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CHAPTER 2

CHAPTER 2. THE TIME VALUE OF MONEY. Overview. Using formulae, tables, financial calculators and spreadsheets to determine the: Future Value of: a single sum an annuity Present Value of: a single sum an annuity a perpetuity a growing perpetuity

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CHAPTER 2

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  1. CHAPTER 2 THE TIME VALUE OF MONEY

  2. Overview • Using formulae, tables, financial calculators and spreadsheets to determine the: • Future Value of: • a single sum • an annuity • Present Value of: • a single sum • an annuity • a perpetuity • a growing perpetuity • a cash flow growing at a constant rate • an uneven cash flow stream

  3. Overview (continued) • Define and calculate an effective rate • Distinguish between nominal and real interest rates • Apply compounding and discounting to complex cash flow streams • Apply time value of money principles to real world problems and the valuation of bonds • Establish the factors that determine the term structure of interest rates

  4. Future Value • Future value is the value in dollars that an investment or series of investments will grow over a stated time period at a specified interest rate.

  5. FUTURE VALUES FV: The amount of cash which will have accrued by a given date resulting from earlier lump-sum or periodic investments. PV:The value of an investment at the beginning of a period, sometimes referred to as the principal sum. r: The interest rate, expressed as a decimal fraction. I: The periodic investments or instalments made, excluding single lump-sum investments. n:The number of periods for which the investment is to receive interest.

  6. Future Value - Single Sum PV = 100 FV = ? An amount of R100 is invested for a one year rate of 12% p.a. What is the future value of the investment? FV = PV (1 + r) = R100 (1.12) = R112 0 1

  7. FUTURE VALUES Lump Sum: Multiple periods – Annual Interest Compounded • Example 2.2: Calculating the future value: more than one year • An amount of R100 is invested for 10 years at a rate of 12% p.a. compound interest. What is the future value of this investment at the end of 10 years? • For year one, FV = R100 (1.12) = R112 • For year two, FV = R112 (1.12) = R125.44 • For year three, FV = R125.44 (1.12)= R140.50 etc • This can be generalized to: • FV = PV (1 + r)n(Formula 2.2) • FV = R100 (1.12)10 • = R100 × 3.1058 • = R310.58

  8. FUTURE VALUES Lump Sum: Multiple periods – Annual Interest Compounded • Example 2.3: Calculating the principal • An investor wishes to invest a sum of money which will accumulate to R310.58 in 10 years time. How much must be invested today, if a rate of 12% p.a. is obtained? • Changing the subject of Formula 2.2, it can be stated as: • PV = FV / (1+r)n(Formula 2.3) • = 310.58 / (1.12)10 • = 310.58 / 3.1058 • = R100 • As expected from the results of Example 2.2, the required investment is R100. Table A may clearly be used to determine the calculation of (1.12)10.

  9. FUTURE VALUES Lump Sum: Multiple periods – Annual Interest Compounded • Example 2.4: Calculating the number of periods • An investor is informed that an investment of R100 will grow to R310.58. If it is known that the applied rate is 12%, after how long can the R310.58 be collected? The use of logarithms (or Table A) is required: • (1+r)n =FV / PV (Formula 2.4) • = 310.58 / 100 • = 3.1058 • The number 3.1058 is the factor defined in table A. Because it is known that the interest rate is 12%, it is possible to move down the 12% columnuntil the number nearest to 3.1058 is found. In this example it is found in the 10-period row. • Also: Ln3.1058/Ln1.12 = 10 years

  10. FUTURE VALUES Lump Sum: Multiple periods – Annual Interest Compounded • Example 2.5: Calculating the interest rate • An investor is given the opportunity of investing R100 today with a promised future value of R310.58 in 10 years’ time. • At what rate is the investment accruing interest? •  Developing from equation 2.4, it is possible to make r the subject of the formula as follows: • (1+r)n =FV / PV (Formula 2.5) • r = (FV/PV)(1/n) - 1 • = (310.58/100)(1/10) – 1 • = 0.12 OR 12% • Table A can again be used. This time one would search for a number close to 3.106 by looking along the 10 period row. Once the closest number to 3.106 is located, the column in which it is situated is the required interest rate.

  11. Investing with Warren Buffett • Warren Buffett earned 20.3% per year from 1965 to 2009 (45 yrs). If your grandfather had invested $1000 on your behalf then, how much would you have at the end of 2009? • FV = 1000 (1+0.203)45 = $4 092 166 How much would you have accumulated if he had rather invested in the general share market (S&P500)? [See page 2-7] If you invested with Allan Gray, R10 000 in 1974, how much would you have in 2010? [See page 2-8]

  12. FUTURE VALUES Lump Sum: Multiple periods – Annual Interest Compounded Example 2.6: Future value, interest compounded monthly An investor deposits R100 into an account which offers 12% p.a. interest compounded monthly. Find the value of the investment at the end of one year. 12% over 12 months = 1% interest added every month. At the end of the twelfth month it would be: FV = R100 (1.01)12 = R100 × 1.1268 = R112.68 The effective rate in this example is 12.68%. The formula required to generalize this calculation is as follows: FV = PV (1+ r/m)mn(Formula 2.6)

  13. Single SumMultiple periods and;Non - Annual Compounding Example: An investor deposits R100 into an account which offers 12% p.a. interest compounded monthly. What is the value of the investment at the end of 10 years?

  14. Future Value Calculation Answer: FV = PV (1 + r/m)mn = R100 (1 + 0.12/12)12x10 = R100 x 3.300 = R 330

  15. Annual Effective Rate

  16. Annual Effective Rate Interest rates quoted by three banks: • Bank X: 15%, compounded daily • Bank Y: 15.5%, compounded quarterly • Bank Z: 16%, compounded annually Which bank would you borrow from?

  17. Annual Effective Rate You would borrow from the bank quoting the “highest” interest rate!

  18. What is the future value of an Ordinary Annuity?

  19. Future Value using Tables & Formula • Formula • 100 x 3.3744 = 337.44 • Go to Table B - select factor of for 3 years and 12%

  20. 0 1 2 3 0 1 2 3 Ordinary Annuity or Annuity Due? • Annuity Due • Ordinary Annuity

  21. Future Value of an Annuity Due

  22. What is the Future Value if the annuity is payable in advance? • Using the Formula • Table B - select 4 periods (3+1) and 12% = 4.7793 and then minus 1 = 3.7793

  23. Calculation: The Present Value of a future amount due one year from today • Example 2.11: • An investment offers the opportunity to receive R100 one year from now if R90 is paid immediately. Should an investor who applies a 12% discount rate make the investment? • PV = FV / (1+r)n(using Formula 2.3) • = 100 / (1.12)1 • = R89.29

  24. Calculation:The Present Value of a future amount due more than one year from today • An investment offers the opportunity to receive R100 in ten years’ time. If an investor applies an interest factor of 12%, what is the highest price which will be offered for the investment? • PV = FV / (1+r)n(using Formula 2.3) • = 100 / (1.12)10 • = R32.20

  25. Present Value of an Annuity – using the Formula & Table D • The formula is; • R100 x 2.4018 = R240.18 • Also, use Table D – 3 periods, 12% = 2.4018 The

  26. Present Value of an Annuity Due

  27. PV of an Annuity Due – using a Formula

  28. Deferred Annuity • A deferred annuity commences a number of years in the future. An important example is a Pension.

  29. Present Value of a Perpetuity An investor wants to buy 1000 non-redeemable 9% preference shares of R1 each. If the interest rate which he applies is 12%, what is the present value of the investment? The investor is buying a future cash flow in perpetuity amounting to 9% of R1000, that is R90. Because a 12% return on the investment is expected, this problem requires the principal sum to be determined. PV = CF / r = 90 / 0.12 = R750

  30. Growing Perpetuities PV = PMT0 (1+g) (r – g) If a company has just made a payment of R10 000 and this is expected to grow at the expected inflation rate of 3% per year and the discount rate is 8%, then the present value of this perpetual payment stream is? PV = 10000 (1+0.03) (0.08 – 0.03) = R206000

  31. Growing annuity • PV of a Growing Annuity = PMT (1+g) 1 – (1+g)n (1+r)n r – g • What is the PV of a future salary growing at 6% per year for 40 years, if the required return is 9% per year. = R180 000 (1+0.06) 1 – (1.06)40 (1.09) 40 0.09 – 0.06 = R4 277 276

  32. Example:Calculating the required contribution to a retirement fund • An investor wishes to: • Retire in 25 year's time . Although, the investor is due a pension from her employment, she wishes to augment this by acquiring a further pension of R6000 per month by contributing to a retirement fund. • Receive a monthly pension income of R6000 per month for 12 years • Receive a lump sum payment of one third of the accumulated sum on retirement. • Additional information • The fund is currently earning a return of 12% per annum, interest compounded monthly. The return is expected to remain unchanged and to be sustainable over the next 37 years • Required • Determine the monthly contribution that the investor is required to make to the retirement annuity fund over the next 25 years.

  33. Retirement Funding [Three step calculation] • 1. Calculate the present value required at retirement date to generate an annuity of R6000 per month for 12 years. • 2. R456 823 is 2/3 of required amount. Find the full amount. • Full amount required = R456823 / 0,6667 = R685 200 • The monthly contribution required over the next 25 years to generate an accumulated sum of R685 200 is;

  34. Using Financial Calculators • What do these keys mean? • N = number of periods • I/Y = interest rate as a percentage. Enter a number, so if the rate is 10%, enter 10, not 0.10. Other calculators may reflect the interest rate per period as [i] and the number of periods as [n]. • PV = present value • PMT = annuity payment. Specify this as a zero when working with single sums only • In most cases, three or four inputs will be specified, and the financial calculator will solve for the remaining variable. On some calculators you will need to first press the COMPUTE key prior to pressing the missing input key.

  35. Let’s do some of the previous examples by using an HP-10BII financial calculator. • An investment of R100 invested for 10 years earning 12% per year compound interest will result in a future value of R310.6. Key in the following input values and press the FV key for the solution. • First enter the present value as a negative number, -100 or press 100 followed by (-), depending on the calculator being used, and then press the PV key,. Then enter 10 and press N, enter 12 and press I/Y, enter 0 and press PMT and then press FV (or Comp FV) to find the answer. • What is the interest rate that will achieve a present value of R100 growing to R310.60 within 10 years? Enter the inputs in the following sequence, then press the [I/YR] key to determine the interest rate of 12%.

  36. The role of Interest Rates Interest is a payment for the use of money The supply and demand for money (loans) is determined by 3 main factors: • The time value of money (preferring it now rather than later) • The risk of capital repayment • Expected inflation

  37. Interest Rate Theory • The expectation theory • The slope of the term structure of interest rate depends on the expected future spot rates of interest • The liquidity preference theory • The interest rate risk is greater, the longer the time to maturity • The market segmentation theory • Different investors have different investment preferences as to timing due to legal, regulatory, business and personal motives.

  38. A Typical Yield Curve[The term structure of interest rates] The length of time of the loan The interest rate set by the market Yield % 20 18 16 14 12 10 5 10 15 20 25 Years to Maturity

  39. Applying Time Value of Money Principles to Bonds • A bond is a financial instrument issued by the government and companies to raise funds • The bond will stipulate that the issuer is obliged to pay the bond holder a fixed interest or coupon rate until the maturity of the bond is repaid to bondholders.

  40. Applying Time value of Money principles to bonds Example A Government security which has a fixed coupon rate of 10% per year, coupon interest rate payable semi-annually and maturity date is the end of Oct 2015. Assume the current date is 1 November 2011. The current market yield is 12% p.a. What is the value of the bond? The interest rate per half-year is 6% and there are 8 half-years until maturity.

  41. Example: Valuing bonds Current date 1 November 2011

  42. Volatility of values based on the term Example: Two bonds pay a coupon rate of 7.5% but one is redeemable in a year’s time and the other bond is redeemable in 6 years time. How will values change if interest rates change?

  43. Self-study example: Transfer of Ronaldo to Real Madrid • Cristiano Ronaldo was transferred to Real Madrid for €93 million at the beginning of July 2009. Ronaldo, who is also the captain of Portugal, is reported to have signed a 6-year contract and agreed to be paid €11.11 million in his first year (2009-2010) with this amount rising by 25% annually for the remaining 5 years of his contract. • Required: • What will Ronaldo’s salary be in the final year of his contract (2014-2015)? • At a discount rate of 4% per year, what is the present value to Ronaldo of his future earnings with Real Madrid? Assume his annual earnings are paid on 31 December of each year with his first annual salary due on 31 December 2009 for the 2009-2010 season. Assume the current date is also 31 December 2009 so that the first payment is due right away.

  44. PV of Ronaldo’s contract

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