Question

A venture capitalist, willing to invest $\backslash (1,000,000$, has three investments to choose from. The first investment, a software company, has a $10\%$chance of returning $\backslash )5,000,000$profit, a $30\%$chance of returning $\backslash (1,000,000$profit, and a $60\%$chance of losing the million dollars. The second company, a hardware company, has a $20\%$chance of returning $\backslash )3,000,000$profit, a $40\%$chance of returning $\backslash (1,000,000$profit, and a $40\%$chance of losing the million dollars. The third company, a biotech firm, has a $10\%$chance of returning $\backslash )6,000,000$profit, a $70\%$of no profit or loss, and a $20\%$chance of losing the million dollars.

a. Construct a PDF for each investment.

b. Find the expected value for each investment.

c. Which is the safest investment? Why do you think so?

d. Which is the riskiest investment? Why do you think so?

e. Which investment has the highest expected return, on average?

Short answer

(a) The answer is given in the explanation.

(b) The expected values of investment are given,

A software company$=\$200,000$

A hardware company$=\$600,000$

A Biotech Firm$=\$400,000$.

(c) The third investment, a biotech firm is the safest.

(d) The first investment, a software company is the riskiest.

(e) The Hardware company has the highest expected return.

Step-by-step solution

Given information 

The total amount a venture capitalist is interested to invest$=\$1,000,000$

Part (a) Step 1: Explanation

A software company

$X$	$P\left(X\right)$
$5000000$	$0.10$
$1000000$	$0.30$
$-1000000$	$0.60$
Total	$1$

A hardware company

$X$	$P\left(X\right)$
$3000000$	$0.20$
$1000000$	$0.40$
$-1000000$	$0.40$
Total	$1$

A Biotech Firm

$X$	$P\left(X\right)$
$6000000$	$0.10$
$0$	$0.70$
$-1000000$	$0.20$
Total	$1$

Part (b) Step 1: Explanation

The total amount a venture capitalist is interested to invest$=\$1,000,000$

Part (b) Step 2: Explanation

The expected value for the software company:

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$5000000$	$0.10$	$500000$
$1000000$	$0.30$	$300000$
$-1000000$	$0.60$	$-600000$

$$\begin{gathered} E\left[x\right]=\sum _i^n{x}_{i}P\left(x_i\right) \\ E\left[x\right]=\left(\$5,000,000\times 0.1\right)+\left(\$1,000,000\times 0.3\right)-\left(\$1,000,000\times 0.6\right) \\ =\$200,000 \end{gathered}$$

A hardware company

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$3000000$	$0.20$	$600000$
$1000000$	$0.40$	$400000$
$-1000000$	$0.40$	$-400000$

$$\begin{gathered} E\left[x\right]=\sum _i^n{x}_{i}P\left(x_i\right) \\ E\left[x\right]=\left(\$3,000,000\times 0.2\right)+\left(\$1,000,000\times 0.4\right)-\left(\$1,000,000\times 0.4\right) \\ =\$600,000 \end{gathered}$$

A Biotech Firm

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$6000000$	$0.10$	$600000$
$0$	$0.70$	$0$
$-1000000$	$0.20$	$-200000$

$$\begin{gathered} E\left[x\right]=\sum _i^n{x}_{i}P\left(x_i\right) \\ E\left[x\right]=\left(\$6,000,000\times 0.1\right)+\left(0\times 0.7\right)-\left(\$1,000,000\times 0.2\right) \\ =\$400,000 \end{gathered}$$

Part (c) Step 1: Explanation

The total amount a venture capitalist is interested to invest$=\$1,000,000$

Part (c) Step 2: Explanation

From part (b) we can see the expected return value from the Biotech firm is $\$400,000$. Therefore the third investment, a biotech firm, is the safest of the three because the risk of loss is the lowest.

Part (d) Step 1: Explanation

The total amount a venture capitalist is interested to invest$=\$1,000,000$

Part (d) Step 2: Explanation

From part (b) we can see the expected return value from the software company is $\$200,000$which is the lowest among the three investments. Therefore the first investment, a software company, is the riskiest.

Part (e) Step 1: Explanation

The total amount a venture capitalist is interested to invest$=\$1,000,000$

Part (e) Step 2: Explanation

From part (b) we can see that the Hardware company has the highest expected return of$\$600,000$