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# Fv = Future Value, Pv = Present Value, I = Interest Rate per Period, and N = Number of Periods by elk15834

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Fv = Future Value, Pv = Present Value, I = Interest Rate per Period, and N = Number of Periods document sample

• pg 1
```									                                Variables      Variables        Numbers and      Variables
Mgirvin uses   Excel uses       Formulas         textbook uses
Future Value                    FV             FV               ?                FV or C
Present Value                   PV             PV                       10000    PV or C
Annual Interest Rate            i                                         0.06
Number of compounding
periods per year                n                                          12
Years                           x                                          10
Period Rate                     i/n            rate                              r
Total Number of Periods         x*n            nper or npery                     t
FV = PV*(1+i/n)^((x*n)
FV(rate,nper,pmt,[pv],[type])

Total Interest
Variables      Variables Excel Numbers and      Variables
Mgirvin uses   uses            Formulas         textbook uses
Future Value                    FV             FV              ?                FV or C
Present Value                   PV             PV                      10000    PV or C
Annual Interest Rate            i                                        0.06
Number of compounding
periods per year                n                                         12
Years                           x                                         10
Period Rate                     i/n            rate                    0.005    r
Total Number of Periods         x*n            nper or npery             120    t
FV = PV*(1+i/n)^((x*n)                                           \$18,193.97
FV(rate,nper,pmt,[pv],[type])                                    \$18,193.97

Total Interest     \$8,193.97
PV = Investment =                                             \$100.00
i = Annual Interest Rate =                                         0.1
n = Compounding Periods per Year =                                   1
x = years =                                                          4
Simple Interest =

Year                                                  Interest Earned    Amount in Bank
Year 0
Year 1
Year 2
Year 3
Year 4

FV = Future Value of Investment @ Simple Interest =
PV = Investment =                                             \$100.00
i = Annual Interest Rate =                                         0.1
n = Compounding Periods per Year =                                   1
x = years =                                                          4
Simple Interest =                                              \$10.00

Year                                                  Interest Earned    Amount in Bank
Year 0                                                                           \$100.00
Year 1                                                         \$10.00            \$110.00
Year 2                                                         \$10.00            \$120.00
Year 3                                                         \$10.00            \$130.00
Year 4                                                         \$10.00            \$140.00

FV = Future Value of Investment @ Simple Interest =           \$140.00
PV = Investment =                                               \$100.00
i = Annual Interest Rate =                                           0.1
n = Compounding Periods per Year =                                     1
x = years =                                                            4

Year                                                    Interest Earned    Amount in Bank
Year 0
Year 1
Year 2
Year 3
Year 4

FV = Future Value of Investment @ Compound Interest =

FV = Future Value of Investment @ Compound Interest =

FV = Future Value of Investment @ Simple Interest =
FV = Future Value of Investment @ Compound Interest =
Interest on Interest =
PV = Investment =                                               \$100.00
i = Annual Interest Rate =                                           0.1
n = Compounding Periods per Year =                                     1
x = years =                                                            4

Year                                                    Interest Earned    Amount in Bank
Year 0                                                                             \$100.00
Year 1                                                           \$10.00            \$110.00
Year 2                                                           \$11.00            \$121.00
Year 3                                                           \$12.10            \$133.10
Year 4                                                           \$13.31            \$146.41

FV = Future Value of Investment @ Compound Interest =           \$146.41

FV = Future Value of Investment @ Compound Interest =           \$146.41

FV = Future Value of Investment @ Simple Interest =             \$140.00
FV = Future Value of Investment @ Compound Interest =           \$146.41
Interest on Interest =                                            \$6.41
PV = Investment =                                        \$100.00
i = Annual Interest Rate =                                    0.1               \$300.00
Amount in Bank each Ye
n = Compounding Periods per Year =                              1                     Amount in Bank each Ye
\$250.00
x = years =                                                     4
\$200.00
Amount in Bank each Year with Simple Amount in Bank each Year
Time    Interest                               with Compound Interest     Difference
\$150.00
0                                  \$100.00                    \$100.00       \$0.00
1                                  \$110.00                    \$110.00       \$0.00   \$100.00
2                                  \$120.00                    \$121.00       \$1.00
3                                  \$130.00                    \$133.10       \$3.10    \$50.00
4                                  \$140.00                    \$146.41       \$6.41
5                                  \$150.00                    \$161.05      \$11.05          \$0.00
6                                  \$160.00                    \$177.16      \$17.16
7                                  \$170.00                    \$194.87      \$24.87
8                                  \$180.00                    \$214.36      \$34.36
\$700.00
9                                  \$190.00                    \$235.79      \$45.79
\$600.00
10                                  \$200.00                    \$259.37      \$59.37

Future Value
\$500.00
\$400.00
\$300.00
\$200.00
\$100.00
\$0.00
\$300.00
Amount in Bank each Year with Compound Interest
Amount in Bank each Year with Simple Interest
\$250.00

\$200.00

\$150.00

\$100.00

\$50.00

\$0.00
0 1 2 3 4 5 6 7 8 9 10

\$700.00
0.00% Annual Rate
\$600.00
\$500.00                                                       2.50% Annual Rate
\$400.00                                                       5.00% Annual Rate
\$300.00                                                       7.50% Annual Rate
\$200.00                                                       10.00% Annual Rate
\$100.00
12.50% Annual Rate
\$0.00
15.00% Annual Rate
0   1   2   3    4   5   6   7       8   9   10
17.50% Annual Rate
Time                         20.00% Annual Rate
If we invest \$100,000.00 with an annual rate of 7.00% compounded 12 times a
year for 10 years, what is the future value of the investment?
Present Value = PV                                                \$     100,000.00
Annual Interest Rate = i                                                     7.00%
Number of Compoundin Periods per Year = n                                        12
Years = x                                                                        10
Future Value = FV
Period Rate = i/n
Total Periods = n*x
(1 + i/n)^(n*x)
Future Value = FV
Future Value = FV
Future Value = FV
Total Interest earned = cash put in - cash taken out
Write it in words:
If we invest \$100,000.00 with an annual rate of 7.00% compounded 12 times a
year for 10 years, what is the future value of the investment?
Present Value = PV                                                \$        100,000.00
Annual Interest Rate = i                                                        7.00%
Number of Compoundin Periods per Year = n                                           12
Years = x                                                                           10
Future Value = FV
Period Rate = i/n                                                         0.005833333
Total Periods = n*x                                                                120
(1 + i/n)^(n*x)                                                      2.0096613766956
Future Value = FV                                                 \$        200,966.14
Future Value = FV                                                 \$        200,966.14
Future Value = FV                                                        \$200,966.14
Total Interest earned = cash put in - cash taken out                     \$100,966.14
If we invest \$100,000.00 with an annual rate of 7.00% compou
times a year for 10 years, our future value of the investment
\$200,966.14. Of that amount, \$100,966.14 is the interest earn
Write it in words:                                                                              investment.
ual rate of 7.00% compounded 12
e value of the investment will be
66.14 is the interest earned on the
ent.
If we invest \$15,000.00 with an annual rate of 5.50% compounded 365 times a
year for 10 years, what is the future value of the investment?
Present Value = PV                                                \$       15,000.00
Annual Interest Rate = i                                                      5.50%
Number of Compoundin Periods per Year = n                                       365
Years = x                                                                         10
Future Value = FV
Period Rate = i/n
Total Periods = n*x
Future Value = FV
Future Value = FV
Total Interest earned = cash put in - cash taken out
Write it in words:
If we invest \$15,000.00 with an annual rate of 5.50% compounded 365 times a
year for 10 years, what is the future value of the investment?
Present Value = PV                                                \$         15,000.00
Annual Interest Rate = i                                                         5.50%
Number of Compoundin Periods per Year = n                                          365
Years = x                                                                            10
Future Value = FV
Period Rate = i/n                                                        0.000150685
Total Periods = n*x                                                               3650
Future Value = FV                                                 \$         25,997.72
Future Value = FV                                                          \$25,997.72
Total Interest earned = cash put in - cash taken out                       \$10,997.72
If we invest \$15,000.00 with an annual rate of 5.50% compoun
times a year for 10 years, our future value of the investment
\$25,997.72. Of that amount, \$10,997.72 is the interest earned
Write it in words:                                                                               investment.
al rate of 5.50% compounded 365
e value of the investment will be
7.72 is the interest earned on the
ment.
How much would we have to invest today, if we want to have \$1,000,000.00 in 40
years and we could earn an annual interest rate (discount rate) of 10.00%
compounded 12 times a year?
Present Value = PV
"Annual Interest Rate" = Discount Rate (term used when
doing PV calculations) = i                                                       10.00%
Number of Compoundin Periods per Year = n                                             12
Years = x                                                                             40
Future Value = FV                                               \$          1,000,000.00
Period Payment = PMT
Period Rate = i/n
Total Periods = n*x
(1 + i/n)^(n*x)
Present Value = PV
Present Value = PV
Present Value = PV
Total Interest earned = cash put in - cash taken out
Write it in words:
How much would we have to invest today, if we want to have \$1,000,000.00 in 40
years and we could earn an annual interest rate (discount rate) of 10.00%
compounded 12 times a year?
Present Value = PV
"Annual Interest Rate" = Discount Rate (term used when
doing PV calculations) = i                                                       10.00%
Number of Compoundin Periods per Year = n                                             12
Years = x                                                                             40
Future Value = FV                                               \$          1,000,000.00
Period Payment = PMT
Period Rate = i/n                                                           0.008333333
Total Periods = n*x                                                                  480
(1 + i/n)^(n*x)                                                      53.7006631743289
Present Value = PV                                              \$             18,621.74
Present Value = PV                                              \$             18,621.74
Present Value = PV                                                           -\$18,621.74
Total Interest earned = cash put in - cash taken out            \$            981,378.26
We would have to invest \$18,621.74 today, if we want t
\$1,000,000.00 in 40 years and we could earn an annual int
Write it in words:                                                        (discount rate) of 10.00% compounded 12 times a y
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ould have to invest \$18,621.74 today, if we want to have
00.00 in 40 years and we could earn an annual interest rate
count rate) of 10.00% compounded 12 times a year.
How much would we have to invest today, if we want to have \$150,000.00 (for our
daughter's college tuition) in 18 years and we could earn an annual interest rate (discount
rate) of 6.95% compounded 365 times a year?
Present Value = PV
"Annual Interest Rate" = Discount Rate (term used when doing
PV calculations) = i                                                                   6.95%
Number of Compoundin Periods per Year = n                                                365
Years = x                                                                                 18
Future Value = FV                                                   \$           150,000.00
Period Payment = PMT
Period Rate = i/n
Total Periods = n*x
(1 + i/n)^(n*x)
Present Value = PV
Present Value = PV
Present Value = PV
Total Interest earned = cash put in - cash taken out
Write it in words:
How much would we have to invest today, if we want to have \$150,000.00 (for our
daughter's college tuition) in 18 years and we could earn an annual interest rate (discount
rate) of 4.00% compounded 365 times a year?
Present Value = PV
"Annual Interest Rate" = Discount Rate (term used when doing
PV calculations) = i                                                                     4.00%
Number of Compoundin Periods per Year = n                                                  365
Years = x                                                                                   18
Future Value = FV                                                  \$               150,000.00
Period Payment = PMT
Period Rate = i/n                                                    0.000109589041095890
Total Periods = n*x                                                                       6570
(1 + i/n)^(n*x)                                                             2.0543521665502
Present Value = PV                                                 \$                73,015.72
Present Value = PV                                                 \$                73,015.72
Present Value = PV                                                                (\$73,015.72)
Total Interest earned = cash put in - cash taken out               \$                76,984.28

We would have to invest \$73,015.72 today, if we want to
(for our daughter's college tuition) in 18 years and we co
Write it in words:                                                     interest rate (discount rate) of 4.00% compounded 36
e to invest \$73,015.72 today, if we want to have \$150,000.00
hter's college tuition) in 18 years and we could earn an annual
e (discount rate) of 4.00% compounded 365 times a year?
If you want to buy a \$350,000.00 C & C Router Machine to improve manufacturing efficiency and you
have \$200,000.00 today that you can invest at an annual rate of 8.50% compounded 12 times a year, how
long do you have to wait (be careful about what period you need to make the calculation and what
period you need for the answer) until you can afford the machine? (Assume the \$350,000.00 is the price
in the future).
PV                                    \$         200,000.00
i                                                     8.50%
n                                                         12
FV                                    \$         350,000.00
x*n                                                          months
x/n                                                          years
Write it in words:

35 = 20*(1.0071)^(12*x)
35/20 = (1.0071)^(12*x)
LN(35/20) = LN((1.0071)^(12*n))
LN(35/20) = 12*n*LN(1.0071)
LN(35/20)/LN(1.0071) = 12*n
LN(35/20)/LN(1.0071)/12 = n
6.6 = n = years                             6.591550252

FV/PV                                              1.75
(1+i/n)                             1.007083333333330
LN(FV/PV)                                  0.559615788
LN((1+i/n))                                0.007058364
LN(FV/PV)/LN((1+i/n))                      79.28406056 months
79.2840605560736/n = x =                   6.607005046 years
If you want to buy a \$350,000.00 C & C Router Machine to improve manufacturing efficiency and you
have \$200,000.00 today that you can invest at an annual rate of 8.50% compounded 12 times a year, how
long do you have to wait (be careful about what period you need to make the calculation and what period
you need for the answer) until you can afford the machine? (Assume the \$350,000.00 is the price in the
future).
PV                                    \$        200,000.00
i                                                    8.50%
n                                                        12
FV                                    \$        350,000.00
x*n                                           79.28406056 months
x/n                                           6.607005046 years

If you have \$200,000.00 to invest today at 8.50% compounded 12 a year and you
Write it in words:                     need \$350,000.00 to buy the machine, you would have to wait 6.607 years

35 = 20*(1.0071)^(12*x)
35/20 = (1.0071)^(12*x)
LN(35/20) = LN((1.0071)^(12*n))
LN(35/20) = 12*n*LN(1.0071)
LN(35/20)/LN(1.0071) = 12*n
LN(35/20)/LN(1.0071)/12 = n
6.6 = n = years                             6.591550252

FV/PV                                               1.75
(1+i/n)                              1.007083333333330
LN(FV/PV)                                   0.559615788
LN((1+i/n))                                 0.007058364
LN(FV/PV)/LN((1+i/n))                       79.28406056 months
79.2840605560736/n = x =                    6.607005046 years
nded 12 a year and you
to wait 6.607 years
The Higher The Discount Rate, The Lower The Present Value
\$120.00

\$100.00

\$80.00                                                                   0.00% Annual Rate
2.50% Annual Rate
Present Value

5.00% Annual Rate

\$60.00                                                                   7.50% Annual Rate
10.00% Annual Rate
12.50% Annual Rate
15.00% Annual Rate
\$40.00
17.50% Annual Rate
20.00% Annual Rate

\$20.00

\$0.00
0     1    2     3    4     5     6    7    8     9    10
Time
If you want to buy a \$350,000.00 C & C Router Machine to improve manufacturing efficiency
and you can invest \$250,000.00 today for the next 5 years (compounding 2 times a year), what
annual interest rate (APR) do you need to find (be careful about periods) so that you can afford
the machine? (Assume the \$350,000.00 is the price in the future).
PV                           \$        250,000.00
n                                                 2
x                                                 5
FV                           \$        350,000.00
i/n
i
Write it in words:

FV/PV                                        1.4
FV/PV^(1/(n*x))                     1.034219694
FV/PV^(1/(n*x))-1                   0.034219694 Half year rate
0.0342196941293802*2                0.068439388 Annual Rate
If you want to buy a \$350,000.00 C & C Router Machine to improve manufacturing efficiency
and you can invest \$250,000.00 today for the next 5 years (compounding 2 times a year), what
annual interest rate (APR) do you need to find (be careful about periods) so that you can afford
the machine? (Assume the \$350,000.00 is the price in the future).
PV                           \$         250,000.00
n                                                 2
x                                                 5
FV                           \$         350,000.00
i/n                                         3.42%
i                                           6.84%
If you want to buy a \$350,000.00 C & C Router Machine to improve
manufacturing efficiency and you can invest \$250,000.00 today for the next 5
years (compounding 2 times a year), the annual interest rate (APR) you need to
Write it in words:                                             find is 6.84%.

FV/PV                                       1.4
FV/PV^(1/(n*x))                    1.034219694
FV/PV^(1/(n*x))-1                  0.034219694 Half year rate
0.0342196941293802*2               0.068439388 Annual Rate
Rule of 72 givens an estimate of what rate you need to
double Money.
PV                                          \$1,000.00
FV                                          \$2,000.00
n                                                   12
x                                                    5
n*x
Notice that this is the number 1.2 not 0.012. To get an estimate of
Rule of 72: i/n aprox = 72/(n*x)                                              Rule of 72), you would have to divide the result by
Notice that this is the number 14.4 not 0.144. To get an estimate of
i = (72/(n*x)*n) =                                                            Rule of 72), you would have to divide the result by
estimate of rate needed to double \$
Using RATE:
i check
FV check                                                 Remember, this is an estimate.
not 0.012. To get an estimate of the real rate (using the
would have to divide the result by 100.
not 0.144. To get an estimate of the real rate (using the
would have to divide the result by 100.
Rule of 72 givens an estimate of what rate you
need to double Money.
PV                                     \$1,000.00
FV                                     \$2,000.00
n                                              12
x                                                5
n*x                                            60
Notice that this is the number 1.2 not 0.012. To get an estimate of the re
Rule of 72: i/n aprox = 72/(n*x)           1.200                     Rule of 72), you would have to divide the result by 100
Notice that this is the number 14.4 not 0.144. To get an estimate of the r
i = (72/(n*x)*n) =                         14.40                     Rule of 72), you would have to divide the result by 100
estimate of rate needed to double \$        0.144
Using RATE:                               13.94%
i check
FV check                               \$2,045.65 Remember, this is an estimate.
To get an estimate of the real rate (using the
to divide the result by 100.
To get an estimate of the real rate (using the
to divide the result by 100.
What is the Annual Interest Rate?
Time Given as:                                     36 months
# of Compounding Periods per Year       n                       2
Present Value                           PV              72,500.00
Future Value                            FV             100,000.00
Years                                   x
Total periods                           n*x
Annual Interest Rate                    i
Period Interest Rate                    i/n

Check:
What is the Annual Interest Rate?
Time Given as:                                     36 months
# of Compounding Periods per Year        n                     2
Present Value                            PV            72,500.00
Future Value                             FV           100,000.00
Years                                    x                     3
Total periods                            n*x                   6
Annual Interest Rate                     i              11.012%
Period Interest Rate                     i/n             5.506%

Check:                                 \$100,000.00
If we invest \$250,000.00 with an annual rate of 5.50% compounded 2 times a year
for 15 years, what is the future value of the investment?
Present Value = PV                                                     -\$250,000.00
Annual Interest Rate = i                                                      5.50%
Number of Compoundin Periods per Year = n                                         2
Years = x                                                                        15
Future Value = FV
Period Rate = i/n
Total Periods = n*x
Future Value = FV

If we borrow \$250,000.00 with an annual rate of 5.50% compounded 2 times a
year for 15 years, what is the future value of the amount we owe?
Present Value = PV                                 \$                   250,000.00
Annual Interest Rate = i                                                    5.50%
Number of Compoundin Periods per Year = n                                        2
Years = x                                                                       15
Future Value = FV
Period Rate = i/n
Total Periods = n*x
Future Value = FV
Check:
\$564,150.43

Check:
(\$564,150.43)
If we invest \$250,000.00 with an annual rate of 5.50% compounded 2 times a year
for 15 years, what is the future value of the investment?
Present Value = PV                                                     -\$250,000.00
Annual Interest Rate = i                                                      5.50%
Number of Compoundin Periods per Year = n                                         2
Years = x                                                                        15
Future Value = FV
Period Rate = i/n                                                            0.0275
Total Periods = n*x                                                              30
Future Value = FV                                                      \$564,150.43

If we borrow \$250,000.00 with an annual rate of 5.50% compounded 2 times a
year for 15 years, what is the future value of the amount we owe?
Present Value = PV                                 \$                   250,000.00
Annual Interest Rate = i                                                     5.50%
Number of Compoundin Periods per Year = n                                        2
Years = x                                                                       15
Future Value = FV
Period Rate = i/n                                                           0.0275
Total Periods = n*x                                                             30
Future Value = FV                                                     -\$564,150.43
Check:
\$564,150.43

Check:
(\$564,150.43)
4.1                             4.2                                    4.3
PV =          1000   Age =                             19    PV =                    \$   1.00
i=           8.00%   Future Age =                      25    n=                              1
n=               1   Age Difference = x =                    x=                             12
x=               4   FV =                    \$ 100,000.00    FV =                    \$   2.00
FV =                 i=                            11.00%    i=
FV =                 n=                                  1   i=
x=                                      Check with Rule of 72
PV =
PV =
4.4
PV =    \$         10,000.00
i=                    7.00%
n=                         1
FV =    \$         20,000.00
x=
x=

4.4
PV =    \$         10,000.00
i=                    7.00%
n=                         1
FV =    \$         30,000.00
n*x =
x=
4.1                              4.2                                    4.3
PV =          1000    Age =                              19   PV =                    \$    1.00
i=           8.00%    Future Age =                       25   n=                               1
n=                1   Age Difference = x =                6   x=                              12
x=                4   FV =                    \$ 100,000.00    FV =                    \$    2.00
FV =    \$ 1,360.49    i=                            11.00%    i=                          5.95%
FV =     \$1,360.49    n=                                  1   i=                          5.95%
x=                                  6   Check with Rule of 72            6
PV =                     53464.08361
PV =                     (\$53,464.08)
FV = PV*(1+i/n)^(n*x)
2 = 1*(1+i)^12
2 = (1+i)^12
2^(1/12)-1 = i
4.4
PV =    \$            10,000.00
i=                       7.00%
n=                            1
FV =    \$            20,000.00
x=                 10.24476835
x=                 10.24476835

4.4
PV =    \$            10,000.00
i=                       7.00%
n=                            1
FV =    \$            30,000.00
n*x =              16.23757367
x=                 16.23757367

FV = PV*(1+i/n)^(n*x)
2 = 1.07^x
LN2/LN1.07 = x

FV = PV*(1+i/n)^(n*x)
3 = 1.07^x
LN3/LN1.07 = x
4.1

4.2

4.3
4.1

4.2

4.3
Compounding is a means to get rich, as long as you have prudent saving habits. The definition of Compounding
is: "The process of accumulating interest in an investment over time to earn more interest."

Discounting is like "interest backwards". If you know a future value amount and you want to know what it is
worth today, given an appropriate discount rate (same as interest rate), you make a Present Value calculation
(PV = FV/((1+i/n)^(n*x))). The Present Value calculation is the Future Value calculation, but backwards. With the
Future Value calculation you add all the interest to the Present Value amount to get the Future Value amount;
time is going forward. However, with the Present Value calculation you take out (remove) all the interest from
the Future Value amount to get the Present Value amount; time is going Backwards. The definition of
Discounting is: "Using the appropriate discount rate, you discount back the future value amount to get the
Present Value amount" or "what is the current value of future cash flows given an appropriate discount rate".
As you increase the length of time involved in a future value calculation (assuming a positive rate of return), the
future value increases or the present value decreases. The time variable, or "input", is the most important
variable in the Future Value and Present Value calculations because it has the most effect on the resultant
number as compared to the other input variables.

As you increase the Period Interest Rate, i/n, the Future Value amount increases; whereas, the Present Value will
decrease.
1st City             2nd City
i                                   0.07   i           0.08
n              1
PV                                  8000   PV         8000
x                                     10   x             10
FV                                         FV
FV

1st City FV
2nd City FV
Extra Interest Due To Compounding
1st City                     2nd City
i                                         0.07    i                  0.08
n                     1
PV                                        8000    PV                8000
x                                            10   x                    10
FV                                  \$13,600.00    FV          \$17,271.40
FV          \$17,271.40

1st City FV                         \$13,600.00
2nd City FV                         \$17,271.40
Extra Interest Due To Compounding    \$3,671.40    textbook answer looks incorrect.
Annual Interest    # Compound
No.   PV                Years        Rate               Periods          FV
1    \$     3,150.00            6               18.00%              1
2    \$     8,453.00           19                6.00%              1
3    \$    89,305.00           13               11.00%              1
4    \$   227,382.00           29                5.00%              1
FV
Annual Interest    # Compound
No.   PV                Years        Rate               Periods          FV
1    \$     3,150.00            6               18.00%              1    \$     8,503.60
2    \$     8,453.00           19                6.00%              1    \$    25,575.39
3    \$    89,305.00           13               11.00%              1    \$   346,796.33
4    \$   227,382.00           29                5.00%              1    \$   935,935.14
FV
\$8,503.60
\$25,575.39
\$346,796.33
\$935,935.14
Annual Interest
No. PV   PV   Years        Rate
1                   12                4.00%
2                    4                9.00%
3                   16               12.00%
4                   21               11.00%
# Compound
Periods          FV
1    \$    17,328.00
1    \$    41,517.00
1    \$   790,382.00
1    \$   647,816.00
Annual Interest
No. PV                   PV                Years        Rate
1       (\$10,823.02)    \$    10,823.02           12                4.00%
2       (\$29,411.69)    \$    29,411.69            4                9.00%
3      (\$128,928.43)    \$   128,928.43           16               12.00%
4       (\$72,388.42)    \$    72,388.42           21               11.00%
# Compound
Periods          FV
1    \$    17,328.00
1    \$    41,517.00
1    \$   790,382.00
1    \$   647,816.00
Annual Interest                          # Compound
No.   PV               Years        Rate              Annual Interest Rate   Periods
1    \$      715.00            6                                                         1
2    \$      905.00            7                                                         1
3    \$   15,000.00           18                                                         1
4    \$   70,300.00           21                                                         1
FV
\$     1,381.00
\$     1,718.00
\$   141,832.00
\$   312,815.00
Annual Interest    Annual Interest    # Compound
No.   PV               Years        Rate               Rate               Periods
1    \$      715.00            6               11.60%             11.60%              1
2    \$      905.00            7                9.59%              9.59%              1
3    \$   15,000.00           18               13.29%             13.29%              1
4    \$   70,300.00           21                7.37%              7.37%              1

FV = PV*(1+i/n)^(n*x)
715 = 1381*(1+i/n)^(6)
(715/1381)^(1/6)-1
FV
\$     1,381.00
\$     1,718.00
\$   141,832.00
\$   312,815.00
Annual Interest    # Compound
No.   PV               Years   Years   Rate               Periods
1    \$      250.00                                9.00%              1
2    \$    1,941.00                                7.00%              1
3    \$   32,805.00                               12.00%              1
4    \$   32,500.00                               19.00%              1
FV
\$     1,105.00
\$     3,700.00
\$   387,120.00
\$   198,212.00
Annual Interest    # Compound
No.   PV               Years        Years        Rate               Periods
1    \$      250.00     17.24506178 17.24506178              9.00%              1
2    \$    1,941.00     9.535063588 9.535063588              7.00%              1
3    \$   32,805.00     21.77872066 21.77872066             12.00%              1
4    \$   32,500.00     10.39415177 10.39415177             19.00%              1
FV
\$     1,105.00
\$     3,700.00
\$   387,120.00
\$   198,212.00
Total Cost Education = FV =   320000
x=                                18
PV =                           50000
n=                                 1
i/n =
i=
Words:
Total Cost Education = FV =                  320000
x=                                                18
PV =                                          50000
n=                                                 1
i/n =                                     10.8633%              10.8633%
i=                                     0.108632935           0.108632935
To allow our current investment of \$50,000.00 to cover the our child's eduction costs in
Words:                                      18 years, we would have to earn an annual return of 10.86%.

FV = PV*(1+i/n)^(n*x)
(FV/PV)^(1/(n*x))-1
the our child's eduction costs in
al return of 10.86%.
Annual Rate = i =   0.07   FV = PV*(1+i/n)^(n*x)
n=                     1   LN(FV/PV)/LN(1+i/n) = n*x
PV1 =                  1
FV1 =                 2    check     FV check
x*n =
x=

Annual Rate = i =   0.07
n=                     1
PV2 =                  1
FV2 =                 4    check     FV check
x*n =
x=

Words:
Words:
LN(1+i/n) = n*x
Annual Rate = i =        0.07
n=                          1
PV1 =                       1
Annual Rate = i =        0.07            FV = PV*(1+i/n)^(n*x)
n=                          1            LN(FV/PV)/LN(1+i/n) = n*x
PV1 =                       1
FV1 =                      2             check     FV check
x*n =               10.24477              10.24477     \$2.00
x=                  10.24477

Annual Rate = i =        0.07
n=                          1
PV2 =                       1
FV2 =                      4             check     FV check
x*n =               20.48954              20.48954     \$4.00
x=                  20.48954

To double my investment at an Annual Rate of 7.00% I would ha
Words:                                  to invest for 10.24years.
To quadruple my investment at an Annual Rate of 7.00% I wou
Words:                               have to invest for 20.49years.
LN(1+i/n) = n*x

nnual Rate of 7.00% I would have
10.24years.
an Annual Rate of 7.00% I would
for 20.49years.
Year 2                       1893
Year 1                       2009
Year Difference = x =
PV =                        \$1.00
FV =                    \$6,450.00
n=                               1   check
i/n =
i=
Year 2                        1893
Year 1                        2009
Year Difference = x =          116
PV =                         \$1.00
FV =                     \$6,450.00
n=                                1   check
i/n =                    7.855186%     0.078552
i=                      0.07855186
Unfunded Pension Liability = FV =                                       \$ 750,000,000.00
Must be paid in = years = x =                                                          25
Discount Rate = i =                                                                  0.08
n=                                                                                      1
n*x =                                                                                  25
i/n =                                                                                0.08

Present Value of Liabilitiy (Lump Sum Value today that stock Analysts
can subtract from current firm market value worth) = PV =

Words:
check
Unfunded Pension Liability = FV =                                       \$ 750,000,000.00
Must be paid in = years = x =                                                          25
Discount Rate = i =                                                                  0.08
n=                                                                                      1
n*x =                                                                                  25
i/n =                                                                                0.08

Present Value of Liabilitiy (Lump Sum Value today that stock Analysts
can subtract from current firm market value worth) = PV =               (\$109,513,428.68)

The financial analyst would like to know what the f
today so it can help them to calculate what the fi
worth today. The Present Value of the future Unfun
Words:                                                                                              \$109,513,428.68.
check

109,513,428.68

st would like to know what the future liability is worth
elp them to calculate what the firm's market value is
esent Value of the future Unfunded Pension Liability is
\$109,513,428.68.
FV = Lottery value in future =   2,000,000.00
x=                                         80
n=                                          1
Discount Rate =                          0.09   Check
PV =
Words:
FV = Lottery value in future =   2,000,000.00
x=                                           80
n=                                            1
Discount Rate =                            0.09                        Check
PV =                               (\$2,027.26)                          2027.263
The present value of this future
Words;                                lottery payout is \$2,027.26.
Part 1                  Part 2
PV =       \$5,000.00    PV =       \$5,000.00
i=              0.105   i=              0.105
n=                  1   n=                  1
x=                 45   x=                 35
FV =                    FV =

Words:

Check                   Check
Part 1                            Part 2
PV =       \$5,000.00              PV =           \$5,000.00
i=              0.105             i=                  0.105
n=                  1             n=                      1
x=                 45             x=                     35
FV =     \$446,963.97              FV =         \$164,683.37

The investment Strategy that this suggests is that the most important variable in
investment strategy is the Total Number of Periods. For 10 extra years, we get
Words:                             \$282,280.60 extra dollars of return.

Check                                Check
446963.9694                           164683.366
PV =          13000    Time Line
x=                 6   0           1   2            3   4
i=              0.09                   \$13,000.00
n=                 1
FV =

Words:

Check:
5   6   7   8            years
\$21,802.30
PV =          13000       Time Line
x=                 6      0             1              2             3              4
i=              0.09                                   \$13,000.00
n=                 1
FV =     \$21,802.30

If we receive \$21,802.30 in 2 years, and then we wait 6 years, our FV will be \$21,802.30
Words:                                    (Remember: 8 - 2 = 6).

Check:
21802.3014
5   6   7   8            years
\$21,802.30
PV =           1
FV =           4
Months =      24
Years = x =
n=             4             (3 month rate) , then 3 + 3 + 3 + 3 = 12 months
n*x =
i/n =              <<== Three Month Rate            Check:     Check:
i=                 <<== Annual Rate

Words:
PV =                 1
FV =                 4
Months =            24
Years = x =          2
n=                   4           (3 month rate) , then 3 + 3 + 3 + 3 = 12 months, so 4 total period in a year
n*x =                8
i/n =          18.921% <<== Three Month Rate            Check:       Check:
i=             75.683% <<== Annual Rate                   0.189207           4

First, if the problems gives us time in months, we have to determine the years.
After we do that the problem is straight forward. The 3-month rate to
period in a year
i/n =        0.35%
n=              12
i=
PV =     \$1,800.00
FV =     \$3,500.00            Check:
n*x =                months
x=                   years

Words:
i/n =           0.35%
n=                  12
i=               0.042
PV =        \$1,800.00
FV =        \$3,500.00                         Check:
n*x =     190.3255241 months                   190.3255
x=        15.86046034 years

It would take 15.86 years for the investment to grow to \$3,500.00 given a monthly
Words:                                      Rate of 0.35%.
FV =     \$75,000.00
n=                12
i/n =         0.42%
i=
x=               10
n*x
PV =

Words:
FV =      \$75,000.00
n=                 12
i/n =          0.42%
i=            5.040%
x=                 10
n*x              120
PV =     (\$45,356.05)

Words:    The Present Value of \$75,000.00 given a monthly rate of 0.42% is \$45,356.05.
1st                         2nd
FV =            1,000,000.00   FV =     1,000,000.00
x=                        45   x=                 45
i=                   11.00%    i=              5.00%
n=                         1   n=                  1
PV =                           PV =

Words:

Check:                       Check:
1st                                       2nd
FV =            1,000,000.00            FV =          1,000,000.00
x=                        45            x=                      45
i=                   11.00%             i=                   5.00%
n=                         1            n=                       1
PV =               -9,129.90            PV =           -111,296.51

I want to be a millionaire when I retire, so with either option, I am going to invest earlier in