Mathematics Assignment #75260

STUDENTS:

Andres Hoyos Joy Hwang Alicia Klassanoff Ljubomir Petrovic

Bronco Vuskovich Maira Alejandra Soto Alice Benatar

ASSIGNMENT 2

Problem 1: There are three major business organization forms from start-up to a major corporation. What are

these three business organization forms? What are advantages and disadvantages for each

organization form?

Advantages and Disadvantages of Business Organizations

Easy to create

Easy to raise

capital

Taxes paid once

as personal income

Limited Liability

Unlimited Life

Easy to transfer

Ownership

Management by the owners

Easy to make

decisions

Sole Proprietorship

Partnership

Corporation

100% possible

50% possible

Difficult or impossible

Problem 2: Based on the chart below, answer the following two questions.

1. If the value of the firm is 10 million, and $F is 12 million, how much will be the payoff to debt holders and how much will be the payoff to shareholders? Debt Holders: $ 10 million Shareholders: $ 0 2. If the value of the firm is 25 million, and $F is 12 million, how much will be the payoff to debt holders and how much will be the payoff to shareholders? Debt Holders: $ 12 million Shareholders: $ 13 million Problem 3: Assume that you are nearing graduation and have applied for a job with a local bank. As part of the bank's evaluation process, you have been asked to take an examination that covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.

a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2, and (2) an uneven cash flow stream of -$50, $100, $75, and $50 at the end of Years 0 through 3.

b. What's the future value of an initial $100 after 3 years if it is invested in an account paying 10% annual interest and compounded annually? How about if the interest is compounded monthly, daily or hourly?

Compounded Annually

Compounded Monthly

Compounded Daily Compounded Hourly

Rate 𝑟 = 0.1 𝑟 = 0.1

12 = 0.0083 𝑟 =

0.1

365 = 0.00027 𝑟 =

0.1

365 × 24 = 0.000011

Period 𝑛 = 3 𝑛 = 3 × 36 𝑛 = 3 × 365 = 1095 𝑛 = 3 × 365 × 24 = 26280

𝑭𝑽 = 𝑷𝑽 × (𝟏 + 𝒓)𝒏 𝐹𝑉 = 100 × (1 + 0.1)3

𝐹𝑉 = 100 × 0.0083)36

𝐹𝑉 = 100 × (1 + 0.1

0.00027 )1095

𝐹𝑉 = 100 × (1 + 0.1

0.000011 )26280

FUTURE VALUE

$133.10 $134.82 $134.98 $134.99

c. What is the present value of $100 to be received in 3 years if the annual interest rate is

10% and compounded annually? How about if the interest is compounded monthly, daily or hourly?

Compounded Annually

Compounded Monthly

Compounded Daily Compounded Hourly

Rate 𝑟 = 0.1 𝑟 = 0.1

12 = 0.0083 𝑟 =

0.1

365 = 0.00027 𝑟 =

0.1

365 × 24 = 0.000011

Period 𝑛 = 3 𝑛 = 3 × 36 𝑛 = 3 × 365 = 1095 𝑛 = 3 × 365 × 24 = 26280

𝑷𝑽 = 𝑭𝑽

(𝟏 + 𝒓)𝒏 𝑃𝑉 =

100

(1 + 0.1)3

𝑃𝑉

= 100

(1 + 0.0083)36

𝑃𝑉 = 100

(1 + 0.00027)1095 𝑃𝑉 =

100

(1 + 0.000011)26280

PRESENT VALUE

$75.13 $74.17 $74.08 $74.08

d. 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?

I/Y PV PMT FV Nper

20% -100 0 200 3.8 YEARS

e. If you want an investment to double in three years, what annual interest rate must it earn?

N PV PMT FV I/Y

3 -100 0 200 25.99%

f. What is the present value of the following uneven cash flow stream? The annual interest rate is 10%, compounded annually. How about the interest is compounded monthly, daily or hourly?

CF1 CF2 CF3 CF4

FV 100 300 300 -50

Period 1 2 3 4

Rate 10% 10% 10% 10%

𝑷𝑽 = 𝑭𝑽

(𝟏 + 𝒓)𝒏 𝑃𝑉

= 100

(1 + 0.1)1

𝑃𝑉

= 300

(1 + 0.1)2

𝑃𝑉

= 300

(1 + 0.1)3

𝑃𝑉

= −50

(1 + 0.1)4

PRESENT VALUE PER

YEAR

$ 90.91

$247.93

$ 225.39

$ 34.15

𝑻𝒐𝒕𝒂𝒍 𝑷𝑽 = 𝟗𝟎. 𝟗𝟏 + 𝟐𝟒𝟕. 𝟗𝟑 + 𝟐𝟐𝟓. 𝟑𝟗 − 𝟑𝟒. 𝟏𝟓 = $𝟓𝟑𝟎. 𝟎𝟖

g. What is the effective annual rate (EAR) for an annual percentage rate (APR) or a nominal

annual rate of 12%, compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily? Compounded hourly?

Compounded Semiannually

Compounded Quarterly

Compounded Monthly

Compounded Daily

Compounded Hourly

Period 𝑛 = 2 𝑛 = 4 𝑛 = 12 𝑛 = 365 𝑛 = 365 × 24 = 8760

Rate 𝑟 =

0.12

2 = 0.06 𝑟 =

0.12

4 = 0.03 𝑟 =

0.12

12 = 0.01 𝑟 =

0.12

365 = 0.00033

𝑟 = 0.12

8760 = 0.0000137

𝑬𝑨𝑹 = (𝟏 + 𝒓)𝒏 − 𝟏

𝐸𝐴𝑅 = (1 + 0.06)2 − 1 = 0.1236

𝐸𝐴𝑅 = (1 + 0.03)4

− 1 = 0.1255

𝐸𝐴𝑅 = (1 + 0.01)12

− 1 = 0.1268

𝐸𝐴𝑅 = (1 + 0.00033)365

− 1 = 0.127475

𝐸𝐴𝑅 = (1 + 0.0000137)8760

− 1 = 0.127496

EAR 12.36% 12.55% 12.68% 12.7475% 12.7496%

h. Will the effective annual rate (EAR) ever be equal to the APR or nominal annual rate (quoted)? Yes, only if the funds are compounded annually for one year. Otherwise, EAR should be greater than APR. i. What is the value at the end of Year 3 of the following cash flow stream if the quoted (nominal) interest rate is 10%, compounded semiannually?

CF1 CF2 CF3

PV 100 100 100

Period Compounded for twice a year for 2 years

𝑛 = 2 × 2 = 4

Compounded for twice a year for 1 year 𝑛 = 2 × 1 = 2

No compounding

Rate 𝑟 =

0.1

2 = 0.05 𝑟 =

0.1

2 = 0.05

N/A

𝑭𝑽 = 𝑷𝑽 × (𝟏 + 𝒓)𝒏

𝐹𝑉 = 100 × (1 + 0.05)4

𝐹𝑉 = 100 × (1 + 0.05)2 = 110.25 $ 100

FUTURE VALUE PER YEAR

$ 121.55 $ 110.25 $ 100

𝑻𝒐𝒕𝒂𝒍 𝑭𝑽 = 𝟏𝟐𝟏. 𝟓𝟓 + 𝟏𝟏𝟎. 𝟐𝟓 + 𝟏𝟎𝟎 = $𝟑𝟑𝟏. 𝟖𝟎

= 100

X 1.05 X 1.05

X 1.05 X 1.05 X 1.05 X 1.05

Sum = Future Value of the annuity

= 110.25

= 121.55

= $ 331.80

j. What is the present value of the above cash flow stream if the quoted (nominal) annual interest rate is 10%, compounded daily?

CF1 CF2 CF3

FV 100 100 100

Period Discounted for 365 times yearly for 1 year

𝑛 = 365

Discounted for 365 times yearly for 2 years

𝑛 = 365 × 2 = 730

Discounted for 365 times yearly for 3

years 𝑛 = 365 × 3 = 1095

Rate 𝑟 =

0.1

365 = 0.000274

𝑟 = 0.000274 𝑟 = 0.000274

𝑃𝑉 = 𝐹𝑉

(1 + 𝑟)𝑛 𝑃𝑉 =

100

(1 + 0.000274)365

= 90.48

𝑃𝑉 = 300

(1 + 0.000274)730

= 81.87

𝑃𝑉

= 300

(1 + 0.000274)1095

= 74.08 PRESENT VALUE

PER YEAR $ 90.48 $ 81.87 $ 74.08

𝑻𝒐𝒕𝒂𝒍 𝑷𝑽 = 𝟗𝟎. 𝟒𝟖 + 𝟖𝟏. 𝟖𝟕 + 𝟕𝟒. 𝟎𝟖 = $𝟐𝟒𝟔. 𝟒𝟑

k. Suppose someone offered to sell you a note calling for the payment of $1,000 in two years (or 730 days). They offer to sell it to you for $850. You have $850 deposit in a bank that pays a 6.76649% nominal rate with daily compounding, 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 effective annual rate on the note versus that of the bank account.

90.49 =

81.87 =

74.08 =

÷ (1.000274)730

= 𝜋𝑟^2

÷ (1.000274)1095

= 𝜋𝑟2

÷ 1.000274

= 𝜋𝑟2

PV of annuity =

246.43

=

(1) FV of $850 in savings:

𝐹𝑉 = 850 × (1 + 0.0676649

365 )730 = $972.89

The note yields a larger return than the savings. Buy the note. (2) PV of the note:

𝑃𝑉 = 1000

(1 + 0.0676649

365 )730

= $873.68

The note is worth $873.68, which is more than the asking price. Buy the note. (3) EAR on note versus the bank account:

𝐸𝐴𝑅 𝑜𝑓 𝑠𝑎𝑣𝑖𝑛𝑔𝑠 = (1 + 0.0676649

365 )365 − 1 = 7%

𝐸𝐴𝑅 𝑜𝑓 𝑛𝑜𝑡𝑒 = ( 1000

850 )

1 2 − 1 = 8.47%

The note yields higher EAR. Buy the note.