FINANCE IN A CANADIAN SETTING Sixth Canadian Edition Lusztig, Cleary, Schwab

CHAPTER FIVE CHAPTER FIVE Time Value of Money

Learning Objectives 1. Describe how compound interest works. 2. Explain what is meant by the time value of money. money. 3. Define discounting and compare it to compounding. compounding. 4. Explain the difference between the nominal and the effective rate of interest. and the effective rate of interest. 5. Discuss how discounting and compounding affect effective yields and payment levels of affect effective yields and payment levels of term loans. term loans.

Introduction n Time value of money – refers to the fact that $1 received today is worth more than $1 received tomorrow n Compounding and discounting form the basis for the valuation process used in finance.

Basic Compounding n The table shows the ending wealth that an investor could have accumulated by the end of 1998 had he invested $1000 in 1938 n Cumulative Wealth ($000s) Stocks1,0912,10310,12827,63951,038193,038485,068 Bonds1,0561,4341,6232,0843,69110,48037,720 T-Bills 1,0061,0581,2441,8933,67811,48923,253 U.S. stocks 1,3442,65115,60244,80466,815303,3222,260,431

Compound and Discounting Variables P = current cash flow F = future cash flow A = the amount of annuity i = the stated (or nominal) interest rate I = dollar amount of interest r = the effective period rate of return n = # of periods under consideration m = # of compounding periods per year PV = present value of a future cash flow(s) FV = future value of a cash flow(s)

Compounding and Discounting F n = P(1 + r) n OR FV = PV(1 + r) n n The equations represent the compounding relationship that is the basis for determining equivalent future and present values of cash flows

Compounding and Discounting PV = FV x 1 (1 + r) n (1 + r) n n Discounting – the process of converting future values of cash flows into their present value equivalents

Cash Flows Across Time Periods n To determine the present and future values associated with multiple cash flows that are paid through time, the following process is used: 1.Choose a point in time as the basis for economic comparison 2.Shift cash flows that occur at different times into equivalent amounts at the chosen point in time through compounding or discounting 3.Add or subtract all of these equivalent cash flows to obtain a net total

Annuities n Annuity – series of payments over a specific period that are for the same amount and are paid at the same interval where the discount rate is applied to all cash flows n Ordinary Annuity – payments that take place at the end of each period n Examples of annuities include interest payments on debt and mortgages

Annuity Formulas n Future Value of an annuity n Present Value of annuity

Annuity Due n Annuity due - payments are made at the beginning of each period Example: leasing arrangements n To compensate for the payment made at the beginning of the time period, multiply the future or present value annuities factors by (1 +r) to shift them by one period (1 +r) to shift them by one period

Perpetuities n Exist when an annuity is to be paid in perpetuity n Present value of Perpetuity

Varying Compound Periods n Any time period can be chosen for compounding n Effective interest rate – actual interest rate earned after adjusting the nominal interest rate for the number of compounding periods

Varying Compound Periods n Effective annual rate formula n Effective period rate formula

Amortization of Term Loans n Compounding and discounting are found in debt financing n Under term loans or mortgages, borrower repays original debt in equal instalments n Instalments consist of two portions: 1. Interest 2. Principle

Amortization of Term Loans n Common computational problems with term loans or mortgages include: 1. What effective interest rate is being charged? 2. Given the effective interest rate, what amount of regular payments have to be made over a given time period, or what is the duration over which payments have to take place given the amount? 3. Given a set of repayments over time, what portion represents interest on principle?represents interest on principle? represents repayment of principle?represents repayment of principle?

Repayment Schedules for Term Loan and Mortgages n Most loans are not repaid on an annual basis n Loans can have monthly, bi-monthly or weekly repayment schedules n In Canada, interest on mortgages is compounded semi-annually posing a problem in calculating the effective period interest rate

Summary 1.Compounding specifies how a given amount of money grows over time at a particular rate of interest. By compounding at a rate that represents the time value of money, we can calculate future values of current cash flows. 2.Discounting is the inverse of compounding; it allows us to calculate present values of future cash flows

Summary 3.Because of time value of money, cash flows that occur at different points in time can be compared only by transforming them through compounding or discounting into equivalent flows with reference to a particular point in time. 4.The basic formulas for compounding and discounting are independent of the choice of the base period. (cont’d)

Summary (cont’d) However, because of institutional conventions, one must adjust the formulas to distinguish between the nominal or quoted rate of interest and the effective rate of interest. 5.Loans with level repayments can be split into interest and principle.