3-1 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Chapter 3 The Time Value of Money: An Introduction to Financial Mathematics

3-2 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Learning Objectives Understand simple interest and compound interest, including accumulating, discounting and making comparisons using the effective interest rate. Value, as at any date, contracts involving multiple cash flows. Calculate and distinguish present and future values of different annuities. Apply knowledge of annuities to solve a range of problems, including problems involving principal-and- interest loan contracts.

3-3 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Fundamental Concepts Cash flows — fundamental to finance, the funds that flow between parties either now or in the future as a consequence of a financial contract. Rate of return — relates cash inflows to cash outflows. (Equation 3.1) Interest rate — special case of rate of return (used when the financial agreement is in the form of debt).

3-4 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Time Value of Money Money received now can be invested to earn additional cash in the future. Relates to opportunity cost of giving up money or resources for a period of time — either forgone investments or consumption. Consider time significance — significant amount of time may elapse between cash outflows and inflows. Cash flows that occur at different points in time cannot simply be added together or subtracted — this is one of the critical issues conveyed in this chapter.

3-5 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Simple Interest Typically used only for a single time period. Interest is calculated on the original sum invested: –Where S is the lump sum payable Present Value: Typically present cash equivalent of an amount to be paid or received at some future date, calculated using simple interest.

3-6 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Compound Interest Compounding involves accumulating interest on previous interest payments, which will generate further interest. This earning of interest on interest is one of the key differences between simple interest and compound interest. The backbone of many time-value calculations are the present value (PV) and future value (FV) based on compound interest. The sum or future value accumulated after n periods is: The present value of a future sum is: Note: The PV and FV formulas are the inverse of each other!

3-7 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Nominal and Effective Interest Rates Nominal rate: –Quoted interest rate where interest is charged or calculated more frequently than the time period specified in the interest rate. Effective rate: –Interest rate where interest is charged at the same frequency as the interest rate quoted. –Used to convert different nominal rates so that they are comparable.

3-8 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Nominal and Effective Interest Rates (cont.) The distinction is important when interest is compounded over a period different from that expressed by the interest rate, e.g. more than once a year. The effective interest rate can be calculated as:

3-9 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Effective Annual Interest Rate Example 3.7: –Calculate the effective annual interest rates corresponding to 12% p.a., compounding: (a) Semi-annually = (b) Quarterly = (12.55%) (c) Monthly = (12.68%) (d) Daily = ( %)

3-10 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Real Interest Rates The real interest rate is the interest rate after taking out the effects of inflation. The nominal interest rate is the interest rate before taking out the effects of inflation. The real interest rate (i*) can be found as follows:

3-11 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Continuous Interest Rates Continuous interest is a method of calculating interest in which it is charged so frequently that the time period between each charge approaches zero. Continuous interest is an example of exponential growth:

3-12 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Geometric Rates of Return The rate of return between two dates, measured by the change in value divided by the earlier value. The average of a sequence of geometric rates of return is found by a process that resembles compounding. Average geometric rate of return is also referred to as the average compound rate of return.

3-13 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Valuation of Contracts with Multiple Cash Flows Value additivity: –Cash flows occurring at different times cannot be validly added without accounting for timing. –Only cash flows occurring at the same time can be added. –Therefore, it is necessary to convert multiple cash flows into a single equivalent cash flow. –Cash flows can be carried either forward in time (accumulated) or back in time (discounted).

3-14 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Valuation of Contracts with Multiple Cash Flows (cont.) Where a cash flow of C dollars occurs on a date t, the value of that cash flow at a future valuation date t* is given by: Measuring the rate of return: –Where there are n cash inflows C t (t = 1,..., n), following an initial outflow of C 0, the internal rate of return is that value of r that solves the equation:

3-15 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Internal Rate of Return (IRR) Consider three cash flows: –$1000 today, +$1120 in 1 year, +$25 in 2 years. What is the average rate of return on the initial investment of $1000, taking into account compounding, i.e. the IRR? The IRR is the r that satisfies the following equation:

3-16 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Internal Rate of Return (cont.) The answer cannot be solved for precisely, as the equation is a quadratic equation. Alternatively, and more generally, trial and error can be used, substituting different values for r. In practice, this would be done with a computer, using a program such as Excel or Lotus.

3-17 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Internal Rate of Return (cont.) The solution is: IRR = 14.19%, substitute back into the equation to confirm: The result is zero, confirming that the IRR is 14.19%.

3-18 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Annuities An annuity is a stream of equal cash flows, equally spaced in time. We consider four types of annuities: –Ordinary annuity. –Annuity due. –Deferred annuity. –Ordinary perpetuity.

3-19 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Ordinary Annuities Annuities in which the time period from the date of valuation to the date of the first cash flow is equal to the time period between each subsequent cash flow. Assume that the first cash flow occurs at the end of the first time period: $C$C$C$C$C$C$C$C$C$C$C$C

3-20 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Valuing Ordinary Annuities Present value (PV) of an ordinary annuity: Using the present value of annuity tables, values of A(n,i) for different values of n and i can be found.

3-21 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Ordinary Annuities Example 3.16: Find the present value of an ordinary annuity of $5000 p.a. for 4 years if the interest rate is 8% p.a: (a) Discounting each individual cash flow.

3-22 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Ordinary Annuities (cont.) Example 3.16 (cont.): Find the present value of an ordinary annuity of $5000 p.a. for 4 years if the interest rate is 8% p.a. (b) Using Equation (c) Using Table 4, Appendix A and Equation P = C x A(n,i) = $

3-23 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Annuity Due Where the first cash flow is to occur immediately: An annuity due of n cash flows is simply an ordinary annuity of (n – 1) cash flows, plus an immediate cash flow of C. The present value of an annuity due: $C$C$C$C$C$C$C$C$C$C$C$C$C$C

3-24 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Deferred Annuity Annuity in which the first cash flow is to occur after a time period that exceeds the time period between each subsequent cash flow: Present value of a deferred annuity: $C$C$C$C$C$C$C$C$C$C$C$C

3-25 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Deferred Annuity (cont.) The present value (PV) of a deferred annuity involves taking the present value of an ordinary annuity. This figure is a present value but, as the annuity is deferred, we need to discount the PV further. If the first cash flow is k periods into the future, we discount the PV by (k – 1) periods.

3-26 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Ordinary Perpetuity An ordinary annuity where the cash flows are to continue forever: The present value of an ordinary perpetuity: $C$C$C$C$C$C$C$C$C$C

3-27 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Ordinary Annuities Future value of an ordinary annuity: Example 3.20: –Starting with his next monthly salary, Harold intends to save $200 each month. –If the interest rate is 8.4% p.a., payable monthly, how much will Harold have saved after 2 years? –Solution: Monthly interest rate is 0.4/12 = 0.7%. Using Equation 3.28, Harold’s savings will amount to:

3-28 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Example: Ordinary Annuities (cont.) Substituting the values we have: Thus, at the end of 2 years, Harold will have saved $

3-29 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Principal-and-Interest Loans An important application of annuities is to loans involving a sequence of equal cash flows, each of which is sufficient to cover the interest accrued since the previous payment and to reduce the current balance owing. Such loans can be referred to as: –Principal-and-interest loans. –Credit loans. –Amortised loans.

3-30 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Principal-and-Interest Loans (cont.) Example 3.22: Borrow $ Make 5 years of annual repayments at a fixed interest rate of 11.5% p.a. What is the annual repayment? Use the PV of annuity formula:

3-31 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Principal-and-Interest Loans (cont.) Example 3.22 (cont.): Substituting values: Thus, annual repayments on this loan are $

3-32 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Balance owing at a given date: –Equals the present value of the then-remaining repayments. Loan term required: –Solving for the required loan term n: Principal-and-Interest Loans (cont.)

3-33 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Changing the interest rate: –In some loans (usually called variable interest rate loans), the interest rate can be changed at any time by the lender. Two alternative adjustments can be made: –The lender may set a new required payment, which will be calculated as if the new interest rate is fixed for the remaining loan term. –The lender may allow the borrower to continue making the same repayment and, instead, alter the loan term to reflect the new interest rate. Principal-and-Interest Loans (cont.)

3-34 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University General Annuities Annuity in which the frequency of charging interest does not match the frequency of payment. Thus, repayments may be made either more frequently or less frequently than interest is charged. Link between short period interest rate (i S ) and long period interest rate (i L ):

3-35 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian National University Summary Fundamental concepts in financial mathematics include rates of return, simple interest and compound interest. Valuation of cash flows: –Present value of a future cash flow. –Future value of a current payment/deposit. Annuities are a special class of regularly spaced fixed cash flows.