Management Consulting/2.4 MANAGEMENT OF FUNDS AND ASSETS
1. Merely increasing the level of current asset holding does not necessarily reduce the riskiness of the firm. Rather, the composition of current assets, whether highly liquid or highly non-liquid, is the important factor to consider. Explain.
2. Since the rights issue allows the ordinary shareholders to purchase the share at price much lower than the current market price, why does not shareholders wealth increase? Illustrate your answer.
3. Should the titles of controller and treasurers be adopted under Indian context? Would you like to modify their functions in view of the company practices in India? Justify your opinion.
4. Why is the consideration of time important in financial decision making? How can time be adjusted? Illustrate your answer.
1.Merely increasing the level of current asset holding does not necessarily reduce the riskiness of the firm. Rather, the composition of current assets, whether highly liquid or highly non-liquid, is the important factor to consider. Explain.
Current asset is an indication to an outside creditor, whether the company has got the ability to meet its immediate payment obligation. As such it is weighed agnist his immediate payment obligation, composition of the current asset in terms of liquidity and immediate realisability etc. Another indicator, is what is the contribution of the promotor in the composition of current asset. The source of funds to maintain current asset is from current liability and part of promotors contribution. If promotor contribution is handsome(say at least 25%, of current asset, it is a healthy sign. Again, the composition of current asset and its component is also important. Normally, Stock and receivables are major component of current asset. The question of whether they are realisable and market able is important. The stock should be marketable and priced at cost price or market price which ever is less.
There shall not be any obsolete stock. The receivables should be current and not overdue for payment or bad debts.
Again a higher current asset also may be an indication of the corporates poor inventory management and inabiilty to recover its dues.
Another important point to be kept in mind is that stock declared in the balance sheet is only a bookish figure. There is no guarantee that actual stock in godown is matching with the stock declared in the balance sheet. At least in indian condition, Chartered accountants who audit their accounts do not vouchsafe the quality or quanity of actual stock lying in the godown. They only assures the arithemetical accuracy of the stock/purchase/sales registers.
It is unfortunate that that physical stock verification is also difficult. There is no practicial way of really assessment of the stock. Methods prescribed in text books are often not practical. Actual valuation is difficult and are not often done..
The magnitude of current assets needed is not always the same, it increases and decreases over time. However, there is always a minimum level of current assets which is continuously required by the firm to carry on its business operations, which is referred to as permanent fixed working capital. The extra working capital, i.e., variable or fluctuating or temporary working capital, needed to support the changing production and sales activities. Depending upon the changes in production and sales, the need for working capital, over and above permanent working capital, will fluctuate. The temporary working capital is created by the firm to meet liquidity requirements that will last only temporarily.
2.Since the rights issue allows the ordinary shareholders to purchase the share at price much lower than the current market price, why does not shareholders wealth increase? Illustrate your answer.
Cash-strapped companies can turn to rights issues to raise money when they really need it. In these rights offerings, companies grant shareholders a chance to buy new shares at a discount to the current trading price. Let's look at how rights issue work, and what they mean for all shareholders.
Defining a Rights Issue and Why It's Used
A rights issue is an invitation to existing shareholders to purchase additional new shares in the company. More specifically, this type of issue gives existing shareholders securities called "rights", which, well, give the shareholders the right to purchase new shares at a discount to the market price on a stated future date. The company is giving shareholders a chance to increase their exposure to the stock at a discount price.
But until the date at which the new shares can be purchased, shareholders may trade the rights on the market the same way they would trade ordinary shares. The rights issued to a shareholder have a value, thus compensating current shareholders for the future dilution of their existing shares' value.
Troubled companies typically use rights issues to pay down debt, especially when they are unable to borrow more money. But not all companies that pursue rights offerings are shaky. Some with clean balance sheets use them to fund acquisitions and growth strategies. For reassurance that it will raise the finances, a company will usually, but not always, have its rights issue underwritten by an investment bank.
How Rights Issues Work
So, how do rights issues work? The best way to explain is through an example.
Let's say you own 1,000 shares in Wobble Telecom, each of which is worth $5.50. The company is in a bit of financial trouble and sorely needs to raise cash to cover its debt obligations. Wobble therefore announces a rights offering, in which it plans to raise $30 million by issuing 10 million shares to existing investors at a price of $3 each. But this issue is a three-for-10 rights issue. In other words, for every 10 shares you hold, Wobble is offering you another three at a deeply discounted price of $3. This price is 45% less than the $5.50 price at which Wobble stock trades.
As a shareholder, you essentially have three options when considering what to do in response to the rights issue. You can (1) subscribe to the rights issue in full, (2) ignore your rights or (3) sell the rights to someone else. Here we look how to pursue each option, and the possible outcomes.
1. Take up the rights to purchase in full
To take advantage of the rights issue in full, you would need to spend $3 for every Wobble share that you are entitled to under the issue. As you hold 1,000 shares, you can buy up to 300 new shares (three shares for every 10 you already own) at this discounted price of $3, giving a total price of $900.
However, while the discount on the newly issued shares is 45%, it will not stay there. The market price of Wobble shares will not be able to stay at $5.50 after the rights issue is complete. The value of each share will be diluted as a result of the increased number of shares issued. To see if the rights issue does in fact give a material discount, you need to estimate how much Wobble's share price will be diluted.
In estimating this dilution, remember that you can never know for certain the future value of your expanded holding of the shares, since it can be affected by any number of business and market factors. But the theoretical share price that will result after the rights issue is complete - which is the ex-rights share price - is possible to calculate. This price is found by dividing the total price you will have paid for all your Wobble shares by the total number of shares you will own. This is calculated as follows:
1,000 existing shares at $5.50 $5,500
300 new shares for cash at $3 $900
Value of 1,300 shares $6,400
Ex-rights value per share $4.92 ($6,400.00/1,300 shares)
So, in theory, as a result of the introduction of new shares at the deeply discounted price, the value of each of your existing shares will decline from $5.50 to $4.92. But remember, the loss on your existing shareholding is offset exactly by the gain in share value on the new rights: the new shares cost you $3, but they have a market value of $4.92. These new shares are taxed in the same year as you purchased the original shares, and carried forward to count as investment income, but there is no interest or other tax penalties charged on this carried-forward, taxable investment income.
2. Ignore the rights issue
You may not have the $900 to purchase the additional 300 shares at $3 each, so you can always let your rights expire. But this is not normally recommended. If you choose to do nothing, your shareholding will be diluted thanks to the extra shares issued.
3 Sell your rights to other investors
In some cases, rights are not transferable. These are known as "non-renounceable rights". But in most cases, your rights allow you to decide whether you want to take up the option to buy the shares or sell your rights to other investors or to the underwriter. Rights that can be traded are called "renounceable rights", and after they have been traded, the rights are known as "nil-paid rights".
To determine how much you may gain by selling the rights, you need to estimate a value on the nil-paid rights ahead of time. Again, a precise number is difficult, but you can get a rough value by taking the value of ex-rights price and subtracting the rights issue price. So, at the adjusted ex-rights price of $4.92 less $3, your nil-paid rights are worth $1.92 per share. Selling these rights will create a capital gain for you.
It is awfully easy for investors to get tempted by the prospect of buying discounted shares with a rights issue. But it is not always a certainty that you are getting a bargain. But besides knowing the ex-rights share price, you need to know the purpose of the additional funding before accepting or rejecting a rights issue. Be sure to look for a compelling explanation of why the rights issue and share dilution are needed as part of the recovery plan. Sure, a rights issue can offer a quick fix for a troubled balance sheet, but that doesn't necessarily mean management will address the underlying problems that weakened the balance sheet in the first place. Shareholders should be cautious.
Shareholders don't like rights issues. Any company that gives its investors the unwelcome choice between stumping up more cash or seeing their existing holding diluted can expect to take a significant hit to its stock.
Mere muttering in the City that a firm is pondering a cash call can severely undermine the share price.
WHICH MEANS YOUR HOLDING WEALTH WILL BE DOWN AND NOT INCREASE.
3.should the titles of controller and treasurers be adopted under indian context? would you like to modify their functions in view of the company practices in india?justify your opinion.
The titles of controller and treasurers are job
Descriptions, highlighting the jobs to be performed
In the organization. This cannot be tempered with
Under any conditions, whether in india or any other
What one should do define the role in the organization
Which takes care of the organization culture and
Employees understand and appreciate the role.
SEE THE DETAILS HEREBELOW.
Position reports to:
Chief Executive, Principal Finance Executive or Operations Executive
Plan, direct and adniiii:ister the treasury activities of the organisation, including placement of borrowings and movement of surplus funds, to acliieve short and long term business objectives.
Direct day to day operations on the money market in accordance with company policy and guidelines.
Direct funding for the company's activities including short, medium and long term lending requirements.
Liaise with merchant banks and other financial institutions to assist in the overall funding of the company's activities.
Provide for the arrangement and administration of terms of credit and working and depository accounts in appropriate banks.
Provide counsel and make recommendations to management with regard to possible changes in borrowing strategy.
Maintain an up to date knowledge of local and international lending trends and advise on short and long term borrowing as it may affect the company's Equidity
Ensure money market activities operate in accordance with Corporate Affairs and Securities Coniiiiission requ~irements and all legislative and other conditions relating to borrowing.
Liaise regularly with The Reserve Bank, the company's bankers, major brokers and lending institutions to ensure the company's interests are served.
Assess recommendations for new funding proposals and employment of surplus funds submissions.
Research and develop new fund raising and employment of surplus funds plans.
Survey new sources of borrowings to meet projected cash requirements by developing existing contacts and making new ones.
Ensure all bank overdraft, overseas borrowing, bill lines and other negotiations are properly and promptly undertaken.
May be responsible for the selection and training of staff.
Typically, incumbents would have an appropriate tertiary quafification and 10 years' experience in finance.
Position reports to:
Principal Finance or Administration Executive, OR may report to the Chief Executive.
Control the finance and accounting methods, practices and procedures of the organisation. Prepare appropriate financial assessments, accounting records and other information, allowing senior management to make informed business decisions and thereby improve the organisation's profitability.
Direct and review the analysis and interpretation of statistical and accounting information for use in management decision making.
Review and analyse operating results in relation to cost, budgets and operating policies; consolidate capital asset, maintenance and operating budgets.
Participate in the development of short and longer term financial policies, objectives and plans, and oversee the implementation of agreed financial plans.
Prepare schedules for financial reporting in accordance with management and statutory requirements.
Ensure major capital expenditure requests comply with budgets and approval systems.
Maintain a regular review of income and expenditure to ensure cash flow is adequate to meet future business needs.
Direct the preparation of required regular reports of liquidity, profit and loss, debtors /creditors, sales and capital expenditure including the preparation of related management information material.
Interpret operating results as they affect the financial position of the corporation and make recommendations for cost reduction and profit improvement.
Supervise maintenance of the company chart of accounts; assign new classifications and cost codes as may be necessary; ensure correct and accurate accounting classification of all expenditures and documents.
Liaise with the organisation's external auditors as required.
Direct and control general accounting functions to optirnise the utilisation of accounting staff
Control the selection and training of finance staff, establish lines of control and delegate responsibilities to subordinate staff.
May direct internal audits, credit control, office equipment purchase and utilisation, manpower statistics, timekeeping and payroll; may supervise investment of the company's short term liquid assets.
Ensure activities comply with relevant Acts, legal demands and professional and ethical standards.
Typically, incumbents Would have 10 to 12 years' accounting experience, with a three year degree or diploma, and responsibility for at least three to four professional accounting staff. The position has functional accountability for Finance and Accounting.
THE CONCEPT OF ROLE
In, any social system, such as the family club, religious community, work
organization, etc., Individuals have certain obligations towards the system, which in
turn gives each one of them .a defined place in the society. This system of mutual
obligations can be called a role and the individual's place, a position or an office. For
example, when one joins a new club, one is admitted as member is defined in terms
of the hierarchical placement and privileges (the power one will enjoy). One also
agrees to abide by ` certain rules, carry out certain activities when required, volunteer
for certain work, etc. The other members of the club expect all this from the
individual, and one also expects to dd. the needful. All these expectations, together
with one's response to them comprise the role. Briefly then, and individual occupies a
hierarchical position in a system, along with the ensuing powers and privileges, and
performs certain functions in response to his and the member's expectations. In this
case the former is the office (or position) and the latter the role.
Role is the position one occupies in a social system, as defined by the f unctions on
performs in response to the expectations of the 'significant' members of social
system, and one's own expectations from that position or office. Role and office ( or
position), though two sides of the same coin, are, however, two separate concepts.
According to Katz and Kahn, "Office is essentially a relational concept, defining each
position in terms of its relationship to other and to the system as a whole".
Office for Positional and Role
- is based on power relations ------- based on mutuality
- has related privieges - --------------has related obligations
- is usually hierarchical --------------- is non-hierarchical
- is created by other ------------------- is created by other and the role
- is part of the structure --------------- is part of the dynamics
- is evaluative - --------------------------is descriptive
While office is a relational and power-related concept, role is an 'obligational"
concept. Office is concerned with the hierarchical position and privileges, while a,
role is concerned with the obligations of position. Exhibit 1 distinguishes between
these two concepts.. While office is a point in the social structure defining an office
holder's power, role is integrated set of behavious expected from a person occupying
that office. An organization can be represented accoring to the offices, or the roles.
Exhibits 2 and 3 represent a part of an organization in two different ways.
An office becomes a role when it is actually defined and determined by the
expectations of other office holders (as reflected in the way an office is discharged by
the concerned office holder). Each role has its own system, consisting of the role
occupant and those who have a direct relationship with him, and thereby, certain
expectations from the role.
Using the currently accepted terminology suggested by Kaiz and Kahn, we will term
the "significant' others having expectations from a role as role senders. They ' send'
expectations to the role. The role occupant also has expectations from his role, and in
that sense the role occupant is also a role sender:
Let us take an example. In a family the father has both a position (office) and a role.
Thee father's position defines his authority in the family. In some societies he is the
final decision make and the other members abide by his decisions. There are certain
expectation from the father that define his role - that he would earn for the family,
protect the family against threats, etc. In his position as the head or the family
system, his role is to maintain and protect the family. While the position gives him
some privileges, the role places certain obligations on him.
A role is not defined without the expectations of the role senders, including the role
occupant. The position of a personnel manager may be created in an organizations,
but his role will be defined by the expectations (stated or unstated) that different persons
have from the personnel manager, and the expectations that he in turn, has from the
role. In this sense, the role gets defined in each system by the role senders, including
the role occupant.
However, a question that can be raised is: If the role is defined in each case by the
role senders, how can we talk about a role in general, e.g. the father's role? While
strictly speaking a role in general. does not make much sense, in a lager social system
the expectations from a role are largely shared, and have common elements. These
are generalised, and we therefore, talk about the role of the Indian mother, or the role
of a chairman in a public sector concern, etc.
Confusion sometimes arises because the word role has two different connotation. At
times it denotes the position a person holds in an organization along with the
expectations from that position (e.g. the role of a teacher, a policeman, etc), and
elsewhere it describes only the expected behaviour or activities (for example, a
disciplinarian or an evaluatory role or a teacher, task and maintenance roles, etc.). For
the sake of convenience we shall use the word role for a position a person holds in a
system (organization), as defined by the expectations various 'significant' persons,
including oneself, have from that person. We will use the tem function to indicate a
se inter-relaited expectations from a role. We can therefore say that while 'sales
manager' is a role, developing a sales force and customer contact are the related
Distinction needs to be made amongst certain work-related terms; office, role, job
functions, tasks, etc. Although there are no universally accepted definitions, work is
generally a wider terms, whereas office, role and job are ways or organizing work or
dividing responsibilities. Functions are sub-units of a role. A function can be further
subdivided into tasks. Exhibit 4 provides the definitions of these terms.
Work is a wider concept linking a person with his tools and with others performing a similar activity.
Office or position is a specific point in an organizational structure, defining the power of the
person occupying it.
Role' is the set of obligations generated by the 'significant' others, and the individual
occupying an office.
Job is a specific requirement to produce a product or achieve an objective.
Function is a group of expected behaviors for a role.
Example: An individual X may occupy an office of Branch Y of a bank. As a part of this
office the individual the. individual reports to the Regional Manager. Similarly, a large
number of persons, in turn, report to X. His role is to develop the branch by getting a
successively larger market share of deposits and advances. One of the functions under this
role' is to increase deposits. One task which he performs, as part of this function, is to
undertake a survey of potential depositors, another is to contact the prestigious and 'big'
OFFICE--- ROLE---- JOB
The concept of role is vital for the integration of the individual with an organization.
The individual and organization come together through a role. As shown in Exhibit 5
the organization has its own structure and goals. Similarly, the individual has his
personality and needs (motivations). These interact with each other and to some
extent get integrated in a role. Role is also a central concept in work motivation. It is
only through a role that the individual and an organization interact with each other, as
shown in Exhibit 6.
4.why is the consideration of time important in financial decision making?how can time be adjusted?illustrate your answer.
Money has time value. A rupee today is more valuable than a year hence.
It is on this concept “the time value of money” is based. The recognition of the time value of money and risk is extremely vital in financial decision making.
Most financial decisions such as the purchase of assets or procurement of funds, affect the firm’s cash flows in different time periods. For example,
if a fixed asset is purchased, it will require an immediate cash outlay and will generate cash flows during many future periods. Similarly if the firm borrows funds from a bank or from any other source, it receives cash and
commits an obligation to pay interest and repay principal in future periods.
The firm may also raise funds by issuing equity shares. The firm’s cash
balance will increase at the time shares are issued, but as the firm pays
dividends in future, the outflow of cash will occur. Sound decision-making
requires that the cash flows which a firm is expected to give up over period
should be logically comparable. In fact, the absolute cash flows which differ
in timing and risk are not directly comparable.
Cash flows become logically
comparable when they are appropriately adjusted for their differences in
timing and risk. The recognition of the time value of money and risk is
extremely vital in financial decision-making. If the timing and risk of cash flows is not considered, the firm may make decisions which may allow it to
miss its objective of maximising the owner’s welfare. The welfare of owners
would be maximised when Net Present Value is created from making a
financial decision. It is thus, time value concept which is important for
Thus, we conclude that time value of money is central to the concept of
finance. It recognizes that the value of money is different at different points
of time. Since money can be put to productive use, its value is different
depending upon when it is received or paid.
In simpler terms, the value of
a certain amount of money today is more valuable than its value tomorrow.
It is not because of the uncertainty involved with time but purely on account CHAPTER
Basic Concept of Time Value of Money
of timing. The difference in the value of money today and tomorrow is
referred as time value of money.
REASONS FOR TIME VALUE OF MONEY
Money has time value because of the following reasons:
1. 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. Cash inflows is
dependent out on our Creditor, Bank etc. As an individual or firm is
not certain about future cash receipts, it prefers receiving cash now.
2. 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 a year hence.
3. Consumption: Individuals generally prefer current consumption to
4. 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.
Thus, the fundamental principle behind the concept of time value of
money is that, a sum of money received today, is worth more than if the
same is received after a certain period of time.
For example, if an individual
is given an alternative either to receive ` 10,000 now or after one year, he
will prefer ` 10,000 now. This is because, today, he may be in a position to
purchase more goods with this money than what he is going to get for the
same amount after one year.
Thus, time value of money is a vital consideration in making financial
decision. Let us take some examples:
EXAMPLE 1: A project needs an initial investment of ` 1,00,000. It is
expected to give a return of ` 20,000 per annum at the end of each year, for
six years. The project thus involves a cash outflow of ` 1,00,000 in the ‘zero
year’ and cash inflows of ` 20,000 per year, for six years. In order to decide,
whether to accept or reject the project, it is necessary that the Present Value
of cash inflows received annually for six years is ascertained and compared
with the initial investment of ` 1,00,000.
The firm will accept the project only when the Present Value of cash
inflows at the desired rate of interest exceeds the initial investment or at
least equals the initial investment of ` 1,00,000.
EXAMPLE 2: A firm has to choose between two projects. One involves an
outlay of ` 10 lakhs with a return of 12% from the first year onwards, for
ten years. The other requires an investment of ` 10 lakhs with a return of
14% per annum for 15 years commencing with the beginning of the sixth
year of the project. In order to make a choice between these two projects,
it is necessary to compare the cash outflows and the cash inflows resulting
from the project. In order to make a meaningful comparison, it is necessary
that the two variables are strictly comparable. It is possible only when the
time element is incorporated in the relevant calculations. This reflects the
need for comparing the cash flows arising at different points of time in
TIMELINES AND NOTATION
When cash flows occur at different points in time, it is easier to deal with
them using a timeline. A timeline shows the timing and the amount of each
cash flow in cash flow stream. Thus, a cash flow stream of ` 10,000 at the
end of each of the next five years can be depicted on a timeline like the one
As shown above, 0 refers to the present time. A cash flow that occurs
at time 0 is already in present value terms and hence does not require any
adjustment for time value of money. You must distinguish between a period
of time and a point of time1. Period 1 which is the first year is the portion
of timeline between point 0 and point 1. The cash flow occurring at
point 1 is the cash flow that occurs at the end of period 1. Finally, the
discount rate, which is 12 per cent in our example, is specified for each
period on the timeline and it may differ from period to period. If the cash
flow occurs at the beginning, rather than the end of each year, the timeline
would be as shown in Part B. Note that a cash flow occurring at the end of
the year 1 is equivalent to a cash flow occurring at the beginning of year 2.
Cash flows can be positive or negative. A positive cash flow is called a cash
inflow; and a negative cash flow, a cash outflow.
1.4 VALUATION CONCEPTS
The time value of money establishes that there is a preference of having
money at present than a future point of time. It means
(a) That a person will have to pay in future more, for a rupee received
today. For example : Suppose your father gave you ` 100 on your
tenth birthday. You deposited this amount in a bank at 10% rate of
interest for one year. How much future sum would you receive after
one year? You would receive ` 110
Future sum = Principal + Interest
= 100 + 0.10 × 100
= ` 110
What would be the future sum if you deposited ` 100 for two years?
You would now receive interest on interest earned after one year.
Future sum = 100 × 1.102
= ` 121
We express this procedure of calculating as Compound Value or
Future Value of a sum.
(b) A person may accept less today, for a rupee to be received in the
future. Thus, the inverse of compounding process is termed as
discounting. Here we can find the value of future cash flow as on
1.5 TECHNIQUES OF TIME VALUE OF MONEY
There are two techniques for adjusting time value of money. They are:
1. Compounding Techniques/Future Value Techniques
2. Discounting/Present Value Techniques
The value of money at a future date with a given interest rate is called
future value. Similarly, the worth of money today that is receivable or payable
at a future date is called Present Value.
Compounding Techniques/Future Value Technique
In this concept, the interest earned on the initial principal amount becomes
a part of the principal at the end of the compounding period.
FOR EXAMPLE: Suppose you invest ` 1000 for three years in a saving
account that pays 10 per cent interest per year. If you let your interest
income be reinvested, your investment will grow as follows:
First year : Principal at the beginning 1,000
Interest for the year (` 1,000 × 0.10) 100
Principal at the end 1,100
Second year : Principal at the beginning 1,100
Interest for the year (` 1,100 × 0.10) 110
Principal at the end 1210
Third year : Principal at the beginning 1210
Interest for the year (` 1210 × 0.10) 121
Principal at the end 1331
This process of compounding will continue for an indefinite time period.
The process of investing money as well as reinvesting interest earned
there on is called Compounding. But the way it has gone about calculating
the future value will prove to be cumbersome if the future value over long
maturity periods of 20 years to 30 years is to be calculated.
A generalised procedure for calculating the future value of a single
amount compounded annually is as follows:
Formula: FVn = PV(1 + r)n
In this equation (1 + r)n is called the future value interest factor (FVIF).
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 taking into consideration, the above example, we get the same result.
FVn = PV (1 + r)n
= 1,000 (1.10)3
FVn = 1331
To solve 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 combination of ‘r’ and ‘n’. Exhibit 1.1 presents
one such table showing the future value factor for certain combinations of
periods and interest rates.
Exhibit 1.1 Value of FVIFr, n for various combinations of r and n
n/r 6% 8% 10% 12% 14%
2 1.124 1.166 1.210 1.254 1.300
4 1.262 1.360 1.464 1.574 1.689
6 1.419 1.587 1.772 1.974 2.195
8 1.594 1.851 2.144 2.476 1.853
10 1.791 2.159 2.594 3.106 3.707
12 2.012 2.518 3.138 3.896 4.817
Future Value of A Single Amount (Lumpsum)
The formula for calculating the Future Value of a single amount is as follows:
FVn = PV (1 + r)n
ILLUSTRATION 1: If you deposit ` 55,650 in a bank which is paying a 12
per cent rate of interest on a ten-year time deposit, how much would the
deposit grow at the end of ten years?
SOLUTION: FVn = PV(1 + r)n or FVn = PV(FVIF12%,10 yrs)
FVn = ` 55, 650 (1.12)10
= ` 55,650 × 3.106 = ` 1,72,848.90
1.6 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 = 1
m n r
× ⎛ ⎞ ⎜ + ⎟
where, FVn = Future value after ‘n’ years
PV = 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.
ILLUSTRATION 2: Calculate the compound value when ` 1000 is invested
for 3 years and the interest on it is compounded at 10% p.a. semi-annually.
BASIC CONCEPT OF TIME VALUE OF MONEY 7
SOLUTION: The formulae is
FVn = 1
m n r
× ⎛ ⎞ ⎜ + ⎟
2 3 .10
× ⎛ ⎞ × ⎜ + ⎟
= ` 1340
The compound value of Re. 1 at 5% interest at the end of 6 years is
` 1.340. Hence the value of ` 1000 using the table (FVIFr, n) will be
FVn = 1000 × 1.340
= ` 1,340
ILLUSTRATION 3: Calculate the compound value when ` 10,000 is invested
for 3 years and interest 10% per annum is compounded on quarterly basis.
SOLUTION: The formulae is
FVn = 1
m n r
× ⎛ ⎞ ⎜ + ⎟
4 3 .10
× ⎛ ⎞ ⎜ + ⎟
= 10,000 (1 + 0.025)12
= ` 13,448.89
ILLUSTRATION 4: Mr. Ravi Prasad and Sons invests ` 500, ` 1,000,
` 1,500, Rs 2,000 and ` 2,500 at the end of each year. Calculate the compound
value at the end of the 5th year, compounded annually, when the interest
charged is 5% per annum.
SOLUTION: Statement of the compound value
End of Amount Number of Years Compounded Interest Future
the Year Deposited Compounded Factor (FVIFr, n) Value
(1) (2) (3) (4) (2) × (4)
1 500 4 1.216 608.00
2 1,000 3 1.158 1,158.00
3 1,500 2 1.103 1,654.50
4 2,000 1 1.050 2,100.00
5 2,500 0 1.000 2,500.00
Amount at the end of 5th year is Future Value = 8020.50
8 FINANCIAL MATHEMATICS
1.7 FUTURE VALUE OF MULTIPLE CASH FLOWS
The above illustration is an example of multiple cash flows.
The transactions in real life are not limited to one. An investor investing
money in instalments may wish to know the value of his savings after ‘n’
years. The formulae is
FVn = 1
⎛ ⎞ ⎜ + ⎟
where FVn = Future value after ‘n’ years
PV = Present value of money today
r = Interest rate
m = Number of times compounding is done in a year.
1.8 EFFECTIVE RATE OF INTEREST IN CASE OF
Effective interest rate brings all the different bases of compounding such as
yearly, half-yearly, quarterly, and monthly on a single platform for comparison
to select the beneficial base. Now, the question is which works out highest
interest amount? When interest is compounded on half-yearly basis, interest
amount works out more than the interest calculated on yearly basis. Quarterly
compounding works out more than half-yearly basis. Monthly compounding
works out more than even quarterly compounding. So, if compounding is
more frequent, then the amount of interest per year works out more. Now,
we want to equate them for comparison.
Suppose, an option is given as the following:
Basis of Compounding Interest Rate
Now, the question is which basis of compounding is to be accepted to
get the highest interest rate. The answer is to calculate ‘Effective Interest
The formulae to calculate the Effective Interest Rate is
EIR = 1 1
⎛ ⎞ ⎜ + ⎟ −
where EIR = Effective Rate of Interest
r = Nominal Rate of Interest (Yearly Interest Rate)
m = Frequency of compounding per year
BASIC CONCEPT OF TIME VALUE OF MONEY 9
Take nominal interest rate as the base and find-out the comparable rate
of interest for half-yearly, quarterly and monthly basis and select that which
is most attractive.
(i) A company offers 12% rate of interest on deposits. What is the
effective rate of interest if the compounding is done on
(ii) As an alternative, the following rates of interest are offered for
choice. Which basis gives the highest rate of interest that is to be
Basis of Compounding Interest Rate
(i) The formula for calculation of effective interest is as below:
EIR = 1 1
⎛ ⎞ ⎜ + ⎟ −
(A) When the compounding is done on half-yearly basis:
⎡⎛ ⎞ ⎤ ⎢⎜ + ⎟ − ⎥
⎣⎝ ⎠ ⎦
= 1.1236 – 1
(B) When the compounding is done on quarterly basis
⎡ ⎤ ⎢ + ⎥ − ⎣ ⎦
(C) When the compounding is done on monthly basis
⎡ ⎤ ⎢ + ⎥ − ⎣ ⎦
10 FINANCIAL MATHEMATICS
Basis of Compounding Interest Rate EIR
Yearly 12% 12%
Half-yearly 12% 12.36%
Quarterly 12% 12.55%
Monthly 12% 12.68%
(ii) When the compounding is done on half-yearly basis
⎡ ⎤ ⎢ + ⎥ − ⎣ ⎦
When the compounding is done on quarterly basis:
⎡ ⎤ ⎢ + ⎥ − ⎣ ⎦
When the compounding is done on monthly basis
⎡ ⎤ ⎢ + ⎥ − ⎣ ⎦
Thus, out of all interest rate, interest rate of 11.75% on half-yearly
compounding works out to be the highest effective interest rate i.e.,
12.09% so this option is to be accepted.
ILLUSTRATION 6: Find out the effective rate of interest, if nominal rate of
interest is 12% and is quarterly compounded.
SOLUTION: EIR = 1 1
⎡⎛ ⎞ ⎤ ⎢⎜ + ⎟ − ⎥
⎣⎝ ⎠ ⎦
⎡⎛ ⎞ ⎤ ⎢⎜ + ⎟ − ⎥
⎣⎝ ⎠ ⎦
= [(1 + 0.03)4 – 1]
= 1.126 – 1
= 12.6% p.a.
The compound rate of growth for a given series for a period of time can
be calculated by employing the future value interest factor table (FVIF)
Years Profit (in Lakhs)
How is the compound rate of growth for the above series determined?
This can be done in two steps:
(i) The ratio of profits for year 6 to year 1 is to be determined i.e.,
(ii) The FVIFr,n table is to be looked at. Look at a value which is close
to 1.79 for the row for 5 years. The value close to 1.79 is 1.762 and
the interest rate corresponding to this is 12%. Therefore, the
compound rate of growth is 12 per cent.
1.9 DISCOUNTING OR PRESENT VALUE CONCEPT
Present value is the exact opposite of 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 process of determining present value of a
future payment or receipts or a series of future payments or receipts is called
discounting. The compound interest rate used for discounting cash flows is
also called the discount rate. In the next chapter, we will discuss the net
present value calculations.
1.10 SIMPLE AND COMPOUND INTEREST
In compound interest, each interest payment is reinvested to earn further
interest in future periods. However, if no interest is earned on interest, the
investment earns only simple interest. In such a case, the investment grows
Future value = Present value [1 + Number of years × Interest rate]
For example, if ` 1,000 is invested @ 12% simple interest, in 5 years
it will become
1,000 [ 1 + 5 × 0.12] = ` 1,600
12 FINANCIAL MATHEMATICS
The following table reveals how an investment of ` 1,200 grows over
time under simple interest as well as compound interest when the interest
rate is 12 per cent. From this table, we can feel the power of compound
interest. As Albert Einstein once remarked, “ I don’t know what the seven
wonders of the world are, but I know the eighth – the compound interest.
You may be wondering why your ancestors did not display foresight.
Hopefully, you will show concern for your posterity.”
Value of ` 1,000 invested at 10% simple and compound interest
Year Simple Interest Compound Interest
Starting Balance + Interest Starting Balance + Interest
= Ending Balance = Ending Balance
1 1,000 + 100 = 1,100 1,000 + 100 = 1,100
5 1,400 + 100 = 1,500 1,464 + 146 = 1,610
10 1,900 + 100 = 2,000 2,358 + 236 = 2,594
20 2,900 + 100 = 3,000 6,116 + 612 = 6,728
50 5,900 + 100 = 6,000 1,06,718 + 10672 = 11,7,390
100 10,900 + 100 = 11,000 1,25,27,829 + 12,52,783 = 1,37,80,612
ILLUSTRATION 7: Mr. Rahul has deposited ` 1,00,000 in a saving bank
account at 6 per cent simple interest and wishes to keep the same, for a
period of 5 years. Calculate the accumulated Interest.
SOLUTION: S1 = P0 (I) (n)
where S1 = Simple interest
P0 = Initial amount invested
I = Interest rate
n = Number of years
S1 = ` 1,00,000 × 0.06 × 5 years
S1 = ` 30,000
If the investor wants to know his total future value at the end of ‘n’ years.
Future value is the sum of accumulated interest and the principal amount.
FVn = P0 + P0(I) (n)
S1 + P0
ILLUSTRATION 8: Mr. Krishna’s annual savings is ` 1,000 which is invested
in a bank saving fund account that pays a 5 per cent simple interest. Krishna
wants to know his total future value or the terminal value at the end of a 8
years time period.
SOLUTION: FVn = P0 + P0 (I) (n)
= ` 1000 + ` 1000 (0.05) (8)
= ` 14,000
ILLUSTRATION 9: Suppose Mr. Jai Singh Yadav deposited ` 10,00,000 in
a financial institute which pays him 8 percent compound interest annually
for a period of 5 years. Show how the deposit would grow.
SOLUTION: FV5 = P0 (1 + I)8
FV5 = 10,00,000 ( 1 + 0.08)5
= 10,00,000 (1.469)
FV5 = ` 14,69,000
Note: See compound value of one rupee Table for 5 years at 8% interest.
Variable Compounding Periods/Semi-annual Compounding
ILLUSTRATION 10: How much does a deposit of ` 40,000 grow in
10 years at the rate of 6% interest and compounding is done semi-annually.
Determine the amount at the end of 10 years.
SOLUTION: FV10 =
⎛ ⎞ ⎜ + ⎟
2 10 0.06
Rs. 40,000 1
× ⎛ ⎞ ⎜ + ⎟
= ` 40,000 (1.806)
= Rs. 72,240
Alternatively, see the compound value for one rupee table for year 20
and 3% interest rate.
ILLUSTRATION 11: (Quarterly compounding): Suppose a firm deposits
` 50 lakhs at the end of each year, for 4 years at the rate of 6 per cent
interest and compounding is done on a quarterly basis. What is the compound
value at the end of the 4th year?
SOLUTION: FV4 =
× ⎛ ⎞ ⎜ + ⎟
4 4 6
Rs. 50,00,000 1
× ⎛ ⎞ ⎜ + ⎟
= ` 50,00,000 × 1.267
= ` 63,35,000
14 FINANCIAL MATHEMATICS
Compound Growth Rate
Formula: gr = V0 (1 + r)n = Vn
where, gr = Growth rate in percentages
V0 = Variable for which the growth rate is needed
Vn = Variable value (amount) at the end of year ‘n’
(1 + r)n = Growth rate.
ILLUSTRATION 12: From the following dividend data of a company, calculate
compound rate of growth for 2003–2008.
Year Dividend per Share (`)
SOLUTION: gr = V0 (1 + r)n = Vn
= 21 (1 + r)5 = 31
= (1 + r)5 =
Alternatively, the compound value one Rupee table for 5 years should
be seen till closed value to the compound factor is found. After finding the
closest value, first above it is seen to get the growth rate.