An entrepreneur is looking for financing.

They are sure that their app will reach commercialization in 3.5 years. They estimate the app will be worth $500,000 by that time (Year 2.5).

Arbutus Capital offers financing to the entrepreneur. Today, they are willing to pay $100,000 for 50% of the company. That will give them a payment of 50%*$500,000 = $250,000 at Year 3.5.

What is the closest value to the EAR demanded by Arbutus as compensation for financing the entrepreneur?

Question

An entrepreneur is looking for financing.

They are sure that their app will reach commercialization in 3.5 years. They estimate the app will be worth $500,000 by that time (Year 2.5).

Arbutus Capital offers financing to the entrepreneur. Today, they are willing to pay $100,000 for 50% of the company. That will give them a payment of 50%*$500,000 = $250,000 at Year 3.5.

What is the closest value to the EAR demanded by Arbutus as compensation for financing the entrepreneur?

BlackTom AI · Accepted Answer

We start by identifying the cash flows and the investment horizon. The entrepreneur receives $100,000 today in exchange for 50% of the company. In 3.5 years, Arbutus would own 50% of a company valued at $500,000, which is $250,000 in payoff at Year 3.5.

Option 1: 25.74%. To check this, compute the effective annual rate (EAR) using the formula EAR = (Ending value / Beginning value)^(1/n) - 1. Here, Ending/Beginning = 250,000 / 100,000 = 2.5, and n = 3.5 years. So EAR = 2.5^(1/3.5) - 1. The natural log approach gives ln(2.5) ≈ 0.9163; dividing by 3.5 yields ≈ 0.2618; exponentiating gives ≈ 1.2997, and subtracting 1 yields ≈ 0.2997 or 29.97%. This is higher than 25.74%, so option 1 is not correct.

Option 2: 35.72%. Using the same calculation, EAR ≈ 29.97%, which is noticeably lower than 35.72%. Therefore option 2 overstates the required return and is incorrect.

Option 3: 29.93%. This matches the computed EAR from the cash-flow setup: EAR ≈ 2.5^(1/3.5) - 1 ≈ 0.2997, i.e., about 29.97%. Rounding to the nearest hundredth or thousandth aligns with 29.93% as the closest value among the choices.

In summary, only the third option aligns with the mathematics of converting a 2.5x total return over 3.5 years into an annualized rate, while the first two options either understate or overstate the required compensation when annualized. The core idea is to treat the investment as an initial outlay of $100,000 growing to $250,000 over 3.5 years and derive the EAR from that growth factor.

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