1. Business Funding & Financial Awareness Time Value of Money – The Role of Interest Rates in Decision Taking J R Davies May 2011

2. Time Value of Money – The Role of Interest Rates in Decision Taking May 2011 Dick Davies

3. 3 Investment/Financing Decisions The time dimension – An investment decision involves the commitment of resources on the expectation of future benefits 0 1 2 3 Costs Benefits Benefits Benefits…… Time Many financial and investment decisions involve costs and benefits spread out over time This implies it is necessary to allow for The time value of money The impact of risk and uncertainty

4. 4 Time Value of Money A pound today is worth more than a pound to- morrow………even in the absence of –Risk and uncertainty –Inflation The time value of money stems from the interest rate – effectively the price that balances the supply and demand for loans- and this will positive in a world of constant prices and no uncertainty

5. 5 Determining Interest Rates – A Simple Model Saving = Deposits = Lending Investment = Borrowing Saving and Investment r Interest rate

6. 6 Interest rates adjust to changes in savings and investment – eg an increase in savings Saving = Deposits = Lending Investment = Borrowing Saving and Investment rr1rr1 Interest rate

7. 7 Interest Rates The riskless real rate of interest (r 0 ): the rate of interest that can be expected in the absence of –Risk and uncertainty –Inflation A premium is added to the “real” rate of interest for –Risk and uncertainty – this will vary across borrowers –Inflation.

8. 8 Time Dimension – Investment/Financing Decisions Capital Budgeting (Real Investments) 0 1 2 3 4 5 -C 0 +C 1 +C 2 +C 3 +C 4 +C 5 012 3 4 5 Share Purchase (Financial Investment) Loan (Financing Decision) +C 0 -C 1 -C 2 -C 3 -C 4 -C 5 Time 0 1 2 3 4 5

9. 9 Evaluating Cash Flows Arising at Different Points in Time Cash flows that occur at different points in time cannot be summed to determine the net benefit position If a cash payment is made now on the expectation of receiving cash inflows in the future it is necessary to borrow the funds to make the payment now, and this implies an interest cost will be incurred, or use your own funds - and this implies foregoing the interest income that could have been earned on these funds in either case there is an interest cost to consider.

10. 10 Adjusting Values to Allow for Interest (1) Assume the interest rate is 10 % ie 0.10 What are the equivalent values at the end of one year, year two, etc to a sum of £100 available today ? 100 ? ?? 0 1 23 time

11. 11 Adjusting Values to Allow for Interest (2) Given an interest rate of 10 % what is the equivalent value at the end of one year of £100 that is available today ? 100(10)110?? 0 123 The original sum (£100) plus interest that can be earned over one year (£10). time

12. 12 Adjusting Values to Allow for Interest (3) Given an interest rate of 10 % what is the equivalent value at the end of two years of £100 that is available today ? 100 (10) 110(11)121? 0 1 23 The original sum (£100) plus interest (£10) for year one, and interest of £10 for year two on the initial £100, plus £1 of interest on the interest of £10 earned in period one. Time

13. 13 Adjusting Values to Allow for Interest (4) Given an interest rate of 10 % £100.00 today £110.00 next year £121.00 two years from now £133.10 three years from now all have the same real value (in principle) and are equally acceptable (assume no risk and no inflation for simplicity).

14. 14 Future Value Factors To obtain the equivalent value at a point in time in the future of a sum available today we must multiply this sum by a future value factor – also referred to as a compound interest factor, or more simply as the interest factor – to allow for interest that can be earned on the sum available to-day: FVF n/r = (1 + r) n where r is the rate of interest n is the number of time periods in the future

15. 15 Developing Future Value Factors

16. 16 Developing Future Value Factors - An Example

17. 17 Developing Future Value Factors This implies one more interest factor is introduced for each added time period and the value at the end of period n is given by Multiply together n interest factors

18. Using Future Values (1) What will £800 deposited in a bank account at an interest rate of 12 per cent grow to by the end of year 5 if all interest income is reinvested?.

19. 19 Determining the Future Value of a Sum (2) What will £800 invested at interest rate is 12 per cent grow to by the end of year 5 ? The interest rate for these calculations must be written in decimal form. In principle this implies that £800 today is of equivalent value to £1410 to be received after five years. Use factors taken from table 1

20. 20 Determining the Future Value of a Sum (3) You expect to receive £1000 at the end two years and you expect to be able to invest this to earn an interest rate of 8 per cent. What can the sum be expected to grow to by the end of year 5 ? The interest rate for these calculations must be written in decimal form. In principle this implies that £1000 after two years is of equivalent value to £1259.70 to be received after five years. Use factors taken from table 1

21. 21 Annuities An annuity is a constant payment at the end in each time period for a specified number of periods. A constant periodic NCF 012345678 … AAAAAAAAAAAAAAAA Constant net cash flows

22. 22 An Example of an Annuity 500 500 500 01230123 An annuity of £500 for three years An annuity is a constant payment at the end (or the start) of each time period for a specified number of time periods. A constant periodic NCF Annuities

23. 23 Investment Example and the use of Annuity Factors Future value annuity factor for three years at 10 per cent

24. 24 Future Value Annuity Factors

25. 25 Using Future Value Annuity Factors What will be the accumulated value of annual savings of £1200 deposited in a savings account at the end of each of the next 8 years if the interest rate is 7 percent ? 012 3 45 6 7 8 1200 1200 1200 1200 1200 1200 1200 1200 Accumulated Value ?

26. 26 Using Future Value Annuity Factors (2) What will be the value of annual savings of £1200 for the next 8 years if the interest rate is 7 percent ? (Interest being reinvested at 7 per cent.) Year Opening ValueInterest Closing ValueSavings 10.00 £0.001200 21200.0084.001,284.001200 32484.00173.882,657.881200 43857.88270.054,127.931200 55327.93372.965,700.891200 66900.89483.067,383.951200 78583.95600.889,184.831200 810384.83726.9411,111.761200 912311.76 FV (8) = 1200 times FVAF 8/0.07 = 1200 times 10.2598 = 12311.76

27. 27 Example: Using Time Value Concepts (3) Determining Pension Income An individual pays £3,000 per annum into a pension fund (a defined contribution scheme) for thirty years. The scheme guarantees a minimum return of 5 per cent. How much will have been accumulated in the fund by the end of the 30 year period.

28. 28 Assessing Pension Payments 0 1 2 ………………...…………………… 30 Period for contributions V 30 = £3000 times FVAF 30/.05 = £3000 x 66.4388 = £199,317 3000 3000……. 3000

29. 29 Using Future Value Annuity Factors (5) Hendy Hotels Ltd Hendy Hotels is a family owned concern that avoids the use of external funding. The owners recognise that they will have to undertake a major investment five years from now to meet EU safety regulations. This investment will cost £600,000 and the company’s management intend putting aside funds at the end of each of the next five years so as to be able to cover the expenditure. The funds can be invested at 7 per cent until needed. If the same amount is saved each year how much has to saved on an annual basis to cover the expenditure?

30. 30 Using Future Value Annuity Factors (5) Hendy Hotels Ltd Hendy Hotels is a family owned concern that avoids the use of external funding. The owners recognise that they will have to undertake a major investment five years from now to meet EU regulations. This investment will cost £600,000 and the company’s management intend putting aside funds at the end of each of the next five years so as to be able to cover the expenditure. The funds can be invested at 7 per cent until needed. If the same amount is saved each year how much has to saved on an annual basis to cover the expenditure? FV (5) = X times FVAF 5/0.07 = £600,000 = X times 5.7507 = £600,000 X = 600,000 / 5.7507 = £104,334

31. 31 Present Value or Discount Factors (1) To derive the value today, the present value, of a sum expected in the future this future sum must be multiplied by a present value or discount factor. This has a value of less than one as the denominator (1+r) is greater than one when r is positive, and applying this to a future NCF will allow for the loss of interest as a result of the delay in the receipt of the cash..

32. 32 Discount Rates - Terminology The discount rate The opportunity cost of funds – interest foregone by waiting. The required rate of return. The cost of capital.

33. 33 Present Value Factors Time 012n012n

34. 34 Present Value Factors All financial arithmetic is based on the future value equation. If a future value is known the equivalent value today is derived by multiplying the future value by the discount factor, one over the interest factor i.e.

35. 35 Present Value Factors at 10 % Interest lost in the delay in receiving cash.

36. 36 Determining Present Values What is the equivalent value today of £650 to be received three years from now if the interest rate (discount rate) is 14 percent ?

37. 37 Present Value Factors

38. 38 Net Present Value of an Investment The surplus expected from a project, measured in today’s values ….after appropriate allowances have been made for the –recovery the capital outlay –the interest charges It can also be defined as the increment of wealth generated created by an investment

39. 39 Net Present Value Equation

40. 40 Assessing Investment Proposals Using NPV An investment of 1200 is expected to produce cash flows of 500 at the end of years 1, 2 and 3. The required rate of return is 10 per cent. Determine the investment’s NPV

41. 41 Present Value Annuity Factors at 10% Annuity Factors Discount Factors Years

42. 42 Present Value Annuity Factors As The Sum of Discount Factors

43. 43 Using Present Value Annuity Factors What is the equivalent value today of £840 to be received at the end of each year for the next seven years if the interest rate is 6 percent ?

44. 44 Example: Using Time Value Concepts (1) Arrangements for repaying a bank loan A bank makes a loan at £10,000 at a fixed interest rate of 12 per cent and this is to be repaid in five equal instalments. (Each instalment covers repayment of the loan as well as the interest on the outstanding balance of the loan. Determine the annual instalment. (Convert a capital sum into a constant cash flow.) The instalment is the equivalent constant annual cash flow to a capital sum.

45. 45 Bank Loan – the Required Annual Payments Loan= Present Value of Repayments at 12 per cent 10,000= X. PVAF 5|.12 10,000= X times 3.6048 X= 10,000/3.6048 = 2,774

46. 46 Internal Rate of Return The rate of discount at which the NPV is equal to zero. This may be interpreted as the highest rate of interest that can be paid on a loan used to finance a project without making a loss.

47. 47 Investment Appraisal (IRR)

48. 48 Loan Analysis (2)

49. 49 NPV and IRR NPV PRODUCTIVITY OF CAPITAL (IRR) SIZE OF THE INVESTMENT

50. 50 Investment Appraisal (IRR) Consider the simple investment considered earlier - an outlay of 1200 that is expected to produce three annual NCFs of 500 and a discount rate of 10 per cent. The NPV was 43 and the IRR was 12 per cent. Now double the size of all the NCFs – the NPV doubles but the IRR remains at 12 per cent.

51. Determining the IRR (1) An investment of £400,000 is expected to a constant annual NCF of £102,865 for the next 6 years. Determine the approximate value of the investment’s IRR.

52. Determining the IRR (2) An investment of £750,000 is expected to produce NCFs at the end of years 1 to 5 of £150,000, £200,000, £300,000, £300,000 and £100,000 respectively. Determine the approximate value of the investment’s IRR. Use Excel!!