EXAMPLE.

What is the present worth of an annuity of £120, payable yearly, to continue 50 years, at 4 per cent, compound interest?

Here a=120, t=50, and R=1.04, to find P.

1. Required the present worth of an annuity of £10, for 7 years, payable yearly, at 4 per cent, compound interest.

2. Required the present worth of an annuity of £100, payable yearly, to continue for 20 years, at 4 per cent, compound

interest.

3. Required the present worth of an annuity of £40, payable quarterly, to continue 50 years, at 4 per cent, compound in

terest.

4. What annuity, payable yearly, to continue 20 years, may be purchased for £260. 3s. 24d., at 4 per cent, compound interest?

5. An annuity of £25, payable yearly, is purchased for £416. 11s. 6d.; how many years ought it to continue, interest per cent?

4

6. Required the value of a freehold estate, of £250 per annum, allowing the purchaser 5 per cent for his money.

7. A farmer, on obtaining a lease of a farm for 25 years, pays a fine, or grassum, of £1000; how much additional rent would have been equivalent to the same; interest at 4 per cent ?

REVERSIONARY ANNUITIES.

Let n represent the time after which the annuity is to commence; then the present worth, &c. of any annuity in reversion may be found by the following Theorems :

R-1 1. pax or Log.p=log. (Rt—1)—log. rRt+n+log. a. TRI

2. a: Ri+orlog.a=log.rRt+n—log. (R'—1)+log. p.

2. a=px

3. t=

4. n=

Rt-1

log. a-log. (a-rRnp)
log. R

a

log. (a-log. rp

log. R

If the annuity be to continue for ever, after its commencement, Rt is in that case the same as Rt-1: hence

EXAMPLE.

Required the present worth of an annuity of £80, payable yearly, for 24 years, to commence 8 years hence, reckoning compound interest at 5 per cent ?

Here a = 80, t = 24, n = 8, and R = 1.05, to find p.

Present worth of £1, for (24+8) = 32 years

Deduct for 8 years

EXERCISES.

1. Required the present value of an annuity of £100, payable yearly, for 60 years, to commence 40 years hence, reckoning interest at 3 per cent.

2. What annuity, payable yearly, for 15 years, to commence 5 years hence, may be purchased for £487. 19s. 34d., the purchaser being allowed 5 per cent?

3. If an annuity of £87. 10s., to commence at the expiration of 9 years, can be purchased for £500, required the time of its continuance, interest at 5 per cent.

4. A reversionary annuity of £20, to continue 12 years, may be purchased for £145. 16s. 8d.; when will it commence, reckoning interest at 5 per cent?

5. What annuity, payable half-yearly, for 21 years, and to commence 4 years hence, may be purchased for £500, reckoning interest at 4 per cent ?

MISCELLANEOUS EXERCISES.

1. A person, on obtaining the lease of a farm, for 21 years, pays a fine, or grassum, of £500; what additional annual rent would have been equivalent to that sum, reckoning interest at 4 per cent ?

2. Whether is the reversion of an estate, to continue for ever, after the expiration of 20 years, or a lease of the same estate for 20 years, most advantageous, supposing the annual rent of the estate £600, and compound interest at 4 per cent?

3. A person obtained a lease of a farm for 21 years, 14 of which are to run; what ready money ought he to pay, in order to have 7 years added to the lease, the yearly rent being £300, and reckoning compound interest at 5 per cent?

4. A lease of an estate, to continue 19 years, is offered for £50 per annum, and £400 ready money; but it is proposed

to give an additional rent, instead of advancing the £400, what ought that addition to be?

5. In what time will a sinking fund of 4 millions extinguish a debt of 600 millions of pounds sterling, reckoning interest at 34 per cent?

6. How much of the national debt will a sinking fund of 5 millions of pounds sterling extinguish in 25 years; and what will the fund itself amount to, at the end of that period?

ANNUITIES ON LIVES.

LIFE ANNUITIES are payments made, at regular periods, du ring the life of one or more persons.

The value of these for any proposed life, or lives, depends on two circumstances; the interest of money, and the probability of the duration of the proposed life, or lives.

As the sale and purchase of Life Annuities have now become a considerable branch of business, it is hoped the following practical rules, for performing calculations of this description, will be found to include the most common cases that occur, and prove of considerable use to those who may be concerned in transactions of this kind*.

In order to facilitate computations, respecting Annuities on Lives, tables have been formed, for exhibiting the rate of mortality, at every age; and, consequently, the probability of living to any proposed age.

* Those who wish to be better informed on this subject, may consult Dr. Price on Annuities, and the Practical Calculator, by John Davidsou, A.M.