**Unit 5**

**Time Value of Money**

**5.1 Introduction**

This unit aims at providing basic concepts on the time value of money. This is very important for taking any financial decision. In a business we are investing huge amounts of money today in anticipation of uncertain future returns or revenues. You have already learned that capital is not only scarce but also has cost. Cost in simple terms is nothing but the interest. Suppose you would like to borrow Birr 1, 000 today and return the same after a month without any interest. Do you think some one is going to lend you Birr 1, 000? Definitely no. If you are prepared to pay interest of 3% for a month on the borrowed money, people will come forward to lend you money. The reason is simple money is not available freely and it is capable of earning interest i.e., Birr 30. It is evident that today’s Birr 1, 000 is equivalent to Birr 1, 030 after a month. Here Birr 30 is called cost of capital in financial management.

**5.2 Time value of money**

By experience, we all know that the value of a sum of money received today is more than its value received after some time this is called time value of money. Conversely, the sum of money received in future is less valuable than it is today. The present worth of birr received after some time will be less than a birr received today. Since, a birr received today has more value, individuals, as a rational human beings would naturally prefer current receipt to future receipts.

The time value of money is also known as time preference for money. The time preference for money in business unit normally expressed in terms of rate of return or more popularly as a discount rate. In a business revenues are spread over a period of time i.e., the life of the project. It is nothing but we are trying to calculate the present value versus future value.

**5.3 Techniques of present value Vs future value**

** Compound value:** where a sum of money deposited one time and earns interest for a specified period. The interest is paid on principal as well as on an interest earned but not withdrawn during earlier period is called compound interest.

FV = PV + (interest X principal) For example you deposit

Birr 100 @10% interest

After one year = 100 + (100 x .10)

= 110

After two years = 110 + (110 x .10)

= 121

After three years = 121 + (121 x .10)

= 133.10

FV_{1} = P(1+i)

FV_{2} = P(1+i)^{2} FV_{2} = FV_{1} + F_{1}i P(1+i)^{2}

FV_{3} = P(1+i)^{3} FV_{2} = F_{1}(1+i)P

FV_{2} = P

Fn = P(1+i)^{n}

Here the term (1+i)^{n} is the compounded value factor (CVF) of a lump sum of birr 1. The values may be directly traced from the present value tables. You have already learned the calculation of present value factors in financial accounting II. Hence, you can directly apply the present value factors and find out the values.

The same may be written as below:

FV = P(CVFn . i)

FV = Future value

P = Present value

CVFn = Compounded value factor year

i = rate of interest

Suppose you deposit Br. 55, 650 in a bank which will pay you 12 percent interest for a period of 10 years. How much would the deposit grow at the end of ten year?

FV = P(CVFn . i)

FV = 55, 650 (CVF10 . 12)

FV = 55, 650 (3 . 106)

= 172, 849.90

** Compound value of Annuity:** An annuity is a fixed payment (or receipt) each year for a specified number of years. Assume that a sum of birr 1 is deposited at the end of each year for four years at 6% interest. This implies that

1(1+.06)^{3} 1.191 Birr 1 deposited at the end of year 1 grow for 3 years.

1(1+.06)^{2} 1.124 Birr 1 deposited at the end of year 2 grow for 2 years.

1(1+.06)^{1} 1.06 Birr 1 deposited at the end of year 3 grow for 1 year.

- Birr 1 deposited at the end of year 4 grow for no interest.

4375

FV_{4} = A(1+i)^{3} + A(1+i)^{2} + A(1+i) + A

FV_{4} = A[(1+i)^{3} + (1+i)^{2} + (1+i)+1

FV_{n} = A[ ((1+i)^{n} – 1) / i]

The same may be written as below

FV = A(CVAFn i)

FV = Future value

A = Annuity

CVAFn = Compounded Value Annuity Factor to yare

1 = rate of interest

Assume that you deposit a sum of birr 5, 000 at the end of each year for four years at 6% interest. How much would this annuity accumulate at the end of fourth year.

FV = A(CVAFn i)

= 5, 000 (CVAF4 .06)

= 5, 000 (4.375)

= 21, 875

** Sinking Fund:** This is going to be in reverse to the compounded value annuity factor. Here we proceed that to create certain sum of money, how much we have to set aside every year for a specified period.

FV = A(CVAFn.i)

A = FV( 1/CVAFn.i)

A = FV (SFFn i) SFF = Sinking Fund Factor

A = F [ (i/(1+i)^{n}) - 1]

For instant to clear off a loan of birr 21, 875 after four years, how much we have to set aside?

FV = A(CVAFn .1)

A = FV( 1/CVAFn.i)

= 21, 875 FV91/4.375)

= 21, 875 x .2286

= 5, 000

** Present Value:** here, we calculate the present value of future earnings at a particular rate of interest. This may be further classified into two

- Present value of a lump sum. The present sum of money to be invested today in order to get birr 1 at the end of year 1, 2, 3 so on and so forth at the rate of 10% interest.

We know F_{1 }= P(1+i) at the end of year 1.

1 = P (1+10)

P =1/(1+i)

= 0.909

F_{2} = P(1+i)^{2}

1 = P(1+.10)^{2}

P =1/(1+10)^{2}

= 0.826

F_{3} = P(1+i)^{3}

1 = P(1+.10)^{3}

= 0.751

Fn = P(1+I)n

P = Fn/ (1+i)^{n}

= Fn [(1+i)^{n}]

You wanted to know the present value of birr 50, 000 to be received after 15 years at the rate of interest 9%

PV = FV (PVFn i)

= 50, 000 (PVF15 .09)

= 50, 000 (.275) present value table

= 13, 750

** Present value of Annuity:** An investor some times may receive constant amount for a certain number of years. We may have to calculate the present value of such annuity to be received each year for a specific period.

**Loan Amortization Table**

**Year Ope. Bala. Annu. Install. Interest Principal Closi. Balance**

** Biirrs Birrs Birrs Birrs Birrs**

1 1, 000, 000 298, 312 150, 000 148, 312 851, 688

2 851, 688 298, 312 127, 753 170, 559 681, 129

3 68, 129 298, 312 102, 169 196, 143 484, 986

4 484, 986 298, 312 72, 748 225, 564 259, 422

5 259, 422 298, 312 38, 913 259, 399 23

- Birr 23 left because annuity is taken as 298, 312 instead of 298, 311.

** Multi period Compounding:** Till now, we have seen the cash flows will occur once in a year. But, the cash flows may occur monthly, bi-monthly, quarterly, half yearly and yearly. In such instances we have to apply the following formulae.

Fn = P[1+ (1/m)]^{nxm}

You have deposited a birr of 1, 000 in Commercial Bank of Ethiopia at 12 percent interest per annum. It compound annually, semi-annually, quarterly and monthly for two years. How much does it grow?

- Annual compounding

n = 2 i = .12%

FV = P(CVF_{n} .i)

= 1, 000 (CVF_{2} .12)

= 1, 000 (1.254)

= 1, 254

- Half-yearly n = 2 x 2 = 4 i = 12/2= 6%

FV = 1, 000 (CVF_{4} .06)

= 1, 000(1.262)

= 1, 262

- Quarters n = 4 x 2 = 8 i = 12/4 = 3%

FV = 1, 000 (CVF_{12} .03)

= 1, 000 (1.267)

= 1, 267

- Monthly n = 12 x 2 = 24, i = 12/12 = 1%

FV = 1, 000 (CVF_{24}. .01)

= 1, 000 (1.270)

= 1, 270

**5.5 Summary**

In our day-to-day life we prefer possession of a given amount of cash now, rather than the same amount at future time. This is time value of money or time preference for money, which arises because of (a) uncertainty of cash flows (b) subjective preference for consumption and

< >availability of investments. The last justification is the most sensible justification for the time value of money.

Interest rate or time preference rate gives money its value and facilitates the comparison of cash flows accounting at different time periods. Two alternative procedures can be used to find the value of cash flows: compounding and discounting. In compounding, future values of cash flows at a given interest rate at the end of a given period of time are found. The future value (F) of a lump sum today (p) for ‘n’ period at ‘i’ rate of interest is given by the following formula:

Fn = P(1+i)n

= P(CVFn .i)

The Compound Value Factor (CVFn i) can be found out form the tables.

The future value for annuity for ‘n’ periods at ‘i’ interest may be calculated by the following formula.

Fn = P[ ((1+i)^{n} – 1) / i]

= P (CVAFn .i)

Compounded Value Annuity Factors (CVAFn .i) is also found from the tables