1 4 ARITHMETI OF FINANE Introduction Definitions Present Value of a Future Amount Perpetuity - Growing Perpetuity Annuities ompounding Agreement ontinuous ompounding - Lump Sum - Annuity ompounding Magic? Summary Introduction Time is money. Given the choice of having a dollar today or having that same dollar in the future, who would choose the same dollar in the future? A dollar in the hand now is worth more than one in the future. Many financial agreements are based on a flow of money happening at some point in the future. For example, buying a house, car, stereo or other major item often results in an agreement to pay over time. To make informed financial decisions it is necessary to be able to understand the financial value that time plays in the flow of money. In order to compare one financial arrangement to the next, it is necessary to put both into the same point in time. It would not make sense to compare the future value of one amount of money to the present value of a different amount. We live in the present. It makes the most sense, therefore, to put all cash values into present value terms for the sake of comparison. This chapter discusses the methods for putting cash flows into present value terms.
2 7 Financial Management and Decision Making Definitions There are five components used in understanding the time value of money. They are:, or the Present Value of money, which is the value today of cash flows expected sometime in the future discounted by an appropriate discount rate (sometimes called an expected rate of return ). FV, or the Future Value of money, which is the dollar amount expected to be received in the future adjusted for the time value of money, this adjustment being frequently based on the interest that could be earned on the current dollar amount. r, or the interest rate prevailing at the time, also called the discount rate. n, or the number of periods (years, months, days, etc.) between the present and the future., or the amount of the periodic payment being made between the present and the future. The relationship between these variables is determined by the nature of the cash flow. There are three important relationships between these variables, namely the discounting of: - a series of discrete cash flows, - perpetuities, and - annuities. Present Value of a Future Amount The relationship between today s dollar and next year s dollar is, FV x ( If $ was invested at 0% annual interest, it would be worth $.0 in one year s time. Solving this formula for the Present Value, gives: FV If you are promised $.0 in one year s time, it is worth $.00 in present value dollars. ompound interest rates mean that interest is earned on the interest previously paid. If the $.00 investment were held for two years, it would result in: FV $.00 x (.) x (.)
3 hapter 4: Arithmetic of Finance 7 or $. In general, the future value of an amount of money invested at compound interest rates is given by: FV x ( n and solving for the present value gives: FV ( n If $. is promised in two years time, the present value of that money would be:. (. ) $. 00 In other words, you would be willing to pay someone exactly $.00 today for the right to receive $. in two years time if the interest rate is 0%. Therefore, from this formula, a future amount can be converted to a present value amount by multiplying the future amount by: ( n which is called the present value discount factor or deflator. Knowing the interest rate and the number of periods, it is possible to arrive at the present value discount factor. Appendix 4. includes many of the present value discount factors that may be used. Example You expect to receive $50,000 when your house is sold in five years time. What is the present value of your house if interest rates are 5%? The discount factor of /(.5) 5 is quickly found in Appendix 4. by looking at 5% for five years, giving.497. The present value is therefore: $50,000 x.497 $4,860 (Note: Since Appendix 4. only has four significant figures - and the fourth one is arrived at by rounding from the next significant figure - the answers arrived at by using this table will only be correct to three significant figures. More accurate calculations are necessary when dealing in large sums of money.
4 74 Financial Management and Decision Making The present value of the house in this example is more correctly $4,858.8). Perpetuity What happens to the calculation of present value if the incoming cash were to be promised in perpetuity? This is not uncommon and there are presently various kinds of perpetuities being traded in the London financial markets. The cash flows in each period are adjusted by the present value discount factor appropriate for that period, that is, the cash flow in period is multiplied by the discount factor for period, and so on to infinity. The formula is therefore: Eqn( ): ( ( (... where, and are distinct independent payments received in periods,, and respectively. Notice that cash flows in perpetuities by their nature continue to infinity. If we assume that all payments are equal then can be factored out from the right side of the equation, giving: Eqn( ): ( ( (... and then multiplying both sides of the equation by / gives: Eqn( ): 4... In order to eliminate the infinite series to the right, subtract equation () from equation (): Eqn() - Eqn() x [ - /( ] x [/( ] and solving for the Present Value, we find: /r This means that an infinite flow of money in the amount of has a present value exactly equal to /r where r is the prevailing interest rate. Example What would you pay to receive $50 per year forever if interest rates were at %? $50/. or $46.67.
5 hapter 4: Arithmetic of Finance 75 What if interest rates dropped to 0%? Then $50/. or $ Growing Perpetuity This illustrates two important points about perpetuities: - the value of a perpetuity is inversely proportional to the level of interest rates; and - a drop in interest rates increases the value of the perpetuity. In this case, the amount of the payment being made in perpetuity is growing by g percent per period similar to compound interest on a cash flow. Therefore, the present value of the perpetuity becomes: Eqn( 4): ( ( g) ( ( g) (... Multiplying both sides of this equation by g)/ gives: Eqn(5): g) g) g) g) 4... and subtracting equation (5) from equation (4) gives: Eqn(4) - Eqn(5) x [ - ( g)/( ] x [/( ] and solving for the Present Value gives: /(r - g) (Note: This only holds for g less than Example At the end of the present year, you expect to receive $00 in interest income on your investments. You expect this amount to grow by 5% every year. If this income stream is expected to last forever, what would you be willing to sell it for if present interest rates are %? $00/(. -.05) $00/.07 $,48 Annuities The third form of future cash flows is annuity, in which the payments do not continue forever but last only for a certain number of periods (n). The present value of the cash flows is therefore calculated on the same basis as for perpetuities (equation ()) but only until period n.
6 76 Financial Management and Decision Making Eqn(6):... n n From the discussion on perpetuities, it was shown that: Eqn(7): r... By subtracting equation (6) from equation (7): Eqn(7) - Eqn(6) r n n n n n n... Notice that the right side of this equation is a perpetuity which begins in period n and it has already been shown that the value of a perpetuity beginning in period 0, in this case n, is /r (remembering that the cash flows in each period are equal). Using the same principles established earlier, to discount this perpetuity back to the present we multiply by the present value discount factor: / n which results in: r r r n n This very important relationship connects, r, n, and so that if given any three of these variables, the fourth can be found. However, calculating /r [ - / n ] can be tedious and tables are available giving the calculations for several values of r and n (refer Appendix 4.). The amounts given in the body of the table in Appendix 4. are called the present value annuity factors. Examples. How much money would you be able to borrow if you had to repay the loan in 0 years with equal annual payments at an interest rate of %, if the most you could afford in annual payments is $,600 with the first payment in one year s time?
7 hapter 4: Arithmetic of Finance 77,600. (.) 0 By looking in Appendix 4., we find that the present value annuity factor is Therefore:, $4,690. What would the annual payments be on a $50,000 house mortgage if it was to be repaid in 0 years with 9% interest? In Appendix 4. the present value annuity factor for 9% over 0 years is 9.85 and substituting into the Present Value equation: $50, $5,477. If payments on a stereo were $,500 per year, the price of the stereo was $4,754 and interest rates were 0%, how long would it take to repay the loan? In this case, we know the present value and the payment amount and we want to find the number of years used to calculate the present value annuity factor. Therefore: $4,754 $,500 Annuity Factor for n years at 0% Annuity Factor for n years at 0%.69 Looking in Appendix 4. in the 0% column we find.69 in Row 4. It will therefore take four years to repay the loan. 4. What is the interest rate being charged on a $0,000 loan if annual payments were $4,96 for 6 years? $0,000 $4,96 Annuity factor for 6 years at r% Annuity factor for 6 years at r% Looking in Appendix 4., Row 6, the column having is the 7% column. The interest being charged is 7%. ompounding Agreement There is no mystery in doing these calculations if the period being considered is not annual. The important point is that the interest rate and the length of the period are in agreement. Traditionally interest rates are quoted as annual rates, yet often monthly payments are required. It must be understood if the interest
8 78 Financial Management and Decision Making rate being discussed compounds monthly or annually. If it compounds monthly, then: monthly annually If the annual rate is quoted at 0% with monthly compounding, then the monthly rate is: r monthly (.) / -.5% It is necessary to use.5% in the above formula if monthly payments are being considered. If interest rates are compounding annually, then the annual rate is simply divided by the number of periods in the quoted annual rate. So if an interest rate of 0% is compounded annually, then the monthly rate is 0%/ or.667%. Using fractional interest rates in the days of computers and advanced calculators is not a problem. While the use of the tables has been standard procedure, they are fast becoming obsolete. ontinuous ompounding ontinuous compounding is a term used when taking the limit of the interest rate as the period of time of compounding approaches zero. Rather than annually, monthly, weekly, daily, etc, the length of time before interest is paid (charged) on interest is instantly. Lump Sum For a lump sum: FV e nr where e is the base of the natural logarithms as defined by: e log ( /x) x, as x approaches infinity. (Note that e is an irrational number which to six significant figures is.788.) Example What is the present value of $8,000 to be received in 5 years using 9% continuous discounting?
9 hapter 4: Arithmetic of Finance 79 $8,000 e ,000 e $5,0 Annuity For an annuity: [( - e - rn )/r] Example Determine the present value of receiving $,000 per year for the next 5 years if it is continuously discounted at 5%., ( e ) , $, ompounding Magic? A common approach used to convince investors that a particular investment is the best, is to talk in terms of future values rather than present values. Exhibit 4. illustrates the effect of interest rates and time on future values, according to the formula: FV n Notice that in Exhibit 4., the higher the interest rate or the longer the period of time, the faster the future value becomes a very large number. It is the real rate of interest which is important. For example, telling a graduate student that $,500 saved per year at 0% interest compounded annually will result in $66, in 40 years time overlooks the corresponding inflation rate, but it sounds impressive.
10 80 Financial Management and Decision Making Exhibit 4. Dollars Time (n) or Interest Rate Nevertheless, it is magical to earn (charge) interest on interest. How long would it take to become a millionaire if you could save only $,000 per year and earn 5% interest? Appendix 4. shows the accumulated future value of a fixed dollar invested every year at a given percentage rate. It would only take years. Once it was a dream to be a millionaire. However, during periods of high interest rates and salary levels it becomes a reasonable expectation. Yet the present value of that $m discounted back to the present at 5% over years would only be $,. Summary Since time influences the value of money it is necessary to adjust for this before comparing alternate financing arrangements and investments. The relationship between the present value and the future value of money is the interest rate. The key point is that all dollar amounts must be considered at the same point in time. Living in the present, it makes the most sense to put all values into present dollar terms. Although it is possible to deal in future values it can cause undue misunderstandings because of the size of the dollar amounts which result. Furthermore, we have an intuitive understanding of what today s money is worth and hence it is more effective and appropriate to use today s values rather than a point in the future when comparing relative amounts of money. Transforming future dollar amounts into present dollar value terms can be done whether the amount is a discrete value to be received in the future, a perpetuity, a growing perpetuity, or an annuity. It is helpful to know and understand the formulae for these transformations since the tables provided only cover specific interest rates and specific numbers of periods. More importantly you need to
11 hapter 4: Arithmetic of Finance 8 understand the concepts and relationships which the formulae represent. It is possible to use various techniques to bring future cash flows back to the present. onfusion usually arises over the timing of the cash flows and the best way around this confusion is to draw a time line and place each flow on the line at the point in time where it occurs. ompounding can result in very large future values. When calculating over various time periods, one must always have agreement between the period of time and the interest rate used. Finally, remember that those in the finance and banking industry already know how to do these calculations. This means that there are no bargains in the financing packages offered by various payment arrangements. However, by putting financing packages or investment schemes into present value terms, they can be better understood, compared, and evaluated. Glossary of Key Terms Deflator/Discount Factor The rate that is applied to future cash flows to restate them in year zero dollars. (Present Value) The value today of expected future cash flows discounted at an appropriate rate. FV (Future Value) Expected future dollars adjusted to take account of their interest earning potential between the present time and their expected future incidence. Annuity A series of set cash payments over a set period. ompounding Determining the future value by the use of compounding interest, that is, interest on interest, period by period. Perpetuity A series of equal cash payments continuing into infinity. Selected Readings Brealey, R. & Myers, S., Principles of orporate Finance, Fourth Edition, McGraw-Hill, 99. Brigham, E., Financial Management Theory and Practice, Third Edition, The Dryden Press, 98. Francis, J.., Investments: Analysis and Management, Fifth Edition, McGraw-Hill, New York, 99. Keown, A.J., Scott, D.F., Martin, J.D., and Petty, J.W., Basic Financial
12 8 Financial Management and Decision Making Management, Third Edition, Prentice-Hall, 985. Peirson, G., and Bird, R., Business Finance, Third Edition, McGraw Hill, Sydney, 98. Questions 4. What is meant by the term the time value of money? Why is this concept important in business finance? 4. Explain the relationship between the discount rate and the present value of a sum to be received in the future. 4. What is meant by the term continuous compounding? Why is an investment on which interest is compounded continuously more attractive than one on which interest is compounded semi-annually? 4.4 Illustrate the difference between an annuity and a perpetuity. 4.5 Explain the relationship between the table for the present value of an annuity and that for the present value of a single sum. 4.6 A rich aunt upon hearing that you are about to commence a degree, decides to gift you some money rather than leave it to you in her will. She makes you two alternative offers: a. To give you $0,000 at the end of each of your four years at university; or b. To give you $00,000 at the end of four years. Required: Imagine you are about to commence your degree. Which offer would you accept? (Assume a market interest rate of.5%.) 4.7 What is the maximum price you would pay for an investment that promises a cash flow of $,500 per annum for the next years if you want to earn a 5% rate of return on your investment?
13 hapter 4: Arithmetic of Finance You have borrowed $5,000 from the bank to pay for a skiing holiday in the South Island. If the bank requires you to repay the loan in three annual instalments of $,906, what interest rate are you being charged? 4.9 A friend has just started a new business venture. She claims that if, in one year s time, you were to invest $0,000 in the venture, she would guarantee a return of $8,500 per annum for the succeeding five years. Required: a. If the market rate of return on a similar investment is %, should you accept the offer? b. If the offer were changed so that you were required to invest the $0,000 now and receive $8,500 per annum during years one through to five, would you accept? 4.0 You decide to invest in your friend s venture outlined in 4.9 and raise a loan of $0,000 to be paid in 0 six-monthly instalments. If the interest rate is 0% per annum, what will be the amount of these instalments? 4. The AB Manufacturing ompany has purchased a $50,000 grinding machine and expects a return of $5,74 over the next 5 years. What rate of return does the machine yield? 4. You borrow $50,000 at a rate of 8% for 5 years and repay the loan in equal six-monthly payments. What amount must you repay every six months? If the interest rate were 0% what would the six-monthly repayments be? What would the repayments be if the loan were at 6% for 0 years? 4. You wish to buy a stereo in two year s time and estimate that it will cost you $,00. If you have $,800 now, at what annual rate must it be invested? 4.4 Upon retirement, the manager of Paul s Peanut ompany inquires about the pension he will receive from the superannuation fund to which he has contributed for the past 0 years. The manager of the fund outlines four alternatives that are available: a. per annum for the next 5 years b. per annum for the next 0 years c. per annum for 0 years beginning in 5 years time, or d. A lump sum of $80,000 to be paid in one year s time. Required: Assuming a discount rate of 5%, calculate the value in today s dollars of each of the four
14 84 Financial Management and Decision Making alternatives. 4.5 If the population of New Zealand is currently.5 million and has grown at a constant annual rate of 4%, what was the population five years ago? 4.6 Upon graduating from university in four years time you wish to purchase a new car which you estimate will cost you $7,000. To pay for the car how much must you deposit at the end of each of the next four years in an account bearing interest at 8% per annum compounded annually? 4.7 If an investment has grown fourfold after being invested at 6% per annum compounded semi-annually, for how long has it been invested? 4.8 On December two students bought a new beer fridge at a cost of $,5. They paid $0 deposit and agreed to pay the balance in four equal annual instalments including both the principal and interest at 4% on the declining balance. How much would the students have to repay each year? 4.9 What is the present value of a $,500 perpetuity discounted at.5%? 4.0 If Martin rowe made an average of 6 runs per innings last season, and his average has been improving at a rate of 5% per season, what was the average number of runs he would have made six seasons ago? 4. You are offered $5,000 today, $50,000 in 0 years, or $00,000 in 5 years. Assuming you can earn % on your funds at the local bank, which alternative should you choose? 4. On January 987 you win a large sum of money in a lottery. You decide to continue working for 0 years and retire on your 0th birthday ( January 997). In order to provide for your retirement, you have developed a comprehensive financial plan.. You will deposit a lump sum in a bank account paying 0% interest per annum.. During your retirement up until your death (on January 07), you want to receive an annual income of $00,000 with the first payment made on January 997 and the last payment on January 06.. Along with three friends you intend to build a yacht to participate in the 08 Americas up
15 hapter 4: Arithmetic of Finance 85 hallenge and estimate that to do this you will require an additional $50m on January Upon your death you wish to leave $500,000 to the SPA. Required: How much will you need to deposit in the bank account to provide for your retirement? 4. A friend wishes to borrow $65,600 from you. He agrees to repay the loan in five annual instalments of $6,000, the first being paid in one year s time. What rate of return are you earning on the loan? 4.4 A payment that is to be received in ten years time has a present value of $0,000 based on a discount rate of 5%. What is the value of the payment? 4.5 Assuming a discount rate of 8%, what is the present value of a single investment yielding the following cash flows: a. $0,000 to be paid in five years time b. $50,000 to be received at the end of years five through to ten (inclusive) c. $,000 to be paid today d. $0,000 to be received in eleven years time 4.6 If the average house price today is $90,000 and the average price ten years ago was $45,70, what has been the per annum increase in the average house price? 4.7 An insurance broker offers you a chance to invest $0,000 in a fund which pays no interest but will return $44,05 in ten years time. What rate of return does the fund yield? 4.8 An investment yields a return of $500 per annum forever. What rate of return are you earning if you pay the following amount for the investment? a. $,000 b. $5,000 c. $6, This year the AB ompany paid a dividend of $0.9 per share on its ordinary shares. If the dividend ten years ago was $0.5 what is the annual growth in dividends?
16 86 Financial Management and Decision Making 4.0 If an investor has a choice between an investment earning 8% per annum compounded annually and an investment earning 8% per annum continuously compounded, which should s/he choose? 4. What is the value of a perpetuity which pays $00 per year forever if your required rate of return is 0%? 4. If you invest $,000 today how much will you have: a. In 6 years at 7%? b. In 5 years at 0%? c. In 5 years at 0% (compounded semiannually)? 4. Mary Mills has retired after 5 years with a company. Her total pension funds have an accumulated value of $00,000 and her life expectancy is another 8 years. Her pension fund manager assumes that she can earn an 8% return on her investment. What will her yearly annuity be for the next 8 years? 4.4 You wish to retire in 0 years, at which time you want to have accumulated enough money to receive an annuity of $,000 for 5 years of your retirement. During the period before retirement, you can earn 8% annually and after retirement you can earn 0% on your money. What are your annual contributions to the retirement fund to allow you to receive the $,000 annuity? 4.5 Your younger sister, Susie, will start university in five years. She has just informed your parents that she wants to go to a university which will cost them $5,000 per year for four years (assume this is paid at the beginning of each yea. Anticipating Susie s ambitions, your parents started investing $500 per year five years ago and will continue to do so for five more years. (Use 0% as the appropriate discount rate throughout this problem). Required: a. How much more will your parents have to invest each year for the next five years to have the necessary funds for Susie s education? Now (five years late Susie is 8 years old and she wants to go overseas instead of going to university. Your parents have accumulated the necessary funds for her education. Instead of university fees, your parents pay $5,000 towards her travel expenses and plan to take a year-end vacation themselves costing $8,000 per year each year for the next three years. b. How much money will your parents have at the end of three years to help pay for your masters degree, which you will start then? c. You plan to work on a masters degree and then perhaps a Ph.D. If graduate school costs
17 hapter 4: Arithmetic of Finance 87 $0,000 per year, approximately how long will you be able to continue studying if your parents money is your only source of funds?