5 51 Explain the importance of the time value of money and how it is related to an investors opportunity costs 52 Define simple interest and explain how it works 53 Define compound interest and explain how it works ID: 437022
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Slide1Slide2
Time Value of Money
5
5.1
Explain the importance of the time value of money and how it is related to an investor’s opportunity costs.
5.2
Define simple interest and explain how it works.
5.3
Define compound interest and explain how it works.
5.4
Differentiate between an ordinary annuity and an annuity due, and explain how special constant payment problems can be valued as annuities and, in special cases, as perpetuities.
5.5
Differentiate between quoted rates and effective rates, and explain how quoted rates can be converted to effective rates.
5.6
Apply annuity formulas to value loans and mortgages and set up an amortization schedule.
5.7
Solve a basic retirement problem.
5.8
Estimate the present value of growing perpetuities and annuities.Slide3
5.1 OPPORTUNITY COST
Money is a
medium of exchange
.
Money has a time value because it can be invested today and be worth more tomorrow.The opportunity cost of money is the interest rate that would be earned by investing it.
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© John Wiley & Sons Canada, Ltd.Slide4
5.1 OPPORTUNITY COST
Required rate of return
(
k
) is also known as a discount rate.To make time value of money decisions, you will need to identify the relevant discount rate you should use.
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5.2 SIMPLE INTEREST
Simple interest
is interest paid or received only on the initial investment (
principal
).The same amount of interest is earned in each year.Equation 5-1:
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EXAMPLE: Simple Interest
The same amount of interest is earned in each year.
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5.2 SIMPLE INTERESTSlide7
5.3 COMPOUND INTEREST
Compound interest
is interest that is earned on the principal amount
and
on the future interest payments.The future value of a single cash flow at any time ‘
n’ is calculated using Equation 5.2.
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USING EQUATION 5.2
Given three known values, you can solve for the one unknown in Equation 5.2
Solve for:
FV - given PV, k, n (finding a future value)
PV - given FV, k, n (finding a present value)
k - given PV, FV, n (finding a compound rate)n - given PV, FV, k (find holding periods)
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[
5.2]Slide9
COMPOUND VERSUS SIMPLE INTEREST
Simple interest grows principal in a linear manner.
Compound interest grows exponentially over time.
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EXAMPLE: Compounding (Computing Future Values)
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[
5.2]
5.3 COMPOUND INTERESTSlide11
Compound value interest factor (
CVIF
)
represents the future value of an investment at a given rate of interest and for a stated number of periods.
The CVIF for 10 years at 8% would be:$100 invested for 10 years at 8% would equal:
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5.3 COMPOUND INTEREST
Compounding
(Computing
Future
Values)Slide12
EXAMPLE: Using the CVIF
Find the FV
20
of $3,500 invested at 3.25%.
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5.3 COMPOUND INTERESTSlide13
5.3 COMPOUND INTEREST
Discounting (Computing Present Values)
The inverse of compounding is known as
discounting
.
You can find the present value of any future single cash flow using Equation 5.3.
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[
5.3]Slide14
Present value
i
nterest
f
actor (PVIF) is the inverse of the CVIF.
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5.3 COMPOUND INTEREST
Discounting (Computing Present Values)Slide15
EXAMPLE: Using the
PVIF
Find the
PV
0 of receiving $100,000 in 10 years time if the opportunity cost is 5%.Booth • Cleary – 3rd Edition
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5.3 COMPOUND INTEREST
Discounting (Computing Present Values)Slide16
Solving for Time or “Holding Periods”
Equation 5.3 is reorganized to solve for
n
:
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5.3]
5.3 COMPOUND INTERESTSlide17
EXAMPLE: Solving for ‘n’
How many years will it take $8,500 to grow to $10,000 at a 7% rate of interest?
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5.3 COMPOUND INTERESTSlide18
Solving for Compound Rate of Return
Equation 5.3 is reorganized to solve for
k
:
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5.3]
5.3 COMPOUND INTERESTSlide19
EXAMPLE: Solving for ‘k’
Your investment of $10,000 grew to $12,500 after 12 years. What compound rate of return (k) did you earn on your money?
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5.3 COMPOUND INTERESTSlide20
5.4 ANNUITIES AND PERPETUITIES
An
annuity
is a finite series of equal and periodic
cash flows. A
perpetuity is an infinite series of equal and periodic cash flows.
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© John Wiley & Sons Canada, Ltd.Slide21
An
ordinary annuity offers payments at the end of each period.
An
annuity due
offers payments at the beginning of each period. Booth • Cleary – 3rd Edition
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21© John Wiley & Sons Canada, Ltd.
5.4 ANNUITIES AND PERPETUITIESSlide22
The formula for the compound sum of an
ordinary annuity is:
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[
5.4]
5.4 ANNUITIES AND PERPETUITIESSlide23
EXAMPLE: Find the Future Value of an Ordinary Annuity
You plan to save $1,000 each year for 10 years. At 11% how much will you have saved if you make your first deposit one year from today?
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5.4 ANNUITIES AND PERPETUITIESSlide24
The formula for the compound sum of an
annuity due is:
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[
5.6]
5.4 ANNUITIES AND PERPETUITIESSlide25
EXAMPLE: Find the Future Value of an Annuity Due
You plan to save $1,000 each year for 10 years. At 11% how much will you have saved if you make your first deposit today?
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5.4 ANNUITIES AND PERPETUITIESSlide26
The formula for the present value of an
annuity
is:
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[
5.5]
5.4 ANNUITIES AND PERPETUITIESSlide27
EXAMPLE: Find the Present Value of an Ordinary Annuity
What is the present value of an investment that offers to pay you $12,000 each year for 20 years if the payments start one year from day? Your opportunity cost is 6%.
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5.4 ANNUITIES AND PERPETUITIESSlide28
The formula for the present value of an
annuity
is:
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5.7]
5.4 ANNUITIES AND PERPETUITIESSlide29
EXAMPLE: Find the Present Value of an Annuity Due
What is the present value of an investment that offers to pay you $12,000 each year for 20 years if the payments start one today? Your opportunity cost is 6%.
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5.4 ANNUITIES AND PERPETUITIESSlide30
A perpetuity is an
infinite
series of equal and periodic cash flows.
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5.8]
5.4 ANNUITIES AND PERPETUITIESSlide31
EXAMPLE: Find the Present Value of a Perpetuity
What is the present value of a business that promises to offer you an after-tax profit of $100,000 for the foreseeable future if your opportunity cost is 10%?
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5.4 ANNUITIES AND PERPETUITIESSlide32
5.5 QUOTED VERSUS EFFECTIVE RATES
A
nominal
rate of interest is a ‘stated rate’ or
quoted rate (QR).An
effective annual rate (EAR) rate takes into account the frequency of compounding (m
). Booth • Cleary – 3rd Edition
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[
5.9]Slide33
EXAMPLE: Find an Effective Annual Rate
Your personal banker has offered you a mortgage rate of 5.5 percent compounded semi-annually. What is the effective annual rate charged (EAR)on this loan?
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5.5 QUOTED VERSUS EFFECTIVE RATESSlide34
EXAMPLE: Effective Annual Rates
EARs increase as the frequency of compounding increase.
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5.5 QUOTED VERSUS EFFECTIVE RATESSlide35
5.6 LOAN OR MORTGAGE ARRANGEMENTS
A
mortgage
loan is a borrowing arrangement where the principal amount of the loan borrowed is typically repaid (amortized) over a given period of time making equal and periodic payments. A blended payment is one where both interest and principal are retired in each payment.
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EXAMPLE: Loan Amortization Table
Determine the annual blended payment on a five –year $10,000 loan at 8% compounded semi-annually.
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5.5]
5.6 LOAN OR MORTGAGE ARRANGEMENTSSlide37
EXAMPLE: Loan Amortization Table
The loan is amortized over five years with annual payments beginning at the end of year 1.
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5.6 LOAN OR MORTGAGE ARRANGEMENTSSlide38
EXAMPLE: Mortgage
Determine the monthly blended payment on a $200,000 mortgage amortized over 25 years at a QR = 4.5% compounded semi-annually.
Number of monthly payments = 25 × 12 = 300
Find EAR:
Find EMR:
Determine monthly payment:
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5.6 LOAN OR MORTGAGE ARRANGEMENTSSlide39
EXAMPLE: Mortgage Amortization
The mortgage is amortized over 25 years with annual payments beginning at the end of the first month.
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5.6 LOAN OR MORTGAGE ARRANGEMENTSSlide40
5.7 COMPREHENSIVE EXAMPLES
Time value of money (TMV) is a tool that can be applied whenever you analyze a cash flow series over time.
Because of the long time horizon, TMV is ideally suited to solve retirement problems.
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COMPREHENSIVE EXAMPLE:
Retirement Problem
Kelly, age 40 wants to retire at age 65 and currently has no savings.
At age 65 Kelly wants enough money to purchase a 30 year annuity that will pay $5,000 per month.
Monthly payments should start one month after she reaches age 65.
Today Kelly has accumulated retirement savings of $230,000.
Assume a 4% annual rate of return on both the fixed term annuity and on her savings.
How much will she have to save each month starting one month from now to age 65 in order for her to reach her retirement goal?
*NOTE – these are ordinary annuities
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COMPREHENSIVE EXAMPLE:
Retirement Problem
How much will the fixed
t
erm annuity
cost at age 65?
Steps in Solving the Comprehensive Retirement ProblemCalculate the present value of the retirement annuity as at Kelly’s age 65.
Estimate the value at age 65 of her current accumulated savings.
Calculate gap between accumulated savings and required funds at age 65.
Calculate the monthly payment required to fill the gap.
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COMPREHENSIVE EXAMPLE:
Retirement Problem
Example Solution – Preliminary Calculations
Preliminary calculations Required
Monthly rate of return when annual APR is 4%
Number of months during savings period
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COMPREHENSIVE EXAMPLE: Retirement Problem
Time Line & Analysis Required to Identify Savings Gap
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Age 40 65 95
25 year asset accumulation phase
30 year asset depletion phase (retirement)
30 year fixed-term retirement annuity = 30
×12 =360 months
Existing Savings
Additional monthly savingsSlide45
COMPREHENSIVE EXAMPLE: Retirement Problem
Monthly Savings Required to fill Gap
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Age 40 65 95
25 year asset accumulation phase
Existing Savings
Additional monthly savings
Monthly savings to fill gap?
Your Answer
30 year asset depletion phase (retirement)Slide46
Appendix
5AGROWING ANNUITIES & PERPETUITIES
Growing Perpetuity
A growing perpetuity is an infinite series of periodic cash flows where each cash flow grows larger at a constant rate.
The present
value of a growing perpetuity is calculated using the following formula:
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[
5A-2]Slide47
Growing
Annuity
An annuity is a finite series of periodic cash flows where each subsequent cash flow is greater than the previous by a constant growth rate.
The formula for a growing annuity is:
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[
5A-4]
Appendix
5A
GROWING ANNUITIES & PERPETUITIESSlide48
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