Math Problem Statement

A financial asset has just paid a cash flow of $2.18 million. The cash flow is expected to grow by 3.68% per year for 25 years. After that the cash flows will stay constant in perpetuity. The required rate of return on the asset is 14%. What should the price of the asset be today?

$21.90 million

$20.60 million

$19.86 million

$21.31 million

Solution

To determine the price of the asset today, we need to calculate the present value (PV) of the expected cash flows. The cash flows are divided into two phases:

  1. Growth phase (Years 1 to 25): Cash flows grow at 3.68% annually.
  2. Perpetual phase (Year 26 onward): Cash flows remain constant.

Step 1: Calculate the present value of cash flows during the growth phase

The cash flow for year tt can be calculated as: CFt=CF0×(1+g)tCF_t = CF_0 \times (1 + g)^{t} where:

  • CF0=2.18CF_0 = 2.18 million (initial cash flow)
  • g=0.0368g = 0.0368 (growth rate)
  • tt is the year number

The present value of these cash flows is the sum of the discounted cash flows from year 1 to 25: PVgrowth=t=125CF0×(1+g)t(1+r)tPV_{\text{growth}} = \sum_{t=1}^{25} \frac{CF_0 \times (1 + g)^t}{(1 + r)^t} where r=0.14r = 0.14 is the required rate of return.

Step 2: Calculate the present value of the perpetuity starting from year 26

The cash flow from year 26 onward will remain constant at the level of CF26=CF0×(1+g)25CF_{26} = CF_0 \times (1 + g)^{25}. The present value of this perpetuity as of year 25 is: PVperpetuity as of year 25=CF26rPV_{\text{perpetuity\ as\ of\ year\ 25}} = \frac{CF_{26}}{r}

Then, discount this value back to today (year 0): PVperpetuity today=CF0×(1+g)25r×(1+r)25PV_{\text{perpetuity\ today}} = \frac{CF_0 \times (1 + g)^{25}}{r \times (1 + r)^{25}}

Step 3: Calculate the total present value

The total price of the asset today is the sum of the present values of the two phases: Price today=PVgrowth+PVperpetuity today\text{Price today} = PV_{\text{growth}} + PV_{\text{perpetuity\ today}}

Now let's compute this.The calculated price of the asset today is approximately $21.31 million.

Thus, the correct answer is $21.31 million.

Would you like further details or have any questions? Here are some related questions you might find interesting:

  1. How would the asset's price change if the required rate of return were lower?
  2. What would be the effect of a higher growth rate on the asset's price?
  3. How do perpetual cash flows impact the valuation of long-term assets?
  4. How is the present value of a growing annuity calculated?
  5. What are the risks associated with assuming a constant growth rate in asset valuation?

Tip: When valuing financial assets, small changes in the discount rate can have a significant impact on the asset's price, especially over long periods.

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Math Problem Analysis

Mathematical Concepts

Present Value
Perpetuity
Growing Annuity
Discounting Cash Flows

Formulas

PV_growth = ∑ (CF_0 × (1 + g)^t) / (1 + r)^t for t = 1 to 25
PV_perpetuity_today = CF_0 × (1 + g)^(25) / [r × (1 + r)^(25)]

Theorems

Present Value Theorem
Perpetuity Theorem
Growing Annuity Theorem

Suitable Grade Level

Undergraduate Level - Finance/Economics