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Business Funding & Financial Awareness Time Value of Money – The Role of Interest Rates in Decision Taking J R Davies May 2011 Time Value of Money – The Role of Interest Rates in Decision Taking May 2011 Dick Davies Investment/Financing Decisions The time dimension – Many financial and investment decisions involve costs and benefits spread out over time An investment decision involves the commitment of resources on the expectation of future benefits Costs Benefits Benefits Benefits…… Time 0 1 2 3 This implies it is necessary to allow for •The time value of money •The impact of risk and uncertainty 3 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 4 Determining Interest Rates – A Simple Model Interest rate Saving = Deposits = Lending r Investment = Borrowing Saving and Investment 5 Interest rates adjust to changes in savings and investment – eg an increase in savings Interest rate Saving = Deposits = Lending r r1 Investment = Borrowing Saving and Investment 6 Interest Rates • The riskless real rate of interest (r0): 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. rM r0 f u eal ation for example if the r rate is 4 per cent, the rate of infl he is 6 per cent, and t risk premium is 5 per cent rM r0 f u 0.04 0.06 0.05 0.15 7 Time Dimension – Investment/Financing Decisions Capital Budgeting (Real Investments) -C0 +C1 +C2 +C3 +C4 +C5 Time 0 1 2 3 4 5 Share Purchase (Financial Investment) -C0 +C1 +C2 +C3 +C4 +C5 Time 0 1 2 3 4 5 Loan (Financing Decision) +C0 -C1 -C2 -C3 -C4 -C5 Time 0 1 2 3 4 5 8 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. 9 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 2 3 time 10 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 1 2 3 time The original sum (£100) plus interest that can be earned over one year (£10). 11 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 2 3 Time 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. 12 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). 13 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: FVFn/r = (1 + r)n where r is the rate of interest n is the number of time periods in the future 14 Developing Future Value Factors V1 V0 V0 r V0 (1 r ) V2 V1 V1r V1 (1 r ) V0 (1 r )(1 r ) V0 (1 r ) 2 Vn V0 (1 r ) n 15 Developing Future Value Factors - An Example V1 100 100 x 0.10 100(1 0.10) 110 V2 110 110 x 0.10 110(1 0.10) 100(1 0.10)(1 0.10) 110(1 0.10) 121 2 V3 121 121 x 0.10 121(1 x 0.10) [100(1 0.10)(1 0.10)] (1 0.10) 100(1 0.10) 3 100 x 1.331 133.10 16 Developing Future Value Factors V1 V0 (1 r ) V2 V0 (1 r ) 2 V3 V0 (1 r ) 3 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 Vn V0 (1 r ) n (1 r )(1 r )(1 r )....(1 r ) Multiply together n interest factors 17 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? . 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 ? Vn V0 (1 r ) n V 6 800(1 0.12) 5 800(1.7623) 1409.87 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. 19 Use factors taken from table 1 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 ? V5 V2 (1 r ) 5 2 V5 1000 (1 0.08 ) 3 1000 (1.2597 ) 1259 .70 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. 20 Use factors taken from table 1 Annuities An annuity is a constant payment at the end in each time period for a specified number of periods. A constant periodic NCF Constant net cash flows A A A A A A A A 0 1 2 3 4 5 6 7 8 … 21 Annuities 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 An Example of an Annuity 0 1 2 3 500 500 500 An annuity of £500 for three years 22 Investment Example and the use of Annuity Factors FV (3) V3 500 x (1.1) 2 500 x (1.1)1 500 x (1.0) V3 500 x [(1.1000) 2 (1.1000)1 (1.0000)] V3 500 x [(1.2100) (1.100) (1.000)] V3 500 x [3.3100] 1655 Future value annuity factor for three years at 10 per cent 23 Future Value Annuity Factors 24 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 ? 1200 1200 1200 1200 1200 1200 1200 1200 0 1 2 3 4 5 6 7 8 Accumulated Value ? 25 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.) Opening Closing Year Value Interest Value Savings 1 0.00 0.00 £0.00 1200 2 1200.00 84.00 1,284.00 1200 3 2484.00 173.88 2,657.88 1200 4 3857.88 270.05 4,127.93 1200 5 5327.93 372.96 5,700.89 1200 6 6900.89 483.06 7,383.95 1200 7 8583.95 600.88 9,184.83 1200 8 10384.83 726.94 11,111.76 1200 9 12311.76 FV (8) = 1200 times FVAF8/0.07 26 = 1200 times 10.2598 = 12311.76 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. 27 Assessing Pension Payments Period for contributions 3000 3000……. 3000 0 1 2 ………………...…………………… 30 V30 = £3000 times FVAF30/.05 = £3000 x 66.4388 = £199,317 28 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? 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 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 FVAF5/0.07 = £600,000 = X times 5.7507 = £600,000 X = 600,000 / 5.7507 = £104,334 30 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. 1 (1 r ) n 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.. 31 Discount Rates - Terminology • The discount rate • The opportunity cost of funds – interest foregone by waiting. • The required rate of return. • The cost of capital. 32 Present Value Factors 1 PVFn / r (1 r ) n Time 0 1 2 n 33 Present Value Factors All financial arithmetic is based on the future value equation. Vn V0 (1 r ) n 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 1 Vn x V0 (1 r ) n i.e. 1 V0 Vn (1 r ) n 34 Present Value Factors at 10 % 1.2 1 0.8 Interest lost in the 0.6 delay in receiving cash. 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 YEARS 35 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 ? 1 V0 Vn (1 r ) n 1 V0 650 (1 0.14) 3 650 0.6750 438.73 36 Present Value Factors 37 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 38 Net Present Value Equation 1 1 NPV I 0 C1 C2 .... (1 r ) (1 r ) 2 39 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 Time NCF PVF(10%) PV 0 -1,200 1.000 -1,200.0 1 500 0.909 454.5 2 500 0.826 413.0 3 500 0.751 375.5 NPV = 43.0 40 Present Value Annuity Factors at 10% 12.0000 Annuity Factors 10.0000 8.0000 6.0000 Discount Factors 4.0000 2.0000 0.0000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Years 41 Present Value Annuity Factors As The Sum of Discount Factors Present Value Sum of Present Year Factor 10% Value Factors 1 0.9091 0.9091 2 0.8264 1.7355 3 0.7513 2.4869 4 0.6830 3.1699 5 0.6209 3.7908 6 0.5645 4.3553 7 0.5132 4.8684 8 0.4665 5.3349 9 0.4241 5.7590 10 0.3855 6.1446 42 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 ? Cash Present Value Year Flow Factor 6% Present Value 1 840 0.9434 792.45 2 840 0.8900 747.60 3 840 0.8396 705.28 4 840 0.7921 665.36 5 840 0.7473 627.70 6 840 0.7050 592.17 7 840 0.6651 558.65 Present Value = 4689 20 Using PVAF 1 to 7 840 5.5824 4689.20 43 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 44 flow to a capital sum. Bank Loan – the Required Annual Payments Loan = Present Value of Repayments at 12 per cent 10,000 = X . PVAF5|.12 10,000 = X times 3.6048 X = 10,000/3.6048 = 2,774 45 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. 1 1 NPV 0 I 0 C1 C2 .... (1 i) (1 i ) 2 46 Investment Appraisal (IRR) Time NCF PVF12%) PV 0 -1,200 1.000 -1,200.0 1 500 0.893 446.4 2 500 0.797 398.6 3 500 0.712 355.9 NPV = 0 47 Loan Analysis (2) Period Loan at Interest Loan at End Repayment Outset (12%) of Year 1 1200 144 1,344 500 2 844 101 945 500 3 445 55* 500 500 Surplus 0 48 NPV and IRR PRODUCTIVITY OF CAPITAL (IRR) NPV SIZE OF THE INVESTMENT 49 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. Time NCF PVF(12%) PV Time NCF PVF(10%) PV 0 -2,400 1.0000 -2,400.0 0 -2,400 1.000 -2,400 1 1000 0.8928 892.8 1 1000 0.909 909 2 1000 0.826 826 2 1000 0.7972 797.2 3 1000 0.751 751 3 1000 0.7118 711.8 NPV = 0 NPV = 86 50 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. NPV 0 400,000 102,863 x PVAF |i 6 PVAF 400,000/102,863 3.8887 6/i PVAF6/0.14 3.8887 t (Look up able, row 6) i 0.14 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. NPV 0 750,000 150,000 x PVF|i 250,000 x PVF|i 300,000 x PVF|i 300,000 x PVF|i 100,000 x PVF|i 1 2 3 4 5 1 1 1 1 1 NPV 0 750,000 150,000 x 250,000 x 300,000 x 300,000 x 100,000 x (1 i )1 (1 i ) 2 (1 i ) 3 (1 i ) 4 (1 i )5 Time NCF 0 -750,000 1 100,000 2 250,000 3 300,000 4 300,000 Use Excel!! 5 100,000 IRR = 12%

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