Accounting

With simple interest,

Ct = C0 × [(1 + (r)(t)]

1,100 = 600 x [1+(r)(12)]

1,100/600 = 1+ 12(r)

1.8333 = 1+12(r)

1.8333-1 = 12(r)

r = 0.8333/12 = 0.0694 = 6.94%

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Lecture Problem 1-c

If Ben can earn simple interest of 8.6% per year, then how much would Ben need to invest today to have $1,100 in 12 years?

C0 = Ct / [(1 + (r)(t)] = 1,100/[1+(0.86)(12)] = 541.34

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Lecture Problem 2

If Martha invests $500 today in an account that earns 12.34% per year in compound interest, how much will she have in 12 years?

With compound interest, FVt = C0 × (1 + r)t

In this case:

C0 = 500

r = .1234

t = 12

FV12 = 500 × (1.1234)12 = $2,020.15

Martha would have $2,020.15 in 12 years

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Lecture Problem 3

How much will you have in 9 years if you invest $1,234 in 2 years and your account has a return of 2.36% per year for each of the next 14 years?

FVt = Ck × (1 + r)t-k

Note: in 9 years means in 9 years from today

t = 9

k = 2

t – k = 9 – 2 = 7

r = .0236

C2 = 1,234

FVt = Ck × (1 + r)t-k

FV9 = C2 × (1 + r)7

FV9 = 1,234 × (1.0236)7 = $1,452.87

Mode is not relevant, since PMT = 0

Enter 7 2.36 -1,234 0

N I% PV PMT FV

Solve for 1,452.87

You would have $1,452.87

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Lecture Problem 4

If Maria plans to invest $800 in 3 years in an account that has an expected return of 7.46% per year and JoJo plans to invest $1,100 in 5 years in an account that has an expected return of 4.52% per year, then who is expected to have more money in 11 years?

Maria

FVt = Ck × (1 + r)t-k

t = 11; k = 3; t – k = 11 – 3 = 8

r = .0746

C3 = 800

FV11 = C3 × (1 + r)8

= 800 × (1.0746)8 = $1,422.54

Mode is not relevant, since PMT = 0

Enter 8 7.46 -800 0

N I% PV PMT FV

Solve for 1,422.54

Maria is expected to have $1,422.54 in 11 years

JoJo

FVt = Ck × (1 + r)t-k

t = 11; k = 5; t – k = 11 – 5 = 6

r = .0452

C5 = 1,100

FV11 = C5 × (1 + r)6

= 1,100 × (1.0452)6 = $1,434.13

Mode is not relevant, since PMT = 0

Enter 6 4.52 -1,100 0

N I% PV PMT FV

Solve for 1,434.13

JoJo is expected to have $1,434.13 in 11 years

Answer: JoJo is expected to have more money in 11 years

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Lecture Problem 5

If Maria plans to invest $800 in 3 years in an account that has an expected return of 7.46% per year and JoJo plans to invest $1,100 in 5 years in an account that has an expected return of 4.52% per year, then who is planning to invest in a riskier account?

The expected return or interest rate associated with a given investment or asset is related to the risk of the investment’s or asset’s cash flows

Recall that in FNAN 303, we impose the simplifying assumption of a “flat yield curve” which means that the rate (expected return interest rate, etc.) associated with any set of cash flows depends only on the risk of those cash flows.

Maria’s investment has an expected return of 7.46%

JoJo’s investment has an expected return of 4.52%

Answer: Maria is planning to invest in a riskier account

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Lecture Problem 6-a

I Scream Ice Cream Company just bought 1 ton of sugar from Sweet Cane, I Scream has been offered the following 2 options, and the discount rate is 7.20% per quarter.

Option A: I Scream Ice Cream will pay $1,260 to Sweet Cane in 2 quarters

Option B: I Scream Ice Cream will pay $1,340 to Sweet Cane in 3 quarters

Which option, A or B, is better for I Scream Ice Cream?

To answer this question, we need to find the present value of options A and B from the perspective of I Scream Ice Cream, which would be making a payment to Sweet Cane

Option A:

PV0 = Ct ÷ (1+r)t

r = .0720

t = 2

C2 = -1,260

PV0 = C2 ÷ (1+r)2

= -1,260 ÷ (1.0720)2

= -1,096.43

Option B:

PV0 = Ct ÷ (1+r)t

r = .0720

t = 3

C3 = -1,340

PV0 = C3 ÷ (1+r)3

= -1,340 ÷ (1.0720)3

= -1,087.73

The better option for I Scream Ice Cream is the one with the higher present value, which is option B: -1,087.73 > -1,096.43

The better option for I Scream Ice Cream is B, which is equivalent in value to paying $1,087.73 today. This is better than option A, which is equivalent in value to paying $1,096.43 today.

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Lecture Problem 6-b

I Scream Ice Cream Company just bought 1 ton of sugar from Sweet Cane, I Scream has been offered the following 2 options, and the discount rate is 7.20% per quarter.

Option A: I Scream Ice Cream will pay $1,260 to Sweet Cane in 2 quarters

Option B: I Scream Ice Cream will pay $1,340 to Sweet Cane in 3 quarters

Which option, A or B, is better for Sweet Cane?

To answer this question, we need to find the present value of options A and B from the perspective of Sweet Cane, which would be receiving a payment from I Scream Ice Cream

Option A:

PV0 = Ct ÷ (1+r)t

r = .0720

t = 2

C2 = 1,260

PV0 = C2 ÷ (1+r)2

= 1,260 ÷ (1.0720)2

= 1,096.43

Option B:

PV0 = Ct ÷ (1+r)t

r = .0720

t = 3

C3 = 1,340

PV0 = C3 ÷ (1+r)3

= 1,340 ÷ (1.0720)3

= 1,087.73

The better option for Sweet Cane is the one with the higher present value, which is option A: 1,096.43 > 1,087.73

The better option for Sweet Cane is A, which is equivalent in value to receiving $1,096.43 today. This is better than option B, which is equivalent in value to receiving $1,087.73 today.

======================

Lecture Problem 7-a

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a discount rate is 8%, for an expected cash flow of $800,000 in 3 years. What is the value of plant A? The plants are expected to produce no cash flows other than the cash produced when they are sold.

PV0 = Ct / (1 + r)t

t = 3

r = .06

C3 = 800,000

PV0 = 800,000 / (1.06)3 = 671,695

Mode is not relevant, since PMT = 0

Enter 3 6 0 800,000

N I% PV PMT FV

Solve for -671,695

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Lecture Problem 7-b

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a discount rate is 8%, for an expected cash flow of $800,000 in 3 years. What is the value of plant B? The plants are expected to produce no cash flows other than the cash produced when they are sold.

PV0 = Ct / (1 + r)t

t = 3

r = .08

C3 = 800,000

PV0 = 800,000 / (1.08)3 = 635,066

Mode is not relevant, since PMT = 0

Enter 3 8 0 800,000

N I% PV PMT FV

Solve for -635,066

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Lecture Problem 7-c

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a discount rate is 6%, for an expected cash flow of $800,000 in 5 years. What is the value of plant C? The plants are expected to produce no cash flows other than the cash produced when they are sold.

PV0 = Ct / (1 + r)t

t = 5

r = .06

C5 = 800,000

PV0 = 800,000 / (1.06)5 = 597,807

Mode is not relevant, since PMT = 0

Enter 5 6 0 800,000

N I% PV PMT FV

Solve for -597,807

=================================

Lecture Problem 8-a

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a cost of capital is 8%, for an expected cash flow of $800,000 in 3 years. Which plant, A or B, is riskier, or are they equally risky? The plants are expected to produce no cash flows other than the cash produced when they are sold.

Plant B is riskier than plant A, because plant B has a higher cost of capital than plant A. Recall that riskier cash flows are associated with higher discount rates, as investors demand greater reward for bearing more risk.

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Lecture Problem 8-b

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a cost of capital is 8%, for an expected cash flow of $800,000 in 3 years. Which plant, A or B, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.

Plant A: PV0 = 800,000 / (1.06)3 = 671,695

Plant B: PV0 = 800,000 / (1.08)3 = 635,066

The present value of plant A (671,695) is greater than the present value of plant B ($635,066), so plant A is worth more despite the fact that both plants are expected to be sold in 3 years for $800,000. Recall that a higher cost of capital leads to a lower present value, all else equal (cash flow amount and timing).

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Lecture Problem 8-c

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years. Which plant, A or C, is riskier, or are they equally risky? The plants are expected to produce no cash flows other than the cash produced when they are sold.

Plants A and C are equally risky, because they have the same cost of capital. Recall that riskier cash flows are associated with higher discount rates, as investors demand greater reward for bearing more risk. Since the costs of capital are the same, the cash flows, and thus the plants, are equally as risky.

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Lecture Problem 8-d

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years. Which plant, A or C, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.

Plant A: PV0 = 800,000 / (1.06)3 = 671,695

Plant C: PV0 = 800,000 / (1.06)5 = 597,807

The present value of plant A (671,695) is greater than the present value of plant C (597,807), so plant A is worth more despite the fact that both plants have the same cost of capital (and level of risk) and are expected to be sold for $800,000. Recall that a longer time leads to a lower present value, all else equal (cash flow amount and discount rate).

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Lecture Problem 8-e

I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years and plant D which has a cost of capital is 6%, for an expected cash flow of $900,000 in 5 years. Which plant, C or D, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.

Plant C: PV0 = 800,000 / (1.06)5 = 597,807

Plant D: PV0 = 900,000 / (1.06)5 = 672,532

The present value of plant D (672,532) is greater than the present value of plant C (597,807), so plant D is worth more despite the fact that both plants have the same cost of capital (and level of risk) and are expected to be sold at the same time (in 5 years). Recall that a larger expected cash flow leads to a higher present value, all else equal (time until cash flow and discount rate).

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Lecture Problem 9

What is the cost of capital of the I Scream Ice Cream Company plant in Texas if it is worth $1,000,000 and is expected to produce no cash flows other than the cash produced when it is sold in 4 years for an expected cash flow of $1,400,000?

PV0 = Ct / (1 + r)t

t = 4

PV0 = 1,000,000

C4 = 1,400,000

Can be solved algebraically, but much easier with financial calculator

Mode is not relevant, since PMT = 0

Enter 4 -1,000,000 0 1,400,000

N I% PV PMT FV

Solve for 8.78

Confirm

PV0 = Ct / (1 + r)t

= 1,400,000 / (1.0878)4

= $999,843.02 ≈ $1,000,000 ☺

(Difference due to rounding)

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Lecture Problem 10

In how many years from today is the I Scream Ice Cream Company plant in Florida expected to be sold if it is worth $1,000,000, has a cost of capital of 8.20 percent, and is expected to produce no cash flows other than the cash produced when it is sold for an expected cash flow of $1,300,000?

PV0 = Ct / (1 + r)t

PV0 = 1,000,000

Ct = 1,300,000

r = .0820

Can be solved algebraically, but much easier with financial calculator

Mode is not relevant, since PMT = 0

Enter 8.20 -1,000,000 0 1,300,000

N I% PV PMT FV

Solve for 3.33

The plant will be sold in 3.33 years

Confirm

PV0 = Ct / (1 + r)t

= 1,300,000 / (1.0820)3.33

= $999,923.04 ≈ $1,000,000 ☺

(Difference due to rounding)

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Lecture Problem 11-a

Three years ago, Pablo invested $2,000. In 2 years, he expects to have $2,850. If Pablo expects to earn the same annual rate of return after 2 years from today as the annual rate implied from the past and expected values given in the problem, then how much does he expect to have in 5 years from today?

To solve:

1) Find the implied return over the 5 year period from 3 years ago to 2 years from today

2) Use the implied return to determine how much he’ll have 5 years from today

1) Find the implied return over the 5 year period from 4 years ago to 1 year from today

Mode is not relevant, since PMT = 0

Enter 5 -2,000 0 2,850

N I% PV PMT FV

Solve for 7.34

2) Use the implied return to determine how much he’ll have 5 years from today

Mode is not relevant, since PMT = 0

Enter 3 7.34 -2,850 0

N I% PV PMT FV

Solve for 3,524.76

Pablo would have $3,524.76 in 5 years from today, which is 3 years from 2 years from today

(Solutions may differ somewhat due to rounding annual rate of return)

Alternatively

Mode is not relevant, since PMT = 0

Enter 8 7.34 -2,000 0

N I% PV PMT FV

Solve for 3,524.70

Pablo would have $3,524.70 in 5 years from today, which is 8 years from 3 years ago

(Solutions may differ somewhat due to rounding annual rate of return)

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Lecture Problem 11-b

Three years ago, Pablo invested $2,000. In 2 years, he expects to have $2,850. If Pablo expects to earn the same annual rate of return after 2 years from today as the annual rate implied from the past and expected values given in the problem, then in how many years from today does he expect to have exactly $4,000?

To solve:

1) Find the implied return over the 5 year period from 3 years ago to 2 years from today

2) Use the implied return to determine when goal will be reached relative to one of the given values and then relative to today

1) Find the implied return over the 5 year period from 4 years ago to 1 year from today

Mode is not relevant, since PMT = 0

Enter 5 -2,000 0 2,850

N I% PV PMT FV

Solve for 7.34

2) Use the implied return to determine when goal will be reached relative to one of the given values and then relative to today

Mode is not relevant, since PMT = 0

Enter 7.34 -2,850 0 4,000

N I% PV PMT FV

Solve for 4.79

Pablo would have $4,000 in 4.79 years from 2 years from today

Therefore, Pablo would have $4,000 in 6.79 years from today

(Solutions may differ somewhat due to rounding annual rate of return)

Alternatively

Mode is not relevant, since PMT = 0

Enter 7.34 -2,000 0 4,000

N I% PV PMT FV

Solve for 9.79

Pablo would have $4,000 in 9.79 years from 3 years ago

Therefore, Pablo would have $4,000 in 6.79 years from today

(Solutions may differ somewhat due to rounding annual rate of return)

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