Laurence Booth Sean Cleary. LEARNING OBJECTIVES Time Value of Money 5 5.1 Explain the importance of...

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 © 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.


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:


)() (time Value kPnPn

EXAMPLE: Simple InterestThe same amount of interest is earned in each year.


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.


)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)


)1( 0n

n kPVFV [5.2]

COMPOUND VERSUS SIMPLE INTEREST• Simple interest grows principal in a linear manner.• Compound interest grows exponentially over time.


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)


)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


• 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:


)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%.


$6,964.26

1.99$3,500

0.035)1(500,3$

20

%5.3,20020

knCVIFPFV


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.


)1(

00 nk

FVPV

[5.3]

Present value interest factor (PVIF) is the inverse of the CVIF.


)1(

1 ??, nkn k

PVIF


EXAMPLE: Using the PVIFFind the PV0 of receiving $100,000 in 10 years time if the opportunity cost is 5%.


$61,391.33

0.6139$100,000 629.1

1$100,000

0.05)1(

1000,100$

10

%5,10100

knPVIFFVPV


Solving for Time or “Holding Periods”Equation 5.3 is reorganized to solve for n:


)1(

00 nk

FVPV

[5.3]

1ln

/ln 0

k

PVFVn n


EXAMPLE: Solving for ‘n’

How many years will it take $8,500 to grow to $10,000 at a 7% rate of interest?


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


Solving for Compound Rate of ReturnEquation 5.3 is reorganized to solve for k:


)1(

00 nk

FVPV

[5.3]

1 /1

0

n

n

PV

FVk


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?


%88.101877.0

125.11000,10$

500,12$

1

083.012

1

/1

0

k

k

PV

FVk

n

n


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.


• An ordinary annuity offers payments at the end of each period.

• An annuity due offers payments at the beginning of each period.



The formula for the compound sum of an ordinary annuity is:


1)1(

k

kPMTFV

n

n [5.4]


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?


01.722,16$722.16000,1$

11.0

111.1000,1$

11000,1$

10

10

10

10

,10

FVA

FVA

k

kFVA

FVIFAPMTFVAn

kn


The formula for the compound sum of an annuity due is:


k)(11)1(

k

kPMTFV

n

n [5.6]


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?


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


The formula for the present value of an annuity is:


)1(

11

0

kk

PMTPVn

[5.5]


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%.


06.639,137$47.11000,12$

06.0)06.1(

11

000,12$

0

20

0

06.0,200

PVA

PVA

PVIFAPMTPVA kn


The formula for the present value of an annuity is:


k)(1)1(

11

0

kk

PMTPVn

[5.7]


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%.


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


A perpetuity is an infinite series of equal and periodic cash flows.


0 k

PMTPV [5.8]


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%?


000,000,1$1.0

000,100$ 1

0 k

PPV


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).


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?


5.58%1-0275.1

1-)2

0.055(11-)1(

2

2m

EAR

m

QREAR


EXAMPLE: Effective Annual Rates

EARs increase as the frequency of compounding increase.


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


5.6 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.


EXAMPLE: Loan Amortization TableDetermine the annual blended payment on a five –year $10,000 loan at 8% compounded semi-annually.


$2,515.143.9759

$10,000PMT

0816.0

)0816.01(1

1 PMT$10,000

)1(1

1

5

0

kk

PMTPVn

[5.5]


EXAMPLE: Loan Amortization TableThe loan is amortized over five years with annual payments beginning at the end of year 1.


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


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:


%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


EXAMPLE: Mortgage AmortizationThe mortgage is amortized over 25 years with annual payments beginning at the end of the first month.


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


5.7 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.


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 © John Wiley & Sons Canada, Ltd.


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.



Example Solution – Preliminary Calculations

Preliminary calculations Required• Monthly rate of return when annual APR is 4%

• Number of months during savings period


%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


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


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:


)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:


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


http://wileyfinanceupdates.ca/

http://www.wiley.com/go/boothcanada

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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