## What is the Time Value of Money?

Time Value of Money (TVM) is considered to be a core principle in financial management. TVM is a concept that states that a specific amount of cash is worth more in the present than it will be in the future. It is because of the money’s potential earning capacity. In other words, if the money is invested today, it can grow in the future to be a higher value. TVM can be used to identify the future amount or to identify the present value of the future amount. Therefore, TVM plays a crucial role in not just investment decisions but also financial decisions.

For example, assume that an individual has the opportunity to receive $1,000 today or a year later. By choosing to receive the $1,000 today and depositing it in a savings account for a year at a 5% interest rate, the value becomes $1,050 a year later. Therefore, by choosing to accept the money today, the individual earns an excess of $50. By choosing to receive $1,000 a year later, the individual loses the opportunity to earn $50. The excess amount lost by choosing to receive the same sum in the future is called the opportunity cost.

## Characteristics

Some of the characteristics are

**Present Value of Annuity (PVA)****Future value (FV)****Time Value of Money (TVM)**

The present value of annuity refers to either the current amount held or the discounted value of the future amount. On the other hand, future value refers to the amount earned by investing the present value for a specific period. In the above example, $1,000 is the PVA while $1,050 is the FV. Both characteristics are extremely important in understanding the time value of money.

### Interest or discount rate and TVM

Interest rate is used to calculate the excess amount earned by investing the amount available today. The interest rate on investment could be either simple interest or compound interest. On the other hand, the discount rate refers to the amount used to discount the future value to identify the present value of funds. Both interest rate and discount rate are ideally the same however the term ‘interest rate’ is used to calculate the FV while the term ‘discount rate’ is used to calculate the PV.

### Compounding periods and TVM

To identify the accurate future value of money, it is necessary to know not just the present value and interest rate but also how the interest is compounded. Interest could be compounded monthly, quarterly, semi-annually, or annually. The future value of the investment will be high if the number of compounding periods is more. The interest rate must be adjusted in the formula as per the number of compounding periods. The compounding period also refers to the number of interest installments paid or received in the investment. It is generally referred to as PMT as it is used in calculating the periodical interest payments.

### Time period and TVM

The number of years the investment is made is called the time period. In simple terms, the amount of time it takes for the present value to become a future value is called the time period. It should not be confused with compounding periods. While the time period indicates the lifetime of the investment whereas the compounding periods refer to the number of times interest is paid each year.

## The formula to incorporate the time value of money

The time value of money can be incorporated either by discounting the future value or by using the present value to find the future value.

The fundamental formula to identify the future value is as follows:

$\mathrm{FV}=\mathrm{PV}\times \left[1+{\left(\frac{\mathrm{i}}{\mathrm{n}}\right)}^{(\mathrm{n}\times \mathrm{t})}\right]$

The fundamental formula to calculate the present value is as follows:

$\mathrm{PV}=\frac{\mathrm{FV}}{{\left[1+({\displaystyle \raisebox{1ex}{$\mathrm{i}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n})$}\right.}\right]}^{\mathrm{n}\times \mathrm{t}}}$

where,

PV = Present value of money

FV = Future value of money

i = Interest or Discount rate

n = Number of compounding periods

t = Number of Years

### Sample Problems

The following examples indicate how crucial the time value of money is in the case of both investment decisions and financial decisions.

### Example 1

Assume that Mr. X is investing $200,000 for 3 years at the interest rate of 4% compounded semi-annually. What would be the amount received by Mr. X three years later?

The value provided in the question are as follows:

PV = $200,000

i = 4%

t = 3

n = 2

The amount to be received at maturity of the investment is calculated as follows,

$\mathrm{FV}=\mathrm{PV}{\left[1+\left(\frac{\mathrm{i}}{\mathrm{n}}\right)\right]}^{\left(\mathrm{n}\times \mathrm{t}\right)}\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$200,000{\left[1+\left(\frac{0.04}{2}\right)\right]}^{\left(2\times 3\right)}\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$200,000{\left[1.02\right]}^{\left(6\right)}\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$200,000\times 1.1261624\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$225,232.48$

Therefore, by investing $200,000 today, Mr. X can earn $225,232.48 at the end of three years.

### Example 2

If Mr. A wants to buy a car for $100,000 two years from now and if the savings account is given an interest rate of 6% annually, how much should Mr. A have today?

The value provided in the question are as follows:

FV = $100,000

i = 6%

t = 2

n = 1

The amount currently required by Mr. A can be calculated as follows,

$\mathrm{PV}=\frac{\mathrm{FV}}{{\left[1+\left({\displaystyle \frac{\mathrm{i}}{\mathrm{n}}}\right)\right]}^{\left(\mathrm{n}\times \mathrm{t}\right)}}\phantom{\rule{0ex}{0ex}}\mathrm{PV}=\frac{\$100,000}{{\left[1+\left({\displaystyle \frac{0.06}{1}}\right)\right]}^{\left(1\times 2\right)}}\phantom{\rule{0ex}{0ex}}\mathrm{PV}=\frac{\$100,000}{{\left[1.06\right]}^{\left(2\right)}}\phantom{\rule{0ex}{0ex}}\mathrm{PV}=\frac{\$100,000}{1.1236}\phantom{\rule{0ex}{0ex}}\mathrm{PV}=\$88,999.64$

Therefore, Mr. A currently requires $88,999.64 today to be able to buy a car next year for $100,000.

## Practical applications of the time value of money

The time value of money can be practically applied in the following cases.

### Project selection

Time value of money is most importantly used in discounting cash flow analysis. Before selecting any project, the cash flows that will be generated during the lifetime of the project are listed. These cash flows include both cash inflows and cash outflows. The future cash flows are discounted using the TVM formula and the present value of these cash flows are identified. The initial investment and the discounted cash flows are added to identify the Net Present Value (NPV) of the project. This may be done for multiple projects and the project with the highest net present value is selected by companies. Therefore, TVM is important in management decision-making as well.

While NPV is one of the methods used to select a project, companies may also use the internal rate of return method (IRR). In the IRR method, the cash flows are discounted similarly to NPV. The rate of return is identified by equating the sum of discounted cash flows and initial investment to zero.

### Sinking fund

Finance managers of companies may decide to set aside an amount in case of redemption of debentures in the future. The future value that is required for redemption is set. However, the funds that must be periodically set aside for the sinking fund must be decided based on the compounding interest rate. This amount can be calculated using the formula of the time value of money.

### Capital recovery

The loans obtained by companies must be repaid in specific installments. Upon identifying the number of installments, the size of installments must be identified. That is, the amount that will be paid back in each installment must be calculated. This can once again be done using the formula of the time value of money.

### Deferred payment

A company after obtaining a loan need not start paying interest immediately. The interest can accumulate, and the company can start repayment even after two years. This delayed payment of interest is called the deferred payment. The loan obtained would change in value over two years because of the time value of money. Therefore, to calculate the annual installment amount, the formula of TVM can be used.

### Implicit rate of return

Finance companies offer certain schemes where a large amount of money is invested at the beginning of a period and the return on investment is given back to the investor periodically in the form of an annuity. The time value of money can be used in calculating the value of the annuity and the interest rate.

## Context and Applications

This topic is significant in undergraduate, graduate courses such as

- Masters in Business Administration (Finance)
- Bachelors in Finance
- Masters in Finance

## Practice Problems

**Question 1:** The number of times interest is paid on an investment in a year is called the ________________.

- Time period
- Compounding Period
- Net Present Value
- None of the above

**Answer:** (b)

**Explanation:** Compounding period. Interest earned on investment could be put back into investment monthly, quarterly, semi-annually, or annually. This is called the compounding period.

**Question 2:** What is the rate at which the future value is converted to the present value called?

- Interest rate
- Discounting rate
- Internal rate of return (IRR)
- Holding Period Return (HPR)

**Answer:** (b)

**Explanation:** Discounting rate. The future value is discounted using the discount rate to attain the present value.

**Question 3:** A is investing $80,000 for 10 years. If the interest rate is 4% and the interest is compounded monthly, what is the FV of the investment?

- $119,267
- $93,577
- $88,648
- $114,589

**Answer:** (a)

**Explanation:** $119,267. The calculation of FV as follows,

$\mathrm{FV}=\mathrm{PV}\times {\left[1+(\mathrm{i}/\mathrm{n})\right]}^{(\mathrm{n}\times \mathrm{t})}\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$80,000\times {\left[1+(0.04/12)\right]}^{(12\times 10)}\phantom{\rule{0ex}{0ex}}\mathrm{FV}=\$119,267$

**Question 4:** The cash inflows of Project A are $1,500 in year 1 and $2,300 in year 2. What is the discounted present value of these cash flows if the discount rate is 4.5%?

- $1,435 and $2,106
- $1,000 and $2,000
- $1,300 and $1,935
- $1,200 and $2,546

**Answer:** (a)

**Explanation:** $1,435 and $2106. The calculation of discounted cash flows are as follows,

$\text{PV}=\frac{\mathrm{FV}}{{\left[1+(\mathrm{i}/\mathrm{n})\right]}^{(\mathrm{n}\times \mathrm{t})}}\phantom{\rule{0ex}{0ex}}\text{PVofcashflow1}=\frac{\$1,500}{{\left[1+(0.045/1)\right]}^{(1\times 1)}}=\$1,435\phantom{\rule{0ex}{0ex}}\text{PVofcashflow2}=\frac{\$2,300}{{\left[1+(0.045/1)\right]}^{(1\times 2)}}=\$2,106\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

**Question 5:** Why is the TVM formula used in the case of deferred payment?

- It is used to calculate the NPV.
- It is used to calculate the value of an annuity.
- It is used to calculate the annual installment.
- It is used to calculate the holding period.

**Answer:** (c)

**Explanation:** It is used to calculate the annual installment. Since the installment paid on loan starts at a later date, the TVM formula is used to calculate the future annual installment.

## Common Mistakes

It is incorrect to assume that compounding does not have an impact on the future value if the present value, interest rate, and the number of years are the same. For example, assume that Mr. A and B are investing $1,000 for 3 years at the interest rate of 8%. Mr. A’s investment is compounded annually while Mr. B’s investment is compounded monthly. The future value of both investments can be calculated as follows,

$\begin{array}{rcl}\text{FVofMr.A'sinvestment}& =& \$1000{\left[1+(0.08/1)\right]}^{(1\times 3)}\\ & =& \$1,259.71\\ & & \\ \text{FVofMr.B'sinvestment}& =& \$1000{\left[1+(0.08/12)\right]}^{(12\times 3)}\\ & =& \$1,270.24\\ & & \\ & & \end{array}$

Mr. A makes fewer returns compared to Mr. B. Therefore, from the above example, it can be confirmed that compounding frequency plays an important role in calculating FV.

## Related concepts

While studying about the time value of money, it is important to read the following to get a better knowledge:

- Principles of finance
- Capital Budgeting
- Investments

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