Part A

Given the following cash inflow at the end of each year, what is the future value of this cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year?

Year 1$15,000Year 2$20,000Year 3$30,000Years 4 through 6$0Year 7$150,000

Part B

County Ranch Insurance Company wants to offer a guaranteed annuity in units of $500, payable at the end of each year for 25 years. The company has a strong investment record and can consistently earn 7% on its investments after taxes. If the company wants to make 1% on this contract, what price should it set on it? Use 6% as the discount rate. Assume that it is an ordinary annuity and the price is the same as present value.

Part C

A local government is about to run a lottery but does not want to be involved in the payoff if a winner picks an annuity payoff. The government contracts with a trust to pay the lump-sum payout to the trust and have the trust (probably a local bank) pay the annual payments. The first winner of the lottery chooses the annuity and will receive $150,000 a year for the next 25 years. The local government will give the trust $2,000,000 to pay for this annuity. What investment rate must the trust earn to break even on this arrangement?

Part D

 Your dream of becoming rich has just come true. You have won the State of Tranquility’s Lottery. The State offers you two payment plans for the $5 million jackpot. You can take annual payments of $250,000 for the next 20 years or $2,867,480 today.

a. If your investment rate over the next 20 years is 8%, which payoff will you choose?
b. If your investment rate over the next 20 years is 5%, which payoff will you choose?
c. At what investment rate will the annuity stream of $250,000 be the same as the lump sum payment of $2,867,480?

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Rubric Name: Assignment 4 Rubric

CriteriaExemplarySatisfactoryUnsatisfactoryUnacceptableCriterion ScorePart A – Future Value25 points

Student accurately calculates the future value of the cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year.

20 points

Student mostly accurately calculates the future value of the cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year.

15 points

Student partially or incorrectly calculates the future value of the cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year.

0 points

Student did not calculate the future value of the cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year.

Score of Part A – Future Value,/ 25Part B – Guaranteed Annuity25 points

Student accurately calculates the price that should be set on the contract.

20 points

Student mostly accurately calculates the price that should be set on the contract.

15 points

Student partially or incorrectly calculates the price that should be set on the contract.

0 points

Student does not calculate the price that should be set on the contract.

Score of Part B – Guaranteed Annuity,/ 25Part C – Investment Rate25 points

Student accurately calculates the investment rate the trust must earn to break even.

20 points

Student mostly accurately calculates the investment rate the trust must earn to break even.

15 points

Student partially or incorrectly calculates the investment rate the trust must earn to break even.

0 points

Student did not calculate the investment rate the trust must earn to break even.

Score of Part C – Investment Rate,/ 25Part D – (a.) Lottery – Payoff at 8%10 points

Student accurately calculates the payoff at 8%.

7 points

Student mostly accurately calculates the payoff at 8%.

4 points

Student partially or incorrectly calculates the payoff at 8%.

0 points

Student does not calculate the payoff at 8%.

Score of Part D – (a.) Lottery – Payoff at 8%,/ 10Part D – (b.) Lottery – Payoff at 5%10 points

Student accurately calculates the payoff at 5%.

7 points

Student mostly accurately calculates the payoff at 5%.

4 points

Student partially or incorrectly calculates the payoff at 5%.

0 points

Student does not calculate the payoff at 5%.

Score of Part D – (b.) Lottery – Payoff at 5%,/ 10Part D – (c.) Lottery – Investment Rate5 points

Student accurately calculates the investment rate.

4 points

Student mostly accurately calculates the investment rate.

2 points

Student partially or incorrectly calculates the investment rate.

0 points

Student did not calculate the investment rate.

Score of Part D – (c.) Lottery – Investment Rate,/ 5

What This Guide Covers

This guide explains how to calculate future values, present values of annuities, required investment rates, and lottery payout comparisons using financial formulas. It demonstrates how time value of money concepts are applied in finance and investment decision making.

What the Assignment Is Actually Testing

This assignment evaluates your understanding of financial mathematics, time value of money, future value calculations, annuity valuation, and investment decision analysis. It tests your ability to apply financial formulas accurately and interpret the meaning of the results.

Section 1: Part A – Future Value of Uneven Cash Flows

The future value formula is:

FV=PV(1+r)nFV = PV(1+r)^nFV=PV(1+r)n

PVPVPV

r (%)r\,(\%)r(%)

nnn24681012141618205001000150020002500$2,653.30

The cash flows are:

Year 1 = $15,000
Year 2 = $20,000
Year 3 = $30,000
Years 4–6 = $0
Year 7 = $150,000

Each amount must be compounded to the end of Year 7.

Future Value at 6%

Year 1:

15000(1.06)6=21277.8515000(1.06)^6 = 21277.8515000(1.06)6=21277.85

Year 2:

20000(1.06)5=26764.5120000(1.06)^5 = 26764.5120000(1.06)5=26764.51

Year 3:

30000(1.06)4=37874.7930000(1.06)^4 = 37874.7930000(1.06)4=37874.79

Year 7 already occurs at the end of Year 7:

$150,000

Total Future Value at 6%:

FV=21277.85+26764.51+37874.79+150000=235917.15FV = 21277.85 + 26764.51 + 37874.79 + 150000 = 235917.15FV=21277.85+26764.51+37874.79+150000=235917.15

The future value at 6% is approximately $235,917.15.

Future Value at 9%

Year 1:

15000(1.09)6=25154.3215000(1.09)^6 = 25154.3215000(1.09)6=25154.32

Year 2:

20000(1.09)5=30772.5620000(1.09)^5 = 30772.5620000(1.09)5=30772.56

Year 3:

30000(1.09)4=42346.8330000(1.09)^4 = 42346.8330000(1.09)4=42346.83

Total Future Value at 9%:

FV=25154.32+30772.56+42346.83+150000=248273.71FV = 25154.32 + 30772.56 + 42346.83 + 150000 = 248273.71FV=25154.32+30772.56+42346.83+150000=248273.71

The future value at 9% is approximately $248,273.71.

Future Value at 15%

Year 1:

15000(1.15)6=34603.1615000(1.15)^6 = 34603.1615000(1.15)6=34603.16

Year 2:

20000(1.15)5=40227.6020000(1.15)^5 = 40227.6020000(1.15)5=40227.60

Year 3:

30000(1.15)4=52470.1930000(1.15)^4 = 52470.1930000(1.15)4=52470.19

Total Future Value at 15%:

FV=34603.16+40227.60+52470.19+150000=277300.95FV = 34603.16 + 40227.60 + 52470.19 + 150000 = 277300.95FV=34603.16+40227.60+52470.19+150000=277300.95

The future value at 15% is approximately $277,300.95.

Section 2: Part B – Guaranteed Annuity Price

The present value formula for an ordinary annuity is:

PV=PMT×1−(1+r)−nrPV = PMT \times \frac{1-(1+r)^{-n}}{r}PV=PMT×r1−(1+r)−n​

PMTPMTPMT

r (%)r\,(\%)r(%)

nnn246810121416182020004000600080001000012000$12,462.21

Where:

PMT = $500
r = 6% = 0.06
n = 25 years

Substituting values:

PV=500×1−(1.06)−250.06PV = 500 \times \frac{1-(1.06)^{-25}}{0.06}PV=500×0.061−(1.06)−25​

Calculation:

PV=500×12.7834=6391.70PV = 500 \times 12.7834 = 6391.70PV=500×12.7834=6391.70

The company should set the annuity price at approximately $6,391.70.

Section 3: Part C – Investment Rate Required to Break Even

The trust receives $2,000,000 today and must pay $150,000 annually for 25 years.

We solve for the interest rate using the annuity present value equation:

2000000=150000×1−(1+r)−25r2000000 = 150000 \times \frac{1-(1+r)^{-25}}{r}2000000=150000×r1−(1+r)−25​

Divide both sides by 150,000:

13.3333=1−(1+r)−25r13.3333 = \frac{1-(1+r)^{-25}}{r}13.3333=r1−(1+r)−25​

Using financial interpolation or a financial calculator, the investment rate is approximately:

r≈5.6%r \approx 5.6\%r≈5.6%

The trust must earn approximately 5.6% annually to break even.

Section 4: Part D – Lottery Payoff Decision

Part D(a): Choosing Between Annuity and Lump Sum at 8%

The annuity pays $250,000 annually for 20 years.

Using the present value formula:

PV=250000×1−(1.08)−200.08PV = 250000 \times \frac{1-(1.08)^{-20}}{0.08}PV=250000×0.081−(1.08)−20​

Calculation:

PV=250000×9.8181=2454525PV = 250000 \times 9.8181 = 2454525PV=250000×9.8181=2454525

Present value of annuity = $2,454,525

Lump sum offered = $2,867,480

Since the lump sum is larger, you should choose the $2,867,480 lump sum payment at 8%.

Section 5: Part D(b) – Choosing Between Annuity and Lump Sum at 5%

Using the present value formula:

PV=250000×1−(1.05)−200.05PV = 250000 \times \frac{1-(1.05)^{-20}}{0.05}PV=250000×0.051−(1.05)−20​

Calculation:

PV=250000×12.4622=3115550PV = 250000 \times 12.4622 = 3115550PV=250000×12.4622=3115550

Present value of annuity = $3,115,550

Since this exceeds the lump sum of $2,867,480, you should choose the annuity payment plan at 5%.

Section 6: Part D(c) – Investment Rate That Makes Both Options Equal

Set the present value equal to the lump sum:

2867480=250000×1−(1+r)−20r2867480 = 250000 \times \frac{1-(1+r)^{-20}}{r}2867480=250000×r1−(1+r)−20​

Divide both sides by 250,000:

11.4699=1−(1+r)−20r11.4699 = \frac{1-(1+r)^{-20}}{r}11.4699=r1−(1+r)−20​

Solving for the interest rate gives:

r≈6.3%r \approx 6.3\%r≈6.3%

The investment rate at which both options are equal is approximately 6.3%.

Section 7: Conclusion (How to Write It)

This assignment demonstrates how future value, present value, and annuity calculations support financial decision making. The results show that higher interest rates increase future values while lower discount rates increase the present value of annuities. The calculations also demonstrate how investment rates influence whether individuals should select lump sum payments or annuity streams. Understanding these financial principles is essential for evaluating investments, retirement plans, insurance contracts, and long term financial opportunities.

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