(Solved) UNIT III STUDY GUIDE Time Value of Money Course Learning Outcomes for Unit III Upon completion of this unit, students should be able to: Apply time...

Sarah Wiggum would like to make a single investment and have $1.6 million at the time of her retirement in 35 years. She has found a retirement fund that will earn 3% annually. How much will Sarah have to invest today? If she earned an annual return of 20%, how soon could she then retire?
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Question 2
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If you deposit $17,000 today in an account earning an annual rate of return of 10%, how much interest would be earned in the third year? How much would this amount differ from simple interest?
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Question 3
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To buy a new house, you must borrow $150,000. To do this, you take out a $150,000, 20year, 10% mortgage. Your mortgage payments, which are made at the end of each year (one payment each year), include both principle and 9% interest on the declining balance. What amount will your annual payment be?
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Question 4
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What is the present value of a perpetual stream of cash flow that pays $80,000 at the end of one year and grows at a rate of 5% indefinitely? The rate of interest used to discount the cash flows is 10%. What is the present value of the growing perpetuity?
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Question 5
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How much do you have to deposit today so that, beginning 11 years from now, you can withdraw $9,000 a year for the next 8 years (periods 11 through 18) plus an additional amount of $18,000 in the last year (period 18)? Assume an interest rate of 6%.
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UNIT III STUDY GUIDE Time Value of Money Course Learning Outcomes for Unit III
Upon completion of this unit, students should be able to:
3. Apply time value of money techniques to various pricing (valuation) and budgeting problems.
3.1 Explain compounding and discounting to calculate future and present values.
3.2 Determine annuities from a single sum.
3.3 Demonstrate how to apply time value of money concepts to complex cash streams. Reading Assignment
Chapter 5:
Time Value of Money  The Basics, pp. 126150
Chapter 6:
The Time Value of Money  Annuities and Other Topics, pp. 156179 Unit Lesson
Creating value hinges on timing of cash flows. In finance, understanding time value of money is critical to
value creation because cash flows resolve when an organization can create value. Discounting and
compounding are key ingredients for looking at how time affects value creation. Take the case of Brite
Phuteur, who has started a career and needs a new car.
Phuteur has looked at different cars and has a good idea what he wants to buy. After looking at several
dealerships, Phuteur noted pricing did not necessarily line up with his ideas about paying a fair price for the
value he thought he would receive. Phuteur found dealerships varied their pricing on identical cars.
Despite not understanding price variances between dealers, Phuteur set aside money from his pay each pay
period for a good down payment. Interest rates on savings at his local bank amounted to roughly 2%. If
Phuteur set aside $100 a month at the end of the year, he wanted to know how much he would have after
earning interest. This scenario is Phuteur's first experience where he needed to understand compounding.
Phuteur asked his banker, Cash Kownter, for help. Kownter explained that compound interest adds interest
earned to principal used when calculating future interest. For example, Phuteur started by putting $1,000 in
savings in his bank, which pays 2% interest a year compounded monthly. Since Phuteur's local bank figures
interest monthly, his monthly interest rate is just .16667% but compounds each of the 12 months during the
year. Kownter showed Phuteur a formula to calculate the amount he would have by year's end. A general
future value formula solves this calculation as follows (Titman, Keown, & Martin, 2014, p. 130):
FV = PV(1 + i)n, where FV equals future value, PV equals present value, i equals an interest rate, and n is the
number of periods. Applying this equation, Phuteur can substitute his numbers into the formula as follows:
FV = $1,000(1 + .16667%)12
Thus,
FV = $1,000(1.0016667)12
FV = $1,020.18 BBA 3301, Financial Management 1 Phuteur observed he has more than simple interest, which simply adds 2% to UNIT
$1,000,
which only
accounts for
x STUDY
GUIDE
$1,020.00 of the $1,020.18 earned. Kownter showed Phuteur how to solve this
problem in Excel as follows:
Title Despite compound interest exceeding simple interest, Phuteur noted the amount earned is not much more
than he put in his savings to start. Phuteur asked Kownter what he would earn if he left the money in the bank
for five years. Kownter changed his calculation as follows:
FV = $1,000(1 + .16667%)12
Thus,
FV = $1,000(1.0016667)12*5
FV = $1,105.08
Again Kownter made the calculation in Excel as follows: This time, Phuteur noted more interest earned by comparing $1,105.08 with $1,020.00 principal plus simple
interest. Phuteur earned $85.08 in interest compared with just $.18 for just a year. Phuteur inferred the more
periods the greater the interest earned. Kownter explained compounding calculates interest on interest each
period. Instead of simply computing interest on $1,000.00, compounding recalculates interest each period (in
this case 60 periods), which grows interest earned. BBA 3301, Financial Management 2 Although compound interest showed a nice improvement in interest earned, Phuteur
what he could
UNIT x wondered
STUDY GUIDE
earn if he also saved $100 a month. Kownter told Phuteur, "Now you are talking
about an annuity." Kownter
Title
explained an annuity is a series of equal payments for a specific time. Again, Kownter showed Phuteur how
to calculate compound interest as follows:
FVn = PMT(1 + i)n1 + PMT(1 + i)n2...+ PMT(1 + i)1 + PMT(1 + i)0
Alternatively, one can apply a future value annuity factor (FVA) to an annuity payment (PMT) using the
formula as follows:
(1+) âˆ’1 FVA = = [ Applying this factor to the payment results in the following formula:
(1+) âˆ’1 FVn = PMT = [ Using $100 a month savings deposit Kownter filled in this formula as follows:
FVA = [(1.001667)60/.001667]
FVA = 63.04734
FV60 = 100(63.04734)
FV60 = 6304.74
Alternatively, Kownter again substitutes in Excel as follows: BBA 3301, Financial Management 3 With both an initial deposit of $1,000 in savings and an added $100 a month, the
result
is as follows:
UNIT
x STUDY
GUIDE Title Phuteur felt better about saving for a down payment after calculating what he would have. Interest earned
over five years is $409.82 ($105.08 plus $304.74) plus principal of $7,000 ($1,000 plus $6,000 or 60
payments times $100). Compound interest allowed Phuteur to grow his money by adding interest on interest
earned.
Besides resolving he could have a large down payment, Phuteur still needed to know what he would have to
pay monthly on a car. Phuteur looked at several cars on the Internet and settled on a car that sells for
$25,000. Phuteur figured he would have a balance of $17,509.81 ($25,000 minus $7,409.81) he would need
to finance. Phuteur asked at the bank what rate he would have to pay on a car loan. The bank gave Phuteur a
rate of 4.5%. Phuteur asked Kownter how he could calculate his payments. Kownter responded by saying it
might benefit Phuteur to prepare an amortization table to break down his payments between principal and
interest.
Kownter explained first Phuteur would need to calculate his payment. Kownter showed Phuteur how to
calculate his payment in Excel as follows: BBA 3301, Financial Management 4 Kownter assumed a normal five year loan and interest would compound monthly.
a variant
of the
UNITUsing
x STUDY
GUIDE
formula used to calculate the future value (FV) Kownter calculated Phuteur's payment
Title as follows:
PVn = PMT = [ (1+ 1 Ã—
) / Substituting the above information Kownter solved the equation for the payment. $17,509.81 = PMT[ 1
045 5Ã—12
(1+.
)
12 .045/12 $17,509.81 = PMT(53.63938)
PMT = $17,590.81/$53.64197
PMT = 327.93
Kownter prepared the following amortization table in Excel: Payments
Original balance
Payment
Payment
Payment
Payment
Payment 1
2
3
4
5 Payment
Payment
Payment
Payment
Payment
Payment 55
56
57
58
59
60 327.93
327.93
327.93
327.93
327.93 Principal
Interest Reduction
65.96
64.98
63.99
63.00
62.01 261.97
262.95
263.94
264.93
265.92 327.93
7.28
320.65
327.93
6.08
321.85
327.93
4.87
323.06
327.93
3.66
324.27
327.93
2.45
325.49
327.93
1.23
326.71
19,676.06 2,085.87 17,590.19 Principal
$17,590.19
17,328.22
17,065.27
16,801.33
16,536.40
16,270.47 1,621.39
1,299.53
976.47
652.20
326.71
0.00 With Kownter's table, Phuteur had an idea what interest and principal he would have to pay each month.
Payments for 60 months is an annuity because they are equal for a set time. Phuteur noticed early payments
had higher interest as part of payments, but later payments included more principal.
Because Kownter based payments on suggested retail price as suggested by the manufacturer, Phuteur
wondered if he could buy his car for less. Phuteur began looking at current prices to see what different
dealers offer. In today's prices, a similar car sold for $21,500. Kownter helped Phuteur explained to Kownter
how discounting the payments based on inflation rate of 2% might give him a better idea of the value today.
Kownter first discounted the $7,409.19 down payment to express it in today's dollars as follows: BBA 3301, Financial Management 5 UNIT x STUDY GUIDE Title Kownter also solves for present value using the following formula:
PV = FVx1/(1 + i)n or PV(1+i)n
In Phuteur's case substituting the values looks like the following:
PV = $7,409.19 x 1/(1 + .00166667)60
PV = $7,409.19 x 1/(1.01004)
PV = $7,409.19 x .90491
PV = $6,704.67
Next, Kownter calculated the present value of the payments Phuteur would have to make as follows: BBA 3301, Financial Management 6 Combining the down payment and interest payments Kownter found the valueUNIT
todayx is
as follows:
STUDY
GUIDE Title Comparing the $25,413.85 with what Phuteur projected five years from now it appears $25,000 is a fair price.
Remember, however, Phuteur found one dealer selling the car he wanted for $21,500 at today's prices. This
price suggests the $25,000 price in five years is too high, and Phuteur might well do better than the price
Kownter used to calculate future payments. Assuming $25,000 is more than Phuteur would have to pay in five
years, he likely will have lower payments based on dealer price variation. This revelation excited Phuteur
about saving for his new car.
In summary, compounding cash flows results in future values, while discounting cash flows, brings future
cash flows back to today's dollars for the sake of comparison. Some cash streams are complex and have
bothâ€”a single sum plus an annuity, or series of equal payments for a set time. Phuteur benefited from doing
his homework and reaching out to his friend at the bank. Phuteur did a good job with his homework and is
now ready to buy a car when the time comes.
Reference
Titman, S., Keown, A. J., & Martin J. D. (2014). Financial management: Principles and applications (12th ed.).
Upper Saddle River, NJ: Pearson. Suggested Reading
In order to access the resource below, you must first log into the myCSU Student Portal and access the
ABI/Inform database within the CSU Online Library.
This article further explores the concept of time value of money.
Benshoof, M. (2005). The time value of money. Professional Builder, 70(3), 74. Learning Activities (NonGraded)
The following video tutorials will help you with the concepts covered in the textbook. You are strongly
encouraged to watch these videos prior to starting the unit assessment.
Click here for CheckPoint 5.2 Calculating Future Value
Click here for CheckPoint 5.3 Calculating Future Values Using NonAnnual Compounding Periods
Click here for CheckPoint 5.4 Calculating Present Value
Click here for CheckPoint 5.5 Solving for n
Click here for CheckPoint 5.6 Solving for i
BBA 3301, Financial Management 7 Click here for CheckPoint 5.7 Calculating EAR
UNIT x STUDY GUIDE
Click here for Checkpoint 6.1 Solving for an Ordinary Annuity Payment
Title
Click here for Checkpoint 6.2 Present Value an Ordinary Annuity
Click here for Checkpoint 6.3 Determining the Outstanding Balance on a Loan
Click here for Checkpoint 6.4 Calculation of a Level Perpetuity
Click here for Checkpoint 6.5 Calculation of a Growing Perpetuity
Click here for Checkpoint 6.6 Calculation of a Complex Cash Flow Stream
Nongraded Learning Activities are provided to aid students in their course of study. You do not have to
submit them. If you have questions, contact your instructor for further guidance and information. BBA 3301, Financial Management 8
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