CHAPTER FIVE Time Value of Money J.D. Han

Learning Objectives 1. Explain what is meant by the time value of money. 2. Define Present Value versus Future Value 3. Define discounting versus compounding. 4. Explain the difference between the nominal and the effective rate of interest.

5.1 Introduction Time value of money -$1 received today is worth more than $1 received tomorrow, or vice versa - Present Value of $1 tomorrow is less than $1: $1 discounted by interest rate - Future Value of $1 today is more than $1: $1 compounded by interest rate

*Simple versus Compound Interest Rate Investing $1000 at 3% per year for 4 years According to Simple Interest: 4 years later The total future cash flow is F = $1000x (1 + 0.8x4)=$1,320 According to Compound Interest: F = $ 1,000 (1+0.08)(1+0.08)(1.08)(1.08) =$1,000(1.08)4 =$1,360.49

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

Compounding and Discounting For now i = r with an annual compounding and annual payment Fn = P(1 + r)n OR FV = PV(1 + r)n The equations represent the compounding relationship that is the basis for determining equivalent future and present values of cash flows

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

Annuities Annuity – series of payments over a specific period that are of the same amount and are paid at the same interval where one discount rate is applied to all cash flows Examples of annuities: interest payments on debt and mortgages

Future Value of an Annuity 1) Numerical Illustration: i= 10%; annuity of 4 years Year 1 2 3 4 $1000 ------------------------------ 1,331 $1000---------------------  1,210 $1000----------->1,100 $1,000 Total FV = 4,641

2) Formula FV = A(1+r)n-1+A(1+r)n-2 + …..+A(1+r)+ A

Present Value of an Annuity Year 1 2 3 4 5 1000 1000 1000 1000 909 826 751 683 Total PV = $3,169

2) Formula PV = A/(1+r)+A/(1+r)2 + …..+A/(1+r)n

*Annuity Due Annuity due - payments are made at the beginning of each period (Example: leasing arrangements) PV = A + A/(1+r)+A/(1+r)2 +…..+A/(1+r)n-1 Formula: multiply the future or present value annuities factors by (1 +r)

** Relationship between FV and PV Formulas FV = A(1+r)n-1+ A(1+r)n-2…..+A/(1+r) + A PV = A/(1+r)+A/(1+r)2 + …..+A/(1+r)n FV = PV/(1+r)n

Perpetuities Exist when an annuity is to be paid in perpetuity Present value of Perpetuity Example: Equities

5.3 Effective Interest Rate: Varying Compound Periods Quoted or Nominal interest rate : i period interest rate x # of periods in a year i annua = i sub-period times # of sub-periods in a year im = i / m

One Complication which creates the gap between r and i: The effective annual interest rate is not simply liner times of the effective sub-annual interest rate; It is a compounded one: m=1, 2, 4, 12, or 365 times compounding by lender(bank) Effective Annual interest rate: r actual interest rate earned/payable after adjusting the nominal/quoted interest rate for the number of compounding periods (1+rannual) = (1+ i/m)m

Second Complications The frequency of interest payments by borrower (1+Effective Annual Interest Rate) =(1 + Effective Semi-annual Interest Rate)2 =(1 + Effective Quarterly Interest Rate)4 =(1 + Effective Monthly Interest Rate)12 = (1 + Effective Daily Interest Rate)365 = ( 1 + r f ) r

Formulas: Effective Interest Rates Effective annual rate formula m = # of compounding by lender per year Effective period rate formula f= # of payments by borrower in a year; 1, 2, 4, or 12

*Numerical Examples Savings at (the ‘quoted’ annual interest rate of) 12% ‘compounded quarterly; Effective Quaterly interest rate = 12%/ 4 = 3%; Effective annual interest rate rannual: rannual = (1+0.03)4 -1= 0,1255 or 12.55%; Effective montly interest rate rmonthly : (1+ rmonthly)12 = 1.1255% rmonthly = (1.1255)1/12-1 = 0.009901or 0.9901%

5.4 Amortization of Term Loans 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 repayment of principle?

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

Numerical Example Question: What is the Canadian monthly payment of a $ 100,000 mortgage with an amortization period of 25 years, a quoted rate of 12 %? Answer: m=?; f=12 Effective annual interest rate= Effective monthly interest rate= PV=?; new ‘n’=old ‘n’ times 12 Which formula to use?

Numerical Example Answer: m=2; f=12 Effective annual interest rate= (1+0.12/2)2-1 Effective monthly interest rate= [(1+0.12/2)2]1/12-1 PV=100,000; n = 300