Time Value of Money

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What is Time Value of Money?

One of the most fundamental concepts in finance is that money has “time value.”

That is to say that money in hand today is worth more than the money that is expected to be received in the future. It is because money today helps an individual to buy whatever he wants today. If an individual behaves rationally, he would not value the opportunity to receive a specific amount of money now equally with the opportunity to have the same amount at some future date.

Individuals normally value the opportunity to receive money now higher than waiting for one or more periods to receive the same amount. This is called time preference for money.

Time preference of money is an individual’s preference for ownership of a given amount of money now, rather than the same amount at some future time. This reflects an important principle that the value of money is time dependent.

Following are four important reasons for the time preference of money:

  • Risk and Uncertainty: Future is always uncertain and risky. Outflow of cash is in our control as payments to parties are made by us. There is no certainty for future cash inflows. As an individual or firm is not certain about future cash receipts, it prefers receiving cash now. Following this, Rs 1 now is certain, whereas Rs 1 receivable tomorrow is less certain. This principle is also referred as bird- in- the- hand.

  • Inflation: In an inflationary economy, the money received today has more purchasing power than the money to be received in future. In other words, a rupee today represents a greater real purchasing power than a rupee at later period.

  • Consumption: Individuals generally prefer current consumption to future consumption. Thus, individuals give more value to the received money today as compared to be received in future.

  • Investment opportunities: An investor can profitably employ a rupee received today to give him a higher value to be received tomorrow or after a certain period of time. For example, an investor can deposit Rs 1000/- in the Bank and can earn 8% return after a fixed period say one year.

    Thus, Rs 1000 today Valuation : Basic Concepts (present value) will become Rs 1080 (future value) after one year. Due to above reasons, time value of money is a vital consideration in making a financial decisions.

Techniques of Time Value of Money

The preceding discussion has revealed that in order to have logical and meaningful comparisons between cash flows that result in different time periods it is necessary to convert the sums of money to a common point in time.

There are two approaches for adjusting time value of money. These are:

  1. Compounding Techniques/Future Value Techniques
  2. Discounting/Present Value Techniques

Compounding Techniques/Future Value Techniques

Compounding techniques are used to calculate the future value of present cash flows. This concept is based on the principle of compound interest. Under this principle, the interest earned on the initial principal amount becomes a part of the principal at the end of the compounding period.

EXAMPLE: Suppose you invest Rs 1000 for three years in a saving account that pays 10 percent interest per year. If you let your interest income be reinvested, your investment will grow as follows:

First yearPrincipal at the beginning
Interest for the year (Rs 1,000 × 0.10)
Principal at the end of First year
1,000
100
1,100
Second yearPrincipal at the beginning
Interest for the year (Rs.1, 100 × 0.10)
Principal at the end
1,100
110
1,210
Third yearPrincipal at the beginning
Interest for the year (Rs.1210 × 0.10)
Principal at the end
1,210
121
1,331

Calculation of Future Value

To know the future value of a sum, the principle of compounding is used. Essentially, practical problem related to present value calculation can be categories as follow:

  • Future Value of a Single Amount
  • Future Value of a Series of Payment
  • Future Value of an Annuity

Future Value of a Single Amount

The Future value (FV) of an investment with compound interest i earned in a given period of n number of years can be calculated using the compound interest principle.

A generalized procedure for calculating the future value of a single amount compounded annually is as follows :

FVn = PV (1 + r)(n)

Where, FVn = Future value of the initial flow n year hence
PV = Initial cash flow
r = Annual rate of Interest
n = number of years

By considering the above example , we get the same result.

FVn = PV (1 + r)(n)
= 1,000 (1.10)(3)
FVn = 1331

In order to solve the future value problems, we consult a future value interest factor (FVIF) table. The table shows the future value factor for certain combinations of periods and interest rates.

To simplify calculations, this expression has been evaluated for various combinations of ‘r’ and ‘n’. Exhibit presents the sample of one such table showing the future value factor for certain combinations of periods and interest rates.

Future Value Interest Factor Table

Example: If you deposit Rs. 50,000 in a bank which is paying a 10 per cent rate of interest on a five-year time deposit, how much would the deposit grow at the end of five years?

Solution Using the formula for calculating the future value of a single amount compounded annually, we can solve as follows:-

FVn = PV (1 + r)(n)

FVn = PV(FVIF10%,5 yrs)
FVn = 50000 (1.10)5
= 50000 × 1.611
= Rs 80550

Multiple Compounding Periods: Interest can be compounded monthly, quarterly and half-yearly. If compounding is quarterly, annual interest rate is to be divided by 4 and the number of years is to be multiplied by 4. Similarly, if monthly compounding is to be made, annual interest rate is to be divided by 12 and number of years is to be multiplied by 12.

The formula to calculate the compound value is

Fvn = Pv (1+ r/m)mn

Where,
FVn = Future value after ‘n’ years
PV = Present value of cash flow today
r = Interest rate per annum
m = Number of times compounding is done during a year
n = Number of years for which compounding is done.

Future Value of a Series of Payment

So far we have considered only the future value of a single payment at time zero. In many instance, we may be interested in the future value of a series of payments made at different time periods.

For example, Mr. A deposit at the end of each year Rs 1,000 Rs. 2,000 Rs.3,000 Rs.4,000 in his saving bank account .The Interest rate is 5% . He wishes to find the future value of his deposits at the end of the 4th year. Table below present the calculation required to determine the sum of money he will have.

YearAmount
deposited
compounded
Number of
year
Compounded
interest factor
Future
value
11,00031.2161,216
22,00021.1582,316
33,00011.1033,309
44,000014,000
TOTAL10,841

Future Value of an Annuity

An annuity is defined as a series of equal payments (fixed) or receipts that occur at evenly spaced intervals. Lease and rental payments are examples. The payments or receipts occur at the end of each period for an annuity.

The Future Value of an Ordinary Annuity (FVoa) is the value that a stream of expected or promised future payments will grow to after a given number of periods at a specific compounded interest.

The Future Value of an Annuity could be solved by calculating the future value of each individual payment in the series using the future value formula and then summing the results.

A more direct formula is:

FVoa = CIFAm*A

Where
A = amount of annuity (fixed payment)
CIFA = compounded interest factor of an annuity for n years at a interest rate of r
r = rate of interest
n = number of years

Example: What amount will accumulate if we deposit Rs.5,000 at the end of each year for the next 5 years? Assume an interest of 6% compounded annually.

Solution:

A = 5000
n = 5 years
r = 6%
Using the formula FVoa = CIFAm*A, we can find the tuture value :
FV = 5.637×5000 = 28,185

Thus, total accumulated amount of 5000 in 5 years with 6 percent interest
rate will be Rs.28,185

Discounting/Present Value Techniques

In the above section, we have discussed how compounding techniques can be used to adjust the time value of money and helps in determining the future value of an investment decision. The concept of present value is exactly opposite of that of compound value (future value).

The present value of a future cash inflow or outflow is the amount of current cash that is of equivalent value to the decision maker. The present value is always less than or equal to the future value because money has interest-earning potential.

The process of determining present value of a future payment or receipts or a series of future payments or receipts is called discounting. Let us illustrate the discounting procedure by using an example.

Example: Mr. A has an opportunity to receive Rs. 1060 one year later. He knows that he can earn 6% interest on his investment. What amount will he be prepared to invest for this opportunity?

Solution:
To answer this question, we must determine how many rupees Mr. A must invest at 6% today to have Rs. 1060 one year afterwards.

Let us assume that P is this unknown amount and by using following formula

Present value = Future value / (1 + r)n

Where,
r = Rate of Interest
n = Number of years
P = 1060/ (1+.06)1
= 1060/1.06
= 1000

Mr. A must invest Rs 1000 today in order to get Rs 1060 after 1 year with 6% rate of interest.

Calculation of Present Value

Practical problem related to present value calculation can be categories as follow

  • Present value of a single cash flow
  • Present value of a series of cash flows
  • Present value of an annuity

Present value of a single cash flow

We will first look at discounting a single cash flow or amount. The cash flow can be discounted back to a present value by using a discount rate that accounts for the factors mentioned above (present consumption preference, risk, and inflation).

Conversely, cash flows in the present can be compounded to arrive at an expected future cash flow.

The present value of a single cash flow can be written as follows:

PV = FVn / (1 + i)n

Where:
PV = the present value (or initial principal)
FVn =future value at the end of n periods
i = the interest rate paid each period
n = the number of periods

Present Value of a Series of Cash Flows

Till we have considered only the present value of a single receipt at some future date. In many situations, especially in capital budgeting decision, we may be interest in the present value of a series of receipts receiving by a firm at a different time period.

For calculating present value of series of cash flow we need to determine the present value of each future payment and then aggregates them to find the total present value.

where,
PV = Present Value of a series of cash flow
C1, C2, C3, Cn = Cash flow in time records, 1,2,3 and n year.
i = rate of Interest for each year
t = number of year extending fram year 1 to n.

The present value interest factor (PVIF) table is used for simplifying the process of calculating the present value of a serious of cash flow. This table is used to find the present value per rupee of cash flows based on the number of periods and rate per period.

Once the value per Rupee of cash flows is found, the actual periodic cash flows can be multiplied by the Rupee amount to find the present value.

Present Value of an Annuity

As discussed earlier, an annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Lease and rental payments are examples. The payments or receipts occur at the end of each period.

The present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, however, it is more convenient to use the ‘one step’ formulas given below.

Where
i is the interest rate per compounding period;
n is the number of compounding periods; and
A is the fixed periodic payment

The present value intrest factor of annuity (PVIFA) can be used to simplify the process of calculating the present value of an annuity. This table is used to find the present value of per rupee annuity based on the number of periods and rate per period.

Once the value per rupee annuity is found, the actual cash flows of annuity can be multiplied by the per rupee amount to find the present value of an annuity.

PV of an ordinary Annuity = A x PVIFA in
A = Amount of Annuity
PVIFA in = Present Value factor of an Annuity for year and interest rate i.

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