5 Chapter 5 – Time value of Money valuing cash flow streams

Scenario

A company is considering an investment that will generate the following cash inflows:

•Year 1: $5,000

•Year 2: $7,000

•Year 3: $6,000

The company requires a return of 8% (discount rate). Calculate the Present Value (PV) of the cash flow stream to determine whether the investment is worth pursuing.

Solution:

1. Apply the PV formula:

$$PV = \frac{5,000}{(1 + 0.08)^1} + \frac{7,000}{(1 + 0.08)^2} + \frac{6,000}{(1 + 0.08)^3}$$

2. Discount each cash flow:

• • Year 1:
    • $$PV_1 = \frac{5,000}{1.08} \approx 4,629.63$$
  • Year 2:
    • $$PV_2 = \frac{7,000}{1.1664} \approx 6,002.40$$
  • Year 3:
    • $$PV_3 = \frac{6,000}{1.2597} \approx 4,761.90$$

3. Sum the discounted values:

PV = 4,629.63 + 6,002.40 + 4,761.90 = 15,393.93

Result:

The Present Value of the cash flow stream is $15,393.93.

Future Value of the Same Cash Flow Stream

Let’s calculate the Future Value (FV) of these cash flows at the end of Year 3:

1. Apply the FV formula:

$$FV = 5,000 \times (1 + 0.08)^2 + 7,000 \times (1 + 0.08)^1 + 6,000$$

2. Compound each cash flow:

• • Year 1 (compounded for 2 years):
    • $$FV_1 = 5,000 \times 1.1664 = 5,832.00$$
  • Year 2 (compounded for 1 year):
    • $$FV_2 = 7,000 \times 1.08 = 7,560.00$$
  • Year 3 (no compounding):
    • $$FV_3 = 6,000$$

3. Sum the Future Values:

$$FV = 5,832.00 + 7,560.00 + 6,000 = 19,392.00$$

Result:

The Future Value of the cash flow stream is $19,392.00.

This is the value of the cash flows if the company waits until the end of Year 3 to collect all payments.

Scenario:

You are considering purchasing a business that will generate the following cash flows over the next 4 years:

• Year 1: $10,000
• Year 2: $12,000
• Year 3: $15,000
• Year 4: $18,000

The required discount rate is 6%. Calculate the Present Value (PV) of this cash flow stream.

Solution:

1.Apply the PV formula:

$$PV = \frac{10,000}{(1 + 0.06)^1} + \frac{12,000}{(1 + 0.06)^2} + \frac{15,000}{(1 + 0.06)^3} + \frac{18,000}{(1 + 0.06)^4}$$

2.Compute the discounted cash flows:

• Year 1:
  • $$PV_1 = \frac{10,000}{1.06} \approx 9,433.96$$
• Year 2:
  • $$PV_2 = \frac{12,000}{1.1236} \approx 10,676.53$$
• Year 3:
  • $$PV_3 = \frac{15,000}{1.1910} \approx 12,594.70$$
• Year 4:
  • $$PV_4 = \frac{18,000}{1.2625} \approx 14,271.24$$

3.Sum the discounted values:

$$PV = 9,433.96 + 10,676.53 + 12,594.70 + 14,271.24 = 46,976.43$$

Result:

The Present Value of the business is $46,976.43.

Scenario:

A university endowment fund generates $500,000 annually to support scholarships. If the discount rate is 5%, calculate the Present Value of the endowment fund.

Solution:

1.Identify the variables:

• • C = 500,000
  • r = 0.05

2.Apply the formula:

$$PV = \frac{C}{r} = \frac{500,000}{0.05} = 10,000,000$$

Result:

The Present Value of the endowment fund is $10,000,000. This means the university would need an initial investment of $10,000,000 at a 5% return to provide perpetual annual payments of $500,000.

Scenario:

A company issues preferred stock paying an annual dividend of $5 per share, which grows at 2% per year. If investors require a 6% return, calculate the Present Value of the stock.

Solution:

1.Identify the variables:

• • C = 5
  • r = 0.06
  • g = 0.02

2.Apply the formula:

$$PV = \frac{C}{r – g} = \frac{5}{0.06 – 0.02} = \frac{5}{0.04} = 125$$

Result:

The Present Value of the preferred stock is $125 per share.

Scenario:

A trust fund is set up to pay $20,000 annually to a beneficiary forever. The discount rate is 4%.

1. 1. Calculate the Present Value of the trust fund.
   2. What happens if the trust fund’s annual payment grows by 1% each year?

Solution:

1. 1. For constant payments:

• • C = 20,000, r = 0.04

$$PV = \frac{C}{r} = \frac{20,000}{0.04} = 500,000$$

1. 2. For growing payments:

• C = 20,000, r = 0.04, g = 0.01

$$PV = \frac{C}{r – g} = \frac{20,000}{0.04 – 0.01} = \frac{20,000}{0.03} = 666,667$$

Results:

The Present Value with constant payments is $500,000.

The Present Value with 1% growth is $666,667.

Background:

Harvard University, one of the world’s most prestigious institutions, relies heavily on its endowment fund to finance a significant portion of its operations. The endowment fund is structured to provide an indefinite income stream to support scholarships, faculty salaries, and campus infrastructure. This setup models a perpetuity, where the university receives annual payments that are expected to last forever.

Objective:

To better understand how the concept of perpetuities applies to real-world financial scenarios, students will analyze the present value of Harvard’s endowment fund using the perpetuity formula.

Scenario:

Harvard’s endowment fund provides an annual cash flow of $400 million. The discount rate used by financial analysts for such investments is 5%.

Task:

• • Calculate the present value (PV) of Harvard’s endowment fund using the perpetuity formula:

$$PV = \frac{C}{r}$$

Where:

• • • C = $400 million (annual cash flow)
    • r = 0.05 (discount rate)

Questions for Analysis:

• • What is the present value of Harvard’s endowment fund given the $400 million annual cash flow and a 5% discount rate?
  • How would the present value change if the discount rate were to increase to 6%?
  • Discuss how changes in the cash flow amount or discount rate could affect the sustainability of the university’s finances.

Reference:

For more information on university endowment structures and their importance, visit Harvard’s Financial Overview.

Discussion Points:

• • How do perpetuities help universities plan for long-term financial stability?
  • What are the risks associated with relying on a perpetuity model for income?
  • Explore other real-world applications of perpetuities, such as preferred stock and ground rent in real estate.

Scenario

You plan to retire in 10 years and want to receive $10,000 per year for 15 years after retirement. The interest rate is 6% per year. How much money do you need to invest today to fund your annuity payments?

Step-by-Step Solution

1. Set Up the Formula:

Use the PV formula for an ordinary annuity:

$$PV_{\text{annuity}} = P \times \frac{1 – (1 + r)^{-n}}{r}$$

2. Substitute the Values:

• • P = 10,000
  • r = 0.06
  • n = 15

$$PV_{\text{annuity}} = 10,000 \times \frac{1 – (1 + 0.06)^{-15}}{0.06}$$

3. Calculate Step-by-Step:

• • $$(1 + 0.06)^{-15} = 0.4173$$
  • $$1 – 0.4173 = 0.5827$$
  • $$0.5827 / 0.06 = 9.7117$$
  • $$PV_{\text{annuity}} = 10,000 \times 9.7117 = 97,117$$

Result

You need to invest $97,117 today to fund 15 years of $10,000 annual payments at 6% interest.

Scenario

You save $5,000 annually for 10 years in an account earning 5% interest per year. What will be the Future Value of your savings?

Step-by-Step Solution

1. Set Up the Formula:

Use the FV formula for an ordinary annuity:

$$FV_{\text{annuity}} = P \times \frac{(1 + r)^n – 1}{r}$$

2. Substitute the Values:

• • P = 5,000
  • r = 0.05
  • n = 10

$$FV_{\text{annuity}} = 5,000 \times \frac{(1 + 0.05)^{10} – 1}{0.05}$$

3. Calculate Step-by-Step:

• • [latex](1 + 0.05)^{10} = 1.6289[/latex]
  • [latex]1.6289 - 1 = 0.6289[/latex]
  • [latex]0.6289 / 0.05 = 12.578[/latex]

$$FV_{\text{annuity}} = 5,000 \times 12.578 = 62,890$$

Result

Your savings will grow to $62,890 after 10 years.

Background:

Maria, a 45-year-old teacher, is planning for her retirement. She wants to retire at 65 and receive a steady income for 20 years after retirement. To achieve this, Maria is considering investing in an ordinary annuity that will start making annual payments at the end of each year once she retires.

Maria’s financial advisor has proposed an annuity plan where she will make equal annual contributions for the next 20 years, with an interest rate of 5% per year. The annuity will pay her $40,000 per year during retirement.

Objective:

Maria needs to evaluate the present value of the annuity to understand its current worth and determine if her contributions will be sufficient to meet her retirement needs.

Task:

• • Calculate the total present value of the ordinary annuity payments she will receive during retirement using the following formula:

$$PV_{\text{annuity}} = P \times \frac{1 – (1 + r)^{-n}}{r}$$

Where:

• • • P = $40,000 (payment per period)
    • r = 0.05 (annual interest rate)
    • n = 20 (number of payment periods)
  • Discuss how Maria’s financial situation might change if the interest rate fluctuates over time or if she decides to increase her annual contributions.

Questions for Analysis:

• • What is the present value of Maria’s future annuity payments at the current 5% interest rate?
  • If the interest rate were to decrease to 4%, how would that affect the present value of her annuity?
  • How do ordinary annuities compare to annuities due in terms of payment structures and present value?

Reference:

To learn more about retirement annuities and their benefits, read Investopedia’s Guide on Retirement Annuities.

Discussion Points:

• • Why might Maria consider an annuity due instead of an ordinary annuity for her retirement?
  • How would the present value change if she opted for a variable annuity that depended on market performance?
  • How do fixed and variable annuities align with different risk tolerances?

Scenario:

You borrow $50,000 for a car loan at 6% annual interest, to be repaid over 5 years in equal annual payments. Calculate the annual payment amount.

Hint: Use the formula for PV of an ordinary annuity:

$$PV = P \times \frac{1 – (1 + r)^{-n}}{r}$$

Steps:

1. 1. PV = 50,000, r = 0.06, n = 5.
   2. Rearrange the formula to solve for P:

$$P = \frac{PV \cdot r}{1 – (1 + r)^{-n}}$$

Solution:

1. 1. Calculate the denominator:

$$](1 + 0.06)^{-5} = 0.7473$$

1 – 0.7473 = 0.2527

$$0.2527 / 0.06 = 4.2117$$

1. 2. Solve for P:

$$P = \frac{50,000}{4.2117} \approx 11,877.47$$

Answer:

Your annual payment is approximately $11,877.47.

Scenario:

A stock pays a dividend of $5 per year, which is expected to grow by 3% annually. Investors require a 10% return to hold the stock. Calculate the Present Value of the stock (its intrinsic value).

Step-by-Step Solution:

1.Identify the variables:

• • C = 5
  • r = 0.10
  • g = 0.03

2.Apply the formula:

$$PV_{\text{growing perpetuity}} = \frac{C}{r – g} = \frac{5}{0.10 – 0.03} = \frac{5}{0.07} = 71.43$$

Result:

The stock’s intrinsic value is $71.43.

Scenario:

A university endowment must provide $1,000,000 annually, growing at 2% per year to account for inflation. If the discount rate is 6%, calculate the initial fund required.

Step-by-Step Solution:

1.Identify the variables:

• • C = 1,000,000
  • r = 0.06
  • g = 0.02

2.Apply the formula:

$$PV_{\text{growing perpetuity}} = \frac{C}{r – g} = \frac{1,000,000}{0.06 – 0.02} = \frac{1,000,000}{0.04} = 25,000,000$$

Result:

The university needs $25,000,000 to fund the perpetuity.

Scenario:

You plan to contribute $5,000 in the first year to a retirement account. Contributions will grow by 3% annually for 15 years, and the account earns an annual return of 7%. Calculate the Present Value of your contributions.

Step-by-Step Solution:

1. Identify the variables:

• • P = 5,000
  • r = 0.07
  • g = 0.03
  • n = 15

2. Apply the formula:

$$PV_{\text{growing annuity}} = 5,000 \times \frac{1 – \left(\frac{1 + 0.03}{1 + 0.07}\right)^{15}}{0.07 – 0.03}$$

3. Simplify step-by-step:

• • $$\frac{1 + g}{1 + r} = \frac{1.03}{1.07} = 0.9626$$
  • Raise to the power of n: $$0.9626^{15} = 0.7417$$
  • Subtract from 1: $$1 – 0.7417 = 0.2583$$
  • Divide by r – g: $$0.2583 / 0.04 = 6.4575$$
  • Multiply by P: $$5,000 \times 6.4575 = 32,287.50$$

Result:

The Present Value of your contributions is approximately $32,287.50.

Scenario:

A property generates $10,000 in rent during the first year, growing at 2% annually. The lease lasts for 10 years, and the discount rate is 5% per year. Calculate the Present Value of the rental income.

Step-by-Step Solution:

1.Identify the variables:

• • P = 10,000
  • r = 0.05
  • g = 0.02
  • n = 10

2.Apply the formula:

$$PV_{\text{growing annuity}} = 10,000 \times \frac{1 – \left(\frac{1 + 0.02}{1 + 0.05}\right)^{10}}{0.05 – 0.02}$$

3.Simplify step-by-step:

• • $$\frac{1 + g}{1 + r} = \frac{1.02}{1.05} = 0.9714$$
  • Raise to the power of n: $$0.9714^{10} = 0.7812$$
  • Subtract from 1: $$1 – 0.7812 = 0.2188$$
  • Divide by r – g: $$0.2188 / 0.03 = 7.2933$$
  • Multiply by P: $$10,000 \times 7.2933 = 72,933$$

Result:

The Present Value of the rental income is $72,933.

Scenario:

You are considering investing in a project with the following cash flows:

• • Year 1: $2,000
  • Growing by 5% annually for the next 8 years.

The required rate of return is 8%. Calculate the Present Value of the project’s cash flows.

Hint: Use the growing annuity formula:

$$PV_{\text{growing annuity}} = P \times \frac{1 – \left(\frac{1 + g}{1 + r}\right)^n}{r – g}$$

Scenario:

You want to save $100,000 in 10 years by making equal annual contributions to an account earning 5% interest per year. How much do you need to contribute each year?

Step-by-Step Solution:

1.Identify the variables:

• • FV = 100,000
  • r = 0.05
  • n = 10

2.Use the FV formula for an ordinary annuity:

$$\large P = \frac{FV}{\frac{(1 + r)^n – 1}{r}}$$

3.Substitute the values:

$$\large P = \frac{100,000}{\frac{(1 + 0.05)^{10} – 1}{0.05}}$$

4.Simplify step-by-step:

$$(1 + 0.05)^{10} = 1.6289$$

$$1.6289 – 1 = 0.6289$$

$$0.6289 / 0.05 = 12.578$$

$$P = 100,000 / 12.578 = 7,952.29$$

Result:

You need to save $7,952.29 annually to reach your goal.

Scenario:

You invest $10,000 today, and it grows to $20,000 in 8 years. What is the annual rate of return?

Step-by-Step Solution:

1.Identify the variables:

• • PV = 10,000
  • FV = 20,000
  • n = 8

2.Use the formula:

$$
\Large r = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} – 1
$$

3.Substitute the values:

$$\Large r = \left(\frac{20,000}{10,000}\right)^{\frac{1}{8}} – 1 = (2)^{\frac{1}{8}} – 1$$

4.Calculate:

$$\Large 2^{\frac{1}{8}} = 1.0905$$

1.0905 – 1 = 0.0905

Result:

The annual rate of return is 9.05%.

Scenario:

You invest $5,000 at 6% annual interest, and you want it to grow to $10,000. How many years will it take?

Step-by-Step Solution:

1.Identify the variables:

• • PV = 5,000
  • FV = 10,000
  • r = 0.06

2.Use the formula:

$$n = \frac{\ln\left(\frac{FV}{PV}\right)}{\ln(1 + r)}$$

3.Substitute the values:

$$n = \frac{\ln\left(\frac{10,000}{5,000}\right)}{\ln(1 + 0.06)}$$

4.Calculate step-by-step:

• • $$\ln(10,000 / 5,000) = \ln(2) = 0.6931$$
  • $$\ln(1 + 0.06) = \ln(1.06) = 0.0583$$
  • $$n = 0.6931 / 0.0583 = 11.89$$

Result:

It will take approximately 11.89 years for your investment to double.

1. Solving for Cash Flow

You want to accumulate $50,000 in 5 years in an account earning 4% interest. How much must you deposit annually?

2. Solving for Rate of Return

You invest $25,000 today and want it to grow to $50,000 in 10 years. What annual return is required?

3. Solving for Number of Periods

You deposit $2,000 at 5% annual interest. How long will it take to grow to $4,000?

Scenario

A real estate developer is assessing a potential project that requires an initial investment of $10,000. The project is expected to generate annual cash inflows of $3,000 for the next 5 years. The developer uses a discount rate of 8%, reflecting the opportunity cost of capital and the risk of the investment. The objective is to calculate the Net Present Value (NPV) to determine if the project is profitable and worth pursuing.

Key Information

• Initial investment (C_0): $10,000
• Annual cash inflows (CF_t): $3,000 for 5 years
• Discount rate (r): 8% or 0.08
• Time period (t): 5 years

Step-by-Step Calculation of NPV

To calculate NPV, use the formula:

$$NPV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t} – C_0$$

Where:

• CF_t = Cash flow in year t
• r = Discount rate
• t = Time period (1 to 5 years)
• C_0 = Initial investment

1. Discount Each Annual Cash Flow to Present Value (PV):

$$PV_1 = \frac{3,000}{(1 + 0.08)^1} = \frac{3,000}{1.08} = 2,777.78$$

$$PV_2 = \frac{3,000}{(1 + 0.08)^2} = \frac{3,000}{1.1664} = 2,571.43$$

$$PV_3 = \frac{3,000}{(1 + 0.08)^3} = \frac{3,000}{1.2597} = 2,380.95$$

$$PV_4 = \frac{3,000}{(1 + 0.08)^4} = \frac{3,000}{1.3605} = 2,205.13$$

$$PV_5 = \frac{3,000}{(1 + 0.08)^5} = \frac{3,000}{1.4693} = 2,042.64$$

2. Sum the Present Values of All Cash Inflows:

$$\text{Total PV of Cash Inflows} = 2,777.78 + 2,571.43 + 2,380.95 + 2,205.13 + 2,042.64 = 11,977.93$$

3. Subtract the Initial Investment (C_0):

$$NPV = 11,977.93 – 10,000 = 1,977.93$$

Conclusion

The Net Present Value (NPV) of the project is $1,977.93. Since the NPV is positive, the project is expected to generate more value than it costs. This indicates that the project is profitable and should be considered a worthwhile investment.

Analysis and Discussion Points

1. Impact of a Higher Discount Rate:

• If the discount rate increases to 10%, how would the NPV change? (Answer: It decreases because the present value of future cash flows would shrink.)

2. Sensitivity to Cash Flow Estimates:

• What happens if the annual cash inflows are lower than expected, say $2,500 instead of $3,000?
• How does a delay in cash inflows (e.g., starting in Year 2) affect the NPV?

3. Other Factors to Consider:

• Non-financial factors like project feasibility, market conditions, and regulatory risks.
• Alternative investment opportunities with similar risks and returns.