Finance

Management Science-II

Prof. R.Madumathi

MODULE 2 Finance – An Introduction The functions of finance in an organization is interlinked with other managerial responsibilities and in many instances, the finance manager could also done the role of a managing director. For the smooth functioning as well as to achieve excellence, organizations have to concentrate on the financial impact of a decision and its consequences. This also helps the organization to aim at a desired competency level against its competitors.

Basic Concept In Finance •

In organizations, flow of money occurs at various points of time. In order to evaluate the worth of money, the financial managers need to look at it from a common platform, namely one time duration. This common platform enables a meaningful comparison of money over different time periods.

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An important principle in financial management is that the value of money depends on when the cash flow occurs – which implies Rs.100 now is worth more than Rs.100 at some future time.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Time Value Of Money

Time Value Of Money The Time-Value Of Money Money like any other desirable commodity has a price. If you own money, you can, 'rent' it to someone else, say a banker, who can use it to earn income. This 'rent' is usually in the form of interest. The investor's return, which reflects the time-value of money, therefore indicates that there are investment opportunities available in the market. The return indicates that there is a –

risk-free rate of return rewarding investors for forgoing immediate consumption

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compensation for risk and loss of purchasing power.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Time Value Of Money •

Risk: An amount of Rs.100 now is certain, whereas Rs.100 receivable next year is less certain. This 'uncertainty' principle affects many aspects of financial management and is termed as risk value of money.

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Inflation: Under inflationary conditions, the value of money, expressed in terms of its purchasing power over goods and services, declines. Hence Rs.100 possessed now is not equivalent to Rs.100 to be received in the future.

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Personal consumption preference: Most of us have a strong preference for immediate rather than delayed consumption. As a result we tend to value the Rs.100 to be received now more than Rs.100 to be received latter.

Future Value Vs. Present Value Future value (FV) and present value (PV) adjust all cash flows to a common time. This is relevant when we want to compare the cash flows occurring at different periods of time. Either in terms of projects, performance or turnover, the cash flows accrue to the company at different stages. The evaluation of all these cash flows are true when they are all brought to the same base period.

Computing Present Value In financial parlance, a value of currency is not kept idle. The amount, if invested would certainly bring additional returns in the future. This future expectation from the present investment is termed as the future value.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Let us assume x amount is invested now and the investor expects r% to accrue on the investment one year ahead. This is translated into present and future values as follows: PV = Rs. x FV = Rs. x + (r * x)

Computing Future Value – Example Let us assume Rs.1,000 is invested now and the investor expects 5% to accrue on this investment one year ahead. This is translated into present and future values as follows: PV = Rs.1,000 FV = Rs.1,000 + (.05 * 1,000) = Rs.1,050.

Computing Future Value This can be restated as FV = PV * (1+r)

This relationship leads to the following concept of discounting the future value to arrive at the present value i.e., PV = FV / (1 + r)

This is the formula for equating the future value that is associated at the end of 1st year. Now the concept of time over a longer duration can be easily brought into the above equation, where 'n' defines the time duration after which the cash flows are expected.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Computing Present Value – Example Let us assume that Rs.1,000 is to be received at the end of 1 year from now and the investor expects 5% rate of return on this investment. Here FV = Rs.1,000 Hence the present value is computed as: PV = FV / (1 + r) = Rs.1000 / (1.05) = Rs.952.

Value With And Without Compounding •

Interest without compounding is a simple interest formula i.e., Pnr/100 Where: P is the principle, n is the number of years and r is the

interest rate.

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Interest with annual compounding adds the interest received earlier to the principle amount and increases the final amount that is received from the investment. Hence, the FV of an investment for a two year duration with annual compounding would be: FV = PV * (1+r)* (1+r) = PV * (1+r)^2.

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Hence Present Value is: PV = FV / (1+r)^2.

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This equation can be generalized for 'n' years as: PV = FV / (1 + r)^n

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Future Value With And Without Compounding

Compound Value In compounding, it is assumed that a certain sum accrues at the end of a time duration, which is again reinvested. In short, when a sum is invested in a year, it will yield interest and the interest is reinvested for the next year and so on till the time when withdrawal is made. The 3 year or 4 year bank deposit is a typical example of this annual interest compounding. Here:

FV = Principal + interest FV = P(1+r)^n

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

The term (1+r)^n is the compound value factor (CVF) of a lump sum of Re.1, and it always has a value greater than 1 for positive r, indicating that CVF increases as r and n increase.

Compound Value – Example Assume a lump sum of Rs.1,000 is deposited in a bank fixed deposit for 3 years for an interest rate of 10% per annum. FV = Principal + interest FV = P(1+r)^n FV = 1000 x (1+.10)^3 = 1000 x 1.331 = Rs.1,331.

Compounding In Less Than A Duration •

Usually, it is common practice to compound the interest on a yearly basis. But, there are instances when compounding is done on a half-yearly, quarterly, monthly or a daily basis. The half-yearly interest rates indicate that interest is payable semiannually, i.e., interest is received r%/2 twice every year. When the principle of compounding is applied, this implies that the r%/2 received twice an year will yield an actual rate which is higher than the declared (r%) rate. This actual rate is called the effective annual rate.

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For instance, let us take an illustration of a banker declaring a 10% p.a. interest payable semiannually. This implies that at the end of the year the amount received for every one rupee will be 1 * (1+[10%/2]) * (1+[10%/2]) i.e., (1.05) * (1.05) = (1.05)^2 = 1.1025.

Indian Institute of Technology Madras

Management Science-II

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Prof. R.Madumathi

The Effective interest rate is 10.25%

Effective Interest Rate The effective interest rate in the previous example was computed as 1.1025 - 1 = .1025 and in percentage terms it will be 10.25%. The effective rate of interest is hence 10.25% and not 10%. This can be expressed through the following formula: FV = PV (1+ r/m)^(m*n) where m is the number of times within a year interest is paid. When half-yearly interest payments are made 'm' will be 12/6 i.e., 2. When quarterly interest payments are made 'm' will be 12/3 i.e., 4. When monthly compounding is done then 'm' will be 12/1 i.e., 12. Compounding on a daily basis, 'm' will be 365/1 i.e., 365. This is referred to as multi-period compounding.

Continuous Compounding Sometimes compounding may be done continuously. For example, banks may pay interest continuously; they call it continuous compounding. It can be mathematically proved that the continuous compounding function will reduce to the following: FV = PV x {e^x} When x = (r * n) and e is mathematically defined as equal to 2.7183.

Continuous Compounding – Example The present value of an investment is Rs.1,000. At 10% p.a. interest rate at the end of 5 years, the future value of this investment with continuous compounding will be: FV = 1,000 x {e^.5} = Rs.1,648.72 When x = (r * n = .1 x 5 = .5) and e is mathematically defined as equal to 2.7183.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Similarly, the present value of a future flow of Rs.100 at 10% p.a. interest rate to be received 5 years hence with continuous compounding will be PV = FV / {e^.5} = 100 / {e^.5} = Rs.60.65.

Annuity There can be a uniform cash flow accrual every year over a period of 'n' years. This uniform flow is called "Annuity". An annuity is a fixed payment (or receipt) each year for a specified number of years. The future compound value of an annuity as follows:

FV = A {[(1+r)^n - 1]/ r}

The term within the curly brackets {} is the compound value factor for an annuity of Re.1, and A is the annuity. The present value of an annuity hence will be

PV = A {[1 - 1/(1+r)^n]/r}

Annuity – Example The Future value of Rs.10 received every year for a period of 5 years at an assumed interest rate of 10% per annum will be

FV = 10 {[(1+0.1)^5 - 1]/ 0.1} = Rs.61.051

The Present value of Rs.100 to be received every year in the next five years at an assumed interest rate of 10% per annum will be

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

PV =100{[1 - 1/(1+0.1)^5]/0.1}=Rs.379.08

Resent Value Of Perpetuity Perpetuity is an annuity that occurs indefinitely. In perpetuity, time period, n, is so large (mathematically n approaches infinity) that the expression (1+r)^n in the present value equation tends to become zero, and the formula for a perpetuity simply condenses into:

PV = A/r

where A is the annuity amount occurring indefinitely and r is the interest rate.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Regular Annuity Vs. Annuity Due •

When an annuity's cash payments are made at the end of each period, it is referred as regular annuity. On the other hand, the annual payments/receipt can also be made at the beginning of each period. This is referred to as annuity due.

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Lease is a contract in which lease rentals (payment) are to be paid for the use of an asset. Hire purchase contract involves regular payments (installments) for acquiring (owning) an asset. A series of fixed payments starting at the beginning of each period for a specified duration is called an annuity due.

Annuity Due The formula for computing value of an annuity due is: FV = A[(1 + r) + (1+r)^2+ (1+r)^3 +....+ (1+r)^n-1] FV = A {[(1+r)^(n-1) -1] / r} Hence, PV = A {[1 - 1/(1+r)^n]/r } * (1+r) PV = A(PVRA,r)*(1+r) Where PVAR is present value of regular annuity and r is the interest rate.

Annuity Due – Example The future value of Rs.10 received in the beginning of each year for a 5 year duration at an assumed rate of 10% p.a. will be: FV = 10 {[(1+0.1)^(5-1) -1] / 0.1} = Rs.46.41. The present value of Rs.100 received in the beginning of each year for 5 years at an assumed interest rate of 10% p.a. will be: PV = 100 {[1 - 1/(1+1.1)^5]/0.1 } x (1+0.1)= Rs.416.98.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

Multi Period Annuity Compounding The compound value of an annuity in case of the multi-period compounding is given as follows: FV = A ‹{[(1+r/m)^(n x m)] -1 } /(r/m)› PV = A ‹{1 -[1/(1+r/m)^(n x m)]} / (r/m)› In all instances, the discount rate will be (r/m) and the time horizon will be equal to (n x m).

PRESENT VALUE OF A GROWING ANNUITY An annuity may not be a constant sum through the time duration, it may also grow at a rate of g% every year. This is referred as a growing annuity. When there is a growth for specific number of years, the present value of an annuity is stated using the following formula:

Present Value Of A Growing Annuity – Example An annuity of Rs.100 is expected to grow at a rate of 2% every year. Assuming the interest rate as 10% per annum the present value for this growing annuity for a 5 year duration will be: PV = 100 x {(1/0.08)-[(1/0.08)*(1.02)^5/(1.1)^5]} = Rs.393.07.

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

FUTURE VALUE OF A GROWING ANNUITY Future value of a growing annuity can be defined by the following formula:

Future Value Of A Growing Annuity - Example Future value of an annuity of Rs.10 growing at 2% every year with an assumed rate of interest at 10% for five years is computed as: FV = 10 x {[1.1^5/0.08]-[1.02^5/0.08]} = Rs.63.30

Present Value Of A Growing Annuity Perpetuity In financial decision-making there are number of situations where cash flows may grow at a compound rate. Here, the annuity is not a constant amount A but is subject to a growth factor 'g'. When the growth rate 'g' is constant, the formula can be simplified very easily. The calculation of the present value of a constantly growing perpetuity is given by the following equation: PV = A/(1+r) + A(1+g)/(1+r)^2 + A(1+g)^2/(1+r)^3 + ..... This equation can be simplified as: PV = A / (r - g)

Present Value Of A Growing Annuity Perpetuity In financial decision-making there are number of situations where cash flows may grow at a compound rate. Here, the annuity is not a constant amount A but is subject to a growth factor 'g'. When the growth rate 'g' is constant, the formula can be simplified very easily. The calculation of the present value of a constantly growing perpetuity is given by the following equation:

Indian Institute of Technology Madras

Management Science-II

Prof. R.Madumathi

PV = A/(1+r) + A(1+g)/(1+r)^2 + A(1+g)^2/(1+r)^3 + ..... This equation can be simplified as: PV = A / (r - g)

Present Value Of A Growing Annuity Perpetuity – Example The present value of an annuity of Rs.10 growing at 2% every year with an assumed rate of interest of 10% to perpetuity is: PV = A / (r - g) PV = 10 / (0.1 - 0.02) = Rs.125.

Indian Institute of Technology Madras