The venture capital fund Techno Fund II made a $4 million investment in Optical
Fibers Corporation five years ago and in return received 1 million shares representing 20 percent of Optical Fibers’ equity. Optical Fibers is now planning an initial public offering in which it will sell 1 million newly created shares for $50 per share. Techno has chosen to exercise its demand registration rights and will sell its shares—alongside the newly created shares—in Optical Fibers’ IPO. The investment banks underwriting Optical Fibers’ IPO will charge a 7 percent underwriting spread, so both the firm and Techno Fund II will receive 93 percent of the $50 per-share offer price. Assuming the IPO is successful; calculate the compound annual return that Techno will have earned on its investment?
To calculate the compound annual return (CAGR) that Techno Fund II will have earned on its investment, we need to consider the initial investment amount, the final investment amount, and the holding period.
First, let's calculate the final investment amount by multiplying the number of shares Techno owns (1 million shares) by the offer price per share ($50). This gives us 1,000,000 * $50 = $50,000,000.
Next, let's calculate the initial investment amount. Techno made a $4 million investment in Optical Fibers five years ago, which means we need to account for the time value of money. We can use the compound interest formula to calculate the present value of the initial investment.
PV = FV / (1 + r)^n
Where:
PV = Present value
FV = Future value (initial investment of $4 million)
r = Compound annual growth rate
n = Number of years (5 years)
Let's rearrange the formula to solve for r:
r = (FV/PV)^(1/n) - 1
r = ($4,000,000 / PV)^(1/5) - 1
To solve for PV, we need to calculate the present value that would grow to $4 million in 5 years. We can use the future value formula:
FV = PV * (1 + r)^n
Rearranging the formula to solve for PV, we have:
PV = FV / (1 + r)^n
Since we know the future value (FV) is $4 million and the number of years (n) is 5, we can substitute these values into the equation:
$4,000,000 = PV * (1 + r)^5
Now, let's solve for PV:
PV = $4,000,000 / (1 + r)^5
Now we need to substitute the expression for PV back into the equation to calculate r:
r = ($4,000,000 / ($4,000,000 / (1 + r)^5))^(1/5) - 1
Simplifying this equation, we get:
r = (1 + r)^-5 - 1
Solving this equation, we find that r ≈ 0.1583, or 15.83%.
So, the compound annual return that Techno Fund II will have earned on its investment is approximately 15.83%.