**“Application of Mathematics in Business”**

**Business Mathematics **

**Mathematics of Finance in Business Sector**

Many transactions in every day life involve making a series of equal payments over a period of time, such as mortgage, rent, etc. In genera], a sequence of fixed annual payments (or receipts) made at uniform (or equal) time intervals is called an annuity. The time between payments is called the period, and the time from the beginning of the first period to the end of the last period is called the term of the annuity. Annuity may be classified into two categories:

**Future Value & Present Value Annuity **

(a) Annuity due: The annuity in which the payment is made at the beginning of each period, i.e.; all payments are to be made at the beginning of successive intervals, for example, rent or leases, is called an annuity due.

(b) Annuity ordinary (or immediate): The annuity in which the payment is made at the end of period, i.e. all the payments are to be made at the end of successive interval, for example, mortgages or loans, is called simple (or immediate) annuity.

**Application of Mathematics of Finance in Business Sector**

- Calculate the present value of a cash flow.
- Calculate the interest rate given a cash flow equation.
- Calculate the maturity date given a cash flow equation.
- Calculate the future value of an ordinary annuity.
- Calculate the periodic interest rate on a cash flow.
- Calculate the effective annual rate on a cash flow.
- Calculate the future value of an uneven cash flow.
- Convert a fractional time compounded period cash flow to its annual effective rate.
- Calculate the future value of a cash flow for fractional time periods.
- Calculate payments on amortized loans.
- Construct a loan amortization table.

**Future and Present Values annuity**

• Identify reasons to calculate the Future and Present Values annuity.

• Construct a financial time line.

• Identify the equation for calculating the future value of a cash flow.

• Calculate the future value of a cash flow.

• Identify the equation for calculating the present value of a cash flow.

• Calculate the present value of a cash flow.

• Identify the variables needed to calculate the interest rate given a cash flow equation.

• Calculate the interest rate given a cash flow equation.

• Identify the variables needed to calculate the maturity date given a cash flow equation.

• Calculate the maturity date given a cash flow equation.

• Simulation Overview:

**Annuities and Perpetuities**

• Construct an ordinary annuity time line.

• Identify the equation for calculating the future value of an ordinary annuity.

• Calculate the future value of an ordinary annuity.

• Identify the equation for calculating the present value of an ordinary annuity.

• Calculate the present value of an ordinary annuity.

• Construct an annuities due time line.

• Identify the equation for calculating the future value of an annuity due.

• Calculate the future value of an annuity due.

• Identify the equation for calculating the present value of an annuity due.

• Calculate the present value of an annuity due.

• Identify the equation for calculating the present value of a perpetuity..

• Calculate the present value of a perpetuity

• Simulation Overview:

**Interest Rates, Uneven Cash Flows, and Amortized Loans**

• Identify three types of interest rates.

• Calculate the periodic interest rate on a cash flow.

• Calculate the effective annual rate on a cash flow.

• Calculate the future value of an uneven cash flow.

• Calculate the present value of an uneven cash flow.

• Convert a fractional time compounded period cash flow to its annual effective rate.

• Calculate the future value of a cash flow for fractional time periods.

• Identify the equation for calculating payments on amortized loans.

• Calculate payments on amortized loans.

• Construct a loan amortization table.

The present value of an annuity is the current value of the total amount of annuity at the end of the given period. In other words, present value of a given sum if money due at the end of a certain period of time is the sum of principal amount plus interest accumulated at the given rate for the same period,

**Some Rules of Future Value & Present Value Annuity**

**Present value **

Single method :

P.V=F.V/ (1+i)?

**Annuity:**

P.V (due) =A/i [1-(1+i) ¯?] x(1+i)

P.V(Ordinary)= A/i[1-(1+i)¯?]

**Future value **

Single method.

F.V=P.V(1+i)?

**Annuity:**

F.V(due)=A/i(1+i)[(1+i)?-1]

F.V(Ordinary)= A ? i[(1+i)?-1]

**Example-1**

M/s. Rahim Textile Mills Limited, Purchase a wagon on installment basis, Such that TK. 5000 is to paid on the signing at the contract and four yearly installments of TK.3000 each payable at the end of the first , second, third and fourth year. It is interest is charged at 5% p.a What would be the cash down price?

**Solution:**

We know that,

Here,

P.V= A ? i[1-1/(1+i)?] A= Amount of installment =3000

I= interest Rate=5%=0.05

N = no. of year = 4

=3000 ? 0.05 [1-1/(1+.05)4] P.V= ?

=10637.85

So, the present value of wagon =(5000+10637.85)=15637.85

**Analysis & Comments of the results in the view of company decision**

From the above results we can decide that if we purchase the mentioned wagon in future it will cost Tk.5000+12,000=17,000/- including principle & interest where as if we purchase the same wagon at present it cost Tk. 15637.85/- Now management come into decision whether they purchase the wagon at present or future.

So we see that such type of business problem can be solved through these mathematical finance formulae easily.

**Example -2 **

A machine costs the company Tk.97000 and its effective life is estimated to be 12 years. If the scrap realizes Tk. 2000 only, what amount should be retained out of profits at the end of each year to accumulate at compound interest at 5% per annum? Solution: Given F= Cost -Scrap Value = 97000 – 2000 = 95000, n = 12,/ = 0.05.

We know that

F.V= A ? i[(1+i)?-1]

Or, 95000=A[(1+0.05)^{12}-1]

Or, A=5964.34

**Comments:**

From the above equation we find that if we deposit Tk. 5964.34/- Per Year From just now in any account, only then a new machine will belongs to the concerned company easily out of interest.