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Time Value

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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%

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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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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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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)

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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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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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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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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 1.315932Note 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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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.

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Time ValueAnnuities

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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Time ValueAnnuities

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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Time ValueAnnuities

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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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

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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)

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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

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• 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 ValueAnnuities/Bonds

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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.

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Time ValueBonds

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

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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

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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

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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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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