Mathematics of finance. Simple and compound interest. Conversion period. Amount of an annuity. Present value.

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Simple and compound interest

Def. Interest. Money paid by an individual or organization for the use of a sum of money called the principal. Interest is usually paid at the ends of specified equal intervals of time e.g. annually, semiannually, quarterly, monthly, daily.

Def. Amount. The sum of the principal plus interest.

Def. Interest rate. The percentage of a sum of money that is charged for the use of the money for a particular period of time (generally one year).

Simple interest. The interest due at the end of a certain period is simple interest if the interest is computed on the original principal only during the entire period. (Simple interest contrasts with compound interest, which includes interest on accrued interest.)

The simple interest I on a principal P for n years at an interest rate of r percent per year is given by the formula

I = Prn

The amount (principal plus interest) A is given by

A = P(1 + rn)

Compound interest. If the interest when due is added to the principal and thereafter earns interest, the interest (calculated in this way) is compound interest. The interest is computed on the principal for the first period, on the principal and the first principal’s interest for the second period, on the second principal and the second principal’s interest for the third period, etc. with the principle plus interest becoming a new principal at the end of each period. The interval of time between successive conversions of interest into principal is called the interest period or conversion period and is usually either three months, six months or one year, in which cases interest is said to be compounded quarterly, semiannually, or annually respectively. The total amount due at any time is called the compound amount.

The nominal rate of interest is the stated yearly rate. The actual rate of interest will depend on the length of the conversion period. For example, if we state that the nominal rate is 6% compounded quarterly, then the conversion period is 3 months and the interest rate is ¼(6%) = 1.5 % for each conversion period.

If P is the original principal, i the rate of interest per conversion period and n the number of conversion periods, then the compound amount A at the end of these n conversion periods is given by

A = P(1 + i)ⁿ

Problem. A person invests $1000 at 6% compounded quarterly. Find the compound amount A and compound interest I after 3 years.

Solution.

P = 1000,

i = 0.25 (6%) = .25×.06 = .015

n =12 (four periods per year for three years)

A = 1000(1 + .015)¹² = 1000(1.015)¹² = $1195.62

I = A - P = 1195.62 - 1000 = $195.62

A principal P placed at a rate of interest r for n years accumulates to an amount A_n as follows:

At simple interest: A_n = P(1 + nr)

At interest compounded annually: A_n = P(1 + r)ⁿ

Present or discounted value of a future amount

Def. Present value. A sum of money which, with accrued interest, will increase to a specified sum at some specified future time. In other words, the present value P of a sum of money A is that principal which will, with accrued interest, accumulate to A at some specified future time.

The formula for present value is obtained by solving the equation A = P(1 + r)ⁿ for P.

The principal P which in n years will accumulate to the amount A_n at the rate of interest r is:

Def. True discount. The true discount is

D = A_n - P

Problem. Find the present value of $1000 due in 4 years at 6% compounded quarterly.

Solution.

A_n = $1000

r = 6% = .06

q = 4

n = 4

The true discount is

D = $1000 - 788.03 = $211.97

Annuities

Annuity. An annuity is a series of equal payments at regular intervals. The payment period or payment interval of an annuity is the length of time between two successive payments. The term of an annuity is the total time between the beginning of the first payment period and the end of the last one. The annual rent is the sum of the payments made in one year.

During the term of the annuity one payment is made at the end of each payment period, but, unless otherwise stated, no payment is made at the beginning of the first payment period. The term is thus equal to the payment period multiplied by the number of payments.

Amount of an annuity. The amount of an annuity is the total amount that would be accumulated at the end of the term if each payment were invested at compound interest at the time of payment.

An annuity in which the periodic payment at the end of each of n equal periods is one dollar and whose interest rate per period is i has an amount given by

The amount of an annuity in which the periodic payment is R dollars is given by

Derivation

Problem. A person deposits $1000 in a savings account at the end of each year for ten years. The money is compounded annually at 6% interest. What will the account be worth at the end of ten years?

Solution.

R = 1000

i = .06

n = 10

Accumulated value of an annuity. The accumulated value of an annuity at a given date is the total amount that would be accumulated by that date if each payment were invested at compound interest at the time of payment.

Present value of an annuity. Suppose you deposit a sum of money P into a savings account and from this money (plus accumulated interest) you make n payouts of R dollars each at the end of each year for n years, exhausting the account on the last payout. What is the formula for the amount P that will just support these payments? View P as being partitioned into n parts p₁, p₂, .... , p_n , where the amount p_i corresponds to the present value of the i-th payout; that is, p_i grows (through accumulating interest) to produce the i-th payout. The n payouts of R dollars each constitute an annuity and the amount of money P that you must initially deposit into the account to support these n payouts is called the present value of the annuity. Thus the present value of an annuity is the sum of the present values of all the payments. An annuity in which the periodic payment at the end of each of n equal periods is one dollar and whose interest rate per period is i has an present value given by

The present value of an annuity in which the periodic payment is R dollars is given by

Derivation

Problem. A man purchases a house for $150,000. He pays $10,000 down. The balance of $140,000 is borrowed and to be paid with interest in a series of semiannual payments over a period of 30 years at an interest rate of 6% per year, compounded semiannually. Find the semiannual payment.

Solution. The 60 semiannual payments are an annuity whose present value is $140,000.

P = 140,000

n = 60

i = .03

References

Hawks, Luby, Touton. Second-Year Algebra

Murray R. Spiegel. College Algebra (Schaum)

Raymond W. Brink. A First Year of College Mathematics

James / James. Mathematics Dictionary

Richard Burington. Handbook of Mathematical Tables and Formulas