Question

Investment risk analysis. The risk of a portfolio of financial assets is sometimes called investment risk. In general, investment risk is typically measured by computing the variance or standard deviation of the probability distribution that describes the decision maker’s potential outcomes (gains or losses). The greater the variation in potential outcomes, the greater the uncertainty faced by the decision maker; the smaller the variation in potential outcomes, the more predictable the decision maker’s gains or losses. The two discrete probability distributions given in the next table were developed from historical data. They describe the potential total physical damage losses next year to the fleets of delivery trucks of two different firms.

Firm A				Firm B
Loss Next Year	Probabiity		Loss Next Year	Probability
0	0.01			0	0
500	0.01			200	0.01
1000	0.01			700	0.02
1500	0.02			1200	0.02
2000	0.35			1700	0.15
2500	0.3			2200	0.3
3000	0.25			2700	0.3
3500	0.02			3200	0.15
4000	0.01			3700	0.02
4500	0.01			4200	0.02
5000	0.01			4700	0.01

a. Verify that both firms have the same expected total physical damage loss.

b. Compute the standard deviation of each probability distribution and determine which firm faces the greater risk of physical damage to its fleet next year.

Short answer

a.

For firm the expected value is 2450.

For firm the expected value is 3990.3

b.

The standard deviation of firm A is 661.43

The standard deviation of firmB is 2218.69

FirmB has more risk than firmA

Step-by-step solution

Given information

The variation is seen in potential outcomes of gains or losses.

Calculating the expected total for both firms

a.

For firmthe expected value is

$$\begin{gathered} E\left(x\right)=\left(\begin{array}{cc}0\times 0.01\times 500\times 0.01+1000\times 0.01+1500\times 0.02+2000\times 0.35+2500\times 0.30+3000\times 0.25& \\ +3500\times 0.02+4000\times 0.01+4500\times 0.01+5000\times 0.01& \end{array}\right) \\ =2450 \end{gathered}$$

For firm B the expected value is

$$\begin{gathered} E\left(x\right)=\left(\begin{array}{c}0\times 0+200\times 0.01+700\times 0.02+1200\times 0.02+1700\times 0.15\\ +2200\times 0.30+2700\times 0.30+3200\times 0.15+3700\times 0.02\\ +4200\times 0.02+4700\times 0.01\end{array}\right) \\ =3990.3 \end{gathered}$$

Here, we see that both firms not have same expectation. Firm B has more expectation than firm A.

Finding the standard deviation of the each probability distribution and calculate the greater risk

b.

$$\begin{gathered} E\left(x\right)=\left(\begin{array}{cc}{0}^2\times 0.01\times {500}^2\times 0.01+{1000}^2\times 0.01+{1500}^2\times 0.02+{2000}^2\times 0.35+{2500}^2\times 0.30& \\ {3000}^2\times 0.25+{3500}^2\times 0.02+{4000}^2\times 0.01+{4500}^2\times 0.01+{5000}^2\times 0.01& \end{array}\right) \\ =6440000 \end{gathered}$$

Then the var(x) is given by

$$\begin{gathered} var\left(x\right)=E\left(x^2\right)-E^2\left(x\right) \\ =6440000-{\left(2450\right)}^2 \\ =437500 \end{gathered}$$

The standard deviation is

$$\begin{gathered} sd\left(x\right)=\sqrt{var\left(x\right)} \\ =\sqrt{437500} \\ =661.43 \end{gathered}$$

Similarly, for firm B we calculate is$E\left(x^2\right)$

$$\begin{gathered} E\left(x\right)=\left(\begin{array}{c}{0}^2\times 0+{200}^2\times 0.01+{700}^2\times 0.02+{1200}^2\times 0.02+{1700}^2\times 0.15\\ +{2200}^2\times 0.30+{2700}^2\times 0.30+{3200}^2\times 0.15+{3700}^2\times 0.02+{4200}^2\times 0.02+{4700}^2\times 0.01\\ \end{array}\right) \\ =64950000 \end{gathered}$$$$\begin{gathered} vaa\left(x\right)=E\left(x^2\right)-E^2\left(x\right) \\ =64950000-{\left(3990.3\right)}^2 \\ =492027505.91 \end{gathered}$$

$$\begin{gathered} sd\left(x\right)=\sqrt{var\left(x\right)} \\ =\sqrt{492027505.91} \\ =22181.69 \end{gathered}$$

Thus, the greater risk is associated with the firm B.