1. CHAPTER FIVE
Time Value of Money
J.D. Han

2. 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.

3. 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

4. *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 ( x4)=$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. 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

6. 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

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

8. 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

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

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

11. Present Value of an Annuity
Year
909
826
751
683
Total PV = $3,169

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

13. *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)

14. ** 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

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

16. 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

17. 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

18. 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

19. 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

20. *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 = %
rmonthly = (1.1255)1/12-1 = or %

21. 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?

22. 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

23. 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?

24. 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