## Time Value of Money (TVM)

The determination of an asset’s monetary value is one of the most important functions of financial analysis. This value is measured in part by the income produced over the asset’s lifetime. Since funds can be paid at various times, it may be difficult to compare the prices of different properties. Let us begin with a simple scenario. Would you rather have a ₹1,000-paying asset today or ₹1,000-paying asset a year from now?

Table of Content

- 1 Time Value of Money (TVM)
- 2 Important Concepts in TVM
- 2.1 Types of Cash Flows and Timelines
- 2.2 Simple Interest and Compound Interest
- 2.3 Compounding and Discount
- 2.4 Nominal Interest Rate, Real Interest Rate and Effective Interest Rate (EIR)
- 2.5 Compounding – Finding Future Value
- 2.6 Discounting – Finding Present Value
- 2.7 Discounting a Single Cash Flow
- 2.8 Discounting a Series of Cash Flows

- 3 Calculating Values of Lump-Sum Amounts
- 4 Calculating Values of Annuities
- 5 Practical Applications of Time Value of Money

It turns out that money paid now is preferable to money paid later (we will see why in a moment). The time value of money is the term for this preference for time. The time value of money is at the heart of many financial calculations, particularly those that involve value. What if you could choose between getting a payment of ₹1,000 today and ₹1,100 in a year? The second choice will give you more money but at a later date. In this section, we will see how companies and investors make that comparison.

### Rationale for TVM

The fact that the value of one rupee today is not equal to the value of one rupee at the end of one year or at the end of the second year underpins the entire spectrum of financial decisions (whether funding or investment). In other words, we cannot presume that the rupee’s value would remain constant. This is referred to as the ‘Time Value of Money.’ In financial decision-making, understanding the time value of money is important.

A financial decision made today would have long-term consequences. Any financial decision includes weighing capital outflows (outlays or investment costs) against cash inflows (benefits or earnings after tax but before depreciation). For a meaningful comparison, the two sets of flows must be strictly comparable. The use of time elements in calculations is a basic necessity of comparability.

To put it another way to make a fair and accurate distinction between cash flows that occur over time, the amounts of money must be converted to common points of time. If the timing of cash flows is not taken into account, the firm will make decisions that are not objective.

This may be due to

- Risk or the uncertainty of future receipts
- Inflation, which reduces money’s buying power
- Opportunities to reinvest funds earned early

It could be argued that the risk factor associated with future money receipts could be removed or minimised to a greater degree by appropriate commitments, default insurance, and so on, so that possibility of default (money not to be received in future) becomes quite remote. Naturally, the time value of money then becomes irrelevant. Similarly, if the economy is believed to be free of inflation, the value of money today and tomorrow can be assumed to be the same, and the time value of money becomes meaningless.

Despite these two drastic assumptions, a rupee received today will be preferred over a rupee received tomorrow (i.e. future), since the rupee received today could be invested and its value tomorrow (in future) would be higher (this is due to the fact that the rupee invested will fetch some interest). Future cash flows are only considered less valuable than current cash flows when reinvestment opportunities of funds earned early are considered.

If the funds were earned now, they would gain a rate of return that would be impossible to achieve if they were received later. Understanding the time value of money entails a basic understanding of the mathematical principles of compounding and discounting. These ideas are present in all types of financial decisions.

### Why Future Amount of Money is Worth Less than Same Amount of Present Money

The principle of present value states that a sum of money today is worth more than a sum of money in the future. To put it another way, money earned in the future is not worth the same as money received today. A ₹1,000 gift today is worth more than ₹1,000 in five years. What is the reason for this? An investor can put ₹1,000 in today and expect to make a profit for the next five years.

Any interest rate that an investment can gain is factored into the present value. For instance, if an investor receives ₹1,000 today and earns a 5% annual return, the ₹1,000 today is unquestionably worth more than ₹1,000 five years from now. If an investor waited five years for ₹1,000, he or she will incur opportunity cost or miss out on the five-year rate of return.

## Important Concepts in TVM

To thoroughly understand the concept of TVM, it is essential that you are well-acquainted with the terminology that is used in it. There are two important concepts related to TVM as follows:

- Types of cash flows
- Timeline of cash flows

### Types of Cash Flows and Timelines

The major types of cash flows are as follows:

#### Annuity

An annuity refers to a series of equal amounts of cash flows (payments or receipts) that are paid or received at equal time intervals for a particular number of periods. For example, Tarun leased a factory from Varun for 10 years and promised to make a payment of ₹12,00,000 at the beginning of each period. This can be called as a 10-year, ₹12,00,000 annuity. A series of cash flows is called as an annuity only when the amounts of all the cash flows are identical and the time interval between two cash flows is equal in each case.

There are two types of annuities that are differentiated from each other because of the timing of the first cash flow as follows:

**Regular annuity:**In this annuity, the payment is made at the end of each period.**Annuity due:**In this annuity, the payment becomes due immediately at the start of each period.

#### Graduated Annuity (Growing Annuity)

Graduated annuity is a type of annuity that involves payment or receival of gradually increasing cash flows for a particular number of periods. It must be remembered that the rate of growth remains constant for the life of the annuity.

**Lumpsum:**When a single payment is made, it is called as a lump sum payment.**Perpetuity:**When annuity cash flows continue for an infinite period, it is called as a perpetuity or perpetual annuity.**Uneven cash flows:**Any series of cash flows that is not in line with the definition of an annuity is considered as a case of uneven cash flows. For example, receiving a payment of ₹10,000, ₹20,000, ₹15,000, ₹40,000, is uneven cash flow.

The timeline of cash flows may differ from one case to another. Most frequently used timelines in TVM are: annual, semi-annual, quarterly, and monthly cash flows. When cash is paid once in a year, it is called annual cash flow. When cash is paid twice a year, it is called semi-annual cash flow. When cash is paid once every three months, it is called quarterly cash flow. Lastly, when cash is paid every month, it is called monthly cash flow.

### Simple Interest and Compound Interest

The time value of money is a term used in accounting to describe the relationship between time and money. In other words, a dollar earned today is worth more than a dollar due at a later date. A dollar is worth more today because of the opportunity to invest the dollar and receive interest on the investment. What exactly is interest? Interest is a type of compensation for the use of capital. It is the amount of money collected or returned in excess of the amount lent or borrowed.

Suppose you lend a friend $100 at a 10% annual interest rate. Your friend will owe you $110 after a year ($100 borrowed plus $10 interest). Simple interest and compound interest are the two methods for calculating interest on money. Simple interest is the return on a single period’s principal. Simple interest is demonstrated in the preceding case. Interest is measured on a yearly basis based on the initial sum lent or borrowed.

Compound interest is the amount of money earned in excess of the amount lent or borrowed for two or more time periods. Compound interest calculates interest for the next year based on the accrued balance (original value plus interest to date) at year’s end. Adjust the interest rate in the previous example from simple to compound. You will receive $10 in interest in the first year for a total of $110.

You will collect $11 in interest after the second year for a total balance of $121. The additional dollar comes from the interest paid on the initial balance as well as interest gained in the first year ($110 X 10%).

### Compounding and Discount

The time value of money theory states that the value of a unit of money can change in the future. Simply put, the current value of one rupee will decrease in the future. The whole definition revolves around the current and future value of capital. Compounding and discounting are the two mechanisms for determining the value of money at various points in time. Compounding is a method for calculating the future value of present money.

Discounting, on the other hand, is a method of calculating the present value of future money. Compounding is useful for determining the potential values of a cash flow at the end of a certain time at a certain rate. Discounting, on the other hand, is used to calculate the present value of potential cash flows at a given interest rate. To comprehend the idea of compounding, you must first comprehend the term future value.

Over a certain period of time, the money you invest today will rise and gain interest, changing its value automatically in the future. As a result, the future value of an investment is what it is worth in the future. Compounding is the method of collecting returns on both the principal and earned interest by reinvesting the whole sum to gain even more interest. Compounding is a technique for calculating the potential value of a current investment.

The compound interest formula that can be used to calculate the future value (compounding) is as follows:

Future Value of Single Cash Flow is calculated as follows:

A=P (1 + R)^{t}

Where,

t = number of years

R = Rate of return on investment

Future Value of annuity is calculated as follows:

FV = PV (1 + R)

Where,

FV = Future value of money

PV = Present value of money

R = Interest rate

t = Number of years

Present value refers to the present value of a potential value. By applying a discount rate, the discounting method aids in determining the current value of potential cash flows. To calculate the present value of a future cash flow, use the following formula:

If n = number of compounding periods per year; then, FV and PV formulae will be modified as:

### Nominal Interest Rate, Real Interest Rate and Effective Interest Rate (EIR)

The nominal interest rate is the rate of interest until inflation is taken into account. It also refers to the rate stated in the loan contract before compounding is taken into account. The nominal interest rate varies from the real interest rate in terms of inflation adjustment, and the effective interest rate differs from the nominal interest rate in terms of compounding adjustment.

Different factors can influence nominal interest rates, including money demand and supply, federal government intervention, central bank monetary policy, and many others. The short-term nominal interest rate is used by central banks to enforce monetary policy. To stimulate economic activity during a recession, the nominal rate is reduced. The nominal rate is boosted during inflationary times. The rate of time preference for current goods over future goods is reflected in the real interest rate.

The real interest rate of an investment is calculated as the difference between the nominal interest rate and the inflation rate:

**Real Interest Rate = Nominal Interest Rate – Inflation (Expected or Actual)**

The nominal interest rate is the rate charged on a loan or investment, while the real interest rate reflects the shift in buying power gained from an investment or giving up by the borrower. The nominal interest rate is usually the one advertised by the lending or investing institution. Adjusting the nominal interest rate to account for inflation’s impact aids in identifying the change in the buying power of a given amount of capital over time.

There are two types of interest rates: nominal interest rate and effective interest rate. The compounding cycle is not taken into account in the nominal interest rate. When the compounding duration is taken into account, the effective interest rate is a more precise indicator of interest charges. The phrase “interest rate of 10%” means that interest is compounded annually at a rate of 10% each year.

The average annual interest rate in this situation is 10%, and the actual annual interest rate is also 10%. The effective interest rate would be greater than 10% if compounding occurred more than once a year. The higher the effective interest rate, the more often compounding occurs.

The relationship between nominal annual and effective annual interest rates is:

i = [1 + (r / m) ]^{m }– 1

where “i” is the effective annual interest rate, “r” is the nominal annual interest rate, and “m” is the number of compounding periods per year.

### Compounding – Finding Future Value

The compound value of an amount of investment can be computed by the following formula:

FV = PV (1 + R)^{t}

Where,

FV = Future value

PV = Amount invested

R = Interest rate

t = Number of years for which investment is made

If n = compounding frequency; then, FV is calculated as:

FV = PV

nt R

When interest rate is compounded semi-annually or quarterly, the rate of interest must be divided by 2 and 4 respectively and, the number of years must be multiplied by 4 and 2 respectively. For instance, if ₹1,000 is invested for three years at a rate of 10% with quarterly compounding, the FV of ₹1,000 can be calculated as:

### Discounting – Finding Present Value

The time value of money is a basic financial concept that holds that money in the present is worth more than the same sum of money to be received in the future. This is true because money that you have right now can be invested and earn a return, thus creating a larger amount of money in the future.

### Discounting a Single Cash Flow

There are two ways to look at an investment. It can be seen in two ways: as a potential asset or as a current asset. We have shown that if we know the present value of an amount (such as our `100 deposit), computing the sum’s future value in years using equation is a reasonably easy process (1). However, if we only know the future value of an amount and not its present value, we can use the following equation to calculate the present value of any future number.

) = +

FV PV

### Discounting a Series of Cash Flows

Usually, capital expenditure project involves cash inflows for years to come. For example, assume that a company is acquiring a machine which involves cash inflows of ₹5,000 each year for five years. What is the present value of the streams of receipts from the project? As shown in Table, the present value of this stream is ₹21,060 if we assume a discount rate of 6 percent compounded annually, the discount factors used in this exhibit were taken from table.

Two points are important in connection with this Appendix. First, notice that the farther we go forward in time, the smaller the present value of the ₹5,000 earnings. The present value of ₹5,000 received a year from now is ₹4,715.00 as compared to only ₹3, 735 for the ₹5,000 earnings to be received 5 years from now. This point simply underscores the fact that money has a time value. Table showing the value of ₹100 invested at 10 present simple and compound interest.

Simple interest | Compound interest | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Year | Initial Amount | + | Interest | = | Year and amt | Initial Amount | + | Interest | = | Year and amt |

1 | 100 | + | 10 | = | 110 | 100 | + | 10 | = | 110 |

5 | 140 | + | 10 | = | 150 | 146 | + | 15 | = | 161 |

10 | 190 | + | 10 | = | 200 | 236 | + | 24 | = | 260 |

20 | 290 | + | 10 | = | 300 | 612 | + | 61 | = | 676 |

Table showing factors at 6% with present value:

Year | Factor @ 6% (Appendix 2.1) | Interest received | Present value |
---|---|---|---|

1 | 0.943 | 5,000 | 4,715.00 |

2 | 0.890 | 5,000 | 4,450.00 |

3 | 0.840 | 5,000 | 4,200.00 |

4 | 0.792 | 5,000 | 3,690.00 |

5 | 0.747 | 5,000 | 3,735.00 |

Total | 21,060.00 |

The second point is that even though the computations involved in Table are accurate, they involve unnecessary work. The same present value of ` 21,060 could have been obtained more easily by referring to Appendix 2.2. Appendix 2.2 is an annuity table which contains the present value of rupee one to be received each year over a series of years, at various rates of interest. Appendix 2.2 has been derived by simply adding the factors from Appendix 2.1 together.

To illustrate, we use the following factors from Table in the computations in Table.

Year | Factor @ 6% |
---|---|

1 | 0.943 |

2 | 0.890 |

3 | 0.840 |

4 | 0.7921 |

5 | 0.7473 |

## Calculating Values of Lump-Sum Amounts

For a lump sum, the present value is the value of a given amount today. For example, if you deposited ₹5,000 into a savings account today at a given rate of interest, say 6%, with the goal of taking it out in exactly three years, the ₹5,000 today would be a present value-lump sum.

### Future Value of Single Amount

Money that is available now is more important than money that will be available in the future. Suppose you have ₹100 and you deposited this amount in a bank at 10 % rate of interest for 1 year. How much future sum or value would you receive after 1 year? You would receive ₹110:

Future value = Principal + Interest = 100 + (0.10 × 100) = 100 × (1.10) = ₹110

What would be the future value if you deposited `100 for 2 years?

You would now receive interest on interest earned after 1 year:

Future value = [100 + 0.10 × 100) + 0.10[100 + (0.10 × 100)]

= 100 × 1.10 × 1.10

= ₹121

You could similarly calculate future value for any number of years. We can express this procedure of calculating compound, or future, value in formal terms. Let ‘i’ represent the interest rate per period, “n” the number of periods before pay-off, and FV the **future value**, or **compound value**. If the present amount or value PV is invested at i rate of interest for one year, then the future value F1 (viz., principal plus interest) at the end of one year will be Future sum Principal Interest on principal

F_{1} = P + P × i = P (1+i)

The outstanding amount at the beginning of second year is: F_{1} = P (1 + i). The compound sum at the end of second year will be:

F_{2} = F1 + F1i = F1 (1+i)

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

Similarly, F_{3 }= F_{2} (1 + i) = P (1 + i)^{3} and so on. The general form of equation for calculating the future value of a lump sum after n periods may, therefore, be written as follows:

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

The term (1+ i) n is the compound value factor (CVF) of a lump sum of 1, and it always has a value greater than ₹1 for positive i, indicating that CVF increases as i and n increase. Suppose You have ₹1,000 are placed in the savings account of a bank at interest rate of 5 %. How much shall it grow at the end of 3 years?

It will grow as follows:

F1 = ₹1,000 + (1,000 × 5%)

F1 = ₹1,000 + 50 = ₹1,050

F2 = ₹1,050 + (1,050 × 5%)

F2 = ₹1,050 + 52.50 = ₹1,102.50

F3 = ₹1,102.50 + (1,102.50 × 5%)

F3 = ₹1,102.50 + 55.10 = ₹1,157.60

Amount of ₹1,000 will earn interest of 50 and will grow to ₹1,050 at the end of the first year. The outstanding balance of ₹1,050 in the beginning of the second year will earn interest of ₹52.50, thus making the outstanding amount equal to ₹1,102.50 at the beginning of the third year. Future or compound value at the end of third year will grow to ₹1,157.60 after earning interest of ₹55.10 on ₹1,102.50. In compounding, interest on interest is earned.

Thus, the compound value of ₹1,000 in the example can also be calculated as follows:

F_{1} = ₹1,000 × 1.05 = 1,050

F_{2} = ₹1,000 × [1.05 ×1.05]

F_{2} = ₹1,000 × (1.05)^{2}

F_{2} = ₹1,000 × 1.1025

F_{2} = ₹1,102.5

F_{3} = ₹1,000 × {1.05 ×1.05 × 1.05]

F_{3} = ₹1,000 × (1.05)^{3}

F_{3} = ₹1,000 × 1.1576

F_{3} = ₹1,157.60

We can see that the compound value factor (CVF) for a lump sum of one rupee at 5 per cent, for one year is 1.05, for two years 1.1025 and for three years 1.1576. In Figure we show the future values of ₹1 for different interest rates. You can see from the figure that as the interest rate increases, the compound value of ₹1 increases appreciably.

### Present Value of Single Amount

1 is the current value of the future sum of 1.08 if it can be invested at 8% today to become 1.08 in the future. In business decision-making, the current value of potential cash receipts is important. To assess whether an investment should be made or how much should be spent, you must first determine how much potential receipts are worth today. Compounding is the polar opposite of discounting. It entails calculating the present value of a potential sum of money that is supposed to contain interest.

## Calculating Values of Annuities

An annuity is a set of equivalent payments made at regular intervals, with each payment compounded or discounted at the time of payment. Rent is the term used to describe each annuity payment. There are various forms of annuities, with an ordinary annuity paying or receiving rent at the end of each year.

### Future Value of Annuity

If you open a savings account that earns compound interest per month and deposit ₹100 at the end of each month, the deposits are the rentals of an annuity. After a year, you will have 12 deposits of ₹100 each, for a total of ₹1.200, but the account will have more than ₹1,200 in it due to the interest earned on each deposit. Your balance is ₹1233.56 if the interest rate is 6% a year, compounded annually.

The sum accrued in the future from all the rentals charged and the interest received by the rents is known as the future value of an annuity or the total of annuity. To generate a table of potential annuity values, we presume that payments of ₹1 are made per year into a fund that receives 8% compounded interest per year.

An annuity of four instalments of one, each paid at the end of each year, with interest compounded at 8% each year, is depicted in Figure:

Notice that there are four rents and four periods. Each rent is paid at the end of each period. At the end of the first period, ₹1 is deposited and earns interest for three periods. The next rent earns interest for two periods, and so on. The amount at the end of the fourth period can be determined by calculating the future value of ₹1 deposited by each individual as follows:

Future value of 1 at 8% for 3 periods = 1.25971

Future value of 1 at 8% for 2 periods = 1.16640

Future value of 1 at 8% for 1 period = 1.08000

The fourth rent of 1 earns no interest = 1.0000

Total for 4 rents = 4.50611

The general formula for the future value of ₹1, with n representing the number of compounding period is

FV = (1+i)n

You can use this formula for calculating future values for any interest rate and any number of periods. For doing so, multiply the principal amount by the factor for the future value of 1 as follows:

FV = (1+i)n

FV = P (1+i)n

Where i = interest rate

n = number of periods

The formula for the future value of an annuity of ₹1 can be used to produce tables for a variety of periods and interest rates.

) + − =

n

1 i 1 F

### Present Value of Annuity

An annuity’s present value is the sum that must be invested today at compound interest to receive periodic rents in the future. We can geta table for the present value of an ordinary annuity of ₹1 by using the present value of ₹1. The present value of an ordinary annuity of ₹1 is shown in Figure:

When compounding occurs, the number of rentals equals the number of periods since each rent is available at the end of each term. We find the current value of the whole annuity by discounting each potential occurrence to the present.

Present value of 1 discounted for 1 period at 8% = 0.92593

Present value of 1 discounted for 2 periods at 8% = 0.85734

Present value of 1 discounted for 3 periods at 8% = 0.79383

Present value of 1 discounted for 4 periods at 8% = 0.73503

Present value of annuity of 4 rents at 8% = 3.31213

Since it is paid first, the first rent is worth more than the others. To solve problems in this field, a table on the present value of annuities can be used. The following formula was used to construct the table:

## Practical Applications of Time Value of Money

In addition, time value of money has applications in many areas of finance including capital budgeting, bond valuation, and stock valuation. Future value describes the process of finding what an investment today will grow to in the future. This is called compounding.

Some of the major applications of TVM are as follows:

### Sinking Fund Problems

At times, a financial manager may need to calculate the amount of annual payments needed to accumulate a particular amount of money at a later date to redeem an existing debt or substitute an existing asset. For example, a financial manager may have a target to accumulate a sum of ₹50,000 after five years.

When the compound interest rate is given (say 8%); then, what amount shall be allocated or provisioned every year so that at the end of 5th year, the manager will have a sum of ₹50,000. This is usually common in case of redemption of debentures. In such cases, the manager needs to determine the share of profits that should be retained by the company every year and which are invested at a given rate to obtain the sum of ₹50,000 after 5 years.

The formula used in this case is:

Future Value = Annuity Amount × FVIFA (r,n)

FV = A × FVIFA (r, n)

The annuity amount is calculated as:

Let us calculate the annuity amount in our example above:

### Capital Recovery Problems

A financial manager may also be curious about the amount of annual equivalent instalments required to repay a certain amount of loan borrowed from a financial institution at a fixed rate of interest for a set period of time. The amount of instalments that are to be paid by the company can be determined by using the following formula:

For example, a company has taken a loan of ₹10 lacs which is to be repaid in 20 equal instalments. In this case if the compounding rate is 15%; then, the amount of each instalment is calculated as follows:

### Compound Growth Rate Problems

A finance manager may be required to measure the compound rate of growth over a period of time, for example, to calculate revenue or income. He can easily measure such compound rates of growth using Compound Factor Tables. Let us see how using an example. Assume that you need to measure the compound rate of growth in income for ABC Co. Ltd. using the following details.

Year | Profit (in ₹ Lacs) |
---|---|

2003 | 60 |

2004 | 65 |

2005 | 75 |

2006 | 87 |

2007 | 98 |

2008 | 100 |

Here, rate of interest can be determined as follows:

FV = PV × FVIF(r%,n)

Where, FVIFA (r,n) = Future Value Interest Factor at r% interest for n years

Therefore,

100 = 60 × FVIF(r%,n)

FVIF(r%,n) = 1.667

Now, we look up the future value table for 6 years and find that nearest value to 1.667 is 1.677 which occurs at 9% rate of interest.

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