# Contents - NCNU Moodle 1916.xls by zhaonedx

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```									Problems

Problem 4-12
Problem 4-13
Problem 4-14
Problem 4-15
Problem 4-23
Problem 4-24
Problem 4-27
Problem 4-28
Problem 4-29
Problem 4-30
Problem 4-32
Problem 4-34
Problem 4-35
Problem 4-36
Problem 4-44
Problem 4-45
Problem 4-46
Problem 4-12

will be paying you \$10,000 at the end of this year, \$20,000 at the end of the following year,
and \$30,000 at the end of the year after that (three years from today). The interest rate is
3.5% per year.

Interest Rate:                             3.50%

Year
0             1             2              3

Cash flow:                                             \$10,000.00    \$20,000.00    \$30,000.00
a. What is the present value of your windfall?
Present Value
Using formulas                         55,390.33
Using Excel functions                  55,390.33

b. What is the future value of your windfall in
three years (on the date of the last payment)?
Future Value
Using formulas                61,412.25
Using Excel functions         61,412.25
Problem 4-13

You have a loan outstanding. It requires making three annual payments at the end of the
next three years of \$1000 each. Your bank has offered to allow you to skip making the
next two payments in lieu of making one large payment at the end of the loan’s term in
three years. If the interest rate on the loan is 5%, what final payment will the bank require
you to make so that it is indifferent between the two forms of payment?

Interest Rate:                            5.00%

Year
0             1              2

Cash flow:                                           \$1,000.00      \$1,000.00
Future value
ments at the end of the
you to skip making the
nd of the loan’s term in
nt will the bank require
ment?

3

\$1,000.00
3,152.50
Problem 4-14

You have been offered a unique investment opportunity. If you invest \$10,000 today, you will receive \$500 one year from now, \$1500 two years from now, and
\$10,000 ten years from now.

Interest rate                     6.00%
NPV                           (2,609.36)
Year
0           1            2            3             4          5         6        7            8                 9               10

Cash flow:                  (\$10,000.00)      \$500.00     \$1,500.00        \$0.00          \$0.00      \$0.00     \$0.00    \$0.00         \$0.00          \$0.00      \$10,000.00
Discounted value             (10,000.00)      471.70      1,334.99         0.00           0.00       0.00      0.00     0.00          0.00           0.00        5,583.95
NPV                           (2,609.36)

a. What is the NPV of the opportunity if the interest rate is 6% per year? Should you take the opportunity?
At an interest rate of               6.00% the NPV is     (2,609.36) so reject the project.

b. What is the NPV of the opportunity if the interest rate is 2% per year? Should you take it now?
At an interest rate of               2.00% the NPV is       135.43 so accept the project.

The relationship is illustrated by the chart below:
Data for NPV profile:
NPV                                                                         0%           2,000.00
1%           1,018.36
2%             135.43
3%            -659.73
4%           -1,376.75
3,000.00
5%           -2,024.13

2,000.00                                                                                                                        6%           -2,609.36
7%           -3,139.06
1,000.00
NPV

0.00
0%        1%           2%          3%           4%            5%         6%       7%       NPV
-1,000.00

-2,000.00

-3,000.00

-4,000.00
Discount rate
Problem 4-15

Marian Plunket owns her own business and is considering an investment. If she
undertakes the investment, it will pay \$4000 at the end of each of the next three
years. The opportunity requires an initial investment of \$1000 plus an additional
investment at the end of the second year of \$5000. What is the NPV of this
opportunity if the interest rate is 2% per year? Should Marian take it?

Interest rate                      2.00%
NPV using Excel function        5,729.69
Year
0           1             2

Cash flow:                    (\$1,000.00)   \$4,000.00   (\$1,000.00)
Discounted cash flow           (1,000.00)    3,921.57      (961.17)
NPV the "long" way              5,729.69

Marion should take the investment.
an investment. If she
each of the next three
at is the NPV of this
take it?

3

\$4,000.00
3,769.29
Problem 4-23

Your grandmother has been putting \$1000 into a savings account on
every birthday since your first (that is, when you turned 1). The
account pays an interest rate of 3%. How much money will be in the
makes the deposit on that birthday?

PMT                           \$1,000.00
Rate                              3.00%
Years                                 18
FV                           \$23,414.44
Problem 4-24

A rich relative has bequeathed you a growing perpetuity. The first payment will
occur in a year and will be \$1000. Each year after that, you will receive a payment
on the anniversary of the last payment that is 8% larger than the last payment. This
pattern of payments will go on forever. If the interest rate is 12% per year,

a. What is today’s value of the bequest?
First Payment                  \$1,000.00
Growth Rate                        8.00%
Interest Rate                    12.00%
Value                        \$25,000.00

b. What is the value of the bequest immediately after the first payment is made?
2 ways:
Value This Year Times
1 + Growth Rate              \$27,000.00
- OR -
Second Payment                \$1,080.00
Growth Rate                       8.00%
Interest Rate                   12.00%
Value                        \$27,000.00
The first payment will
n the last payment. This
12% per year,

Problem 4-27

Your oldest daughter is about to start kindergarten at a private school. Tuition is \$10,000
per year, payable at the beginning of the school year. You expect to keep your daughter
in private school through high school. You expect tuition to increase at a rate of 5% per
year over the 13 years of her schooling. What is the present value of the tuition payments
if the interest rate is 5% per year?

Look at 4-28 for an explanation. In this problem, the difference is that the first payment
occurs today. (If you want to look at this like 4-28, change that sheet just a little and
you can see the balance go to zero.)

Payment                              \$10,000.00
Years                                         13
Value                               \$130,000.00
This is true only because the growth rate equals the interest rate.
Problem 4-28

A rich aunt has promised you \$5000 one year from today. In addition, each year after that, she has
promised you a payment (on the anniversary of the last payment) that is 5% larger than the last
payment. She will continue to show this generosity for 20 years, giving a total of 20 payments. If the
interest rate is 5%, what is her promise worth today?

Discount rate            5.00%
Number of payments                 20
Payment amount          \$5,000.00
PV =           95,238.10 which is equal to the payment amount times the number
Growth Rate=              5.00% of periods discounted for one period.

Here's why:

In order to have enough to make all her payments to you, auntie would
have to deposit that much now. Here's how that would be depleted:
Amount of
- amount                     principal used
Beginning                   paid out end                  for that
balance       +interest     of year        ending balance payment
1         5000.00     95,238.10      4,761.90      5,000.00      95,000.00
2         5250.00     95,000.00      4,750.00      5,250.00      94,500.00     (500.00)
3         5512.50     94,500.00      4,725.00      5,512.50      93,712.50     (787.50)
4         5788.13     93,712.50      4,685.63      5,788.13      92,610.00   (1,102.50)
5         6077.53     92,610.00      4,630.50      6,077.53      91,162.97   (1,447.03)
6         6381.41     91,162.97      4,558.15      6,381.41      89,339.71   (1,823.26)
7         6700.48     89,339.71      4,466.99      6,700.48      87,106.22   (2,233.49)
8         7035.50     87,106.22      4,355.31      7,035.50      84,426.03   (2,680.19)
9         7387.28     84,426.03      4,221.30      7,387.28      81,260.05   (3,165.98)
10         7756.64     81,260.05      4,063.00      7,756.64      77,566.41   (3,693.64)
11         8144.47     77,566.41      3,878.32      8,144.47      73,300.26   (4,266.15)
12         8551.70     73,300.26      3,665.01      8,551.70      68,413.57   (4,886.68)
13         8979.28     68,413.57      3,420.68      8,979.28      62,854.97   (5,558.60)
14         9428.25     62,854.97      3,142.75      9,428.25      56,569.47   (6,285.50)
15         9899.66     56,569.47      2,828.47      9,899.66      49,498.29   (7,071.18)
16        10394.64     49,498.29      2,474.91     10,394.64      41,578.56   (7,919.73)
17        10914.37     41,578.56      2,078.93     10,914.37      32,743.12   (8,835.44)
18        11460.09     32,743.12      1,637.16     11,460.09      22,920.18   (9,822.94)
19        12033.10     22,920.18      1,146.01     12,033.10      12,033.10 (10,887.09)
20        12634.75     12,033.10        601.65     12,634.75          (0.00) (12,033.10)
Problem 4-29

You are running a hot Internet company. Analysts predict that its earnings will grow at 30% per year for the next five years. After that, as competition increases,
earnings growth is expected to slow to 2% per year and continue at that level forever. Your company has just announced earnings of \$1,000,000. What is the present
value of all future earnings if the interest rate is 8%? (Assume all cash flows occur at the end of the year.)

Starting growth rate                           30.00%
Number of years of high growth                      5
Later growth rate                               2.00%
Discount rate                                   8.00%
Year
0                 1                  2                 3                  4                 5                   6

Cash flow                               \$1,000,000.00     \$1,300,000.00      \$1,690,000.00     \$2,197,000.00      \$2,856,100.00     \$3,712,930.00      \$3,787,188.60
PV(cash flow)                                             \$1,203,703.70      \$1,448,902.61     \$1,744,049.43      \$2,099,318.76     \$2,526,957.77
PV(infinite cash flows)                                                                                                             42,958,282.09
Value of company                        51,981,214.36
Problem 4-30

Your brother has offered to give you \$100, starting next year, and after that growing at 3% for the next 20 years. You would like to
calculate the value of this offer by calculating how much money you would need to deposit in the local bank so that the account will
generate the same cash flows as he is offering you. Your local bank will guarantee a 6% annual interest rate so long as you have money in
the account.

Initial payment                           \$100
Annual growth rate                          3%
Number of periods                            20
Discount rate                            6.00%

a. How much money will you need to deposit into the account today?
PV of
growing
annuity \$     1,456.15

b. Using an Excel spreadsheet, show explicitly that you can deposit this amount of money into the account, and every year
withdraw what your brother has promised, leaving the account with nothing after the last withdrawal.
Amount of
- amount paid out                     principal used for
Beginning balance +interest           end of year        ending balance      that payment
1             100.00           1,456.15              87.37             100.00           1,443.52
2             103.00           1,443.52              86.61             103.00           1,427.13               (16.39)
3             106.09           1,427.13              85.63             106.09           1,406.67               (20.46)
4             109.27           1,406.67              84.40             109.27           1,381.80               (24.87)
5             112.55           1,381.80              82.91             112.55           1,352.16               (29.64)
6             115.93           1,352.16              81.13             115.93           1,317.36               (34.80)
7             119.41           1,317.36              79.04             119.41           1,276.99               (40.36)
8             122.99           1,276.99              76.62             122.99           1,230.63               (46.37)
9             126.68           1,230.63              73.84             126.68           1,177.79               (52.84)
10             130.48           1,177.79              70.67             130.48           1,117.98               (59.81)
11             134.39           1,117.98              67.08             134.39           1,050.66               (67.31)
12             138.42           1,050.66              63.04             138.42             975.28               (75.38)
13             142.58             975.28              58.52             142.58             891.22               (84.06)
14             146.85             891.22              53.47             146.85             797.84               (93.38)
15             151.26             797.84              47.87             151.26             694.45              (103.39)
16             155.80             694.45              41.67             155.80             580.32              (114.13)
17             160.47             580.32              34.82             160.47             454.67              (125.65)
18             165.28             454.67              27.28             165.28             316.67              (138.00)
19             170.24             316.67              19.00             170.24             165.43              (151.24)
20             175.35             165.43               9.93             175.35              (0.00)             (165.43)
Problem 4-32

You are thinking of purchasing a house. The house costs \$350,000. You have
\$50,000 in cash that you can use as a down payment on the house, but you need
to borrow the rest of the purchase price. The bank is offering a 30-year
mortgage that requires annual payments and has an interest rate of 7% per year.

Cost of House           \$350,000.00
Down Payment             \$50,000.00
Number of Years                   30
Interest Rate                 7.00%
Annual Payment
Amount                    (24,175.92)
sts \$350,000. You have
the house, but you need
is offering a 30-year
est rate of 7% per year.
mortgage?
Problem 4-34

You would like to buy the house and take the mortgage described in Problem 32.
You can afford to pay only \$23,500 per year. The bank agrees to allow you to pay
this amount each year, yet still borrow \$300,000. At the end of the mortgage (in 30
years), you must make a balloon payment; that is, you must repay the remaining
balance on the mortgage. How much will this balloon payment be?

Payment                      \$23,500.00
Cost                        \$300,000.00
Term                                  30
Rate                              7.00%
Balloon                       63,848.03
- OR -
PV of Payments              (291,612.47)
PV of difference               8,387.53
FV of difference              63,848.03
escribed in Problem 32.
ees to allow you to pay
d of the mortgage (in 30
ust repay the remaining
ent be?
Problem 4-35

You are saving for retirement. To live comfortably, you decide you will need to save \$2
million by the time you are 65. Today is your 30th birthday, and you decide, starting today
and continuing on every birthday up to and including your 65th birthday, that you will put
the same amount into a savings account. If the interest rate is 5%, how much must you set
aside each year to make sure that you will have \$2 million in the account on your 65th
birthday?

Cumulative
value of account
Your age Payment         (end of year)
30 (20,868.91)        (21,912.36)     Number of payments                      36
31   (20,868.91)      (44,920.34)     Interest rate                       5.00%
32   (20,868.91)      (69,078.71)     Future value                \$2,000,000.00
33   (20,868.91)      (94,445.01)
34   (20,868.91)     (121,079.62)     Payment amount                 (20,868.91)
35   (20,868.91)     (149,045.96)
36   (20,868.91)     (178,410.62)
37   (20,868.91)     (209,243.51)     You'll get the same answer if you calculate
38   (20,868.91)     (241,618.05)     the payment that will accumulate to 2
39   (20,868.91)     (275,611.31)     million and then multiply it by 1.05.
40   (20,868.91)     (311,304.23)
41   (20,868.91)     (348,781.81)
42   (20,868.91)     (388,133.26)
43   (20,868.91)     (429,452.28)
44   (20,868.91)     (472,837.25)
45   (20,868.91)     (518,391.48)
46   (20,868.91)     (566,223.41)
47   (20,868.91)     (616,446.94)
48   (20,868.91)     (669,181.65)
49   (20,868.91)     (724,553.09)
50   (20,868.91)     (782,693.10)
51   (20,868.91)     (843,740.12)
52   (20,868.91)     (907,839.48)
53   (20,868.91)     (975,143.82)
54   (20,868.91)   (1,045,813.37)
55   (20,868.91)   (1,120,016.40)
56   (20,868.91)   (1,197,929.58)
57   (20,868.91)   (1,279,738.42)
58   (20,868.91)   (1,365,637.70)
59   (20,868.91)   (1,455,831.94)
60   (20,868.91)   (1,550,535.90)
61   (20,868.91)   (1,649,975.05)
62   (20,868.91)   (1,754,386.17)
63   (20,868.91)   (1,864,017.83)
64   (20,868.91)   (1,979,131.09)
65   (20,868.91)   (2,000,000.00)
Problem 4-36

You realize that the plan in Problem 35 has a flaw. Because your income will increase ove
be more realistic to save less now and more later. Instead of putting the same amount asid
to let the amount that you set aside grow by 3% per year. Under this plan, how much will
today? (Recall that you are planning to make the first contribution to the account today.)

Cumulative
value of account
Your age   Payment      (end of year)
30 (13,823.91)       (14,515.10)
31 (14,791.58)       (30,772.02)
32 (15,826.99)       (48,928.96)
33 (16,934.88)       (69,157.04)
34 (18,120.32)       (91,641.23)
35 (19,388.75)      (116,581.47)
36 (20,745.96)      (144,193.80)
37 (22,198.18)      (174,711.57)
38 (23,752.05)      (208,386.80)
39 (25,414.69)      (245,491.57)
40 (27,193.72)      (286,319.55)
41 (29,097.28)      (331,187.67)
42 (31,134.09)      (380,437.85)
43 (33,313.48)      (434,438.89)
44 (35,645.42)      (493,588.53)
45 (38,140.60)      (558,315.58)
46 (40,810.44)      (629,082.32)
47 (43,667.17)      (706,386.96)
48 (46,723.87)      (790,766.38)
49 (49,994.54)      (882,798.97)
50 (53,494.16)      (983,107.79)
51 (57,238.75)    (1,092,363.87)
52 (61,245.47)    (1,211,289.80)
53 (65,532.65)    (1,340,663.57)
54 (70,119.93) (1,481,322.68)
55 (75,028.33) (1,634,168.55)
56 (80,280.31) (1,800,171.31)
57 (85,899.93) (1,980,374.81)
58 (91,912.93) (2,175,902.12)
59 (98,346.83) (2,387,961.40)
60 #########   (2,617,852.14)
61 #########   (2,866,971.90)
62 #########   (3,136,823.55)
63 #########   (3,429,023.00)
64 #########   (3,745,307.50)
65 #########   (3,892,899.58)
er. Instead of putting the same amount aside each year, you decide
per year. Under this plan, how much will you put into the account
e first contribution to the account today.)

Number of payments                          36
Interest rate                           5.00%
Present value                     (345,314.83)
Growth rate                             3.00%
PV interest factor (formula)       24.979538

Payment                            (13,823.91)
Problem 4-44

You are thinking of making an investment in a new plant. The plant will generate revenues of \$1 million per year for as
long as you maintain it. You expect that the maintenance cost will start at \$50,000 per year and will increase 5% per year
thereafter. Assume that all revenue and maintenance costs occur at the end of the year. You intend to run the plant as
long as it continues to make a positive cash flow (as long as the cash generated by the plant exceeds the maintenance
costs). The plant can be built and become operational immediately. If the plant costs \$10 million to build, and the
interest rate is 6% per year, should you invest in the plant?

The question is, how long will it take
for                                      \$50,000.00 to equal            \$1,000,000.00 if it grows at   5.00% per year?

You can solve this using logarithms, by solving the following equation for n:
1000000 =                        50000 *                            1.05 ^(n-1)

divide both sides by \$ 50,000.00
1.05 ^(n-1)      =                  \$20.00

(n-1)* ln(1.05)    =                  ln(20)
(n-1)* 0.048790164 =                  2.995732274

Divide both sides by                     0.048790164           and add 1 to get               62.40

Or you can use the Excel function:
62.40
Problem 4-45

You have just turned 30 years old, have just received your MBA, and have accepted your f
must decide how much money to put into your retirement plan. The plan works as follows:
plan earns 7% per year. You cannot make withdrawals until you retire on your sixty-fifth b
point, you can make withdrawals as you see fit. You decide that you will plan to live to 10
you turn 65. You estimate that to live comfortably in retirement, you will need \$100,000 p
the end of the first year of retirement and ending on your one hundredth birthday. You wil
same amount to the plan at the end of every year that you work. How much do you need to

Interest rate assumption                       7.00%
You will need                            \$100,000.00

PV as of age 65 =                      (1,294,767.23)

You can accumulate this amount in                   35

Note: The above uses future value of an annuity.
You can also calculate this using PV of an annuity, by figuring the PV of the FV a
annuity implied by that.

PV as of age 30 =                         121,271.70
PMT                                         9,366.29
ment plan. The plan works as follows: Every dollar in the
ls until you retire on your sixty-fifth birthday. After that
decide that you will plan to live to 100 and work until
retirement, you will need \$100,000 per year starting at
your one hundredth birthday. You will contribute the
you work. How much do you need to contribute each

per year for            35 years

years by paying 9,366.29 per year.

nuity.
nnuity, by figuring the PV of the FV and computing the
Problem 4-46

Problem 45 is not very realistic because most retirement plans do not allow you to spec
contribute every year. Instead, you are required to specify a fixed percentage of your sa
contribute. Assume that your starting salary is \$75,000 per year and it will grow 2% per
Assuming everything else stays the same as in Problem 45, what percentage of your inc
contribute to the plan every year to fund the same retirement income?

Interest rate assumption                                7.00%
You will need                                     \$100,000.00

PV as of age 65=                                 (1,294,767.23)
PV as of age 30=                                   (121,271.70)

Your contributions will grow at an annual
rate of                                                 2.00%

Starting salary                                    \$75,000.00
PV interest factor                                  16.253689
You can accumulate this amount in                           35

Age Contribution
31      \$7,461.18
32      \$7,610.40
33      \$7,762.61
34      \$7,917.86
35      \$8,076.22
36      \$8,237.75
37      \$8,402.50
38      \$8,570.55
39      \$8,741.96
40      \$8,916.80
41    \$9,095.14
42    \$9,277.04
43    \$9,462.58
44    \$9,651.83
45    \$9,844.87
46   \$10,041.77
47   \$10,242.60
48   \$10,447.45
49   \$10,656.40
50   \$10,869.53
51   \$11,086.92
52   \$11,308.66
53   \$11,534.83
54   \$11,765.53
55   \$12,000.84
56   \$12,240.86
57   \$12,485.67
58   \$12,735.39
59   \$12,990.10
60   \$13,249.90
61   \$13,514.90
62   \$13,785.19
63   \$14,060.90
64   \$14,342.11
65   \$14,628.96
rement plans do not allow you to specify a fixed amount to
specify a fixed percentage of your salary that you want to
,000 per year and it will grow 2% per year until you retire.
oblem 45, what percentage of your income do you need to
etirement income?

per year for             35    years

years by paying     7,461.18 per year.
or                      9.95% of your annual income.

Balance
\$7,461.18
\$15,593.87
\$24,448.05
\$34,077.28
\$44,538.91
\$55,894.38
\$68,209.48
\$81,554.70
\$96,005.49
\$111,642.67
\$128,552.80
\$146,828.53
\$166,569.11
\$187,880.78
\$210,877.30
\$235,680.48
\$262,420.72
\$291,237.62
\$322,280.66
\$355,709.83
\$391,696.44
\$430,423.85
\$472,088.36
\$516,900.07
\$565,083.92
\$616,880.65
\$672,547.97
\$732,361.71
\$796,617.13
\$865,630.22
\$939,739.23
\$1,019,306.17
\$1,104,718.50
\$1,196,390.91
\$1,294,767.23

```
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