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Time Value

Time Value. Time Value. What would you prefer to have GBP 1,216,653 in five years time or GBP 1,315,932 in seven years time? Current interest rate 4.5%. Time Value (Future Value). Compounding, interest earned in one period is added to the principal to work out interest for the next period.

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Time Value

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  1. Time Value

  2. Time Value • What would you prefer to have • GBP 1,216,653 in five years time or • GBP 1,315,932 in seven years time? • Current interest rate 4.5%

  3. Time Value(Future Value) • Compounding, interest earned in one period is added to the principal to work out interest for the next period. (It is assumed you do not spend it!)

  4. Time Value(Future Value) • Take an example of GBP 1,000,000 invested at 4% for 5 years Year 1 1,000,000 x .04 = 40,000 Year 2 1,040,000 x .04 = 41,600 We could go on but it is boring

  5. Time Value(Future Value) • Luckily we are able to generalise the step by step approach. • Note at the end of year two the total value (Future Value) is 1,081,600 (1,040,000 + 41,600). • We may get the same result by multiplying today’s principal amount (present value or Po) by (1+.04)(1+.04) • 1,000,000 x 1.0816 • So FV = Po x (1.04)2 Try 5 years. = 1.21665 (1.2167)

  6. Time Value(Future Value) • (1.04)5 = 1.216652902 • Therefore the future value of GBP1,000,000 compounded at 4% per annum will be GBP1,216,653

  7. Time Value(Present Value) • Return to our initial question, ‘which would you prefer, GBP 1,216,653 in five years or GBP 1,315,932 in seven years?’ • Our problem is that different amounts at different times are not comparable directly. • Different amounts are comparable today. • We know that at an interest rate of 4% 1,216,653 in five years has a value today of 1,000,000. This is referred to as the present value.

  8. Time Value(Present Value) • 1,000,000 x (1.04)5 = 1,216,653 • 1,000,000 x 1.216653 = 1,216,653 Therefore the present value may be found • 1,216,653 = 1,000,000 1.216653

  9. Time Value(Present Value) • So all we need do is discount 1,315,932 at 4% for 7 years and see if it gives a present value of more or less than 1,000,000. • (1.04)7 = 1.315932 • 1,315,932 = 1,000,000 1.315932 Note the above is the same as 1,315,932 x 1 = 1,315,932 x .759917686 1.315932 = 1,000,000

  10. Time ValueAnnuities • An Annuity is an ‘investment that pays a predetermined annual (or other time period i.e. monthly) regular income. The amounts are always the same. • Annuities also have present values and future values.

  11. Time ValueAnnuitiesFuture Value • Suppose you have an annuity of GBP 1,000 per annum for three years. Payment is made at year end. Payment at No of Yrs FV at 5% End Value end year interest 1. 1,000 2 1.1025 1,102.5 2. 1,000 1 1.05 1,050.0 3. 1,000 0 1,000.0 3,152.5

  12. Time ValueAnnuitiesPresent Value • As with uneven flows we may use tables of factors to find future values and to produce present values • What investment is needed today to produce an annuity of 1,000 p.a. for three years at 5%? • 1,000 x pv annuity factor • 1,000 x 2.7232 = 2,723.2

  13. Time ValueAnnuitiesPresent Value • Proof • 2,723.2 x 1.05 = 2,860 – 1,000 • 1,860 x 1.05 = 1,953 – 1,000 • 953 x 1.05 = 1,000.65

  14. Time ValueAnnuities • We use annuities where the flows are of the same amount. • As individuals we come across them most frequently with Mortgages but also • Purchasing annuities on retirement • Some loan repayments

  15. Time ValueAnnuities • Example. • You wish to borrow GBP500,000 to buy a one bed room flat in Bath. • Mortgage at 5% repaid over 4 years by 4 annual payments • 500,000 = 141,004 3.5460 (PVAn Factor)

  16. Time ValueAnnuities • Example • A benefactor wishes to reward your first class degree in four years time with a gift of GBP1,000. How much will they need to invest annually to produce this sum at an interest rate of 4%? • 1,000 = 235 4.2465

  17. Time ValueAnnuities/Bonds • Bonds Definition A Bond is a negotiable certificate that evidences indebtedness. Bonds are also referred to as ‘notes’ or ‘debentures’ With a fixed interest rate bond you receive a fixed set of cash flows representing the interest (coupon) flows plus a final cash flow of principal.

  18. Time ValueBonds • A bond is issued at par with a face value of USD 100,0000 at 10%, interest paid annually, repayment in three years.

  19. Time ValueBonds Cash Flows Yr0 Yr1 Yr2 Yr3 -100,000 +10,000 +10,000 + 10,000 +100,000 PVF 1 .9091 .8265 .7513 PV –100,000 +9,091 + 8,265 + 82,643 NPV = -100,000 +100,000 (with a bit of rounding) = zero

  20. Time ValueBonds • But what if the interest rate in the market moves to 7 %? Cash Flows Yr0 Yr1 Yr2 Yr3 -100,000 +10,000 +10,000 + 10,000 +100,000 • PVF 1 .9346 .8734 .8163 • PV –100,000 +9,346 +8,734 + 89,793 • NPV –100,000 + 107,873 = + 7,873

  21. Time ValueAPR • Annual Percentage Rate or Effective Rate • You are charged by your credit card supplier at a rate of 6 % per annum, monthly. • What is this on an annual basis? • 12 • - 1 x 100 = 6.16 % 1+ .06 12

  22. Time Value Real and Nominal rates • If you are offered a rate of return of 10% pa but inflation is at 5% then your real rate, i.e. your increased purchasing power is 4.762 % • Cost of product GBP10 • Price after one year 10 x 1.05 = GBP10.5 • Return on GBP10 after one year 10 x 1.1= 11 • Purchasing power = 11 = 1.047619 or 4.762% • 10.5

  23. Time Value • To obtain a real rate of 5% when inflation is 5 % then the nominal rate must be 1.05 x 1.05 = 10.25 Test 10 x 1.1025 = 11.025 = 1.05 10 x 1.05 = 10.5

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