## Introduction to time value of money

Time value of money is one of the most fundamental and important concepts in the study of finance. It is vital to have a clear understanding of the time value of money before branching out to other advanced financial concepts and theories. In this article, you will be exposed to numerous time value of money examples for a better understanding of this basic financial concept.

### The concept

The idea behind time value of money is pretty simple – what is means is that the value of a dollar today is different than the value of a dollar on an earlier or later date. There are several reasons for this phenomenon.

### Time value of money examples – How inflation or deflation affects value of money

Inflation or deflation simply refers to the increase or decrease in price of goods as time goes by. To illustrate, take for example that the price of cake in your neighborhood bakery today is \$1 each. If you have \$1, you can simply exchange the money for a piece of cake from the bakery.

Let’s say that 1 year later, due to an increase in rental, labor wages and material cost, the bakery increases the price of cake by \$0.50 each to sustain its business operations. The price of each cake is now \$1.50. If you still have only \$1 in your pocket you will find yourself unable to exchange the note for a piece of cake anymore. Inflation has caused the dollar to lose its value over the 1 year period.

Similarly, if the price of cake decreases by \$0.50 each due to over-production or a decrease in consumer numbers, each piece of cake now costs only \$0.50. This means that a \$1 note a year later can be exchanged for 2 pieces of cakes. Deflation has caused the dollar to gain value over the 1 year period.

In reality, you will generally find that the value of money deteriorates over time as most market tends to operate in an inflationary environment. This means that most of the time, the \$1 you have today, is much valuable than the \$1 you have 1 year later.

### Time value of money examples – How interest rate affects value of money

Interest is a sum of money a borrower pays to the lender for the right to use the lender’s asset at the present moment. To illustrate, take for example that that you have \$1 today and you want to spend that \$1 cake sold in the neighborhood bakery.

Your classmate, John, has no money now but wants to buy a bar of chocolate which costs \$1. John then approaches you to borrow the \$1 you have and promises to return you the \$1 tomorrow. Since you have only \$1, you refuse to lend John the money because you already have plans to spend the money and you also worry that John may not keep his promise to return the money tomorrow.

To entice you to lend him the \$1, John then proposes to return you \$1.10 tomorrow. Seeing that you can earn \$0.10 by simply delaying the buying of cake for a day, you agree to lend John the money if he promises to return you \$1.10 tomorrow.

The \$0.10 paid by John is commonly known as the interest. Interest is another reason that the value of money tends to deteriorate over time because money you have today can potentially earn you more interest than the money you have at a later date.

## Calculating present and future value

We have seen a few examples that explain why the value of money can change over time. Now we will take a look at more examples to learn how to calculate several components of the time value of money.

### Time value of money examples – Calculating rate of return

Rate of return is simply the return expressed as a ratio of the principle sum. For example, if you had lent John the \$1 and he returned you the \$1.10 the following day, the rate or return you would have earned is 10% per day. Interest rate can be calculated by dividing the interest amount (\$0.10) over the principal amount (\$1).

Do note that you have to specify the time period when you calculate the rate of return. For example, if bank A pays 4% interest annually for a fixed deposit account, by depositing \$1,000 in that account, you will receive \$1,040 a 1 year later the extra \$40 being interest. If another bank, bank B pays 4% interest bi-annually (every 6 months) for their fixed deposit account, by depositing \$1,000 in that account, you will receive \$1,040 half a year later. And if you let the \$1,040 remain in bank B for another 6 months, you will receive \$1,081.60 at the end of 1 year. Hence, the effective annual rate for bank B (8.16%) is in fact much higher than bank A(4%)!

### Time value of money examples – Calculating the future value (FV)

Future value is simply the value of the principle sum of money at a later date. For example, if you have deposited your \$1,000 in bank A’s fixed deposit which yield 4% annual interest, the FV of your \$1,000 a year later would be \$1,040. This can be simply calculated by adding up your principle sum and the interest you receive. Another method is to multiply the principle sum by (1 + rate of return), in this case it will be \$1,000 x (1 + 0.04) which results in \$1,040.

Do note that you have to specify the time period when you calculate the FV. If you deposited \$1,000 in bank A with a 4% fixed deposit, the FV of your \$1,000 a year later will be \$1,040. FV of your \$1,000 2 years and 3 years later will be \$1,081.6 and \$1,124.86 respectively. The general formula is to multiply the principle sum by (1 + rate of return) to the power of number of period. For example, the FV of the above example 10 years later will be \$1,000 x (1+0.04)10 = \$1,480.24.

### Time value of money examples – Calculating the present value (PV)

Present value is the opposite of future value; it is simply the value of a future principle sum of money in present date. It can be easily calculated by reversing the formula for FV – divide the FV by (1 + rate of return) to the power of number of period. Take for example, John approached you and promised to give you \$2,000 1 year later if you can lend him some money now. As you can simply invest your money in bank A which give u a 4% return and you have worry that John may not keep his promise to return the money, you expect a 5% annual interest rate from him. The amount of money you are willing to lend him in return for the \$2,000 a year later will be \$2,000 / (1.05) = \$1,904.76. If he has promised to return you \$2,000 only 5 years later, the amount of money you are willing to lend him in return for the \$5,000 5 years later will be \$2,000 / (1.05)5 = \$1,567.05.

Do note that in the above example, the \$2,000 represents the FV and 5% represents the annual rate of return. The general relation between rate of return, FV and PV is simply

PV x (1 + rate of return)t = FVt,                      where t = no of time period

## Conclusion

We have learnt that time value of money tends to change over time. Most of the time, the value of a \$1 today tends to be greater than the value of a \$1 in the future.

Value of money can be found by using the below formula with three variables, namely, the rate of return, FV, and PV.

PV x (1 + rate of return)t = FVt,                      where t = no of time period

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