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## FMA101 TOPIC 4- Time Value of Money Assignment Answers

### LEARNING OUTCOMES

After completing this topic, you should be able to:

• Explain the role of time value in finance and the meaning of the concepts of future value and present value for a single amount and the relationship that exists between them.
• Calculate and explain both the present values and future values for single amounts, an ordinary annuity and the annuity due, and the present value of a perpetuity and a mixed stream of cash flows.
• Calculate interest for periods more frequently than annually and distinguish between nominal and effective interest rates and calculate these interest rates.
• Determine deposits needed to accumulate a future sum and prepare a loan amortisation schedule.
• Determine the unknown number of periods in order to generate a given amount of cash flow from an initial amount.

• Gitman et al. (2015: Chapter 5)

Please make sure that you also read this study guide carefully as it contains additional information that is not in the prescribed textbook.

The key concepts that you must focus on are:

• the meaning of a future value and how to calculate it for a single amount and for an annuity
• the meaning of a present value and how to calculate it for a single amount and for an annuity
• the use of financial tables that contain future value and present value interest factors
• different types of cash flows

### THE ROLE OF TIME VALUE IN FINANCE

There is a popular saying among economists that goes like this: “A rand today is worth more than a rand in one year’s time.” What they are saying is that money has value related to a point in time.

Inflation is the cause of money being worth less at some time in the future compared to its value today. Inflation erodes the “purchasing power” of money.

You may already have come to realise that you can buy less for, say, R100 today than what you could one or two years ago. Stating it differently, what costs you R100 today, may only have cost you R80 the year before. Inflation over time has eroded the value of the money.

Financial managers understand this phenomenon and they pay particular attention to it when considering the future cash flows stemming from major investment decisions. They, therefore, bring the time value of money into consideration when evaluating future cash flow streams and making investment decisions.

The following example illustrates the point:

An organisation is considering a major capital expansion project. This entails the purchase of very sophisticated, hi-tech machinery costing R30 million. Having this sophisticated machinery will reduce production costs significantly, increase production capacity and output significantly and improve the competitive position and future sales of products dramatically.

The benefits listed above, however, will only accrue to the company over the next ten years once the machinery is operational. Two important financial issues are now at stake.

Firstly, the organisation will have to spend R30 million using money today (present) and, secondly, they will only recover this expenditure from money received in the future (over the next ten years)! The organisation, therefore, needs to decide whether the future benefits in future money terms will outweigh spending money worth R30 million today.

The management of the organisation also realises that in order to make an informed decision, they have to “compare apples with apples”. Management, therefore, needs to convert the future money back to an equivalent value in present-day terms and then make a comparison.

If the sum total of all future net cash flows earned from the investment in the machinery, expressed in the present value of money, exceeds the R30 million investment to be made at present, then the organisation has protected the future purchasing power of its money. If the future cash flows expressed in present values are less in value than the R30 million, then the purchasing power of the future cash flows have eroded. Investing in the machinery will then be a poor decision as the future benefits will be worth less than what R30 million is worth today.

### What does the “present value of a future amount” mean?

When a future amount is converted back to an equivalent amount in present-day terms, we say that we are “calculating the present value (value at present) of a future amount”.

Present and future values can be calculated using electronic calculators or using pre-prepared tables similar to those provided in the prescribed textbook (Appendix A – Financial tables) and in Annexure B at the end of this study guide.

When you use the interest tables, you must make sure that you know how to read an interest or annuity factor from the relevant table. Figure 5.4 on page 154 of the prescribed textbook explains how to read these tables.

You must also make sure that you use the correct interest table when selecting interest or annuity factors, as different tables have different purposes. The heading of each table describes the type of interest or annuity factors included in the particular table. You must make very sure that you are using the correct table when performing calculations.

If you are using an electronic calculator, you must make sure that you understand how to operate the particular model of the calculator that you are using. Except for very basic calculators, all the more sophisticated calculators have user manuals that explain and demonstrate the various functions and keys to use when operating the calculator. If you do not understand how to use a particular calculator correctly, your calculations will be incorrect.

Please note that a calculator does not operate on its own. It cannot decide on your behalf what type of calculation to perform. You have to select the correct mode of operation and provide the correct input for the calculator to perform the desired calculations correctly.

### DIFFERENT TYPES OF CASH FLOWS

You must be able to distinguish between the three different types of cash flows discussed and explain and use each of the different types.

You should note that in the first part of this chapter the authors explain and demonstrate how the present values and future values of the three different types of cash flows are calculated. The three types are single amounts, annuities and mixed streams.

Read and understand what the differences are between a single amount, an annuity and a mixed stream in terms of cash flows.

### SINGLE AMOUNTS AND ANNUITIES

The key concepts that you must focus on are:

• Single Amount
• Ordinary Annuity
• Annuity Due

In order to understand the concepts and calculations described in the first part of this chapter, it is very important that you:

• understand the formula used or procedure followed to calculate the present or future value for each cash flow type in terms of every component of the formula
• identify the data required to perform the calculation from the information provided

Firstly, it is very important that you carefully work through every example presented in the textbook to make sure that you understand the explanation and application.

Secondly, you must supplement the examples by completing the relevant exercises provided at the end of each chapter. You can check your answers by comparing them against the answers provided in Appendix B – Solutions to self- test problems of the prescribed textbook.

### SINGLE AMOUNTS

You must know how to calculate present and future values for a single amount and then interpret and explain the results.

You must be able to do the calculations using both an electronic calculator and the relevant financial tables.

### ANNUITIES

You must be able to distinguish between an ordinary annuity and an annuity due. You need to understand the effect that these different annuity types have on the calculation of the relevant annuity value very well.

You need to understand how to calculate present and future annuity values for both ordinary annuities and annuities due, and perform and interpret such calculations and explain the results.

You must be able to do the calculations using both an electronic calculator and the relevant financial tables.

### MIXED STREAMS

You need to understand how to calculate the present and future values for a mixed stream of cash flows and interpret such calculations and explain the results.

You must be able to do the calculations using both an electronic calculator and the relevant financial tables.

### COMPOUNDING INTEREST MORE FREQUENTLY THAN ANNUALLY

The key concepts that you must focus on are:

• more frequent time periods
• continuous compounding
• nominal interest and effective interest

Interest can be calculated daily, monthly, quarterly, semi-annually or annually. It is important to be able to perform interest calculations for these different periods.

You should take careful note of how a change in time affects the time factor of a calculation as well as the manner in which the given interest rate is adapted to the relevant time. For example, if interest for a period of 3 years must be calculated monthly instead of annually and the annual interest rate is given as 12% per annum, then the time period is changed from annual to monthly as follows:

• Period to use in the calculation: 3 years × 12 months per year = 36 months. Convert n to months (n = 36).
• Interest rate to use in the calculation: 12% p.a. / 12 = 1% per month; i is therefore converted to months as well (i = 1%).

You are expected to understand and perform different time-based interest calculations and interpret and explain the results.

You must be able to explain continuous compounding and perform continuous compounding calculations.

You must be able to distinguish between nominal and effective interest rates and calculate these interest rates.

Furthermore, you must be able to calculate the effective annual interest rate (EAR) based on the information that is provided.

### SPECIAL APPLICATIONS OF TIME VALUE

The key concepts that you must focus on are:

• deposits needed to accumulate a future sum
• loan amortisation
• finding interest growth rates

### DEPOSITS NEEDED TO ACCUMULATE FUTURE SUMS

Financial managers and individuals should know how to calculate the amounts needed in order to accumulate a larger amount at the end of a given period. You need to be able to perform such calculations by using both financial tables and an electronic calculator.

### LOAN AMORTISATION

A loan consists of two components, namely the loan amount (also referred to as the capital sum or principal) and the regular interest payments made on the loan amount as part of the repayment of the loan.

You need to be able to perform the required calculations in order to prepare a loan amortisation schedule that reflects interest payments and the principal amount at any time during the loan repayment period.

### FINDING INTEREST GROWTH RATES

It often happens that financial managers have to determine interest growth rates in order to do calculations for purposes of financial analysis decision-making. Interest growth rates may not be available and financial managers need to know how to use the data or information which incorporates such interest growth rates in order to be able to calculate the actual interest growth rates.

You will notice that interest growth rates can be calculated only if a minimum of two sets of data or information are available for such purposes.

The different methods that can be used to calculate an interest growth rate are demonstrated and explained in the prescribed textbook.

You must be able to calculate interest growth rates using an electronic calculator or the financial tables.

Access the following websites that contain very useful insights and examples of calculations of different time value concepts:

• David R. Frick & Co. 2008. Time value of money concepts. http://www.frickcpa.com/tvom/default.asp, accessed 27 February
• https://www.listenmoneymatters.com/time-value-of-money/, accessed 27 February
• https://www.extension.iastate.edu/agdm/wholefarm/pdf/c5-96.pdf, accessed 27 February

### SELF-ASSESSMENT EXERCISES

At the end of this chapter, there is a series of different self-test problems, warm- up exercises, problems and a more comprehensive case study. You should attempt to answer these questions, perform the calculations and use them to practise and re-enforce your learning and understanding. You may submit any of your answers to the lecturer for assessment and feedback.

SBS also shares copies of previous exam papers with you during the semester. The questions in these exam papers are very good examples of what you are expected to know and be able to do having studied and mastered the content of chapter 5 of the prescribed textbook and having followed the guidance provided in this topic.

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