Answer Key

Conceptual Questions

1. What is the difference between a perpetuity and an annuity? Provide an example of when each might be used in real life.

•Answer:

A perpetuity is a series of cash flows that continues indefinitely, whereas an annuity is a series of cash flows over a finite period.

•Example of a perpetuity: A university endowment paying annual scholarships forever.

•Example of an annuity: A car loan with equal monthly payments for 5 years.

2. Explain why the discount rate must be greater than the growth rate for valuing growing perpetuities and growing annuities.

•Answer:

If the growth rate equals or exceeds the discount rate, the formula for growing perpetuities or growing annuities does not converge to a finite value. This is because the cash flows grow faster than they are discounted, leading to an infinite or undefined present value.

3. Why do ordinary annuities have a lower present value than annuities due, assuming the same cash flows and discount rates?

•Answer:

In an ordinary annuity, payments occur at the end of each period, meaning they are discounted more heavily. In an annuity due, payments occur at the beginning of each period, so they are discounted less, resulting in a higher present value.

4. Describe how growing cash flows account for inflation in financial planning and investment valuation.

•Answer:

Growing cash flows account for inflation by incorporating a growth rate in the cash flow projections. This ensures that the future payments maintain their purchasing power relative to rising prices, making the valuations more realistic.

Short Calculations

1. Present Value of a Cash Flow Stream

Solution:

[latex]PV = \frac{CF_1}{(1 + r)^1} + \frac{CF_2}{(1 + r)^2} + \frac{CF_3}{(1 + r)^3}[/latex]

•Year 1: [latex]\frac{3,000}{(1 + 0.07)^1} = \frac{3,000}{1.07} \approx 2,803.74[/latex]

•Year 2: [latex]\frac{4,000}{(1 + 0.07)^2} = \frac{4,000}{1.1449} \approx 3,492.05[/latex]

•Year 3: [latex]\frac{5,000}{(1 + 0.07)^3} = \frac{5,000}{1.225} \approx 4,081.63[/latex]

•Total PV: [latex]2,803.74 + 3,492.05 + 4,081.63 = 10,377.42[/latex]

Answer: The present value is approximately $10,377.42.

2. Future Value of an Ordinary Annuity

Solution:

[latex]FV = P \times \frac{(1 + r)^n - 1}{r}[/latex]

•[latex]P = 1,500, r = 0.06, n = 10[/latex]

•[latex]FV = 1,500 \times \frac{(1 + 0.06)^{10} - 1}{0.06} = 1,500 \times \frac{1.7908 - 1}{0.06}[/latex]

•[latex]FV = 1,500 \times 13.18 = 19,770[/latex]

Answer: The future value is $19,770.

3. Valuing a Perpetuity

Solution:

[latex]PV = \frac{C}{r}[/latex]

•[latex]C = 20,000, r = 0.04[/latex]

•[latex]PV = \frac{20,000}{0.04} = 500,000[/latex]

Answer: The present value is $500,000.

4. Valuing a Growing Perpetuity

Solution:

[latex]PV = \frac{C}{r - g}[/latex]

•[latex]C = 10,000, r = 0.06, g = 0.02[/latex]

•[latex]PV = \frac{10,000}{0.06 - 0.02} = \frac{10,000}{0.04} = 250,000[/latex]

Answer: The value of the growing perpetuity is $250,000.

Scenario-Based Problems

1. Retirement Planning with Annuities

a. How much money is needed at retirement?

Solution:

[latex]PV = P \times \frac{1 - (1 + r)^{-n}}{r}[/latex]

•[latex]P = 50,000, r = 0.05, n = 25[/latex]

•[latex]PV = 50,000 \times \frac{1 - (1 + 0.05)^{-25}}{0.05} = 50,000 \times 15.3725 = 768,625[/latex]

Answer: $768,625 is needed at retirement.

b. How much to save annually for 20 years?

Solution:

[latex]P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}[/latex]

•[latex]FV = 768,625, r = 0.05, n = 20[/latex]

•[latex]P = \frac{768,625}{\frac{(1 + 0.05)^{20} - 1}{0.05}} = \frac{768,625}{33.065} \approx 23,247[/latex]

Answer: Save $23,247 annually for 20 years.

2. Growing Annuity for a Lease Agreement

Solution:

[latex]PV_{\text{growing annuity}} = P \times \frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}[/latex]

•[latex]P = 12,000, r = 0.06, g = 0.03, n = 8[/latex]

•[latex]PV = 12,000 \times \frac{1 - \left(\frac{1.03}{1.06}\right)^8}{0.06 - 0.03}[/latex]

•[latex]\frac{1.03}{1.06} = 0.9717, 0.9717^8 = 0.7981[/latex]

•[latex]1 - 0.7981 = 0.2019, 0.2019 / 0.03 = 6.73[/latex]

•[latex]PV = 12,000 \times 6.73 = 80,760[/latex]

Answer: The present value is $80,760.

Case Study: Stock Valuation Using a Growing Perpetuity

1. Intrinsic Value:

[latex]PV = \frac{C}{r - g} = \frac{3}{0.10 - 0.04} = \frac{3}{0.06} = 50[/latex]

Answer: $50 per share.

2. Impact of Changes:

•Higher growth rate: Increases the stock’s value.

•Lower required return: Also increases the stock’s value.

3. Overvalued or Undervalued:

•If the stock is trading at $60, it is overvalued compared to the intrinsic value of $50.

Interactive Challenge

1. Future Value of Option 1:

[latex]FV = PV \times (1 + r)^n[/latex]

• • [latex]50,000 \times (1 + 0.05)^{15} = 50,000 \times 2.0789 = 103,945[/latex]

Answer: $103,945.

2. Annual Savings for Option 2:

[latex]P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}[/latex]

• • [latex]P = \frac{100,000}{\frac{(1.05)^{15} - 1}{0.05}} = \frac{100,000}{20.789} \approx 4,811.41[/latex]

Answer: Save $4,811.41 annually.

3. Which Option:

Option 1 meets the goal without requiring consistent annual savings, but Option 2 allows flexibility and may suit someone unable to invest a lump sum upfront.