Tuesday, January 21, 2014

ERP FOR COMPETITIVE ADVANTAGE

ERP FOR COMPETITIVE ADVANTAGE

Enterprise Resource Planning (ERP) systems are core software programs developed for use by corporations to help them integrate their information system for each and every area of the business. ERP programs are designed to bring and create a common database out of many that are available to them ,to manage a single business process. A business process can be defined as a set of collective activities that takes inputs gathered for processing and delivers output in the form of a report or forecasting of some sort , which becomes meaningful to the user. ERP software’s when integrated supports the most efficient operation any kind of business process by seamlessly integrating various tasks such as marketing and sales , production , transportation and logistics , Accounting and finance , human resource and staffing to name  a few.Moreover ERP brings Automation to most of the processes involved .
But to implement a fully automated ERP platform an organisation needs better trained staff which has the requisite knowledge of using it to derive maximum benefit.
Caution
Having said that the real problem lies in the selection and integration of the ERP in the Long Run. It’s a very arduous task  and the possibility of failing is quite high. What an organisation can and must do in such a scenario is to find the right people to handle the ERP system. There are at least three basic  parameters to be fulfilled to manage  ERP in the best possible manner.  People are at the forefront, then the existing processes and then of course the technology involved. ERP is always a change that’s required but People involved in this needs to be trained so that they understand the change and implement it wholeheartedly.
The financial benefits of an ERP system is always pretty daunting to calculate because sometimes ERP increases sales and decreases expenditure in many ways which can’t be measured in the normal manner. Similarly there will be changes over a long period of time and can’t be tracked and be converted into figures . So there is every chance that initially ERP will be resented by some.
But ERP overall changes the group dynamics and bring very radical ideas to the table.The overall advantages brought in by ERP can be analysed from the following real life cases:

ERP Allows for Expansions
Au Bon Pain, the bakery and café chain based in Boston, operates 200 outlets in the United States and Asia. Over the last three years, Health magazine has named Au Bon Pain one of America’s top five healthiest fast-food restaurant chains. The company had an interest in expanding, and Tim Oliveri, the company’s chief financial officer, wanted Au Bon Pain to be able to react to market changes more rapidly while also reducing its costs.
The company’s existing information systems were holding it back, but the company decided to bring in a new SAP ERP system and helped itself  to achieve these goals.The new ERP system replaced disparate systems, some of which were in paper format. This integrated enterprise system brings the storefront to the back office through a number of different module implementations. With the single system, the company is able to reduce its financial reporting cycle from weeks to days, and the system allows better management of its  workforce through a Web-based portal. In addition, the new system facilitates greater compliance with financial regulations and better sales management, along with the electronic purchasing of all raw materials.
One impetus for installing this ERP system was for management to be able to analyse and digest information on market conditions, and react quickly to open new stores or renovate existing stores. For example, cafés in New York City have recently undergone major renovations, resulting in double-digit sales increases. More cafes are planned to open soon.

Sales and Operations Planning (SOP)
Sales and operations planning (SOP) is typically used in for-profit manufacturing organizations; however it can be applied to others as well.
For example, the non-profit America’s Blood Centres is a network of more than 600 donor centres supplying over 3,500 hospitals and healthcare facilities in North America.
As with any business, a blood centre must quickly identify variations in forecasted supply and demand and take action to lessen the potential negative effects of those variations. For instance, if the demand for SDPs at a blood centre falls below projections, “the excess SDPs may be exported if they have sufficient shelf life remaining, or donors may be rescheduled to reduce supply. If SDP demand spikes, recruitment and collections may have to work overtime to recruit more donors.”
Blood banks may not be for-profit companies, but they cannot operate at a loss either. By using measurements, such as RONA (return on net assets), a blood centre can help blood centres avoid situations in which “the cost per unit (CPU) collected is higher than the sales cost of a unit to the health care provider or what an imported unit would cost.”
One of the major challenges to applying the sales and operations planning process to blood banks is data accuracy. The authors note that it is not uncommon to have data inconsistencies between the blood management system and the financial system. Without accurate and consistent data, the results of the sales and operations planning process would not be valid.
Solving Complexities of Supply Chain Management with ERP
Over the last few decades, a number of industry trends have focused on making supply chains more efficient and effective. The first of these started in the 1980s when techniques developed at the Toyota Motor Company, now commonly referred to as lean manufacturing, started to be accepted and copied by companies around the world. The main idea embodied in lean manufacturing or the Toyota Production System (TPS) is that waste must be eliminated. The TPS identified inventory resulting from “overproduction” as waste. Much of industrial practice, especially in the United States, was focused on the idea that efficiency should be measured by having each machine produce to its maximum capacity. The TPS approach was that a company should produce exactly what is required when it is required. The TPS emphasized “pull” systems, where customer demand triggers production, rather than “push” systems, where planning methods like MRP are frequently used to push material through the supply chain to the customer in anticipation of demand.
A second important trend has been a decrease in vertical integration of companies. It means  the extent to which a company produces the components and assemblies used in the products it manufactures.
A third trend is supply base rationalization, an effort to determine the “appropriate” number of suppliers a company should have. Many large firms have thousands of suppliers. However, the rationalization concept says that better supplier relationships can be developed when there are fewer suppliers to manage, and costs will be reduced when a smaller number of suppliers have higher production volumes. In 2008, Ford’s Senior Vice President of Global Purchasing announced that Ford would be reducing its supply base from over 2,000 suppliers to 700 to 800 over a period of years.
These trends—lean manufacturing, reduced vertical integration, and supply base rationalization—have been implemented by manufacturers in a number of industries over the past 30 years and have resulted in higher efficiencies and reduced cost.
Japanese-based companies such as Toyota were obviously affected by disruptions from time to time, but the integrated nature of global supply chains meant that the effect of the earthquake rippled throughout the world. For example, the three largest shipbuilders in the world are located in Korea and are dependent on Japanese steel, and the Japanese steel industry was significantly impacted by the Fukushima nuclear power plant disaster, which was triggered by the tsunami that followed the earthquake.
Toyota is now planning to lead the way in reducing supply chain risk. In September 2011, they announced plans to  create a robust supply chain that would recover within two weeks in the event of another massive disruption. This plan consists of three fundamental steps. First, Toyota intends to push for further standardization of parts across Japanese automakers so they can share common components, which could, in turn, be manufactured in several locations. The second element in the plan is the development of technology that would provide more options for parts that require unique materials or components that are only available from one source. Finally, Toyota plans to make each region in the world independent in its parts procurement so that a disaster in Japan would not affect production overseas & vice versa.
Conclusions
Today’s Managers think in terms of Business processes integrating all the related functional areas and bringing in efficacy & competitiveness. To integrate and cooperate all the things, there is need for sharing of information between various platforms and relates partners.ERP brings these capacity by integrating all the different sources of information to a common information platform. for example when Marketing and sales people use the same data platform ,efficiency will kick in. There are various fundamental changes happening in the ERP sector as envisaged by the TOYOTA example. The need for doing business in a coherent manner gives rise to new innovations each and every day.
Technologies such as radio frequency identification (RFID) and smartphones are fueling explosive growth in the amount of data available for businesses to process. This data can provide real business value, but the challenge is how to manage and evaluate this data to gain business knowledge. Business intelligence (BI) tools are growing in sophistication and power. Technologies such  as in-memory computing will provide greater speed and flexibility to BI users and will increase the number of employees in a company that can make use of BI.
Mobile computing technology is increasing the use of ERP and BI data by making it more convenient to access data when and where it is needed.  Cloud computing has also come in which is the delivery of a software product to a user via the Internet. The user typically accesses the cloud product through a Web browser or a lightweight (meaning small and simple) application for a computer or mobile device.
All these innovations while being essential and required for cutting edge technology in sales and Marketing they also bring in the advantage of First Mover into the industry. Those who foresees the importance of them earlier than others reaps the benefit more than anybody else in the Market. One has to not only move ahead to fuel growth and sustainability but has to constantly endeavour to remain that way. Unless corporations find newer methods and programmes to redesign and redevelop the competitive advantage can be frivolous in the long Run.

References
1. http://scn.sap.com/community/erp/blog/2013/03/11/a-new-application-that-makes-it-easier-to-find-sap-correction-notes(Kristen Scheffler) March 11,2013
2. http://www.aabschools.com/case_studies/587.html(david west)
3. ERP demystified by Alexis Leon Tata McGraw-Hill Education, 2008

4. ERP: Making It Happen: The Implementers' Guide to Success with Enterprise  By Thomas F. Wallace 2001

5. Enterprise Resource Planning  By Bret Wagner, Ellen Monk,2008




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ACCOUNTING FRAUD AT SATYAM COMPUTERS

ACCOUNTING FRAUD AT SATYAM COMPUTERS
OVERVIEW OF THE COMPANY

Satyam Computers Services Limited (SCSL) was registered as a PVT Ltd organisation in AP in the year 1987. Soon in the year 1991 it became a Limited Company. It became the fourth largest IT services company in India and in the year 1995 it received a certification for ISO 9001. Satyam computers offered a large no of services and expertise in the fields of Development of software’s , System Embedment , various Engineering Services such as CAD and CAM , ERP management , EAI ,CRM ,SCM,E-Commerce and Consulting.  In the year 2001, the company was given IMC Ramkrishna Bajaj National Award for its contribution in the field of Service.
In the year 2002 company expanded its operations into China. In the same year Satyam was cited as 'Top choice for SAP support' by Giga Research group. In 2003 the Group announced a new business center in Singapore and also launched a Global solution center. Satyam was given the prestigious IBM Lotus award for its innovative knowledge & content management solutions in the year 2003. It was also given an Outsourcing contract by WORLD BANK worth $15 million in 2003. It inaugurated a new development center in Mississauga, Canada in 2004. Satyam also acquired Citisoft Plc a UK based company and Dynamics, a leading data warehousing company in Singapore in the year 2005. In 2006 Satyam received the best performing stock from CNBC and Excellence in cost management from ICWAI. In the year 2007, Satyam acquired Nitor Global Solutions Ltd, a UK based company in a deal worth $5.5 million. In the same year it established a new development centre in Brisbane to support the fields of Local Governance, insurance, mining and Finance. Satyam was the proud winner of Corporate Social Responsibility Award in ASIA under the poverty alleviation category. In the year 2008, Satyam Computers (SCSL) became the first company registered in India to list its American Depository Receipts or Shares (ADR) on NYSE and Euronext in Amsterdam. (I M Pandey-2013)

HOW THINGS TURNED OUT ?

SCSL had many fortune 500 companies as clients including Unilever, Nestlé ,Cisco, GE, Sony, World Bank,  GM, United States government etc. in quite a few case SCSL had even a role in the maintenance of its  client companies accounting and finances.
The fraud unveiled at SCSL was India’s largest corporate scandal till date and the first casualty in the wake of Global financial meltdown in 2007-2008. But before the fraud was detected and made public Satyam was always under close scrutiny and watch of concerned agencies for various conflicting reports and seemingly ill directed activities.   

In early October 2008 there was a  report that the company was immediately banned from any further World Bank contracts , when it was reportedly found to have installed spy software’s and cameras in some of the World Banks computers. Saytam denied but late it was confirmed by the Bank In December 2008, Satyam’s shareholders revolted the news that the company has agreed to buy two entities which allegedly had ties to the incumbent Chairman and his Son.  By the end of the year 2008 , the board of directors refused to back the chairman and the CEO on the matter and four Board members resigned by February  2009.

The books of Accounts of Satyam was audited for the last many years by PriceWaterCooper (PwC) and they just overlooked somehow the underlined fraud.
This accounting fraud looked very obvious and basic when the then chairman and Satyam’s founder, Mr Ramalinga Raju admitted (Rishi Mehra -2011) the following facets about the Balance Sheet dated 30th Sept 2008:
(a) About Rs 5040 crores which consisted of almost 95% of the cash at bank as on Sept’08 was fictitious, because of largely inflated profit margins and lots of fictitious Assets.
(b)    There was an item under the heading of Current Assets called accrued interest  to the tune of  Rs. 376 crore which  did not exist and just fabricated.
(c)     There was an understatement of a particular liability to the tune of Rs 1230 crores, which was on account of funds arranged by him and the promoter’s family.
(d)    The balance sheet also showed Debtors position to be in excess by atleast Rs. 490 crore.

It was strange that the extraordinarily large fictitious cash pile of bank deposits claimed by successive Balance sheets were never verified properly and its existence on a long term basis raised enough doubt about the quality of the audit done by internationally known Auditors. At the same time it was also surprising for everybody that the Board of Directors which consisted of very highly skilled and experienced individuals  did nothing substantial to avert the potentially damning situation.

The first level of corporate governance in a company is its board of directors itself and in case of SCSL the board comprised of very big names including Mr Vinod Dham who worked with INTERL corporation, professor Krishna Palepu of HBS and former Cabinet Secretary TR Prasad. It was intriguing that such an assembly of the intellect could not detect the fraud while it was going on .
This scandal also raised questions about the way outsourcing companies are being regulated and particularly audited. The failure of regulatory authority was so complete that it was joked that a blind and deaf man would have easily found it, if he wanted it to.
Starting in the financial year 2006, Mr B Ramalinga Raju raised money by hypothecating or pledging his family’s shareholding in the company which was little over 8% to outside lenders to secure collateral loans. He then reinvested the money so garnered back into the company to cover up the shortfall of revenue and profits.  Then the Global financial meltdown happened, share prices  fell by more than 40% and thus Lenders became jittery and began selling the collateral . This aggravated the already grim situation and at this juncture B Ramalinga Raju tried to buy two other companies promoted by the family into Satyam’s fold for a reported $1.6 billion. This could have wiped the fictitious assets such as cash and had the potential to bring more assets into the fold but the scheme could not go through because of staunch resistance from lenders as well as other investors (particularly institutional investors).
Indian Stock market Regulator (SEBI) launched an investigation into the fraud in December 2008 .At the same time US Authorities raised objections and concerns with the statutory Auditor Price Waterhouse, whose parent company was listed in US , regarding its audit of Satyam,after their visit to the company headquarters in early 2009.
According to Mr B Ramalinga Raju, the cover-up was started as an attempt to keep under disguise the poor Quarterly results of the company. In a letter to the Board of Directors of the company in Early January 2009 he clearly admitted to inflating the latest quarterly result that the company came out with in SEPTEMBER 2008. He admitted that in that Quarter the company reported a revenue of 2700 crores and an operating profit (EBIT) of 649 crores amounting to almost 24% margin where as the actual results were a revenue of 2217 crores and an operating profit (EBIT) of 61  crores only which amounted to only 3% margin. This resulted in an artificial cash inflow of almost 588 crores in the Q2 of 2008-2009 FY.

Fabricated Balance Sheet and Income (P/L Account) Statement of Satyam: As of September 30, 2008
                                                (Rs. in crore)
 Actual       Reported    Difference
Cash and Bank Balances               321           5,361        5,040
Accrued Interest on bank FDs        Nil             376.5        376
Understated Liability                    1,230        None         1,230
Overstated Debtors                      2,161        2,651        490
Total                                         Nil             Nil             7,136
Revenues (Q2 FY 2009)                2,112        2,700        588
Operating Profits                         61             649           588
When further audit were done later it was found out that only the operations of SCSL was continued to be exaggerated and not the same was done with any of the subsidiaries. Over a period of last one year alone revenues were shown in excess by 3,000 crores (Actual  8,392 crores and shown as 11,276 crores) and reserves were inflated as well.
Because of the staggering amount of difference between these two figures , the company had to carry some additional resources and various other non operational assets to justify the exaggerated amount of operations in case there is a probe and thus the operating costs got bigger by the day and accentuated the matter further.
Every attempt was made to reduce and even completely eliminate the gap. But they failed. Because the promoters only held 8% of the stock , they feared the failure to increase revenue will lead to the company being taken over and thus they tried in a last ditch attempt to infuse more assets and slush out the nonexistent cash by bringing in some from outside. That’s when they tried to buy the family owned Maytas infrastructure to its fold and infuse cash. Once Satyams problem was solved they can settle for cash to paid to Maytas, and more importantly the can be made much later without burdening the cash flows. But others resisted.
Immediately after the confessions made by Raju the shares of the plunged  by 80%  from a high of Rs 544 in may 208 to Rs 11.50 in January 2009. In NYSE , Satyam’s  shares was selling at $29.10 in july 2008 but by contrast in March 2009 they drastically fell to US $1.80. This amounted to investors losing more than $2.82 billion in investment.(BBC News, 2009).

WHO WAS IMPLICATED?
a. B. Ramalinga Raju (chairman): He claimed in his letter to the Board that nobody else was aware of the fraud and particularly BOARD was innocent.
b. Audit Firm PwC.
This Audit firm audited Satyam’s books starting from the year 2000 till 2009. Several experts criticized the auditors in very harsh words  for failing to detect the fraud particularly the existing non-interest-bearing deposits of more than $1 billion.
According to many professionals, any worthwhile auditing firm would have discovered and questioned the logic behind having such an enormous amount in non-interest bearing assets.  This should have been seen as a red flag by PWC and it failed to do so. Further, it was apparent that the  auditors did not independently try to verify for a few years with the banks in which the management of Satyam claimed to have these deposits” (Blakely, 2009).
c. DSP Merrill Lynch: This US company advised Satyam for the last ten years regarding issue of new shares and reportedly helped the company’s listing in US.

WHO THE SCANDAL AFFECTED?


Immediately following the declaration by Raju of the fraud, Merrill Lynch terminated its contract  with SCSL , Credit Suisse relegated SCSL to third tier and the license of PWC to operate in India was revoked. All the awards given bestowed on SCSL was stripped off (Agarwal and Sharma, 2009).

Some of the parties affected by the scam, according to Manoharan (2011), were:
• More than 60,000 Employees of Satyam dealt with nightmares, non payment of salaries, the spectrum of losing their jobs.
• Clients of Satyam seemed to lost their Trust in SCSL. Many fortune 500 clients like  Cisco, Telstra and World Bank cancelled their contracts with immediate effect.
• Shareholders who lost their investments lost faith in the system.
• Bankers and lenders were on tenterhooks for the timely recovery of their claims and a possible financial and non-financial exposure.

HOW IT COULD HAVE BEEN AVOIDED?

Accounting frauds are related to manipulative financial reporting with intentional misstatement, missing correct figures knowingly and failing to disclose information correctly and with the sole intention of deceiving the end users. The world has seen its share of accounting frauds but they can be avoided with the help of following precautions:

1.  People in charge must understand in general why employees or managerial personnel create fraudulent behaviour and take part in financial frauds. People commit some because they are under pressure or they do it to relieve pressure without getting caught and have justification to make a fraud. Better internal controls and fine leadership will mitigate the problem.

2. Internal control and audit: 
Internal audits are a constant operation to contain frauds and helps evaluating the processes more precisely and thus assessing the procedures to see whether the control systems in place are adequate or not. If necessary they bring to light the needs of revision and solutions.

3.  Whistleblower Program or internal Watch

In every organisation employees must be encouraged to voice their opinion to a designated committee or seniors. And at the same time it shall be ensured that the grievances are heard and dealt with. If somebody brings up or notices an unethical behaviour , he/she must not be punished. That way internal watch works.


4. Independent Audit

Frauds made by low level employees can be detected through internal audits and higher management checks. But it’s difficult catch the big guns by internal audit alone. Thus there is a need to have independent auditors to check senior management and tricky transactions. 
5. Minimize cash transactions: 
This thing is of utmost importance. Each company shall demarcate the policy by which employees are allowed to handle cash themselves. If the amount exceeds the denoted one then they must get seniors  approval to do it. And in case of directors and others a disclosure to the BOARD of DIRECTORS is a must. At no instance a person disbursing cash shall be from the person in charge of Accounts.

6. Reconcile various Accounts regularly: the Auditors must audit and make reconciliation of statements of cash and bank balances physically and must make sure there is no discrepancy in them. All the bank accounts and credit transactions be reconciled and reported. 

7. Surprise Audits and checks:  Audit committees and independent Auditors must conduct surprise audits of Books of Accounts and transactions which occur more discretely. Surprise checks shall also be conducted for Different branches and offices regularly to keep fraudsters under check.

In more instances internal Employees and officers are the ones who indulge in frauds. This happens because of the oversights on the part of the management and auditors and also because of the lack of internal controls in place.


CONCLUSIONS

The management culture at Satyam, when it was dominated by one man , that of their chairman , symbolized an unethical corporate culture. He not only disregarded but also manipulated the BOD. There was urge on his part to impress the investor and thus compelled him to fudge the results. He was under pressure. But at the same time the independent experts sitting at the Board turned a blind eye to what’s happening around them and did not have a clue.  The board almost connived with his actions and turned itself to a blind spectator, where as they shall have seen the real picture and rectified or advised the true sense into it; the lure of big and fat salary and other form of compensations further encouraged such behaviour. In the end they allowed their behaviour to have destroyed the trust of the investors and the image of nation as well.

Michael Useem, management professor at Wharton noted that Satyam case was little bit unusual in a way that it is listed in NYSE. It’s difficult to get funding at low cost in India thus there was this necessity to issue ADRS in US Market. By entering into the US market Satyam accepted the possibility of greater scrutiny and higher amount ethical governance yet the scandal happened at the company. According to his opinion the issue was indeed disturbing.
But Indian IT sector has come a long way since and overcome the inadvertent situation tactically by being more transparent.

In the month of June ,2009, the company named itself Mahindra Satyam after it was taken over after a SEBI led well rounded bidding process by the MAHINDRA GROUP in the month of April ,2009.Finally the company merged into the flagship MAHINDRA company TECH MAHINDRA in 2013. The eventual merger was much delayed due to a pending income tax claim of  Rs 2700 crores . When merged the new entity created India’s fifth largest technology company with annual revenue of over $3billion.  



SAMARESH CHHOTRAY
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Thursday, January 17, 2013

TIME VALUE OF MONEY


TIME VALUE OF MONEY

Learning Objectives
After studying this chapter, you will be able to understand
♦ The concept of time value of money;
♦ Techniques of Discounting and Compounding;
♦ Identify the equation for calculating the present value of an annuity and calculation of the present value of an annuity; and
♦ Identify the equation for calculating the future value of an annuity and calculation of the future value of an annuity.

Money NOW is worth more than money LATER!
THE CONCEPT OF TVM REFERS TO THE FACT THAT THE MONEY RECD TODAY IS DIFFERENT IN ITS WORTH FROM THE MONEY RECEIVABLE AT SOME OTHER TIME IN THE FUTURE.IN OTHER WORDS THE SAME PRINCIPLE CAN BE APPLIED TO STATE THAT MONEY RECEIVABLE IN FUTURE IS LESS VALUABLE THAN VALUE RECEIVABLE TODAY ITSELF.
Time Preference for Money
Time preference for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time. Three reasons may be attributed to the individual’s time preference for money:
n   risk
n   preference for consumption
n   investment opportunities

For example Rs 1000 now is not equal to Rs 1000 in 1 year’s time. Somebody who does not know finance will also prefer the money now to money anytime in the future. That is time preference for money
Required Rate of Return
The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate.
An investor requires compensation for assuming risk, which is called risk premium.
The investor’s required rate of return is:
          Risk-free rate + Risk premium.

Why TIME?
Why is TIME such an important element in your decision?
TIME allows you the opportunity to postpone consumption and earn INTEREST.

The reason why there is time value of money is as follows:
Opportunity Cost: There are alternative productive uses of money. The cost of any decision includes the cost of the next best opportunity forgone. You can save and invest, get interest and spend.
Inflation: It erodes the value of money.
Risk: There are always financial and non-financial risks involved.
The trade-off between money now and money later depends on, among other things, the rate of interest you can earn by investing. It impacts business finance, consumer finance and government finance. Time value of money results from the concept of interest.
Interest rate is the cost of borrowing money as a yearly percentage. For investors, interest rate is the rate earned on an investment as a yearly percentage.

SIMPLE INTEREST
Interest paid (earned) on only the original amount, or principal, borrowed (lent).
SIMPLE INTEREST IS COMPUTED ON THE PRINCIPAL FOR THE ENTIRE PERIOD IT IS BORROWED.

simple interest   =     
S = P + Prt        
total amount A  in simple interest  = 

Example 1:
Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years.  What is the accumulated interest at the end of the 2nd year?
SI       = P (r)(t) = $1,000(.07)(2)         = $140
Compound Interest:
Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Compound interest means interest calculated on the principal for the first year and then interest is not withdrawn – is reinvested. Interest is added to the principal and interest is calculated again
  1. Compound interest  sum (S)   = 

the amount calculated on the basis of compound interest rate is higher than when calculated with the simple rate. The time interval between successive additions of interests is known as conversion (or payment) period. Typical conversion periods are given below:

Conversion Period
Description
1 day
Compounded daily
1 month
Compounded monthly
3 months
Compounded quarterly
6 months
Compounded semiannually
12 months
Compounded annually

EXAMPLE 2: Rs. 2,000 is invested at annual rate of interest of 10%. What is the amount after 2 years if the compounding is done:
(a) Annually? (b) Semi annually? (c) Monthly? (d) Daily?
Solution
(a) The annual compounding is given by: = 2,000 (1.1)2 = 2,000 × 1.21 =
Rs. 2,420
(b) For Semiannual compounding, n = 2 × 2 = 4, I = 0.1/2 = 0.05
A = 2,000 ( 1 + 0.05)4 = 2,000 × 1.2155 = Rs. 2,431
(c) For monthly compounding, n = 12 × 2 = 24, i = 0.1/12 = 0.00833
A = 2,000 (1.00833)24 = 2,000 × 1.22029 = Rs. 2440.58
(d) For daily compounding, n = 365 × 2 = 730, i = 0.1/(365) = 0.00027
A = 2,000 (1.00027)730 = 2,000 × 1.22135 = Rs. 2,442.70
















COMPOUND INTEREST VERSUS SIMPLE INTEREST
The given figure shows graphically the differentiation between compound interest and simple interest. The top two ascending lines show the growth of Rs. 100 invested at simple and compound interest. The longer the funds are invested, the greater the advantage with compound interest. The bottom line shows that Rs. 38.55 must be invested now to obtain Rs. 100 after 10 periods. Conversely, the present value of Rs. 100 to be received after 10 years is Rs. 38.55.
                              Compound Interest versus Simple Interest
Example 3: Julie wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years.
Calculation based on general formula:  FVn   = P(1+i)n                                            
FV5     = $10,000 (1+ 0.10)5 =  $16,105.10
Calculation based on Table:   FV5     = $10,000 (FVIF10%, 5)
= $10,000 (1.611)       = $16,110           [Due to Rounding]

Example 4: If you invest $1,000 today at an interest rate of 10 percent, how much will it grow to be after 5 year ?
FVn = P (1+i)n
FVn = 1,000(1.10)5
FVn = $1,610.51
Example 5: Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%.
Annual    FV           = 1,000(1+ [.12/1])(1)(2)            = 1,254.40
Semi - annual FV   = 1,000(1+ [.12/2])(2)(2)            = 1,262.48
Qrtly  FV                 = 1,000(1+ [.12/4])(4)(2)          = 1,266.77
Monthly FV            = 1,000(1+ [.12/12])(12)(2)        = 1,269.73
Daily  FV                = 1,000(1+[.12/365])(365)(2)     = 1,271.20

Effective Annual Interest Rate
The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year.
(1 +  [ i / m ] )m – 1)
Example 6: Basket Wonders (BW) has a $1,000 CD at the bank.  The interest rate is 6% compounded quarterly for 1 year.  What is the Effective Annual Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1         = 1.0614 - 1  = .0614 or 6.14%

Example 7: If the interest is 10% payable quarterly, find the effective rate of interest.
E = ( 1 + 10% / 4 )4 - 1         = = 0.1038 or 10.38%

TECHNIQUES OF DISCOUNTING
The present value of a sum of money to be received at a future date is determined by discounting the future value at the interest rate that the money could earn over the period. This process is known as Discounting. The figure below shows graphically how the present value interest factor varies in response to changes in interest rate and time. The present value interest factor declines as the interest rate rises and as the length of time increases.


Present Value
Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision-maker.
n  Discounting is the process of determining present value of a series of future cash flows.
n  The interest rate used for discounting cash flows is also called the discount rate.

Present Value of a Single Cash Flow
n  The following general formula can be employed to calculate the present value of a lump sum to be received after some future periods:
             PV =FV x PVF (r,n)  or  S   [   1/ (1+i)n] or S(1 + i) −n
n  The term in parentheses is the discount factor or present value factor (PVF), and it is always less than 1.0 for positive i, indicating that a future amount has a smaller present value.

Example 8: Suppose that an investor wants to find out the present value of Rs 50,000 to be received after 15 years. Her interest rate is 9 per cent. First, we will find out the present value factor, which is 0.275. Multiplying 0.275 by Rs 50,000, we obtain Rs 13,750 as the present value:     
PV =FV x PVF (r,n)  =  FV x PVF (.09,15)  = 50,000 x .275 = 13,750.
Example 9 : Mr. X has made real estate investment for Rs. 12,000 which he expects will have a maturity value equivalent to interest at 12% compounded monthly for 5 years. If most savings institutions currently pay 8% compounded quarterly on a 5 year term, what is the least amount for which Mr. X should sell his property? Given that (1 + i)n = 1.81669670 for i = 1% and n = 60 and that (1 + i) −n = 0.67297133 for i = 2% and n = 20.
Solution :
It is a two-part problem. First being determination of maturity value of the investment of Rs. 12,000 and then finding of present value of the obtained maturity value.
Maturity value of the investment may be found from An = P (1+i)n
Now, An = 12,000 (1+1%)60 = 12,000 × 1.81669670  = 21,800.36040000 = Rs. 21,800.36
Thus, maturity value of the investment in real estate = Rs. 21,800.36
The present value, P of the amount An due at the end of n interest periods at the rate of i% interest per period is given by P = An (1 + i) −n
Thus, P = 21,800.36 (1+ 2%)−20
= 21,800.36 × 0.67297133 = Rs. 14,671.02
Mr. X should not sell the property for less than Rs. 14,671.02
Future Value of a Single Cash Flow
The general form of equation for calculating the future value of a lump sum after n periods may, therefore, be written as follows:


The term  (1 + i)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 i, indicating that CVF increases as i and n increase.

Example 10: If you deposited Rs 55,650 in a bank, which was paying a 15 per cent rate of interest on a ten-year time deposit, how much would the deposit grow at the end of ten years?


We will first find out the compound value factor at 15 per cent for 10 years which is 4.046. Multiplying 4.046 by Rs 55,650, we get Rs 2,25,135 as the compound value:
FV =  PV  x CVF (r,n)   = PV  x CVF (.15,10)   =    55,650 X 4.045555 = 2,25,135.28





























ANNUITY
An annuity is a stream of regular periodic payment made or received for a specified period of time. A recurring deposit with the bank is typical example of an annuity.
An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
Ordinary Annuity:  Payments or receipts occur at the end of each period.
Annuity Due:  Payments or receipts occur at the beginning of each period.
Examples of Annuities

  • Student Loan Payments
  • Car Loan Payments
  • Insurance Premiums
  • Mortgage Payments
  • Retirement Savings

If you make payments of $2,000 per year into a retirement fund, it is an annuity.
If you receive pension checks of $1,500 per month, it is an annuity.
If an investment provides you with a return of $20,000 per year, it is an annuity.
For the future value of an annuity:
FV = PMT  [(1+i)n - 1]/I]
FV = PMT x FVIFA ( i, n)
Example 11: If you save $50 per month at 12 percent per annum, how much will you have at the end of 20 years?
Note that since time periods are months, i = 12%/12 months = 1% per period, for 240 periods.
    FV = PMT[(1+i)n - 1]/i
    FV = 50[(1.01)240 - 1]/.01
    FV = $49,463
Example 12: If you want to save $500,000 for retirement after 30 years, and you earn 10 percent per annum, how much must you save each year?
FV = PMT[(1+i)n - 1]/i
500,000 = PMT[(1.1)30 - 1]/.1
PMT = $3,040 per year
Example 13: Find the amount of an annuity if payment of Rs. 500 is made annually for 7 years at interest rate of 14% compounded annually.
Solution
Here P = 500, n = 7, i = 0.14
A = Rs. 500 × A (7, 0.14) = 500 × 10.7304915 = Rs. 5,365.25
Present Value of an Annuity
The computation of the present value of an annuity can be written in the following general form:
The term within parentheses is the present value factor of an annuity of Re 1, which we would call PVFA, and it is a sum of single-payment present value factors.
Example 14: An investor deposits a sum of Rs 1, 00,000 in a bank account on which interest is credited @10% per annum. How much amount can be withdrawn annually for a period of 15 years?
Answer:
in  this case the deposit of Rs 1,00,000 can be viewed as the present value of future annuity of 15 years@10%. the situation can be also presented as follows:
Rs 1,00,000 = Annuity x PVAF(10% ,15 years)
                   = annuity  x 7.606
Annuity  = 1,00,000 / 7.606 =13,148.
This means the investor can withdraw Rs 13,148 for the next 15 years.








Future Value of an Annuity
Annuity is a fixed payment (or receipt) each year for a specified number of years. If you rent a flat and promise to make a series of payments over an agreed period, you have created an annuity.
FV =   A x CVAF (r, n)
FVA ( future value of an annuity)  =  S =   
(The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA or CVAF)
Example 14: Suppose that a firm deposits Rs 5,000 at the end of each year for four years at 6 per cent rate of interest. How much would this annuity accumulate at the end of the fourth year? We first find CVFA which is  4.3746. If we multiply 4.375 by Rs 5,000, we obtain a compound value of Rs 21,875:
A x CVAF (r, n)     = 5000  x  CVAF (4,.06) = 5000 x 4.3746   =  21,873
Perpetuities
A perpetuity is simply an annuity that continues forever  perpetually).The formula for finding the present value of a perpetuity is:
PV = PMT/i

A variation to perpetuity problems is the case of growing perpetuities

If an annuity continues forever, and grows in amount each period at a rate g, then
PV = PMT1/(i - g)
Example 15: If you invest in a stock that will pay a dividend of $10 next year and grow at 5 percent per year, and you require a 14 percent rate of return, how much is the stock worth to you today?
   PV = PMT1/(i - g)
   PV = 10/(.14-.05)
   PV = $111.11 
Example 16: Ramesh wants to retire and receive Rs. 3,000 a month. He wants to pass this monthly payment to future generations after his death. He can earn an interest of 8% compounded annually. How much will he need to set aside to achieve his perpetuity goal?
Solution     C = Rs. 3,000
r = 0.08/12 or 0.00667
Substituting these values in the above formula, we get
PV = 
If he wanted the payments to start today, we must increase the size of the funds to handle the first payment. This is achieved by depositing Rs. 4,52,775 which provides the immediate payment of Rs. 3,000 and leaves Rs. 4,49,775 in the fund to provide the future Rs. 3,000 payments.
SINKING FUND
It is the fund created for a specified purpose by way of sequence of periodic payments over a time period at a specified interest rate.
Size of the sinking fund deposit is computed from A=P.A (n,i), where A is the amount to be saved, P, the periodic payment, n, the payment period.
Example 17: How much amount is required to be invested every year so as to accumulate Rs. 3,00,000 at the end of 10 years if the interest is compounded annually at 10%?
Solution
Here, A= 3,00,000 n = 10 i = 0.1
Since, A=P.A (n,i)
3,00,000=P.A(10, 0.1)
= P*15.9374248
Therefore, P = 3,00,000/15.9374 = Rs. 18,823.62
P = Rs. 18,823.62


Summary
  1. FV  of a single amount = PV  x  CVF (r,n)
  2. PV of a single cash flow at the end of year n = FV  x PVF (r,n)
  3. FV of a series of equal payments  or Annuities :    A x CVAF (r,n)
  4. PV of a series of annual payments : FV x PVAF (r,n)

Mathematical formulae’s
  1. present value    (P) =      S (1+i)-n   or      S   [   1/ (1+i)n]
Present value    (P) =         (if compounded continuously)
  1. effective rate of  interest  =(r eff ) = 
    • Annuities are equal payments at equal intervals.
    • Annuity Payments are made at the end of the periods.

  1. FVA ( future value of an annuity)  =  S =   
  2. PVA (present value of an annuity )  = p  =   
  3. uniform payments for capital recovery – amortization   =       R =  
  4. uniform payments for capital formation – sinking fund =  =       R =  

  1. Amortization: A loan with a fixed rate of interest is said to be amortized if both principal and interest are paid by a sequence of equal payments made over equal period of time.
                                                                                      
                 Where R is the equal amt of annual  payment.
  1. Amount of deferred annuity:  deferred annuity is an ordinary annuity in which the first payments are postponed until the expiry of ascertain no of years or payment intervals .if the first payment is being made at the end      of 7 years then it deferred for 6 years.

S =
  1. Perpetuity: Perpetuity is an annuity whose payments being on a fixed date and continues forever.

        
  1. Sinking funds: A sinking fund is a fund that’s accumulated for the purpose of paying off a financial obligation at some future designated date.
                         And   S =
Where R is the periodic payment required to accumulate a sum of S over n periods of time.
The “Rule-of-72”
Approx. Years to Double = 72 / i%

Exhibit 18 : How long does it take to double $5,000 at a compound rate of 12% per year (approx.) ?
Approx. Years to Double = 72 / i%

72 / 12% = 6 Years
[Actual Time is 6.12 Years]
Rule of 69
0.35+    69 / int. rate %

Following previous example,
0.35+69/12 = 0.35+5.75 =6.10
Example 19: If you invest $1,000 today at an interest rate of 12 percent, how much time it will take to double?
FVn = P0(1+i)n
2000 = 1,000 (1.12)n
interpolation method
(1.12)n = 2000 /1000= 2
6Yrs + 1Yr* (1.974 - 2.000/1.974-2.211)
= 6Yrs+ 1yr (-0.026/-0.237) =  6Yrs+0.11Yrs= 6.11 Years
Example 20 : How long will it take for $10,000 to grow to $20,000 at an interest rate of 15% per year?
Rule of 72 =72 / 15   =4.80 years
Rule of 69= .35+69/15=.35+4.6 = 4.95 years

 FVn = P0(1+i)n
 20,000 = 10,000 (1.15)n
 n = 4.96 years
Interpolation method:
(1.15)n   =20000/10000 = 2.000
= 4+1x[1.749-2.000/1.749-2.011]
= 4+1x[0.251/0.262]
= 4+1x[0.958] =4.958 =4.96
Example 21: Julie wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%.
Calculation based on gen formula: PV0  = FVn / (1+i)n 
 PV0   = $10,000 / (1+ 0.10)5          = $6,209.21

Calculation based on Table I: 
PV0    = $10,000 (PVIF10%, 5)  = $10,000 (.621) = $6210.00  
Example 22: Y bought a TV costing Rs. 13,000 by making a down payment of Rs. 3,000 and agreeing to make equal annual payment for 4 years. How much would be each payment if the interest on unpaid amount be 14% compounded annually ?
Solution
In the present case, present value of the unpaid amount was (13,000 – 3,000) = Rs. 10,000. The periodic payment, A may be found from
PV = A X PVAF r%, n = A X PVAF14%,4
10,000 =  A X 2.9103  
A = 10,000/2.9103 = 3,432.05

Example 23: A person is required to pay four equal annual payments of Rs. 5,000 each in his deposit account that pays 8% interest per year. Find out the future value of annuity at the end of 4 years.
SOLUTION
FVA =ANNUITY ( CVAFR%,N) = ANNUITY ( CVAF8%,4)  =Rs. 5,000 (4.507)  = Rs. 22,535

Example 24: Assume you will receive an inheritance of Rs100,000, six years from now.  How much could you borrow from a bank today and spend now, such that the inheritance money will be exactly enough to pay off the loan plus interest when it is received?  Assume the bank charges an interest rate of 12 percent?
ANSWER:

= 1,00,000/1.9738
Example 25: A company offers to refund an amount of Rs 44,650 at the end of 5 years for a deposit of Rs 6,000 made annually. Find out the implicit rate of interest offered by the company.
Answer: in this case the refund amount is the future amount of annuity of Rs 6,000 after 5 years at a particular rate of interest. This can be presented as follows:
Rs 44,650 = 6000 x CVAF(r%,5years)
Or
CVAF(r%,5years)=44,650 /6000 = 7.442
Now in the CVAF table the value 7.442 corresponding to 5 years row is found in 20% column. So the implicit rate of interest is 20%.
Example 26: Assume that a deposit is made at zero years into an account that will earn 8% compounded annually. it is desired to withdraw Rs 5,000 three years from now and Rs 7,000 six years from now. What is the size of the year zero deposit that will produce these future payments?
Answer: let the deposit be the sum of the present values of the later withdrawals by using the present value table:
PV = FV x PVF (r%,n)
= 5000 x PVF (8%,3)+ 7000 x PVF (8%,6)
= 5000 x .794+ 7000 x .630
= 3,970 + 4,410
= 8,380.
Amortizing a Loan Example

Example 27: Julie is borrowing $10,000 at a compound annual interest rate of 12%.  Amortize the loan if annual payments are made for 5 years.
                             Payment
    PV0            = R (PVIFA i%,n)
                              $10,000        = R (PVIFA 12%,5)
                              $10,000        = R (3.605)
                             R = $10,000 / 3.605 = $2,774
Example 28:  Assume that Rs 20,00,000 plant extension is to be financed through the following plan: Pay 15% of the cost as down payment and the balance is to be repaid in 8 equal annual installments beginning 4 years from now. What is the size of the required annual loan payments? Compound interest is 9% per annum.
Answer:
The firm borrows 17,00,000 which is (Rs 20,00,000-Rs 3,00,000).compound interest is 9% per annum.the first payment will be made after 3 years or alternatively first payment  will be made in the fourth year. By compounding we get the balance of loan in the fourth year as follows:
FV = PV (1+r)n
FV = 17,00,000 (1+.09)3 = 22,01,550
Now the FV becomes the present value of the 8 payment annuity discounted at 9%.so compute the equal  yearly payment by using the following equation.
PV = Annuity x PVAF(9%,8)
22,01,550 = Annuity x 5.535
Or Annuity = 22,01,550 / 5.535 = 3, 97,750
So annual payment for the next year is going to be Rs 3, 97,750
Example 29:  A 10 year annuity of Rs 2000 per year is beginning at the end of current year. The payment of retirement annuity is to begin 16 years from now and continue to provide a 20 year payment annuity. Interest is 7% per annum , what will be the size of annuity to be received?
Answer: First find the compounded amount at the end of year 10.
FV =annuity x CVAF (7%,10)
FV = 2,000 X 13.816 = 27,632
The amount of Rs 27,632 is available immediately after the last payment. This amount invested for next 5 years will become,(beginning of 16 year)
FV = PV x CVF (7%,5)
FV =27,632 x CVF (7%,10) =27,632 x1.403 = 38,768
Finally find the retirement annuity,
PV  = Annuity x PVAF (7%,20)
38,768 = Annuity x 10.594
Annuity = 38,768/ 10.594 = 3,659
Example 30: (sinking fund) A machine costs Rs 98,000 and its effective life is estimated at 12 years . it's the effective life is estimated at 12 years. If the scrap value is Rs 3, 000, what should be cut out of the profit at the end of each year to accumulate at compound rate of 5% per annum so that a new machine can be purchased after 12 years ?
Answer: Effective cost of the machine = Rs 98,000 –Rs 3,000 = Rs 95,000.
FV =annuity x CVAF (5%,12)
95,000 = Annuity  X 15.917
So annuity = 95,000/15.917 =Rs 5,968.
Example 31: A finance company offers to give Rs.20,000 after 14 years in return for Rs.5,000 deposited today. Using the rule of 69, figure out the approximate interest rate offered.
Solution:
In 14 years Rs.5,000 grows to Rs.20,000 or 4 times. This is 22 times the initial deposit. Hence doubling takes place in 14 / 2 = 7 years.
According to the Rule of 69, the doubling period is 0.35 + 69 / Interest rate
We therefore have 0.35 + 69 / Interest rate = 7
Interest rate = 69/(7-0.35) = 10.38 %
Example 32: Someone offers to give Rs.80,000 to you after 18 years in return for Rs.10,000 deposited today. Using the rule of 69, figure out the approximate interest rate offered.

Solution:
In 18 years Rs.10,000 grows to Rs.80,000 or 8 times. This is 23 times the initial deposit. Hence doubling takes place in 18 / 3 = 6 years.
According to the Rule of 69, the doubling period is
0.35 + 69 / Interest rate.
We therefore have 0.35 + 69 / Interest rate = 6
Interest rate = 69/(6-0.35) = 12.21 %
Example 33: You can save Rs.5,000 a year for 3 years, and Rs.7,000 a year for 7 years thereafter.
What will these savings cumulate to at the end of 10 years, if the rate of interest is 8 percent?
Solution:
Saving Rs.5000 a year for 3 years and Rs.6000 a year for 7 years thereafter is equivalent to saving Rs.5000 a year for 10 years and Rs.2000 a year for the years 4 through 10.
Hence the savings will cumulate to:
5000 x FVIFA (8%, 10 years) + 2000 x FVIFA (8%, 7 years)
= 5000 x 14.487 + 2000 x 8.923 = Rs.90281
Example 34: Krishna saves Rs.24,000 a year for 5 years, and Rs.30,000 a year for 15 years thereafter. If the rate of interest is 9 percent compounded annually, what will be the value of his savings at the end of 20 years?
Solution:
Saving Rs.24,000 a year for 5 years and Rs.30,000 a year for 15 years thereafter is equivalent to saving Rs.24,000 a year for 20 years and Rs.6,000 a year for the years 6 through 20.
Hence the savings will cumulate to:
24,000 x FVIFA (9%, 20 years) + 6,000 x FVIFA (9 %, 15 years)
= 24,000 x 51.160 + 6, 000 x 29.361 =Rs. 1,404,006
Example 35: You plan to go abroad for higher studies after working for the next five years and understand that an amount of Rs.2,000,000 will be needed for this purpose at that time. You have decided to accumulate this amount by investing a fixed amount at the end of each year in a safe scheme offering a rate of interest at 10 percent. What amount should you invest every year to achieve the target amount?
Solution:
Let A be the annual savings.
A x FVIFA (10%, 5years) = 2,000,000
A x 6.105 = 2,000,000
So, A = 2,000,000 / 6.105 = Rs. 327,600

Example 36: How much should Vijay save each year, if he wishes to purchase a flat expected to cost Rs.80 lacs after 8 years, if the investment option available to him offers a rate of interest at 9 percent? Assume that the investment is to be made in equal amounts at the end of each year.
Solution:
Let A be the annual savings.
A x FVIFA (9 %, 8 years) = 80,00,000
A x 11.028 = 80,00,000
So, A = 80,00,000 / 11.028 = Rs. 7,25,426
Example 37: A finance company advertises that it will pay a lump sum of Rs.100,000 at the end of 5 years to investors who deposit annually Rs.12,000. What interest rate is implicit in this offer?
Solution:
12,000 x FVIFA (r, 5 years) = 100,000
FVIFA (r, 5 years) = 100,000 / 12,000 = 8.333
From the tables we find that
FVIFA (24%, 5 years) = 8.048
FVIFA (28%, 5 years) = 8.700
Using linear interpolation in the interval, we get:
Example 38: Someone promises to give you Rs.5,000,000 after 6 years in exchange for Rs.2,000,000 today. What interest rate is implicit in this offer?
Solution:
2,000,000 x FVIF (r, 6 years) = 5,000,000
FVIF (r, 6 years) = 5,000,000 / 2,000,000 = 2.5
From the tables we find that
FVIF (16%, 6 years) = 2.436
FVIF (17%, 6 years) = 2.565
Using linear interpolation in the interval, we get:
Example 39: At the time of his retirement, Rahul is given a choice between two alternatives:
(a) an annual pension of Rs120,000 as long as he lives, and
(b) a lump sum amount of Rs.1,000,000.
If Rahul expects to live for 20 years and the interest rate is expected to be 10 percent throughout , which option appears more attractive.

Solution:
The present value of an annual pension of Rs.120,000 for 20 years when r = 10% is:
120,000 x PVIFA (10%, 20 years)
= 120,000 x 8.514 = Rs.1,021,680
The alternative is to receive a lumpsum of Rs 1,000,000
Rahul will be better off with the annual pension amount of Rs.120,000.
Example 40: What is the present value of an income stream which provides Rs.30,000 at the end of year one, Rs.50,000 at the end of year three , and Rs.100,000 during each of the years 4 through 10, if the discount rate is 9 percent ?
Solution:
The present value of the income stream is:
30,000 x PVIF (9%, 1 year) + 50,000 x PVIF (9%, 3 years)
+ 100,000 x PVIFA (9 %, 7 years) x PVIF(9%, 3 years)
= 30,000 x 0.917 + 50,000 x 0.772 + 100,000 x 5.033 x 0.0.772 = Rs.454,658.
Example 41: What is the present value of an income stream which provides Rs.25,000 at the end of year one, Rs.30,000 at the end of years two and three , and Rs.40,000 during each of the years 4 through 8 if the discount rate is 15 percent ?
Solution:
The present value of the income stream is:
25,000 x PVIF (15%, 1 year) + 30,000 x PVIF (15%, 2 years)
+ 30,000 x PVIF (15%, 3 years)
+ 40,000 x PVIFA (15 %, 5 years) x PVIF (15%, 3 years)
= 25,000 x 0.870 + 30,000 x 0.756 + 30,000 x 0.658
+ 40,000 x 3.352 x 0.658 = Rs.152,395.
Example 42: What is the present value of an income stream which provides Rs.1,000 a year for the first three years and Rs.5,000 a year forever thereafter, if the discount rate is 12 percent?
Solution:
The present value of the income stream is:
1,000 x PVIFA (12%, 3 years) + (5,000/ 0.12) x PVIF (12%, 3 years)
= 1,000 x 2.402 + (5000/0.12) x 0.712
= Rs.32,069

Example 43: What is the present value of an income stream which provides Rs.20,000 a year for the first 10 years and Rs.30,000 a year forever thereafter, if the discount rate is 14 percent ?
Solution:
The present value of the income stream is:
20,000 x PVIFA (14%, 10 years) + (30,000/ 0.14) x PVIF (14%, 10 years)
= 20,000 x 5.216 + (30,000/0.14) x 0.270
= Rs.162,177
Example 44: Mr. Ganapathi will retire from service in five years .How much should he deposit now to earn an annual income of Rs.240,000 forever beginning from the end of 6 years from now ? The deposit earns 12 percent per year.
Solution:
To earn an annual income of Rs.240,000 forever , beginning from the end of 6 years from now, if the deposit earns 12% per year a sum of
Rs.240,000 / 0.12 = Rs.2,000,000
is required at the end of 5 years. The amount that must be deposited to get this
sum is:
Rs.2,000,000 PVIF (12%, 5 years) = Rs.2,000,000 x 0.567 = Rs. 1,134,000

Example 45: Suppose someone offers you the following financial contract. If you deposit Rs.100,000 with him he promises to pay Rs.50,000 annually for 3 years. What interest rate would you earn on this deposit?
Solution:
Rs.100,000 =- Rs.50,000 x PVIFA (r, 3 years)
PVIFA (r,3 years) = 2.00
From the tables we find that:
PVIFA (20 %, 3 years) = 2.106
PVIFA (24 %, 3 years) = 1.981
Using linear interpolation we get:
Example 46: If you invest Rs.600,000 with a company they offer to pay you Rs.100,000 annually for 10 years. What interest rate would you earn on this investment?
Solution:
Rs.600,000 =- Rs.100,000 x PVIFA (r, 10 years)
PVIFA (r,10 years) = 6.00
From the tables we find that:
PVIFA (10 %, 10 years) = 6.145
PVIFA (11 %, 10 years) = 5.889
Using linear interpolation we get:
Example 47: Suppose you deposit Rs.200,000 with an investment company which pays 12 percent interest with compounding done once in every two months, how much will
this deposit grow to in 10 years?
Solution:
FV10 = Rs.200,000 [1 + (0.12 / 6)]10x6
= Rs.200,000 (1.02)60
= Rs.200,000 x 3.281
= Rs.656,200

Example 48: A bank pays interest at 5 percent on US dollar deposits, compounded once in every six months. What will be the maturity value of a deposit of US dollars 15,000 for three years?
Solution:
Maturity value = USD 15 ,000 [1 + (0.05 / 2)]3x2
= 15,000 (1.025)6
= 15,000 x 1.1597
= 17,395.50

Example 49: You have a choice between Rs.200,000 now and Rs.600,000 after 8 years. Which would you choose? What does your preference indicate if market rate of interest is 14% ?
Solution:
The interest rate implicit in the offer of Rs.600,000 after 8 years in lieu of
Rs.200,000 now is:
Rs.200,000 x FVIF (r,8 years) = Rs.600,000
FVIF (r,8 years) = Rs.600,000/Rs.200,000  = 3.000
From the tables we find that
FVIF (15%, 8years) = 3.059
This means that the implied interest rate is nearly 15%.
I would choose Rs.600,000 after 8 years from now because I find a return of 15% quite attractive.

Example 50: Ravikiran deposits Rs.500,000 in a bank now. The interest rate is 9 percent and compounding is done quarterly. What will the deposit grow to after 5 years? If the inflation rate is 3 percent per year, what will be the value of the deposit after 5 years in terms of the current rupee?

SOLUTION
FV5 = Rs.500,000 [1 + (0.09 / 4)]20
 = Rs.500,000 (1.0225)20
= Rs.500,000 x 2.653
= Rs.780,255
If the inflation rate is 3 % per year, the value of Rs.780,255 5 years from now, in terms of the current rupees is:
Rs.780,255 x PVIF (3%, 5 years)
= Rs.780,255 x 0. 863 = Rs.673,360

Example 51: A person requires Rs.100,000 at the beginning of each year from 2015 to 2019. Towards this, how much should he deposit ( in equal amounts) at the end of each year from 2007 to 2011, if the interest rate is 10 percent.
Solution:
The discounted value of Rs.100,000 receivable at the beginning of each year from 2015 to 2019, evaluated as at the beginning of 2014 (or end of 2013) is:
Rs.100,000 x PVIFA (10%, 5 years)
= Rs.100,000 x 3.791= Rs.379,100
The discounted value of Rs.379,100 evaluated at the end of 2011 is
Rs.379,100 x PVIF (10 %, 2 years)
= Rs.379,100 x 0.826= Rs.313,137
If A is the amount deposited at the end of each year from 2007 to 2011 then
A x FVIFA (10%, 5 years) = Rs.313,137
A x 6.105 = Rs.313,137
A = Rs.313,137/ 6.105 = Rs.51,292

Example 52: You require Rs.250 ,000 at the beginning of each year from 2010 to 2012. How much should you deposit( in equal amounts) at the beginning of each year in 2007 and 2008 ? The interest rate is 8 percent.
Solution:
The discounted value of Rs.250,000 receivable at the beginning of each year from 2010 to 2012, evaluated as at the beginning of 2009 (or end of 2008) is:
Rs.250,000 x PVIFA (8 %, 3 years)
= Rs.250,000 x 2.577= Rs.644,250
To have Rs. 644,250 at the end of 2008, let A be the amount that needs to be deposited at the beginning of 2007 and 2008.We then have
Ax (1+0.08) x FVIFA ( 8%, 2years) = 644,250
A x 1.08 x 2.080 = 644,250 or A = 286,792

Example 53: What is the present value of Rs.120,000 receivable annually for 20 years if the first receipt occurs after 8 years and the discount rate is 12 percent.
Solution:
The discounted value of the annuity of Rs.120,000 receivable for 20 years,
evaluated as at the end of 7th year is:
Rs.120,000 x PVIFA (12%, 20 years) = Rs.120,000 x 7.469 = Rs.896,290
The present value of Rs. 896,290 is:
Rs. 896,290 x PVIF (12%, 7 years)
= Rs. 896,290 x 0.452
= Rs.405,119
Example 54: What is the present value of Rs.89,760 receivable annually for 10 years if the first receipt occurs after 5 years and the discount rate is 9 percent.
Solution:
The discounted value of the annuity of Rs.89,760 receivable for 10 years, evaluated as at the end of 4th year is:
Rs. 89,760 x PVIFA (9%, 10 years) = Rs. 89,760 x 6.418 = Rs.576,080
The present value of Rs. 576,080is:
Rs. 576,080x PVIF (9%, 4 years)
= Rs. 576,080x 0.708
= Rs.407,865

Example 54: Metro Corporation has to retire Rs.20 million of debentures each at the end of 6, 7,and 8 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 10 percent.
Solution:
The discounted value of the debentures to be redeemed between 6 to 8 years evaluated at the end of the 5th year is:
Rs.20 million x PVIFA (10%, 3 years) = Rs.20 million x 2.487
= Rs.49.74million
If A is the annual deposit to be made in the sinking fund for the years 1 to 5,
then
A x FVIFA (10%, 5 years) = Rs.49.74 million
A x 6.105 = Rs.49.74 million
A = Rs.8,147,420

Example 55: Ankit Limited has to retire Rs.30 million of debentures each at the end of 7, 8, 9 and 10 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 12 percent.
Solution:
The discounted value of the debentures to be redeemed between 7 to 10 years evaluated at the end of the 6th year is:
Rs.30 million x PVIFA (12%, 4 years) = Rs.30 million x 3.037
= Rs.91.11 million
If A is the annual deposit to be made in the sinking fund for the years 1 to 6,
then
A x FVIFA (12%, 6 years) = Rs.91.11 million
A x 8.115 = Rs. 91.11 million
A = Rs.11,227,357

Example 56: Mr.Mehta receives a provident fund amount or Rs.800,000. He deposits it in a bank which pays 9 percent interest. If he plans to withdraw Rs.100,000 at the end of each year, how long can he do so ?
Solution:
Let `n’ be the number of years for which a sum of Rs.100,000 can be withdrawn annually.
Rs.100,000 x PVIFA (9%, n) = Rs.800,000
PVIFA (9%, n) = Rs.800,000 / Rs.100,000 = 8 .000
From the tables we find that
PVIFA (9%, 14 years) = 7.786
PVIFA (9%, 15 years) = 8.060
Using a linear interpolation we get

Example 57: Mr. Naresh wants to invest an amount of Rs. 400,000, in a finance company at an interest rate of 12 percent, with instructions to the company that the amount with interest be repaid to his son in equal instalments of Rs.100,000, for his education expenses . How long will his son get the amount ?

Solution:
Let `n’ be the number of years for which a sum of Rs.100,000 can be withdrawn annually.
Rs.100,000 x PVIFA (12%, n) = Rs.400,000
PVIFA (12 %, n) = Rs.400,000 / Rs.100,000 = 4
From the tables we find that
PVIFA (12%, 5 years) = 3.605
PVIFA (12%, 6 years) = 4.111
Using a linear interpolation we get

Example 58: As a winner of a competition, you can choose one of the following prizes:
a. Rs. 800,000 now
b. Rs. 2,000,000 at the end of 8 years
c. Rs. 100,000 a year forever
d. Rs. 130,000 per year for 12 years
e. Rs. 32,000 next year and rising thereafter by 8 percent per year forever.
If the interest rate is 12 percent, which prize has the highest present value?
Solution:
(a) PV = Rs.800,000
(b) PV = 2,000,000PVIF12%,8yrs = 2,000,000 x 0.0.404 = Rs.808,000
(c ) PV = 100,000/r = 100,000/0.12 = Rs. 833,333
(d) PV = 130,000 PVIFA12%,12yrs = 130,000 x 6.194 = Rs.805,220
(e) PV = C/(r-g) = 32,000/(0.12-0.08) = Rs.800,000
Option c has the highest present value viz. Rs.833,333

Example 59: You want to borrow Rs.3,000,000 to buy a flat. You approach a housing company which charges 10 percent interest. You can pay Rs.400,000 per year toward loan amortisation. What should be the maturity period of the loan?
Solution:
Let n be the maturity period of the loan. The value of n can be obtained from
the equation.
400,000 x PVIFA(10%, n) = 3,000,000
PVIFA (10%, n) = 7.5
From the tables we find that
PVIFA (10%,14 years) = 7.367
PVIFA (10 %, 15 years) = 7.606
Using linear interpolation we get

Example 60: You want to borrow Rs.5,000,000 to buy a flat. You approach a housing company which charges 11 percent interest. You can pay Rs.600,000 per year toward loan amortisation. What should be the maturity period of the loan?
Solution:
Let n be the maturity period of the loan. The value of n can be obtained from
the equation.
600,000 x PVIFA(11%, n) = 5,000,000
PVIFA (11%, n) = 8.333
From the tables we find that
PVIFA (11%,20 years) = 7.963
PVIFA (11 %, 25 years) = 8.422
Using linear interpolation we get,

CASE STUDY 1
As an investment advisor, you have been approached by a client called Vikas for your advice on investment plan. He is currently 40 years old and has Rs.600,000 in the bank. He plans to work for 20 years more and retire at the age of 60. His present salary is Rs.500,000 per year. He expects his salary to increase at the rate of 12 percent per year until his retirement.
Vikas has decided to invest his bank balance and future savings in a balanced mutual fund scheme that he believes will provide a return of 9 percent per year. You agree with his assessment.
Vikas seeks your help in answering several questions given below. In answering
these questions, ignore the tax factor.
(i) Once he retires at the age of 60, he would like to withdraw Rs.800,000 per year for his consumption needs from his investments for the following 15 years (He expects to live upto the age of 75 years). Each annual withdrawal will be made at the beginning of the year. How much should be the value of his investments when Vikas turns 60, to meet this retirement need?
(ii) How much should Vikas save each year for the next 20 years to be able to withdraw Rs.800,000 per year from the beginning of the 21st year ? Assume that the savings will occur at the end of each year.
(iii) Suppose Vikas wants to donate Rs.500,000 per year in the last 5 years of his life to a charitable cause. Each donation would be made at the beginning of the year. Further, he wants to bequeath Rs.1,000,000 to his son at the end of his life. How much should he have in his investment account when he reaches the age of 60 to meet this need for donation and bequeathing?
(iv) Vikas is curious to find out the present value of his lifetime salary income. For the sake of simplicity, assume that his current salary of Rs.500,000 will be paid exactly one year from now, and his salary is paid annually. What is the present value of his life time salary income, if the discount rate applicable to the same is 7 percent?
Remember that Vikas expects his salary to increase at the rate of 12 percent per
year until retirement.

ANSWER

(i) This is an annuity due
Value of annuity due = Value of ordinary annuity (1 + r)
The value of investments when vikas turns 60 must be:
800,000 x PVIFA (9%, 15 years) x 1.09
= 800,000 x 8.060 x 1.09 = Rs.7,028,320

(ii) He must have Rs.7,092,800 at the end of the 20th year.
His current capital of Rs.600,000 will grow to:
Rs.600,000 x FVIF (9%, 20yrs) = 600,000 x 5.604
= Rs.3,362,400
So, what he saves in the next 15 years must cumulate to:
7,028,320 – 3,362,400 = Rs.3,665,920
A x FVIFA (9%, 20 yrs) = Rs.3,665,920
A x 51.160 = 3,665,920
A = 3,665,920/51.160 = Rs.71,656

(iii)
To meet his donation objective, Vikas will need an amount equal to:
500,000 x PVIFA (9%, 5years) when he turns 69.
This means he will need
500,000 x PVIFA (9%, 5yrs) x PVIF (9%, 9yrs) when he turns 60.
This works out to:
500,000 x 3.890 x 0.460 = Rs.894,700
To meet his bequeathing objective he will need
1,000,000 x PVIF (15%, 9yrs) when he turns 60
This works out to:
1,000,000 x 0.275 = Rs.275,000
So, his need for donation and bequeathing is: 894,700 + 275,000
= Rs.1,169,700
(iv)







CASE STUDY 2
As an investment advisor, you have been approached by a client called Ravi for  advice on his investment plan. He is 35 years and has Rs.200, 000 in the bank. He plans to work for 25 years more and retire at the age of 60. His present salary is 500,000 per year. He expects his salary to increase at the rate of 12 percent per year until his retirement.
Ravi has decided to invest his bank balance and future savings in a balanced mutual fund scheme that he believes will provide a return of 9 percent per year.
You concur with his assessment.
Ravi seeks your help in answering several questions given below. In answering these questions, ignore the tax factor.
(i) Once he retires at the age of 60, he would like to withdraw Rs. 900,000 per year for his consumption needs for the following 20 years (His life expectancy is
80years).Each annual withdrawal will be made at the beginning of the year.
How much should be the value of his investments when he turns 60, to meet his
retirement need?
(ii) How much should Ravi save each year for the next 25 years to be able to withdraw Rs.900, 000 per year from the beginning of the 26th year for a period
of 20 years?
Assume that the savings will occur at the end of each year. Remember that he already has some bank balance.
(iii) Suppose Ravi wants to donate Rs.600, 000 per year in the last 4 years of his life to a charitable cause. Each donation would be made at the beginning of the year.
Further he wants to bequeath Rs. 2,000,000 to his daughter at the end of his life.
How much should he have in his investment account when he reaches the age of
60 to meet this need for donation and bequeathing?
(iv) Ravi wants to find out the present value of his lifetime salary income. For the sake of simplicity, assume that his current salary of Rs 500,000 will be paid
exactly one year from now, and his salary is paid annually. What is the present value of his lifetime salary income, if the discount rate applicable to the same is
8 percent? Remember that Ravi expects his salary to increase at the rate of 12 percent per year until retirement.

ANSWER

(i)
900,000 x PVIFA ( 9 %, 20 ) x 1.09
900,000 x 9.128 x 1.09
= Rs. 8,954,568
(ii)
Ravi needs Rs. 8,954,568 when he reaches the age of 60.
His bank balance of Rs. 200,000 will grow to : 200,000 ( 1.09 )25
= 200,000 ( 8.623 ) = Rs. 1,724,600
This means that his periodic savings must grow to :
Rs. 8,954,568 - Rs. 1,724,600 = Rs. 7,229,968
His annual savings must be:

(iii)

Amount required for the charitable cause:
600,000 x PVIFA ( 9% , 4yrs ) x PVIF ( 9%, 15yrs )
= 600,000 x 3.240 x 0.275
Rs. 534,600
Amount required for bequeathing
2,000,000 x PVIF ( 9%, 20yrs )
= 2,000,000 x 0.178 = Rs.356,000

(iv)













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