Time value of money is a concept in finance that states that money gains value with time. In other words, money that anyone has today is more valuable than money he/she would receive in future. When money is invested, it earns interests at a given rate. The concept is applicable in many fields dealing with money. For example, in trade, when payments delay, there is an introduction of payment for time value. The concept is directly linked to a phrase, ‘a bird in hand is worth two in the bush”. The idea of time value of money is based on the earning potential of money. In summary, the time value principle asserts that money grows by earning interest. Therefore, any amount of it is not equivalent to the specific amount in future.

Time value of money lies on three major concepts: future value, present value, and compounding value. Future value of money is the value that an asset matches at a given date. It is a financial aspect that measures the future value of a given sum of money that is available at the present. It is based on gaining interest rate and specific time duration. The gaining interest rate has a general meaning of the returning rate. In other words, future value is the value of money which is available at the present, when multiplied by accumulation function. This concept assumes any value that may affect the present value within specific time.

On the general view of money time value, any amount that a person has today will have an additional value after a year or so (Hicken, 2014). For example, when $100 is invested in a bank or any other investment institution and subjected to an interest rate as per the institution, it will grow. Similarly, a pair of trousers was saleable at $50 in 2002. Today, the same trousers is marketable at $90. The price changes due to inflation in the economy. Money at the present always has two options: either money is put to immediate use, or it is invested. Financial compensation of investing money is adding value to the invested amount unless it is supposed to be used at the present.

Evaluation of the real value of present amount after some time depends on the rate at which it compounds. Tentatively, the most used interest rate is risk-free. Therefore, it is advantageous to the investor. It provides the least rate guaranteed as stated by a bank’s saving account. For example, a measure of change in purchasing power is calculated by use of actual interest rate. Capitalization is interested in evaluating the actual value of the present amount after a given time. The aspect is also applicable in calculating the present value, given interest rate, time, and the future value. It is termed as discounting. The formula is reversed. It is based on argument of choosing between receiving given amount of money today and receiving the same amount in future. The time chosen is really important. For example, if a hundred pounds is offered today, its value remains constant at the present. However, if received after a year, it will be $105 when interest rate is assumed to be 5% per annum. Therefore, the concept also helps in making wise financial decisions.

The value of a given amount of money after duration varies from bank to bank. The reason is that banks offer different interest rates. While one bank uses simple interest rates in evaluating the future of a given amount of money, other uses compound interest rates. Therefore, there are different financial formulas of determining the future value of money. For example, the formula used while determining the future value (FV) using simple interest rate is:

**FV= PV * (1+ rt),**

Where PV is the principle or the present value, t is time (in years), and r represents interest rate per annum.

Most banks do not apply the use of simple interest rate because it is procedural. If an amount is invested in bank which offers this type of rate, the interest obtained after the first year will not be subjected to value addition as in the case of compound value.

The formula used in getting the future value by use of compound interest is:

**FV = PV * (1+r) t**

Just like in the case of simple interest, PV is the principle or the present value, t stands for time (in years), and r represents interest rate per annum.

## Example

An individual deposited $1000 in a bank that offers a compound interest at a rate of 5% per month. The money stayed in bank for one year. Here, PV = $ 1000, FV =? , r = 5% (12), and time t = 1 year.

In determining FV of his money, we have to put the values into the formula as:

**FV = $1000 * (1+0.05)12 = $1600**

Therefore, the future value of his money will be $1600 after one year of a 5% interest compounding rate. Supposedly, if the same person is promised $1600 and wants to know its present value, he/she will reverse the formula.

The present value of future money is the value of expected amount coming after a given period of time. Unless there are cases of negative interest, present value is always less than the expected amount. Every amount of money has time value characteristic which is based on the expression ‘a dollar today has more value compared to a dollar tomorrow’. Therefore, the present value is lesser. The evaluation of present value of the future amount of money uses the same formula as in calculating future value of the amount. However, the present value calculations use the reverse mode.

An example in a real life situation is when I sold my house at $15,000 in October, 2011. My brother has just sold similar house today at $18,000. Fortunately, I have been investing the money at an interest rate of 4% per annum. Currently, the money has compounded to $17,000. I consider my brother’s choice valuable because my brother’s value would have been $15386.48, in 2011. It is demonstrated by calculations below.

If $15,000 is invested for four years at an interest rate of 4%, it will grow to $17000. That is:

**FV = PV * (1+0.04)4 = $15000 * (1+0.04)4 = $17,000.**

On the other hand, the present value of $18000 after 4 years is:

**PV = FV * (1+0.04)4 = $ 18,000 * (1+0.04)4 = $15,386.48**

The financial aspect is applicable in the situation where one wants to know the exact amount of loan he or she might have borrowed some time ago. Probably, compounding is the third concept in terms of time value of money. Compounding is the process of accumulating time value of money in future situation. It is a situation where the interest earned after, for instance, one year in a period of two years, is also subjected to interest rate and gain more value. When banks give out a loan, the interest obtained at the end of the year is added to the principle to form a larger base for accumulating future earnings. Such base has potentiality to grow faster. The longer the loan in duration is, the more money is gained from accumulation. Furthermore, the accumulation rete varies directly with the amount invested (Luhby, 2014).

For example, receiving a loan of $10,000 from a bank at an interest rate of 8% means that a person will have to pay $21,589. $10,000 earns approximately $800 interest after every year at an interest rate of 8%. The interest adds up to $8000 after ten years. This is true if the interests are withdrawn at the end of every year and added separately. Hence, there is an indication of $3, 589. So the compounding interest is $3,500. The concept uses earnings to generate more earnings. It illustrates how future value can be formed through compounding.

The illustrations of the above concepts lead to the tools used in finance also referred to as financial tool box. A financial tool box involves future value of single amount and present value of perpetuity. It also illustrates how to handle present value of annuity and future value of it. The future value of a present single sum of money is the amount that will be obtained when single present amount is invested at a specific date and at a specific interest rate. It is the sum of the invested money in addition to the interest gained. It is calculated by the formula below:

**FV = PV * (1+i) n**

Where i represents the interest rate while n represents time in years.

For example, the amount of $10,000 was invested on January 1st, 2013 at an annual interest rate of 8%. Considering the value of the investment on Dec 31st, 2015, if compounding is done on quarterly basis, will be expressed by the following formula:

FV = PV = ($10,000) * (1+0.08/4)3x4,

Given that PV is $10,000, compounding periods are 3 × 4 = 12, and interest rate is 8/4 = 2%; hence:

FV = $10,000 * (1 + 0.02)12 = $10,000 * 1.0212 ≈

≈ $10,000 * 1.268242 ≈ $12,682.42

The same formula is applicable in getting present value of a single amount. However, it is reversed. The value obtained when a single sum of money is discounted at a given interest rate from a specific date represents the present value of it. In getting the present value of a single amount, the formula above is rearranged. Hence,

PV = FV / (1+i) n,

For instance, I won a price of $10,000 after participating in Safari.com promotion games. However, I was supposed to be awarded the prize in two-year time. Due to the financial concept, I decided to transfer the price to my dad’s name, who gave me $8,000 at the expense of the prize. I decided to invest money at an annual interest rate of 12%. After calculations, I realized that I had decided wisely. Thus, assuming that future value is $10,000, compounding period is 24(2x12), and interest rate is 12%, present value is obtained by the formula:

PV = $10,000 / (1 + 1%) 24 = $10,000 / 1.0124 ≈

≈ $10,000 / 1.269735≈ $7,875.66

The present value of $7.875.66 is lower than the father’s buying price which is $ 8,000. Therefore, my choice was wise.

The third financial tool describes the future value of an annuity. The latter is a fixed payment required to be paid at a given frequency and at a given time period (Bernard, 2014). There are two types of annuities. The first type is an ordinary annuity, where payments are made after the end of every period. The second one is when an annuity is paid at the beginning of every period. These two types of annuities have different formulas in calculation.

Future value formula of an ordinary annuity helps in determining the future value of money that one has chosen to invest at his/her rate. It is also important in determining the total cost of a loan. It involves calculation of each cash flow values.

For example, let us consider someone paying a loan by giving $1,000 every year for 5 years. After paying, his/her payments are invested at an interest rate of 5% per annum. The earnings of his/her money will be as follows:

1st year =$1000 x (1.05)0 = $1000.00

2nd year =$1000 x (1.05)1 = $1050.00

3rd year =$1000 x (1.05) 2= $1102.50

4th year =$1000 x (1.05)3 = $1157.63

5th year =$1000 x (1.05)4 = $1215.51

Hence, FV = $5525.64

The formula of calculating a bond pricing is similar to that of evaluating the present value of an annuity. Obtaining the total discounted value, one takes the present value of each future payment and sums up the cash flow. An example of the bond formula usage is demonstrated as follows:

PV = C * ( –n)

Here, C stands for cash flow per period, and i is an interest rate of payment.

Hence,

PV = $1000 * ( –5) = $1000 * (4.33) = $4,329.48

This formula can also be applicable while calculating future value of annuity. However, payment frequencies and time of payments may vary.

A bond is an example of an ordinary annuity. Its calculations can also be used to demonstrate present and future value of a single amount. From an ordinary annuity perspective, bonds are paid at the end of every six months. Bond maturity amount is a single amount that occurs on the maturity of the bond, probably at the end of six months. Determining the present value of a bond requires implication of market interest rate. The latter discounts maturity amount and interest payments.

In a case where a 9% $100,000 bond is prepared in January, 2014, the market interest rate has risen to 10% by the time the bond is offered to investors on December 1st, 2015. The date of the bond is December 1, 2015, and it matures on January 31st, 2019. The interest or the bond at the time of payment is $4,500 (9% * $100,000 * 6/12 of 1 year) on every January 31st and June 30th.

Calculation of an approximate price to be paid by the investor for the corporation's bond on December 1, 2015 starts from calculating the present value of the bond. The present value here is the sum obtained at the time of maturity. The present value of the bond's interest payments compounds after every six months of a year. Hence, the calculations will be based on six-month basis as demonstrated below:

Interests $ 4,500 $ 4,000 $ 4,000 $ 4,000 $ 4,000 $ 4,000

6 months 6 months 6months 6 months 6 months 6 months

01/04/15 06/30/15 12/31/15 06/30/16 06/31/16 06/30/18 12/31/15

0 1 2 3 4 5 6

For cases of five-year bond life, the number of years (5) is multiplied by two because of the semi-annual periods. The letter n is used as a repetitive of these periods. Therefore, each semi-annual payment occurring at the end of the ten semi-annuals is $4,500 ($100,000 * 9% * * 6/22).

Therefore,

PVO =PMT * (PVOA) = $4,500x7.722 = $38,749.

Where n = 10 and i =5% for every semi-annual period.

However, this type is limited only to calculating interest payments of streams only.

Looking at present value of the maturity amount of a bond, the second step of calculating the maturity amount is carried out. It can also be represented in a timeline as:

Interests $ 4,500 $ 4,000 $ 4,000 $ 4,000 $ 4,000 $ 4,000

6 months 6 months 6months 6 months 6 months 6 months

01/04/15 06/30/15 12/31/15 06/30/16 06/31/16 06/30/18 12/31/15

0 1 2 3 4 5 6

Based on the formula, present value of bond’s maturity amount can be calculated as:

PV = $100,000 * 0.164= $16,400

Remember, PV = FV * PV for factors where n=10 and i=5%

Another field that requires a lot of financial concepts is non-constant growth. Many businesses enjoy consistence growth. The progress may be the result of introduction of a new product, technology advancements, or increase in market demand. However, these situations do not last forever due to fluctuations and competitions in the market. Hence, such businesses cannot be valued by the use of a simple constant growth stock valuation. Besides, it can be obtained through the equation below:

Po = + ( ) (1+r)-T,

Where Po is a stock price at the time, DT is an expected dividend, t or T is the number of years of constant growth, and r is the required return on the stock (gc < r)

The last tool within the financial tool box is present value of perpetuity. Perpetuity means making constant payments for a given period of time. It is an annuity without ending. The total value of perpetuity is infinite. However, its present value is finite. Based on the principle of money time value, present value of perpetuity is the total of each periodic payment of it. The discounted value reduces with time that means it is finite.

Present value of perpetuity is obtained by the formula below:

PV = A

r

Where A is a fixed periodic payment, and r is an interest rate per discounting period.

For example, if A = $1000 and i = 0.8%, then

PV = $1,000 / 0.08% = ($1,000 * 1,000/8) = $125,000