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:
- Growth phase (Years 1 to 25): Cash flows grow at 3.68% annually.
- 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 can be calculated as: where:
- million (initial cash flow)
- (growth rate)
- is the year number
The present value of these cash flows is the sum of the discounted cash flows from year 1 to 25: where 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 . The present value of this perpetuity as of year 25 is:
Then, discount this value back to today (year 0):
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:
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:
- How would the asset's price change if the required rate of return were lower?
- What would be the effect of a higher growth rate on the asset's price?
- How do perpetual cash flows impact the valuation of long-term assets?
- How is the present value of a growing annuity calculated?
- 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
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