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Project4ReviewandPracticeGuide.pdf
UMGC MBA 620: Financial
Decision Making
Project 4: Review and
Practice Guide
Project 4: Review and
Practice Guide
Time Value of Money and Annuities
Contents Topic 1: Time Value of Money ...................................................................................................................... 3
Outline ...................................................................................................................................................... 3
Time Value of Money (TVM) ..................................................................................................................... 3
Using a Timeline ........................................................................................................................................ 3
Terminology .............................................................................................................................................. 4
Compounding-Terminology ...................................................................................................................... 4
Future Value Equation .............................................................................................................................. 4
Compounding Interest .............................................................................................................................. 5
Compound Interest ............................................................................................................................... 5
Compounding More Than Once a Year ..................................................................................................... 5
Effect of Compounding Frequency ........................................................................................................... 6
Using Excel ................................................................................................................................................ 6
Excel Functions: FV.................................................................................................................................... 6
Continuous Compounding ........................................................................................................................ 7
Discounting-Terminology .......................................................................................................................... 7
Present Value and Discounting ................................................................................................................. 7
Excel Functions: PV and NPV .................................................................................................................... 8
Finding the Interest Rate .......................................................................................................................... 8
Excel Functions: RATE ............................................................................................................................... 9
Finding the Number of Periods ................................................................................................................. 9
Summary of Key Concepts ........................................................................................................................ 9
Topic 2: Annuity & Perpetuity ..................................................................................................................... 10
Outline .................................................................................................................................................... 10
The Analysis of Multiple Cashflows ........................................................................................................ 10
Future Value of Multiple Cash Flows .................................................................................................. 10
Present Value of Multiple Cash Flows ................................................................................................ 10
Level Cash Flows: Annuities and Perpetuities ..................................................................................... 10
Annuities ................................................................................................................................................. 11
Types of Annuities ................................................................................................................................... 11
Present Value of an Annuity ................................................................................................................... 11
Annual Payment of an Annuity ............................................................................................................... 11
Excel Functions: PMT .............................................................................................................................. 12
Interest Rate of an Annuity ..................................................................................................................... 12
Perpetuity ............................................................................................................................................... 12
APR and EAR: Two Ways of Quoting Interest Rates ............................................................................... 12
Annual Percentage Rate (APR) ................................................................................................................ 13
Effective Annual Interest Rate (EAR) ...................................................................................................... 13
Problems/Exercises ..................................................................................................................................... 14
What to do .............................................................................................................................................. 14
Self Study Problems ................................................................................................................................ 14
Questions and Problems ......................................................................................................................... 14
Exercise: 6.40 .......................................................................................................................................... 14
Solution: 6.40 .......................................................................................................................................... 15
Exercise: 6.41 .......................................................................................................................................... 15
Solution: 6.41 .......................................................................................................................................... 16
Exercise 6.42: Notes on the Solution ...................................................................................................... 16
Solution: 6.42 .......................................................................................................................................... 17
References .............................................................................................................................................. 17
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Topic 1: Time Value of Money
Outline • What is time value of money?
• Terminology
• Compounding
• Discounting
Time Value of Money (TVM) • Consume today or tomorrow?
o TVM is based on the belief that people prefer to consume goods today rather than wait to
consume the same goods tomorrow (Parrino et al., 2012).
▪ An apple today is more valuable to us than an apple we can have in one year.
▪ Money has a time value because buying an apple today is more important than buying
an apple in one year.
• For financial managers
o Determine today's value of a future cash flow, or a series of cash flows, mostly in financing
or investment decisions (Parrino et al., 2012).
Time Value of Money Explained
Using a Timeline
Adapted from Parrino et al. (2012)
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Terminology
Compounding-Terminology
Future Value Equation Equation to find a future value
FV𝑛 = PV × (1 + i) 𝑛
FVn = future value of investment at end of period n
PV = original principal (P0) or present value
i = the rate of interest per period
n = the number of periods, often in years
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Compounding Interest
Compound Interest How to Compound Interest: Growth of $1,000 at 10%
The amount of simple interest earned on $1,000 invested at 10% remains constant at $100 per year, but
the amount of interest earned on interest increases each year as more and more interest builds.
Compounding accelerates the growth of the total interest earned.
Based on an image from Parrino et al. (2012)
Compounding More Than Once a Year o The more frequently interest is compounded, the larger the future value of $1 at the end of
a given time period
o If compounding occurs m times within a period, the future value equation becomes
o 𝐹𝑉𝑛 = 𝑃𝑉 × (1 + 𝑖 𝑚⁄ ) 𝑚𝑛
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Effect of Compounding Frequency Future value of $100 after 10
years, by compounding frequency
Daily: $271.79
Monthly: $270.70
Quarterly: $268.51
Yearly: $259.37
Using Excel • Enter the formulas
• Use the built-in functions
o Link to the Excel functions by category
o https://support.office.com/en-us/article/Excel-functions-by-category-5f91f4e9-7b42-46d2-
9bd1-63f26a86c0eb
Excel Functions: FV FV(rate,nper,pmt,[pv],[type]) calculates the future value of an investment based on a constant interest
rate (Microsoft, n.d.).
Rate Required. The interest rate per period.
Nper Required. The total number of payment periods in an annuity.
Pmt Required. The payment made each period; it cannot change over the life of the annuity.
Typically, pmt contains principal and interest but no other fees or taxes. If pmt is omitted, you
must include the pv argument.
Pv Optional. The present value, or the lump-sum amount that a series of future payments
is worth now. If pv is omitted, it is assumed to be 0, and you must include the pmt argument.
Type Optional. Indicates when payments are due—0 for end of period or 1 for beginning of
period. If type is omitted, it is assumed to be 0.
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Continuous Compounding • Continuous Compounding
o When compounding occurs on a continuous basis, the future value equation becomes
e = 2.71828, the base of the natural logarithm
Discounting-Terminology
Present Value and Discounting • Present Value Equation
o General equation to find present value
o (1 + i)n is the present value factor or discount factor.
o The interest rate i is the discount rate.
• Present Value Concepts
o Time and the discount rate affect present value:
▪ The greater the time before a cash flow is to occur, the smaller the present value of the
cash flow.
▪ The higher the discount rate, the smaller the present value of a future cash flow (Parrino
et al., 2012).
𝐹𝑉𝑛 = 𝑃 × 𝑒𝑖 × 𝑛
PV = FV𝑛
(1 + i)𝑛
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Excel Functions: PV and NPV PV(rate,nper,pmt,[fv],[type]) calculates the present value of a loan or an investment, based on a
constant interest rate (Microsoft, n.d.).
Rate Required. The interest rate per period.
Nper Required. The total number of payment periods in an annuity.
Pmt Required. The payment made each period; it cannot change over the life of the annuity.
Typically, pmt contains principal and interest but no other fees or taxes. If pmt is omitted, you
must include the pv argument.
Fv Optional. The future value or a cash balance you want to attain after the last payment is
made. If fv is omitted, it is assumed to be 0 (the future value of a loan, for example, is 0). If fv is
omitted, you must include the pmt argument.
Type Optional. Indicates when payments are due, 0 for end of period or 1 for beginning of
period. If type is omitted, it is assumed to be 0.
NPV( rate,value1,[value2],...) calculates the net present value of an investment by using a discount rate
and a series of future payments (negative values) and income (positive values) (Microsoft, n.d.).
Rate Required. The interest rate per period.
Value1, Value2 Value1 is required; other values are optional.
1 to 254 Arguments representing the payments and income.
Value1, Value2 Must be equally spaced in time and occur at the end of each period.
NPV uses the order of Value1, Value2 to interpret the order of cash flows. Be sure to enter your payment and income values in the correct sequence.
Arguments that are empty cells, logical values, or text representations of numbers, error values,
or text that cannot be translated into numbers are ignored.
If an argument is an array or reference, only numbers in that array or reference are counted.
Empty cells, logical values, text, or error values in the array or reference are ignored.
Finding the Interest Rate • Many situations require a time value of money calculation to determine a rate of change or
growth rate (Parrino et al., 2012)
• An investor or analyst may want to know
o growth rate in sales
o rate of return on an investment
o effective interest rate on a loan
How to solve for i
FV/PV = (1 + i)n
(FV/PV)1/n = (1 + i)
i = (FV/PV)1/n – 1
FV𝑛 PV × (1 + i ) 𝑛
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Excel Functions: RATE RATE(nper, pmt, pv, [fv], [type], [guess]) returns the interest rate per period of an annuity (Microsoft,
n.d.).
Nper Required. The total number of payment periods in an annuity.
Pmt Required. The payment made each period. It cannot change over the life of the annuity.
Typically, pmt includes principal and interest but no other fees or taxes. If pmt is omitted, you
must include the fv argument.
Pv Required. The present value—the total amount that a series of future payments is
worth now.
Fv Optional. The future value, or a cash balance you want to attain after the last payment is
made. If fv is omitted, it is assumed to be 0 (the future value of a loan, for example, is 0). If fv is
omitted, you must include the pmt argument.
Type Optional. Indicates when payments are due, number 0 for end of period or 1 for
beginning of period. If type is omitted, it is assumed to be 0.
Guess Optional. Your guess for what the rate will be. If you omit guess, it is assumed to be 10
percent. If RATE does not converge, try different values for guess. RATE usually converges if
guess is between 0 and 1.
Finding the Number of Periods
Solve for n using the following equations ("l," as in "ln" below, stands for logarithm):
FV/PV = (1 + i)n
ln(FV/PV) = n × ln(1 + i)
n = ln(FV/PV) / ln(1+ i)
Or, PDURATION (rate, pv, fv)
Rate Required. Rate is the interest rate per period.
Pv Required. Pv is the present value of the investment.
Fv Required. Fv is the desired future value of the investment.
Summary of Key Concepts 1. What is time value of money and why is it important?
2. What do future value, principal amount, simple interest and compound interest mean? How is
the future value formula used to make business decisions? Provide an example.
3. Explain present value and how it relates to future value. How is the present value formula used
to make business decisions? Provide an example.
FV𝑛 = PV × (1 + i) 𝑛
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Topic 2: Annuity & Perpetuity
Outline • The Analysis of Multiple Cashflows
• Annuity
• Perpetuity
• APR and EAR
The Analysis of Multiple Cashflows
Future Value of Multiple Cash Flows
Based on an image from Parrino et al. (2012)
Present Value of Multiple Cash Flows
Based on an image from Parrino et al. (2012)
Level Cash Flows: Annuities and Perpetuities
• Annuity
o A series of equally-spaced and level cash flows extending over a finite number of periods
• Perpetuity
o A series of equally-spaced and level cash flows that continue forever
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Annuities
Types of Annuities • Ordinary Annuity
o Cash flows occur at the end of a period
▪ mortgage payment
▪ interest payment to bondholder
• Annuity Due
o Cash flows occur at the beginning of a period
▪ lease
Present Value of an Annuity Example
• amount needed to produce the annuity
• current fair value or market price of the annuity
• amount of a loan that can be repaid with the annuity
Annual Payment of an Annuity Calculate the annual payment of an annuity
PVA = [CF × 1
1 + 𝑖 ] + [CF ×
1
(1 + 𝑖)2 ] + ⋯ + [CF ×
1
(1 + 𝑖)𝑛 ]
= CF
𝑖 × [1 −
1
(1 + 𝑖)𝑛 ]
= CF × 1 − 1/(1 + 𝑖)𝑛
𝑖
PVA = CF × 1 −
1 (1 + 𝑖)𝑛
𝑖
CF = PVA 1 −
1 (1 + 𝑖)𝑛
𝑖 ⁄
= PVA × 𝑖
1 − 1/(1 + 𝑖)𝑛
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Excel Functions: PMT PMT(rate, nper, pv, [fv], [type])
rate Required. The interest rate for the loan.
nper Required. The total number of payments for the loan.
pv Required. The present value, or the total amount that a series of future payments is
worth now; also known as the principal.
fv Optional. The future value, or a cash balance you want to attain after the last payment is
made. If fv is omitted, it is assumed to be 0, meaning the future value is 0.
type Optional. The number 0 or 1 indicates when payments are due.
0 or omitted—Payments are due at the end of the period.
1—Payments are due at the beginning of the period.
Interest Rate of an Annuity Find out the interest rate of an annuity:
Cannot solve for i analytically,
Use Excel:
RATE(nper, pmt, pv, [fv], [type], [guess])
Perpetuity A stream of equal cash flows that goes on forever
Preferred stock and some bonds are perpetuities
APR and EAR: Two Ways of Quoting Interest Rates o The most common way to quote interest rates is in terms of annual percentage rate (APR),
which does not incorporate the effects of compounding
o The most appropriate way to quote interest rates is in terms of effective annual rate (EAR),
which incorporates the effects of compounding (Parrino et al., 2012)
PVA = CF × 1 − 1/(1 + 𝑖)𝑛
𝑖
𝑃𝑉𝑃0 = 𝐶𝐹 × [ 1 −
1 (1 + 𝑖)∞
𝑖 ] = 𝐶𝐹 ×
(1 − 0)
𝑖
= 𝐶𝐹
𝑖
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Annual Percentage Rate (APR) • APR = periodic rate × m
m—the number of periods in a year
• Does not account for the number of compounding periods or adjust the annualized interest rate
for the time value of money
• Not an exact measurement of borrowing/investing rates
Effective Annual Interest Rate (EAR) • EAR accounts for the number of compounding periods and adjusts the annualized interest rate
for the time value of money
• EAR is a more accurate measure of the rates involved in lending and investing (Parrino et al.,
2012)
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Problems/Exercises
What to do Complete the practice exercises from the book and the custom exercise that follows in this guide to
gain the knowledge and skills needed to complete the final Project 4 deliverable.
The answers are provided, so you can check your work.
Self Study Problems • Chapter 5 Self Study problems (all)
• Chapter 6 Self Study problems (all)
Questions and Problems • Chapter 6
o 6.40
o 6.41
o 6.42
Exercise: 6.40 Trevor Diaz wants to purchase a Tesla. The total cost is $129, 482. Trevor plans to put down $20,000 and
pay the rest using a 5.75% five-year bank loan. What is the monthly payment on this auto loan?
o Cost of new car = $129,482
o Down payment = $20,000
o Loan amount = $129,482 − $20,000 = $109,482
o Interest rate on loan, or i = 5.75%
o Term of loan, or n = 5 years
o Frequency of payment, or m = 12
o Monthly payment on loan = PMT
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Solution: 6.40
Financial Calculator Solution
60 5.75/12 109,482 0 0
N i PV PMT FV
2,103.89
Exercise: 6.41 The Sundarams are buying a new 3,500 sq. ft. house in Kentucky and will borrow $237,000 from a bank
at a rate of 6.375% for 15 years. What will their monthly mortgage be?
o Home loan amount = $237,000
o Interest rate on loan , or i = 6.375%
o Term of loan, or n = 15 years
o Frequency of payment, or m = 12
o Monthly payment on loan = PMT
n
n
12 5
1 1
(1 ) PVA PMT
$109, 482 $109, 482 PMT
1 52.0379 1
0575 1
12
0.0575
12
− +
=
= =
− +
=
i
i
$2, 103.89
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Solution: 6.41
Financial Calculator Solution
60 6.375/12 -237,000
0
N i PV PMT FV
2,048.27
Exercise 6.42: Notes on the Solution If the student is 21, then the first withdrawal will be 40,000(1+.05)45 = $359,400.
Withdrawals grow at 5% per year thereafter (a spreadsheet is helpful here).
Next, find the NPV of the anticipated retirement withdrawals at each anticipated interest rate. This is
the lump sum needed at retirement. Use a spreadsheet or the formula for the present value of a
growing annuity (see next slide) to find the present value of withdrawals:
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Solution: 6.42
𝑃𝑉𝐴𝑛 = 359,400.312
. 1 − .05 𝑥 [1 − (
1.05
1.1 ) 20
] = 4,353,086.79
Finally, find the annuity payment that will generate the lump sum at the end of the student's working
life. Assume 40 years of saving, 35 years of saving, and 30 years of saving.
Retirement Analysis Summary
Investment Age = 25 Investment Age = 30 Investment Age = 35
Rate of
return 8% 10% 15% 8% 10% 15% 8% 10% 15%
Inflation
rate 5%
Retirement
income
level
$40,000
Lump sum
needed at
age 65
$5,160,266
$4,353,087
$3,011,353
$5,160,266
$4,353,087
$3,011,353
$5,160,266
$4,353,087
$3,011,353
Annuity
payment
needed
$19,919
$9,835
$1,693
$29,946
$16,062
$3,417
$45,552
$26,463
$6,927
References Microsoft. (n.d.). Excel functions (by category). Retrieved July 22, 2021, from
https://support.microsoft.com/en-us/office/excel-functions-by-category-5f91f4e9-7b42-46d2-9bd1-
63f26a86c0eb
Parrino, R., Kidwell, D. S., & Bates, T. W. (2012). Fundamentals of corporate finance. Wiley.
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Now that you have read this Review and Practice Guide and completed the exercises, you are ready to participate in the assignment in Step 3.
