Time value of Money

We have seen in previous unit that Wealth maximisation is more important than profit maximisation. •Wealth maximisation considers time value of money, which translates cash flows occurring at different periods into a comparable value at zero period. •Financial decisions are generally made by comparing the cash inflows (benefits/returns) and cash outflows (outlays). •Since these two components occur at different time periods, there should be a comparison between the two. •In order to have a logical and a meaningful comparison between cash flows occurring over different intervals of time, it is necessary to convert the amounts to a common point of time.

Definition

•“Time value of money” is the value of a unit of money at different time intervals. •The value of money changes over a period of time. Since a rupee received today has more value, rational investors would prefer current receipts over future receipts. •Some important factors contributing to this are:

–Investment opportunities

–Preference for consumption

–Risk

The time preference for money is generally expressed by an interest rate, which remains positive even in the absence of any risk. It is called the risk free rate.

Required rate of return = Risk free rate + Risk premium

•There are two methods by which the time value of money can be calculated: Compounding technique & Discounting technique

•The future values of all cash inflows at the end of the time horizon at a particular rate of interest are calculated. •The compounding of interest can be calculated by the following equation: A = P (1+i)n Where, A = Amount at the end of the period, P = Principle at the end of the year, i = Rate of interest, n = Number of years.

•In the discounting technique, the present value of the future amount is determined.

•Time value of the money at time 0 on the time line is calculated.

•This technique is in contrast to the compounding approach where we convert the present amounts into future amounts. •Mathematically,

P = A {1 / (1+i)n}

Where P is the present value for the future sum to be received, A is the sum to be received in future, i is the interest rate and n is the number of years.

FVn = PV (1+i)n

Where, FVn = future value of the initial flow in n years hence, PV = initial cash flow, i = annual rate of interest and n = life of investment.

•The expression (1+i)n represents the future value of the initial investment of Re. 1 at the end of n number of years. It is called the Future Value Interest Factor (FVIF).

•‘Rule of 72’: This rule states that the period within which the amount doubles is obtained by dividing 72 by the rate of interest.

For instance, if the given rate of interest is 10%, the doubling period is 72/10, that is, 7.2 years.

•‘Rule of 69’: is expressed as 0.35+69/interest rate. Going by the same example given above, we get the number of years as 7.25 years {0.35 + 69/10}

This method is known to be more accurate.

•When interest is compounded more frequently – half-yearly or quarterly, the future value is calculated as below:

FVn = PV (1+i/m)m*n

Where, FVn = future value after n years, PV = cash flow today

i = nominal interest rate per annum, m = number of times compounding is done during a year, n = number of years for which compounding is done

•general relationship between the effective and nominal rates of interest is as follows:

r = {(1+i/m)m } – 1

Where, r = Effective rate of interest, i = Nominal rate of interest

m= Frequency of compounding per year.

•An investor may be interested in investing money in instalments and wish to know the value of his savings after n years. •To determine the accumulation of multiple flows as at the end of a specified period, –Ascertain the accumulations of each of these flows using the appropriate FVIF –Sum up these accumulations. •Example: to determine the accumulated sum at the end of 3 years, flows being 1st year = 2000, 2nd year = 3000, 3rd year = 4000, at a specified interest rate (i), FV = FVIF(i, 3) X (2000) + FVIF(i, 3) X (3000) + FVIF(i, 3) X (4000).

•Annuity refers to the periodic flows of equal amounts. These flows can be either termed as receipts or payments.

•The future value of a regular annuity for a period of n years at “i” rate of interest can be summed up as under:

FVAn = A {(1+i)n - 1} / i

Where, FVAn = Accumulation at the end of n years, i = Rate of interest, n = Time horizon or no. of years, A = Amount invested at the end of every year for n years.

•The term [{(1+i)n - 1} / i ] is called the Future Value Interest Factor for Annuity (FVIFA).

•This represents the accumulation of Re.1 invested at the end of every year for n number of years at “i” rate of interest.

•Sinking fund is a fund which is created out of fixed payments each period, to accumulate for a future sum after a specified period.

•The sinking fund factor is useful in determining the annual amount to be put in a fund, to repay bonds or debentures or to purchase a fixed asset or a property at the end of a specified period.

A = FVA X i/ {(1+i)n - 1}

•i/ {(1+i)n - 1} is called the Sinking Fund factor.

Given the interest rate, compounding technique can be used to compare the cash flows separated by more than one time period.

•With this technique, the amount of present cash can be converted into an amount of cash of equivalent value in future.

•Likewise, we may be interested in converting the future cash flows into their present values.

•The “Present Value” (PV) of a future cash flow is the amount of the current cash that is equivalent to the investor.

•The process of determining present value of a future payment or a series of future payments is known as discounting

•We can determine the PV of a future cash flow or a stream of future cash flows using the formula:

PV = FVn / (1+i)n

Where, PV = Present Value, FVn = Amount, i = Interest rate and n = Number of years

•The PV of a series of cash flows can be represented by the following formula:

PVAn = A {(1+i)n – 1 / i (1+i)n }

•The term {(1+i)n – 1 / i (1+i)n } is known as the Present Value Interest Factor Annuity (PVIFA). It represents the PVIFA of Re. 1 for the given values of i and n.

•A person may like to find out the present value of his investment assuming he will receive a constant return year after year.

•The PV of perpetuity is calculated as:

PV = A / i

Where, PV of perpetuity is simply equal to the constant annual payment (A) divided by the interest rate (i).

•In some investment decisions of a firm, the returns may not be constant. In such cases, the PV is calculated as follows.

P = A1 PVIF (i, 1) + A2 PVIF (i, 2) + A3 PVIF (i, 3) + A4 PVIF (i, 4) +……………..…. + An PVIF (i, n)

•Capital recovery factor is the annuity of an investment for a specified time at a given rate of interest.

•The reciprocal of the present value annuity factor is called capital recovery factor.

A = PVAn {i (1+i)n / (1+i)n – 1}

•The term {i (1+i)n / (1+i)n – 1} is known as the Capital Recovery Factor.

We have seen in previous unit that Wealth maximisation is more important than profit maximisation. •Wealth maximisation considers time value of money, which translates cash flows occurring at different periods into a comparable value at zero period. •Financial decisions are generally made by comparing the cash inflows (benefits/returns) and cash outflows (outlays). •Since these two components occur at different time periods, there should be a comparison between the two. •In order to have a logical and a meaningful comparison between cash flows occurring over different intervals of time, it is necessary to convert the amounts to a common point of time.

Definition

•“Time value of money” is the value of a unit of money at different time intervals. •The value of money changes over a period of time. Since a rupee received today has more value, rational investors would prefer current receipts over future receipts. •Some important factors contributing to this are:

–Investment opportunities

–Preference for consumption

–Risk

**Future Value**The time preference for money is generally expressed by an interest rate, which remains positive even in the absence of any risk. It is called the risk free rate.

Required rate of return = Risk free rate + Risk premium

•There are two methods by which the time value of money can be calculated: Compounding technique & Discounting technique

*Compounding technique*•The future values of all cash inflows at the end of the time horizon at a particular rate of interest are calculated. •The compounding of interest can be calculated by the following equation: A = P (1+i)n Where, A = Amount at the end of the period, P = Principle at the end of the year, i = Rate of interest, n = Number of years.

*Discounting technique*•In the discounting technique, the present value of the future amount is determined.

•Time value of the money at time 0 on the time line is calculated.

•This technique is in contrast to the compounding approach where we convert the present amounts into future amounts. •Mathematically,

P = A {1 / (1+i)n}

Where P is the present value for the future sum to be received, A is the sum to be received in future, i is the interest rate and n is the number of years.

**Future value of a single flow (lump sum)**•The process of calculating future value will become very cumbersome if they have to be calculated over long maturity periods of 10 or 20 years. •A generalised procedure of calculating the future value of a single cash flow compounded annually is as follows:FVn = PV (1+i)n

Where, FVn = future value of the initial flow in n years hence, PV = initial cash flow, i = annual rate of interest and n = life of investment.

•The expression (1+i)n represents the future value of the initial investment of Re. 1 at the end of n number of years. It is called the Future Value Interest Factor (FVIF).

*Doubling period:*There are two ways to determine how long it will take for an amount invested to double for a given rate of interest:•‘Rule of 72’: This rule states that the period within which the amount doubles is obtained by dividing 72 by the rate of interest.

For instance, if the given rate of interest is 10%, the doubling period is 72/10, that is, 7.2 years.

•‘Rule of 69’: is expressed as 0.35+69/interest rate. Going by the same example given above, we get the number of years as 7.25 years {0.35 + 69/10}

This method is known to be more accurate.

*Increased frequency of compounding*•When interest is compounded more frequently – half-yearly or quarterly, the future value is calculated as below:

FVn = PV (1+i/m)m*n

Where, FVn = future value after n years, PV = cash flow today

i = nominal interest rate per annum, m = number of times compounding is done during a year, n = number of years for which compounding is done

*Effective vs. Nominal rate of interest*•general relationship between the effective and nominal rates of interest is as follows:

r = {(1+i/m)m } – 1

Where, r = Effective rate of interest, i = Nominal rate of interest

m= Frequency of compounding per year.

**Future value of series of cash flows**•An investor may be interested in investing money in instalments and wish to know the value of his savings after n years. •To determine the accumulation of multiple flows as at the end of a specified period, –Ascertain the accumulations of each of these flows using the appropriate FVIF –Sum up these accumulations. •Example: to determine the accumulated sum at the end of 3 years, flows being 1st year = 2000, 2nd year = 3000, 3rd year = 4000, at a specified interest rate (i), FV = FVIF(i, 3) X (2000) + FVIF(i, 3) X (3000) + FVIF(i, 3) X (4000).

**Future Value of an Annuity**•Annuity refers to the periodic flows of equal amounts. These flows can be either termed as receipts or payments.

•The future value of a regular annuity for a period of n years at “i” rate of interest can be summed up as under:

FVAn = A {(1+i)n - 1} / i

Where, FVAn = Accumulation at the end of n years, i = Rate of interest, n = Time horizon or no. of years, A = Amount invested at the end of every year for n years.

•The term [{(1+i)n - 1} / i ] is called the Future Value Interest Factor for Annuity (FVIFA).

•This represents the accumulation of Re.1 invested at the end of every year for n number of years at “i” rate of interest.

**Sinking Fund**•Sinking fund is a fund which is created out of fixed payments each period, to accumulate for a future sum after a specified period.

•The sinking fund factor is useful in determining the annual amount to be put in a fund, to repay bonds or debentures or to purchase a fixed asset or a property at the end of a specified period.

A = FVA X i/ {(1+i)n - 1}

•i/ {(1+i)n - 1} is called the Sinking Fund factor.

**Present Value**Given the interest rate, compounding technique can be used to compare the cash flows separated by more than one time period.

•With this technique, the amount of present cash can be converted into an amount of cash of equivalent value in future.

•Likewise, we may be interested in converting the future cash flows into their present values.

•The “Present Value” (PV) of a future cash flow is the amount of the current cash that is equivalent to the investor.

•The process of determining present value of a future payment or a series of future payments is known as discounting

**Discounting or present value of a single flow**•We can determine the PV of a future cash flow or a stream of future cash flows using the formula:

PV = FVn / (1+i)n

Where, PV = Present Value, FVn = Amount, i = Interest rate and n = Number of years

**Present value of even series of cash flows**•The PV of a series of cash flows can be represented by the following formula:

PVAn = A {(1+i)n – 1 / i (1+i)n }

•The term {(1+i)n – 1 / i (1+i)n } is known as the Present Value Interest Factor Annuity (PVIFA). It represents the PVIFA of Re. 1 for the given values of i and n.

**Present value of perpetuity**•A person may like to find out the present value of his investment assuming he will receive a constant return year after year.

•The PV of perpetuity is calculated as:

PV = A / i

Where, PV of perpetuity is simply equal to the constant annual payment (A) divided by the interest rate (i).

**Present value of an uneven periodic sum**•In some investment decisions of a firm, the returns may not be constant. In such cases, the PV is calculated as follows.

P = A1 PVIF (i, 1) + A2 PVIF (i, 2) + A3 PVIF (i, 3) + A4 PVIF (i, 4) +……………..…. + An PVIF (i, n)

**Capital Recovery Factor**•Capital recovery factor is the annuity of an investment for a specified time at a given rate of interest.

•The reciprocal of the present value annuity factor is called capital recovery factor.

A = PVAn {i (1+i)n / (1+i)n – 1}

•The term {i (1+i)n / (1+i)n – 1} is known as the Capital Recovery Factor.