Learning ObjectivesLearning 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 Introduction5.1 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 *Simple versus Compound Interest RateInterest Rate
Investing $1000 at 3% per year for 4 years
According to Simple Interest:4 years later The total future cash flow isF = $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 5.2 Compound and Discounting VariablesDiscounting Variables
P = current cash flowF = future cash flowPV = 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 periodn = # of periods under consideration
A = the amount of annuity
Compounding and DiscountingCompounding and DiscountingCompounding :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
DiscountingDiscounting
PV = FV
(1 + r)n
Discounting – the process of converting future values of cash flows into their present value equivalents
AnnuitiesAnnuities
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 AnnuityFuture 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
Present Value of an AnnuityPresent Value of an Annuity
Year
1 2 3 4 5
1000 1000 1000 1000
909
826
751
683
Total PV = $3,169
*Annuity Due*Annuity DueAnnuity 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)
)()(
rr
rAPV
n
1
1
111
** 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
PerpetuitiesPerpetuities
Exist when an annuity is to be paid in perpetuity
Present value of PerpetuityExample: Equities
r
APV
5.3 Effective Interest Rate: 5.3 Effective Interest Rate: Varying Compound PeriodsVarying Compound Periods
Quoted or Nominal interest rate : iperiod 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 One Complication which creates the gap between r and the gap between r and i: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 ComplicationsSecond 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 RatesFormulas: Effective Interest Rates
Effective annual rate formulam = # of compounding by lender per year
Effective period rate formula
f= # of payments by borrower in a year; 1, 2, 4, or 12
11
m
m
iannualr
11
f
m
m
ieffectiver
*Numerical Examples*Numerical ExamplesSavings 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 Loans5.4 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 Repayment Schedules for Term Loan and MortgagesTerm 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 ExampleNumerical 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=12Effective annual interest rate=Effective monthly interest rate=PV=?; new ‘n’=old ‘n’ times 12Which formula to use?
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