© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Decision making in finance 1 Author: Glyn Davis

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Learning Objectives 2 On successful completion of the module, you will be able to:  Understand the key terms used within financial mathematics.  Solve simple and compound interest problems.  Solve problems involving depreciation.  Calculate the value of an investment when a sum is added on a regular basis.  Calculate the future value of an ordinary annuity (sinking funds).  Understand the concept of present value and discounting.  Calculate the amount invested to be able to withdraw a sum on a regular basis until the amount is zero (trust funds and loan repayments or present value of an ordinary annuity).  Calculate the net present value of a series of cash flows.  Calculate the present value of an annuity and perpetuity.  Calculate the internal rate of return (IRR) for a series of cash flows.  Calculate the modified internal rate of return (MIRR) where negative and positive cash flows have different interest rates.  Calculate the cost of a mortgage.  Solve problems using the Microsoft Excel spreadsheet. Author: Glyn Davis

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Simple interest 3 Author: Glyn Davis Do you remember Equation (10.1)? It tells us how much simple interest (SI) we would earn if we deposited a principle of £P in a bank for time T years, and if the bank paid an interest rate at R % per annum. Example 10.1 If the bank paid 8 % per annum on deposits, then a deposit of £250 left in the bank for 4 years would earn: Figure 10.1 illustrates the Excel solution The future value of the investment after 4 years which earned simple interest at 8% is £330. The simple interest earned is £20 per year with a total interest earned of £80 over the four years.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Compound interest and depreciation (1/5) 4 Author: Glyn Davis But suppose we did not withdraw our interest from the bank. If this is so, then the interest on deposit would itself earn interest. We would then be earning what is called compound interest Example 10.2 Let us now compare the two methods of earning interest on a year by year basis where the initial deposit is £250 at an interest rate of 8% per annum as illustrated in Table 10.1: Year Simple InterestCompound Interest Future Value (£)Interest earned (£) Future Value (£)Interest earned (£) * 0.08 = * 0.08 = * 0.08 = * 0.08 = * 0.08 = * 0.08 = * 0.08 = * 0.08 = Total Interest Earned

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Compound interest and depreciation (2/5) 5 Author: Glyn Davis So we see that there is a considerable difference between the two methods as illustrated in Figure Compound interest is the method that is invariably used in the business world. Figure 10.3 illustrates the effect of time on the growth rates of investments which are invested using simple compared to compound interest. We observe from Figure 10.3 the large difference between the future value based on simple interest (£ ) and compound interest (£ ) after 50 years.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Compound interest and depreciation (3/5) 6 Author: Glyn Davis FV = P(1 + r) n We can show that the future value of an investment with compound interest is given by Equation 10.4: Figure 10.4 Calculating future value using Equation (10.4).

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Compound interest and depreciation (4/5) 7 Author: Glyn Davis Figure 10.5 Calculating future value using Excel functions FV() and IMPT(). From Excel, the future value of £250 over 4 years compounded at 8% will be worth £ The interest earned would be I = FV - P = £90.12 to the nearest penny.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Compound interest and depreciation (5/5) 8 Author: Glyn Davis Effective annual interest rate (EAIR) and can be calculated using Equation (10.6): Where r = nominal interest rate, f = number of times interest is added per year. If we have one interest period then we can show from Equation (10.6) that the effective annual interest rate is equal to the nominal interest rate (EAIR = r). Example 10.3 A sum of money is deposited now at 10% per annum. How long will it take for the sum to double? Suppose that the sum invested is £1000, then after n years we require the sum to be £2000. Figure 10.8 illustrates the Excel solution (pv is negative in the Excel NPER function given that we deposit this amount). From Excel, the sum will double after 7.27 years if the interest rate was 10% per annum.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Increasing the sum invested 9 Author: Glyn Davis So far, we have examined how an initial deposit would grow if it earned compound interest. But suppose we added to the amount deposited at the end of each year (called increasing the sum invested). Equation (10.8) gives the sum left on deposit if we withdraw fixed amounts from the bank each year. Example 10.8 So if we deposit £1000, and add £100 to our deposit at the end of each year, and if interest is compounded at 10% p.a. then the sum available after four years is given by Equation (10.8). Figure illustrates the Excel solution. From Excel, future value of the investment is £ after 4 years.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Sinking funds or future value of an ordinary annuity (1/3) 10 Author: Glyn Davis A common form of investment made by investors is an annuity. The investor can purchase the annuity either by a single payment or a series of payments over the lifetime of the investment. The investor will then receive regular payments each year, either until the investor’s death or for a guaranteed minimum number of years. A further example would be when a company wishes to set aside a sum of money at the end of each year to replace an asset. We can use Equation (10.8) to solve this problem: The problem involves a zero initial investment, P = 0.0. Substituting P = 0 into Equation (10.8) gives Equation (10.10): If we now make ‘a’ the subject of this formula, we will have an expression telling us how much we must set aside at the end of each year to achieve a specified future value of an ordinary annuity (or sinking fund) formula (final payment at start of year n):

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Sinking funds or future value of an ordinary annuity (2/3) 11 Author: Glyn Davis Example Suppose a machine is expected to last 8 years and its replacement price is estimated at £5000. What annual provision must be made to ensure sufficient funds are available if money can be invested at 8% per annum with payment at the end of the year? Figure illustrates the Excel solution. From Excel, £ deposited at the end of the year would be sufficient to yield the required future value.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Sinking funds or future value of an ordinary annuity (3/3) 12 Author: Glyn Davis Now suppose (as is more likely) the firm wishes to start the fund now and add to it at annual intervals then Equation (10.9) cannot be used to calculate the future value. We will have to use the expression FV = a(1+r) n + a(1+r) n-1 + a(1+r) n a(1+r) Now it can be shown that this expression is equal to Equation (10.12): Rearrange Equation (10.12) to make ‘a’ the subject to give Equation (10.13): Example Repeat the above example, this time assuming that the sum is invested at the beginning of each year. Find £435 deposited at the beginning of the year would be sufficient to yield the required future value – see Excel solution (Figure 10.17).

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance The concept of present value (1/2) 13 Author: Glyn Davis Suppose you were offered the choice of receiving £1000 now or £1000 in twelve months - which would you choose? It is almost certain that you would take the money now, even if you had a cast iron guarantee of receiving the money in the future. It would appear that we have strong preference for holding cash now against receiving cash in the future, and economists call this preference 'liquidity preference'. Now why is this preference so universally held? Almost certainly, inflation will have something to do with it. After all, if prices are rising then £1000 in 1 year will buy less than it will now, and so it will have less value than it has now. Equation (10.3) can be used to calculate how a sum invested now would grow under compound interest to a future value of FV = P(1 + r) n. If FV is the sum receivable in the future, then P must be its present value as given by Equation (10.14):

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance The concept of present value (2/2) 14 Author: Glyn Davis Example What is the present value of £1000 receivable in 2 years if money can be invested at 10% per annum compounded? Figure illustrates the Excel solution. From Excel, £1000 receivable in 2 years has a present value of £ now at the current interest rate it would grow to £1000 in two years. We should be indifferent between receiving £1000 in 2 years and £ now.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Trust funds and loan repayments or present value of an ordinary annuity (1/2) 15 Author: Glyn Davis Let us suppose that we deposit a certain sum of money now, and from this we wish to withdraw at the end of each year a fixed amount. We will continue to withdraw until nothing is left on deposit. In equation (10.5): The terminal future value FV would be zero, and because we are withdrawing ‘a’ would be negative. Therefore, Equation (10.8) can be written as follows: Simplifying this equation gives Equations (10.15) and (10.16):

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Trust funds and loan repayments or present value of an ordinary annuity (2/2) 16 Author: Glyn Davis Example You have decided to set up a trust fund for your son. You require the fund to pay him £2000 per year for the next 10 years. How much will this fund cost you if money can be invested at 10% per annum compounded with payment at the end of the year? This problem involves finding P, the initial investment, so Equation (10.16) is the one required. Figure illustrates the Excel solution From Excel, you will need to pay in £12, now for him to receive £2000 per year for the next 10 years if we assume money can be invested at 10% per annum.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance The present value and net present value of a stream of earnings (1/2) 17 Author: Glyn Davis The present value of a stream of earnings in the future is the amount you would pay now to provide the sum at a future date. The net present value (NPV) of a project or investment is the total of the present values of all the monetary inflows and outflows from the project or investment. Example Let us suppose that we have been promised £2000 in one year, £3000 in two years, £4000 in three years, and £3000 in four years. Equation (10.17) represents the present value of a series of payments:

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance The present value and net present value of a stream of earnings (2/2) 18 Author: Glyn Davis Figure illustrates the Excel solution From Excel, we find the present value = £ (using Equation 10.17) and the net present value is - £ (= , using Equation 10.18). From this calculation we would conclude that this would be a bad investment given that we would pay £ more than we would receive from the investment.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Internal rate of return and investment decisions (1/5) 19 Author: Glyn Davis The internal rate of return (IRR) of a series of cash flows is the discount rate that sets the net present value of the cash flows equal to zero. Example To aid understanding of the internal rate of return (IRR) consider how we calculated the net present value (NPV) in Example with an initial investment of £10,000: We can show that this equation gives NPV = - £ With the internal rate of return we would like to know at what discount rate, r, would the net present value (NPV) become exactly zero. In general, we can write the expression as follows:

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Internal rate of return and investment decisions (2/5) 20 Author: Glyn Davis In this case we need to solve this equation for r given that the equation is non- linear in the unknown variable, r. Figure represents a graph of the variation of NPV against different values of the discount rate, r As we can see the NPV value becomes zero between a discount rate of r = 6% and r = 8%, where the NPV changes from a positive number to a negative value. Figure illustrates the Excel solution From Excel, the internal rate of return (IRR) is 7.18%.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Internal rate of return and investment decisions (3/5) 21 Author: Glyn Davis While the internal rate of return (IRR) assumes the cash flows from a project are re-invested at the internal rate of return (IRR), the modified internal rate of return (MIRR) assumes that positive cash flows are re-invested at the firm's cost of capital, and the initial outlays are financed at the firm's financing cost. Rather than using the IRR we can use the modified internal rate of return (MIRR) to account for both interest rates. Example For example, consider a five year project with an initial outlay of £120,000 and associated cost of capital of 10% will return £39,000 in year 1, £30,000 in year 2, £21,000 in year 3, £37,000 in year 4, and £46,000 in year 5. To find the IRR of the project so that the net present value (NPV) = 0 we solve the equation: This problem can be solved using the IRR Excel function.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Internal rate of return and investment decisions (4/5) 22 Author: Glyn Davis Now, if we assume that the negative cash flows cost a finance rate of 10% and the positive cash flows are re-invested at 12% then we can calculate a modified internal rate of return (MIRR) to represent the interest rate using Equation (10.20): Figure illustrates the Excel solution From Excel, internal rate of return (IRR) = 13% and modified internal rate of return (MIRR) = 12.61%. You can see here that the 12.61% MIRR is lower than the IRR of 13%. In this case, the IRR gives a too optimistic picture of the potential of the project, while the MIRR gives a more realistic evaluation of the project.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Internal rate of return and investment decisions (5/5) 23 Author: Glyn Davis Example Consider the problem of calculating the compound annual growth rate that consists of an initial expenditure of €4500 and results in a series of interest payments on the dates specified in table 10.9 below. Date21/2/20085/4/200822/8/20081/10/20081/2/2009 Payment (€) The CAGR can be calculated for this problem using the Excel function XIRR. CAGR = XIRR(values, dates, guess) CAGR = XIRR(B6:B10,C6:C10) From Figure 10.31, compound annual growth rate is 11.7% with the present value of the schedule of cash flows equal to €

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Calculating the cost of a mortgage (1/2) 24 Author: Glyn Davis Most people will be familiar with mortgages, where a property can be purchased by buying a financial product from a financial services company. If we borrow £P over m months at an interest rate, r, and then the mortgage is repaid at a constant monthly rate of £a, paid at the end of the month with the value given by Equation (10.21): Example In this example we will assume that your mortgage application has been agreed for £200,000 to be repaid over eight years at 12% interest per annum. The mortgage agreement assumes that the payment will be made at the end of the year. How much will this yearly payment be at the end of the year and the effective interest rate?

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Calculating the cost of a mortgage (2/2) 25 Author: Glyn Davis Figure illustrates the Excel solution From Excel, the amount to pay back at the end of each year will be £40, with an effective annual interest rate of 12%. The effective annual interest rate (EAIR, see Cell C23) for this mortgage is simply the annual internal rate of return of its payments (see Cells C21 and C22) given that the mortgage consists of annual payments.

© Oxford University Press, All rights reserved. Chapter 10: Decision making in finance Conclusion 26 In this presentation we explored a range of topics, including: Author: Glyn Davis