378.60÷÷600–$10.631 value of an annuity of $1 will be

of

time=50 yr.

$6378.60, which is $10.361; looking in the table under 6%, we find this number opposite 50 years; hence A's age was 50 years. Hence the following

Rule. Divide the present value of the annuity by the annuity, find this amount in the table under the given rate, and the corresponding time will be the required age.

2. It costs me $6847.50 to buy an annuity of $500, at 5%; what is my age? Ans. 38 years.

3. A lady having quarreled with her relatives, in order to prevent them from inheriting her property, spent the whole

amount, which was $47,600, in buying an annuity of $10,000, at 6%; what was her age? Ans. 75 years.

CASE V.

899. Given, the annuity, its present value, and the age of the annuitant, to find the rate.

1. A gentleman 50 years old bought an annuity of $750 for $9651.765; what was the rate per cent.?

SOLUTION.-If $9651.765 is the present value of an annuity of $750, the present value of an annuity of $1 is of $9651.765, which is

OPERATION.

$9651.765-750=$12.86902 Rate=4%

$12.86902; looking in the table opposite 50 years, we find this amount under 4%; hence the rate was 4%. Hence the

Rule.-Divide the present value of the annuity by the annuity, find this amount in the table opposite the given time, and the corresponding rate will be the rate required.

2. A lady 45 years of age buys an annuity of $400 for $4158.80; what is the rate? Ans. 7%.

3. Mr. Wiggins wishes to purchase an annuity of $850; it costs him $7213.95, his age being 62 years; what is the rate per cent.? Ans. 5%.

NOTE. For a further discussion of the theory of annuities, and general formulas for working them, see Manual.

INSURANCE.

900. Insurance is a contract of indemnity for loss or damage within a given time. It is of two kinds: Property Insurance and Personal Insurance.

901. Property Insurance is security against loss by fire or transportation. Insuring anything is called "taking a risk."

902. Property Insurance includes Fire Insurance, Marine Insurance, Transit Insurance, and Stock Insurance. Transit Insurance is security against loss by transportation by land, or by both land and water; Stock Insurance is indemnity for loss of cattle, etc.

903. Personal Insurance includes Life Insurance, Accident Insurance, and Health Insurance.

Accident Insurance is indemnity for casualties to travellers and others; Health Insurance secures a weekly allowance in case of sickness.

904. The Insurer, or Underwriter, is the party or company taking the risk. The Insured, or Assured, is the party protected.

905. The Policy is the written agreement or contract between the insurers and the insured.

906. The Premium is the sum charged for insurance; it is a certain rate per cent. of the amount insured.

FIRE AND MARINE INSURANCE.

907. Fire Insurance is security against loss by fire; Marine Insurance is security against loss by navigation.

908. The Sum Covered by insurance is the amount insured on a property.

909. The Base is the amount insured on a property. The Rate varies with the risk.

The Rate of insurance is quoted as so so much per cent. Policies are renewed and the premium is paid in advance.

annum.

table:

many cents on the $100, or as annually or at stated periods, Risks are usually rated per The rate for more than 1 yr. is determined by the following

Insurance is generally done by stock companies. When an individual takes a risk, it is called an out-door" business. A Mutual Insurance Company is one in which the profits and losses are shared by those who are insured.

To prevent fraud, companies will seldom insure the full value of property. In cases of loss, the underwriters may either replace the property insured, or pay its value. Only the amount of actual loss can be recovered; aud often claims are adjusted for a part of the amount insured.

Before issuing a policy, the company has the property carefully surveyed and described with respect to the dangers to which it is exposed; and any deception in this respect, or subsequent change which increases the risk, vitiates the policy.

910. Short Rate Tables are tables prepared for reckoning the insurance when the time is less than one year.

The rate in short periods is quoted per annum, and the actual rate for a short period is given in the table. Such a table is given in the appendix, and is used in solving some of the problems in Cases I. and IV.