SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

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


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 An as follows:

 

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

 

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


ole.gif





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 An at the rate of interest r is:


ole1.gif


ole2.gif


ole3.gif




Def. True discount. The true discount is


            D = An - P



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


Solution.


            An = $1000

            r = 6% = .06

            q = 4

            n = 4


             ole4.gif


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


             ole5.gif


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


             ole6.gif


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

 

             ole7.gif




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, .... , pn , where the amount pi corresponds to the present value of the i-th payout; that is, pi 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


             ole8.gif


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

 

             ole9.gif


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


             ole10.gif





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



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]