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Laurence Booth
Sean Cleary
LEARNING OBJECTIVESLEARNING OBJECTIVES
Time Value of Money5
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.
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.
Booth • Cleary – 3rd Edition Page 3© John Wiley & Sons Canada, Ltd.
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.
Booth • Cleary – 3rd Edition Page 4© John Wiley & Sons Canada, Ltd.
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:
Booth • Cleary – 3rd Edition Page 5© John Wiley & Sons Canada, Ltd.
)() (time Value kPnPn
EXAMPLE: Simple InterestThe same amount of interest is earned in each year.
Booth • Cleary – 3rd Edition Page 6© John Wiley & Sons Canada, Ltd.
Example
You invest $500 today for five years and receive 10 percent annual simple interest.Annual interest = $500 × 0.1 = $50 per year
Year Beginning Amount Ending Amount1 $500 $5502 $550 $6003 $600 $6504 $650 $7005 $700 $750
Simple Interest
5.2 SIMPLE INTEREST
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.
Booth • Cleary – 3rd Edition Page 7© John Wiley & Sons Canada, Ltd.
)1( 0n
n kPVFV
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)
Booth • Cleary – 3rd Edition Page 8© John Wiley & Sons Canada, Ltd.
)1( 0n
n kPVFV [5.2]
COMPOUND VERSUS SIMPLE INTEREST• Simple interest grows principal in a linear manner.• Compound interest grows exponentially over time.
Booth • Cleary – 3rd Edition Page 9© John Wiley & Sons Canada, Ltd.
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48$0
$20,000
$40,000
$60,000
$80,000
$100,000
$120,000
Compound interest Simple interest
Year
Futu
re V
alue
EXAMPLE: Compounding (Computing Future Values)
Booth • Cleary – 3rd Edition Page 10© John Wiley & Sons Canada, Ltd.
)1( 0n
n kPVFV [5.2]
Example
You invest $500 today for five years and receive 10 percent annual compound interest.
Year Beginning Amount Interest Ending Amount1 $500 $500 × 0.1 = $50 $5502 $550 $550 × 0.1 = $55 $6053 $605 $605 × 0.1 = $60.50 $6664 $666 $666 × 0.1 = $66.66 $7325 $732 $732 × 0.1 = $73.20 $805
Compound Interest
5.3 COMPOUND INTEREST
• 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:
Booth • Cleary – 3rd Edition Page 11© John Wiley & Sons Canada, Ltd.
)1( ??,n
kn kCVIF
2.1589)08.01( 1008.0,10 knCVIF
$215.89 2.1589100$)08.01(100$ 1010 FV
5.3 COMPOUND INTERESTCompounding (Computing Future Values)
EXAMPLE: Using the CVIF
Find the FV20 of $3,500 invested at 3.25%.
Booth • Cleary – 3rd Edition Page 12© John Wiley & Sons Canada, Ltd.
$6,964.26
1.99$3,500
0.035)1(500,3$
20
%5.3,20020
knCVIFPFV
5.3 COMPOUND INTEREST
5.3 COMPOUND INTERESTDiscounting (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.
Booth • Cleary – 3rd Edition Page 13© John Wiley & Sons Canada, Ltd.
)1(
00 nk
FVPV
[5.3]
Present value interest factor (PVIF) is the inverse of the CVIF.
Booth • Cleary – 3rd Edition Page 14© John Wiley & Sons Canada, Ltd.
)1(
1 ??, nkn k
PVIF
5.3 COMPOUND INTERESTDiscounting (Computing Present Values)
EXAMPLE: Using the PVIFFind the PV0 of receiving $100,000 in 10 years time if the opportunity cost is 5%.
Booth • Cleary – 3rd Edition Page 15© John Wiley & Sons Canada, Ltd.
$61,391.33
0.6139$100,000 629.1
1$100,000
0.05)1(
1000,100$
10
%5,10100
knPVIFFVPV
5.3 COMPOUND INTERESTDiscounting (Computing Present Values)
Solving for Time or “Holding Periods”Equation 5.3 is reorganized to solve for n:
Booth • Cleary – 3rd Edition Page 16© John Wiley & Sons Canada, Ltd.
)1(
00 nk
FVPV
[5.3]
1ln
/ln 0
k
PVFVn n
5.3 COMPOUND INTEREST
EXAMPLE: Solving for ‘n’
How many years will it take $8,500 to grow to $10,000 at a 7% rate of interest?
Booth • Cleary – 3rd Edition Page 17© John Wiley & Sons Canada, Ltd.
years 2.406766.0
1625.0
ln[1.07]
]ln[1.17647
07.1ln
500,8$/000,10$ln
1ln
/ln 0
n
n
k
PVFVn n
5.3 COMPOUND INTEREST
Solving for Compound Rate of ReturnEquation 5.3 is reorganized to solve for k:
Booth • Cleary – 3rd Edition Page 18© John Wiley & Sons Canada, Ltd.
)1(
00 nk
FVPV
[5.3]
1 /1
0
n
n
PV
FVk
5.3 COMPOUND INTEREST
EXAMPLE: Solving for ‘k’Your investment of $10,000 grew to $12,500 after 12
years. What compound rate of return (k) did you earn on your money?
Booth • Cleary – 3rd Edition Page 19© John Wiley & Sons Canada, Ltd.
%88.101877.0
125.11000,10$
500,12$
1
083.012
1
/1
0
k
k
PV
FVk
n
n
5.3 COMPOUND INTEREST
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.
Booth • Cleary – 3rd Edition Page 20© John Wiley & Sons Canada, Ltd.
• 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 Page 21© John Wiley & Sons Canada, Ltd.
5.4 ANNUITIES AND PERPETUITIES
The formula for the compound sum of an ordinary annuity is:
Booth • Cleary – 3rd Edition Page 22© John Wiley & Sons Canada, Ltd.
1)1(
k
kPMTFV
n
n [5.4]
5.4 ANNUITIES AND PERPETUITIES
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?
Booth • Cleary – 3rd Edition Page 23© John Wiley & Sons Canada, Ltd.
01.722,16$722.16000,1$
11.0
111.1000,1$
11000,1$
10
10
10
10
,10
FVA
FVA
k
kFVA
FVIFAPMTFVAn
kn
5.4 ANNUITIES AND PERPETUITIES
The formula for the compound sum of an annuity due is:
Booth • Cleary – 3rd Edition Page 24© John Wiley & Sons Canada, Ltd.
k)(11)1(
k
kPMTFV
n
n [5.6]
5.4 ANNUITIES AND PERPETUITIES
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?
Booth • Cleary – 3rd Edition Page 25© John Wiley & Sons Canada, Ltd.
43.561,18$11.1722.16000,1$
)11.1(11.0
111.1000,1$
)1(11
000,1$
1
10
10
10
10
,10
FVA
FVA
kk
kFVA
kFVIFAPMTFVAn
kn
5.4 ANNUITIES AND PERPETUITIES
The formula for the present value of an annuity is:
Booth • Cleary – 3rd Edition Page 26© John Wiley & Sons Canada, Ltd.
)1(
11
0
kk
PMTPVn
[5.5]
5.4 ANNUITIES AND PERPETUITIES
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%.
Booth • Cleary – 3rd Edition Page 27© John Wiley & Sons Canada, Ltd.
06.639,137$47.11000,12$
06.0)06.1(
11
000,12$
0
20
0
06.0,200
PVA
PVA
PVIFAPMTPVA kn
5.4 ANNUITIES AND PERPETUITIES
The formula for the present value of an annuity is:
Booth • Cleary – 3rd Edition Page 28© John Wiley & Sons Canada, Ltd.
k)(1)1(
11
0
kk
PMTPVn
[5.7]
5.4 ANNUITIES AND PERPETUITIES
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%.
Booth • Cleary – 3rd Edition Page 29© John Wiley & Sons Canada, Ltd.
40.897,145$06.147.11000,12$
)06.1(06.0
)06.1(1
1000,12$
1
0
20
0
,0
PVA
PVA
kPVIFAPMTPVA kn
5.4 ANNUITIES AND PERPETUITIES
A perpetuity is an infinite series of equal and periodic cash flows.
Booth • Cleary – 3rd Edition Page 30© John Wiley & Sons Canada, Ltd.
0 k
PMTPV [5.8]
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Present Value of a PerpetuityWhat 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%?
Booth • Cleary – 3rd Edition Page 31© John Wiley & Sons Canada, Ltd.
000,000,1$1.0
000,100$ 1
0 k
PPV
5.4 ANNUITIES AND PERPETUITIES
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 Page 32© John Wiley & Sons Canada, Ltd.
11
m
m
QRkEAR [5.9]
EXAMPLE: Find an Effective Annual RateYour 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?
Booth • Cleary – 3rd Edition Page 33© John Wiley & Sons Canada, Ltd.
5.58%1-0275.1
1-)2
0.055(11-)1(
2
2m
EAR
m
QREAR
5.5 QUOTED VERSUS EFFECTIVE RATES
EXAMPLE: Effective Annual Rates
EARs increase as the frequency of compounding increase.
Booth • Cleary – 3rd Edition Page 34© John Wiley & Sons Canada, Ltd.
Example
QR = 8%
Frequency of Compounding
Effective Annual Rate
Annual 8.0%Semi-annual 8.16%
Quarterly 8.24322%Monthly 8.29995%
Daily 8.32776%Continuous 8.32781%
Effective Annual Rates
5.5 QUOTED VERSUS EFFECTIVE RATES
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.
Booth • Cleary – 3rd Edition Page 35© John Wiley & Sons Canada, Ltd.
EXAMPLE: Loan Amortization TableDetermine the annual blended payment on a five –year $10,000 loan at 8% compounded semi-annually.
Booth • Cleary – 3rd Edition Page 36© John Wiley & Sons Canada, Ltd.
$2,515.143.9759
$10,000PMT
0816.0
)0816.01(1
1 PMT$10,000
)1(1
1
5
0
kk
PMTPVn
[5.5]
5.6 LOAN OR MORTGAGE ARRANGEMENTS
EXAMPLE: Loan Amortization TableThe loan is amortized over five years with annual payments beginning at the end of year 1.
Booth • Cleary – 3rd Edition Page 37© John Wiley & Sons Canada, Ltd.
Example 5 -4
QR = 8% Principal Borrowed = $10,000EAR = 8.16% Amortization Period = 5 years
(1) (2) (3) (4) (5)
PeriodBeginning Principal PMT Interest
Principal Repayment
Ending Principal
1 $10,000 $2,515 $816 $1,699 $8,3012 $8,301 $2,515 $677 $1,838 $6,4633 $6,463 $2,515 $527 $1,988 $4,4754 $4,475 $2,515 $365 $2,150 $2,3255 $2,325 $2,515 $190 $2,325 $0
Loan Amortization Table
5.6 LOAN OR MORTGAGE ARRANGEMENTS
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:
Booth • Cleary – 3rd Edition Page 38© John Wiley & Sons Canada, Ltd.
%550625.41)2
045.01( 2
%3715318.0
)1(04550625.1
1)1(%550625.4
121
12
EMR
EMR
EMR
$1,106.85
003715.0
)003715.01(1
1
$200,000PMT
300
5.6 LOAN OR MORTGAGE ARRANGEMENTS
EXAMPLE: Mortgage AmortizationThe mortgage is amortized over 25 years with annual payments beginning at the end of the first month.
Booth • Cleary – 3rd Edition Page 39© John Wiley & Sons Canada, Ltd.
Example 5 -5
QR = 4.5% Principal Borrowed = $200,000EAR = 4.55% Amortization Period = 25 yearsEMR =0.372%
(1) (2) (3) (4) (5)
MonthBeginning Principal PMT Interest
Principal Repayment
Ending Principal
1 $200,000 $1,107 $743 $364 $199,6362 $199,636 $1,107 $742 $365 $199,2713 $199,271 $1,107 $740 $366 $198,9054 $198,905 $1,107 $739 $368 $198,5375 $198,537 $1,107 $738 $369 $198,168
Mortgage Amortization Table
5.6 LOAN OR MORTGAGE ARRANGEMENTS
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.
Booth • Cleary – 3rd Edition Page 40© John Wiley & Sons Canada, Ltd.
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 $230,000.• Assume a 4% annual rate of return on both the fixed term
annuity and on her savings.• How much will she have to save each month starting one
month from now to age 65 in order for her to reach her retirement goal?
*NOTE – these are ordinary annuitiesBooth • Cleary – 3rd Edition Page 41© John Wiley & Sons Canada, Ltd.
COMPREHENSIVE EXAMPLE:Retirement Problem
How much will the fixed term annuity cost at age 65?Steps in Solving the Comprehensive Retirement Problem
1. Calculate the present value of the retirement annuity as at Kelly’s age 65.
2. Estimate the value at age 65 of her current accumulated savings.
3. Calculate gap between accumulated savings and required funds at age 65.
4. Calculate the monthly payment required to fill the gap.
Booth • Cleary – 3rd Edition Page 42© John Wiley & Sons Canada, Ltd.
COMPREHENSIVE EXAMPLE:Retirement Problem
Example Solution – Preliminary Calculations
Preliminary calculations Required• Monthly rate of return when annual APR is 4%
• Number of months during savings period
Booth • Cleary – 3rd Edition Page 43© John Wiley & Sons Canada, Ltd.
%326.0
00326.1)04.1(04.11
1)1(%4
083.12
1
12
m
m
m
k
k
k
3001225 n
COMPREHENSIVE EXAMPLE: Retirement ProblemTime Line & Analysis Required to Identify Savings Gap
Booth • Cleary – 3rd Edition Page 44© John Wiley & Sons Canada, Ltd.
Age 40 65 95
25 year asset accumulation phase
30 year asset depletion phase
(retirement)
30 year fixed-term retirement annuity = 30 ×12 =360 months
Existing Savings
Additional monthly savings
142,613$
)04.1(000,230$)1( 2525025
annualkPFV
k
kPMTFVA
n 1)1(25
586,920$
142,613$728,533,1$
GAP
728,533,1$
COMPREHENSIVE EXAMPLE: Retirement Problem
Monthly Savings Required to fill Gap
Booth • Cleary – 3rd Edition Page 45© John Wiley & Sons Canada, Ltd.
142,613$
)04.1(000,230$)1( 2525025
annualkPFV
46.813,1$64.507
586,920$00326.0
1)00326.1(
586,920$
1)1( 30025
kk
FVAPMT n
Age 40 65 95
25 year asset accumulation phase
Existing Savings
Additional monthly savings
586,920$
142,613$728,533,1$
GAP Monthly
savings to fill gap?
Your Answer
30 year asset depletion phase
(retirement)
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:
Booth • Cleary – 3rd Edition Page 46© John Wiley & Sons Canada, Ltd.
)1(
100 gk
PMT
gk
gPMTPV
[5A-2]
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:
Booth • Cleary – 3rd Edition Page 47© John Wiley & Sons Canada, Ltd.
1
11 1
0
n
k
g
gk
PMTPV [5A-4]
Appendix 5AGROWING ANNUITIES & PERPETUITIES
WEB LINKS
Wiley Weekly Finance Updates site (weekly news updates): http://wileyfinanceupdates.ca/
Textbook Companion Website (resources for students and instructors): www.wiley.com/go/boothcanada
Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd. Page 48
Copyright © 2013 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein.
COPYRIGHTCopyright © 2013 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein.
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