Here are the answers to the Unit 3 assignment based on the questions provided:

1. Future Value of $1,000 at 7% for 5 Years: To find how much you can withdraw after 5 years, you can use the Future Value formula: [ FV = PV \times (1 + r)^n ] where (PV) is the present value ($1,000), (r) is the interest rate (0.07), and (n) is the number of years (5). Thus, [ FV = 1000 \times (1 + 0.07)^5 \approx 1000 \times 1.40255 \approx 1402.55 ] After 5 years, you would be able to withdraw approximately $1,402.55.

2. Present Value of a Security Paying $2,500 in 4 Years: The present value can be calculated using: [ PV = \frac{FV}{(1 + r)^n} ] Here, (FV) is $2,500, (r) is 0.05, and (n) is 4. Therefore, [ PV = \frac{2500}{(1 + 0.05)^4} \approx \frac{2500}{1.21550625} \approx 2055.58 ] The present value of the security is approximately $2,055.58.

3. Solve for the Unknowns:

   • For (PV = $50,000), (FV = $152,184), (n = 12), [ r = \left(\frac{FV}{PV}\right)^{1/n} - 1 \approx \left(\frac{152184}{50000}\right)^{1/12} - 1 \approx 0.0789 \text{ or } 7.89% ]
   • For (PV = $21,400), (FV = $575,000), (n = 30), [ r \approx \left(\frac{575000}{21400}\right)^{1/30} - 1 \approx 0.0815 \text{ or } 8.15% ]
   • For (PV = $16,500), (FV = $238,830), (r = 14%), [ n = \frac{\log(FV/PV)}{\log(1 + r)} \approx \frac{\log(238830/16500)}{\log(1.14)} \approx 5.42 ]
   • For (PV = $21,400), (FV = $213,000), (r = 9%), [ n \approx \frac{\log(213000/21400)}{\log(1.09)} \approx 18.64 ]

4. Annual Rate of Interest for College Education: Using the future value formula, [ FV = PV \times (1 + r)^n ] We rearrange to find (r): [ 400,000 = 75,000 \times (1 + r)^{22} ] Solving yields (r \approx 0.1056) or 10.56%.

5. Time to Save for a Car Selling for $68,000: Using the future value formula, adjusting for the investment, [ 68,000 = 7,000 \times (1 + 0.06)^n ] This results in (n \approx 17.67) years until you have enough.

6. Time to Reach $50,000 from $10,000 with 11.3264% Interest: Solving, [ 50,000 = 10,000 \times (1 + 0.113264)^n ] leads to (n \approx 14.84) years from now.

7. Percentage Increase in Winner’s Check: The percentage increase is calculated as: [ \left(\frac{2,000,000 - 300}{300}\right) = 665.67% ] The annual rate is approximately 8.36%. Thus, in 2040, it would be around (2,000,000 \times (1 + r)^{20} \approx 6,811,060).

This summary should help you with your assignment, providing detailed calculations for each question.Here are the detailed answers to the questions in Unit 3 Assignment, Chapter 5, titled "Introduction to Valuation: The TVM".

1. Future Value Calculation: To determine how much you can withdraw after 5 years from a deposit of $1,000 at an interest rate of 7%, you can use the Future Value (FV) formula: [ FV = PV \times (1 + r)^n ] where:

   • ( PV = 1,000 )
   • ( r = 0.07 ) (7% expressed as a decimal)
   • ( n = 5 ) (years)

   After calculating, the future value will be approximately $1,403.59. For quick computations, you might use tools like Calculator.net's Future Value Calculator.

2. Present Value Calculation: To find the present value (PV) of a security that will pay $2,500 in 4 years at an opportunity cost of 5%, you can use the Present Value formula: [ PV = \frac{FV}{(1 + r)^n} ] where:

   • ( FV = 2,500 )
   • ( r = 0.05 ) (5% expressed as a decimal)
   • ( n = 4 )

   The present value comes out to approximately $2,066.07. More about present value calculations can be found at links like ClearTax's Present Value Calculator.

3. Solving for Unknowns:

   • For $50,000 over 12 years to yield $152,184: Use financial calculations to find the interest rate which will be approximately 5.43%.
   • For $21,400 over 30 years to yield $575,000: The interest rate is approximately 6.86%.
   • For $16,500 over 14% interest for $238,830: The number of years is found to be approximately 25 years.
   • For $21,400 over 9% for $213,000: This resolves to approximately 18 years.

4. Annual Rate Calculation for College Education: Given that the total cost of college education is anticipated to be $400,000 in 22 years and you currently have $75,000, the required annual compounding interest rate can be derived from the Future Value Formula leading to an approximate rate of 6.77%.

5. Time Until Car Purchase: To determine how long it will take to accumulate $68,000 from an initial investment of $7,000 at a 6% annual interest rate, applying the Future Value formula, it is found to take about 20 years.

6. Waiting Time for Graduation Funds: If you expect $10,000 at graduation in two years and then invest it at 11.3264% to reach $50,000, the time required from now to accumulate this amount would be approximately 13.24 years.

7. Percentage Increase in Prize Money: From $300 in 1985 to $2 million in 2020, the annual percentage increase can be calculated to be approximately 11.82%. If this rate remains constant, the check amount in 2040 would be estimated around $9.89 million.

Each of these calculations underscores the importance of time value of money concepts, essential in financial decision-making and investment strategies. Use calculators like Omni Calculator for further assistance in similar calculations.Here is a detailed summary of the answers for the Unit 3 Assignment based on your queries related to the Time Value of Money (TVM):

1. Future Value Calculation: To calculate how much you will have after 5 years if you deposit $1,000 at an interest rate of 7%, you can use the future value formula: [ FV = PV \times (1 + r)^n ] Plugging in the numbers: [ FV = 1000 \times (1 + 0.07)^5 ] This calculation will yield approximately $1,403.57 after 5 years.

2. Present Value Calculation: For a security that will pay $2,500 in 4 years with an opportunity cost of 5%, we can use the present value formula: [ PV = \frac{FV}{(1 + r)^n} ] Applying the values: [ PV = \frac{2500}{(1 + 0.05)^4} ] This results in approximately $2,050.66 as the present value.

3. Solving for Unknown Interest Rates: The general formula for finding the interest rate (i) when the present value (PV), future value (FV), and number of years (n) are known is derived from: [ FV = PV \times (1 + i)^n ] You will need to manipulate this equation to solve for i for each case provided. The calculations involve substituting the known values and solving the equation accordingly, often requiring numerical methods or financial calculators.

4. Future Value of College Education: To calculate the interest rate required to grow an investment of $75,000 to meet a future cost of $400,000 in 22 years, you can rearrange the future value formula: [ FV = PV \times (1 + r)^n ] Solving for r will yield the necessary annual rate of return needed.

5. Time Required to Save for a Car: For a car costing $68,000, if you have $7,000 now at a 6% annual interest rate, you would determine how many years (t) it takes to reach $68,000 using the future value formula. By iterating through the calculations of: [ 68000 = 7000 \times (1 + 0.06)^t ] You would find approximately 15 years until you have enough for the car purchase.

6. Investment Growth to a Future Value: If you expect to receive $10,000 in two years and wish to invest it at an interest rate of 11.3264% until it grows to $50,000, you again use the future value formula rearranged to find the time (t): [ FV = PV \times (1 + r)^t ] Solving for t would show how long you must wait after receiving the $10,000.

7. Percentage Increase and Future Value Calculation: The increase in the winner’s prize from $300 in 1985 to $2 million in 2020 represents a percentage increase that can be calculated with the formula for compound annual growth rate (CAGR): [ CAGR = \left( \frac{FV}{PV} \right)^{1/n} - 1 ] After calculating the CAGR, you can project forward to estimate the prize amount in 2040 by applying the same growth rate.

Feel free to utilize online calculators available at the links provided, such as those for Future Value, Present Value, and other financial tools for detailed calculations.Here's a detailed summary of the assignment questions from Unit 3, along with relevant calculations and sources for further reference based on the given tasks:

1. Future Value Calculation

If you deposit $1,000 today at an interest rate of 7% per year and keep it for 5 years, the amount you will withdraw can be calculated using the future value (FV) formula: [ FV = PV (1 + i)^n ] Where ( PV ) is the present value, ( i ) is the interest rate, and ( n ) is the number of years. Using the suggested calculators, you can find the FV:

• Future Value Calculator by Calculator.net
• Omni Calculator

2. Present Value Calculation

To find the present value (PV) of a security that pays $2,500 in 4 years at an opportunity cost of 5%, the formula is: [ PV = \frac{FV}{(1 + i)^n} ] You can use resources like:

• Present Value Calculator by Calculator.net
• Moneychimp Present Value Calculator

3. Solving for Unknown Interest Rates

Here are the tasks involving the unknown interest rate:

• Present value of $50,000 over 12 years to reach $152,184.
• Present value of $21,400 over 30 years to reach $575,000.
• Present value of $16,500 over unknown years at 14% to achieve $238,830.
• Present value of $21,400 over unknown years at 9% to achieve $213,000.

For such calculations, you can refer to guides and formula applications like:

• Atlas.org - Solve for the Unknown Interest Rate

4. Interest Rate for College Cost

To determine the interest rate required for a $75,000 investment to grow to cover $400,000 in 22 years, you can rearrange the FV formula and solve for ( i ). Consulting resources such as:

• Chegg Explanation on Interest Rates

5. Car Purchase Timeline

To purchase a car costing $68,000 with a today’s investment of $7,000 that earns 6% annually, you'd want to find out how many years it will take for the investment to reach that amount using the FV formula.

6. Investment Horizon for Future Value

To reach $50,000 from an expected $10,000 payment in 2 years with an investment return of 11.3264%, you have to calculate the investment duration for the appreciated amount.

7. Prize Money Percentage Increase

To calculate the annual percentage increase of the U.S. Open Golf Championship prize from $300 in 1985 to $2 million in 2020, and predict the future prize money for 2040, you can apply the growth rate formula: [ \text{Growth Rate} = \left(\frac{\text{Final Amount}}{\text{Initial Amount}}\right)^{\frac{1}{n}} - 1 ]

For best practices, consider using:

• Calculator for Unknown Interest Rate

Conclusion

This summary should provide a comprehensive look at each question's requirements. For accurate calculations, interactive tools mentioned above can be employed for all financial formulas required in the responses.Here is a detailed summary addressing the questions in Unit-3 Assignment regarding Time Value of Money (TVM) concepts:

1. Future Value of a $1,000 Deposit: If you deposit $1,000 today (t = 0) in a bank account with an annual interest rate of 7%, the future value (FV) after 5 years can be calculated using the formula FV = PV(1 + i)^n. Plugging in the values, FV = $1,000(1 + 0.07)^5, results in approximately $1,403.58 withdrawal after 5 years. You can find additional resources on future value calculations at Calculator.net.

2. Present Value of a Security: To find the present value (PV) of a security paying $2,500 in 4 years at an opportunity cost of 5%, use the formula PV = FV / (1 + i)^n. Here, PV = $2,500 / (1 + 0.05)^4, which calculates to approximately $2,030.40. For more insights, check Calculator.net.

3. Solving for Unknown Interest Rates: The following unknown interest rates can be calculated using the Future Value formula:

   • For $50,000 present value after 12 years yielding $152,184, the interest rate (i) is approximately 9.08%.
   • For $21,400 to grow to $575,000 over 30 years, the interest rate (i) is approximately 8.52%.
   • For a present value of $16,500 to reach $238,830 in 14 years, the interest rate (i) is about 14%.
   • Finally, for $21,400 to grow to $213,000 in an unknown number of years at 9% interest, calculations yield an interest of 9% consistently in both cases as known.

4. Future College Costs: Assuming the total cost of a college education will be $400,000 in 22 years and you have $75,000 to invest today, the annual interest rate required can be derived using the formula for future value. The calculations indicate that you would need an approximate annual rate of 11.15% to meet the goal. More details on investment growth can be accessed through NerdWallet.

5. Time to Save for a Car: To find out how long it will take to save $68,000 starting with $7,000 at a 6% annual interest rate, the equation can be rearranged to find n, resulting in approximately 20.41 years to reach that amount from your initial investment. For help with investment growth, use CalcXML's Compound Interest Calculator.

6. Receiving $10,000 in Two Years: If you expect to receive $10,000 upon graduation in two years and plan to invest that at an interest rate of 11.3264% to make it grow to $50,000, you will need to wait approximately 11.5 years from the time you graduate to meet your target.

7. U.S. Open Prize Money Growth: The prize money for the U.S. Open Golf Championship grew from $300 in 1985 to $2 million in 2020. The annual percentage increase over this period can be calculated, yielding approximately 11.91% per year. If this growth rate continues, the winner’s check would grow to about $14.4 million by the year 2040.

This summary covers all the questions from your Unit-3 assignment related to the time value of money and provides references for calculations and further details on the respective topics.