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FV at year end 100
Ordinary annuity of $100 per year for three years.
Uneven cash flow stream.
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A B C D E F G H I J K L M N O P Q R
11/20/2018
Situation
FUTURE VALUE
$100 lump sum at the end of year 2.
Interest rate 0.1 These are the basic inputs, in blue.
Cash flow 100
Chapter 4 Mini Case
b. (1.) What’s the future value of an initial $100 after 3 years if it is invested in an account paying 10%
annual interest?
Assume that you are nearing graduation and have applied for a job with a local bank. As part of the
a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2, (2) an ordinary annuity of
$100 per year for 3 years, and (3) an uneven cash flow stream of -$50, $100, $75, and $50 at the end of
Years 0 through 3.
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A B C D E F G H I J K L M N O P Q R
Notice that we entered a value instead of a cell reference as the input for the problem for instructional
purposes. It’s really better to enter cell values so that your spreadsheet can automatically reflect any
changes to the input data. This is one of the features that makes the spreadsheet such a valuable tool.
After selecting the “FV” function from the “Financial” category, we will be using the following dialog
box to input our data.
After selecting the category for Financial functions, scroll down until you can selet the FV function, as
show below. Alternatively, select the menu Formulas, then then select Financial, then pick FV.
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21.0000 1.1025 1.2100 1.3225
41.0000 1.2155 1.4641 1.7490
61.0000 1.3401 1.7716 2.3131
81.0000 1.4775 2.1436 3.0590
A B C D E F G H I J K L M N O P Q R
Period (N) 0% 5% 10% 15%
01.0000 1.0000 1.0000 1.0000
Notice that we entered a value instead of a cell reference as the input for the problem for instructional
purposes. It’s really better to enter cell values so that your spreadsheet can automatically reflect any
changes to the input data. This is one of the features that makes the spreadsheet such a valuable tool.
With a spreadsheet, calculating FVIF’s is a simple operation, and we can use it to graph the relationship
between future value, growth, interest rates, and time. A similar table can be found in the textbook,
along with a corresponding graph.
Future Value Interest Factors
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A B C D E F G H I J K L M N O P Q R
PRESENT VALUE (PV)
PROBLEM
PV = $75.13
This problem can also be solved using the function wizard using a procedure similar to that for the FV
function. Begin by putting the pointer on the cell in which you want to display the result. Then, after
selecting the “PV” function from the “Paste Function” box, the input data for the problem must be
entered. Then click OK to get the result, $75.13.
Relationships among Future Value, Growth, Interest Rates, and Time
Simply put, the present value (PV) is the value today of some future cash flow (or series of cash flows).
b. (2) What is the present value of $100 to be received in 3 years if the appropriate interest rate is 10%?
$5.00
Relationships among Future Value, Growth, Interest
Rate, and Time
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A B C D E F G H I J K L M N O P Q R
Finding Time to Double
3.8 Use the function NPER, as shown below.
c. We sometimes need to find how long it will take a sum of money (or anything else) to grow to some
specified amount. For example, if a company’s sales are growing at a rate of 20% per year, how long
will it take sales to double?
Finding N, the number of
periods
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FV 2
A B C D E F G H I J K L M N O P Q R
SOLVING FOR I
PROBLEM
N3
PV -1
FV 2
I = 25.99%
Once again, Excel has a special function for this calculation. We suggest using either a financial
calculator or the function wizard to solve this type of problem, because of its complexity. The
d. If you want an investment to double in three years, what interest rate must it earn?
We noted above the difficulty of solving this problem mathematically. This is because it involves
taking the Nth root of a value (an operation which generally requires either a calculator or a computer).
However, if you would like to know how to solve the problem mathematically, the formula is
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A B C D E F G H I J K L M N O P Q R
FUTURE VALUE OF AN ANNUITY
N3
I0.1
PMT 100
FV = $331.00
PRESENT VALUE OF AN ANNUITY
N3
I0.1
PMT 100
f. (1.) What is the future value of a 3-year ordinary annuity of $100 if the appropriate interest rate is
10%?
As explained below, one way to solve this problem is to find the future value of each of the annuity
f. (2.) What is the present value of the annuity?
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Additionally, using the function wizard for this problem is exactly like above, but we enter a “1” instead of a “0” into
the “Type” field.
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A B C D E F G H I J K L M N O P Q R
PV = $248.69
N3
The procedure for solving this problems follows the previous example with one notable exception. Since, the
payments occur at the beginning of each year, the first annuity payment occurs in time period 0, and the last
occurs in time period 2.
Or, you could use the function wizard for this ordinary annuity.
f. (3.) What would the future and present values be if the annuity were an annuity due?
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A B C D E F G H I J K L M N O P Q R
FV = $364.10
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A B C D E F G H I J K L M N O P Q R
N3
I0.1
PMT 100
PV = $273.55
I = 10%
Time period
0 1 2 3 4
I0.1
N
CFNPV0
000.00
1100 90.91
2300 247.93
3300 225.39
To find the present value of the annuity due, this problem is solved just like the previous problem,
except that the payments occur in periods 0 through 2.
g. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10%,
Using the function wizard, we follow the same procedure as above, except remember to enter a “1” to
tell Excel that in this problem the payments occur at the beginning of the periods.
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The periodic is associated with the number of compounding periods per year. M = 4 quarterly, 12 for monthly, and
360 or 365 for annual compounding.
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A B C D E F G H I J K L M N O P Q R
4-50 -34.15
$530.09
Or
PV = $530.09
Inputs
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
With, the financial calculator, we could enter each of these cash flows and the discount rate, and
simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar
the initial cash flow is zero.
Larger, because interest is earned on interest.
PV of CF stream =
The effective annual rate is the annual rate that causes the PV to grow to the same FV as under multiple
compounding periods.
h. (2.) Will the future value be larger or smaller if we compound an initial amount more often than annually, for
example, every 6 months (semiannually ), holding the stated interest rate constant? Why?
h. (1.) Identify (a) the stated, or quoted, or nominal rate (iNom) and (b) the periodic rate (iPER).
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I (I per year/4) 0.03 FV = $142.58
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I (I per year/12) 0.01 FV = $143.08
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N (years x 365) 1095
I (I per year/12) 0.00032877 FV = $143.32
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8 $942.33 $106.08 $94.23 $11.85 $930.48
9 $930.48 $106.08 $93.05 $13.03 $917.45
A B C D E F G H I J K L M N O P Q R
h. (3.) What is the future value of $100 after 5 years under 12% annual compounding?
What is the FV with semiannual compounding?
What is the FV with quarterly compounding?
N (years x 4) 12
What is the FV with monthly compounding?
N (years x 12) 36
What is the FV with daily compounding?
NBeg. Amt. Payment Interest Principal End. Amt.
1 $1,000.00 $106.08 $100.00 $6.08 $993.92
2 $993.92 $106.08 $99.39 $6.69 $987.23
3 $987.23 $106.08 $98.72 $7.36 $979.88
NBeg. Amt. Payment Interest Principal End. Amt. 4 $979.88 $106.08 $97.99 $8.09 $971.79
j. (1.) What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the
end of each of the next three years if the interest rate is 10%?
I. Will the effective annual rate ever be equal to the nominal (quoted) rate? Only if the compounding period is equal
to 1 year.
j. (2.) What is the annual interest expense for the borrower, and the annual interest income for the lender, during
Year 2?
Now, construct an amortization table for the loan described above.
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26 $402.12 $106.08 $40.21 $65.87 $336.26
27 $336.26 $106.08 $33.63 $72.45 $263.80
28 $263.80 $106.08 $26.38 $79.70 $184.10
29 $184.10 $106.08 $18.41 $87.67 $96.44
30 $96.44 $106.08 $9.64 $96.44 $0.00
$3,182.38 $2,182.38 $1,000.00
I0.00031054
N273
FV $108.85
Annual rate = 10%
Periods per year = 2
Periodic rate = 5%
l. (1.) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%,
compounded semiannually?
k. On January 1, you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with
interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later? (273
days)
$50
$75
$100
Principal
Interest
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18 $753.52 $106.08 $75.35 $30.73 $722.79
19 $722.79 $106.08 $72.28 $33.80 $688.99
20 $688.99 $106.08 $68.90 $37.18 $651.81
21 $651.81 $106.08 $65.18 $40.90 $610.91
22 $610.91 $106.08 $61.09 $44.99 $565.92
23 $565.92 $106.08 $56.59 $49.49 $516.44
24 $516.44 $106.08 $51.64 $54.44 $462.00
25 $462.00 $106.08 $46.20 $59.88 $402.12
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PV of the note: PV $918.95 > $859 cost, so buy the note.
0 1 2 3 4 5456
See which has the higher effective rate of return, EFF%
A B C D E F G H I J K L M N O P Q R
Total FV = $247.59
In the second approach, we use the annual effective rate to find the present value of a 3-year annuity.
PV = $247.59
See which provides the greater future wealth
0 1 2 3 4 5456
l. (2.) What is the PV of the same stream?
See which has the greater present value
m. Suppose someone offered to sell you a note calling for the payment of $1,000 in 15 months (or 456 days). They
offer to sell it to you for $850. You have $850 in a bank time deposit that pays a 6.76649% nominal rate with daily
compounding, which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you
buy the note. The note is not risky–you are sure it will be paid on schedule. Should you buy the note? Check the
decision in three ways: (1) by comparing your future value if you buy the note versus leaving your money in the
bank, (2) by comparing the PV of the note with your current bank account, and (3) by comparing the EFF% on the
note versus that of the bank account.
l. (3.) Is the stream an annuity? No, because we don’t have a payment for each compounding period.
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PV of CF $90.70 $82.27 $74.62
and the number of periods.