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1 Time Value

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2 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%

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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!)

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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

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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 todays principal amount (present value or P o ) by (1+.04)(1+.04) 1,000,000 x 1.0816 So FV = P o x (1.04) 2 Try 5 years. = 1.21665 (1.2167)

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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

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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.

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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

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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

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10 Time Value Annuities 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.

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11 Time Value Annuities Future 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

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12 Time Value Annuities Present 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

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13 Time Value Annuities Present 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

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14 Time Value Annuities 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

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15 Time Value Annuities 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)

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16 Time Value Annuities 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

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17 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. Time Value Annuities/Bonds

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

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19 Time Value Bonds 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

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20 Time Value Bonds 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

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21 Time Value APR 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

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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

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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) -1 x 100 = 10.25 Test 10 x 1.1025 = 11.025 = 1.05 10 x 1.05 = 10.5

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