Question

Question: You are considering a\(500,000investment in the fast-food industry and have narrowed your choice to either a McDonald’s or a Penn Station East Coast Subs franchise. McDonald’s indicates that, based on the location where you are proposing to open a new restaurant, there is a 25 percent probability that aggregate 10 -year profits (net of the initial investment) will be \)16 million, a 50percent probability that profits will be \(8 million, and a 25percent probability that profits will be \)1.6million. The aggregate 10 -year profit projections (net of the initial investment) for a Penn Station East Coast Subs franchise is\(48million with a 2.5percent probability,\)8million with a 95percent probability, and $48million with a2.5percent probability. Considering both the risk and expected profitability of these two investment opportunities, which is the better investment? Explain carefully

Short answer

Answer

McDonald's has a lower standard deviation than the Subs Franchise, which makes McDonald’s a safer bet.

Step-by-step solution

Standard Deviation

Thestandard deviation is a statistical measure of how far data are spread out around their mean. As a result, the standard deviation is linked to risk.

A return with a larger standard deviation is more volatile and hence regarded riskier than one with a lower volatility or standard deviation.The standard deviation equation is:$\sigma \sqrt{{\sigma }^2}$

Evaluating the equation 

As an investor, it is considered investingin a fast food chain, and you have two options: a McDonalds franchise or a Penn Station East Coast Subs franchise, both of which have a-year chance of return on investment as given in the table below:

The anticipated return value of any firm may be found by taking into account the risk of each investment. The expected value can be computed using the following equation:

${\text{E(x) =q}}_{\text{1}}{\text{x}}_{\text{1}}{\text{+q}}_{\text{2}}{\text{x}}_{\text{2}}\text{+}......{\text{+q}}_{\text{n}}{\text{x}}_{\text{n}}.............\text{(1)}$

The value of are the probability of outcomes.

The value of x is the expected value.

Step 3: Evaluating the expected values

The estimated profit of both enterprises may be computed by substituting the data from the table above in equation one as:

$$\begin{gathered} ExpectedvalueofMcDonalds: \\ E\left(x\right)=25\%\left(16M\right)+50\%\left(8M\right)+25\%(\$-1.6M) \\ E\left(x\right)=0.25\left(16M\right)+0.5\left(8M\right)+0.25(\$-1.6M) \\ E\left(x\right)=4+4-0.4 \\ =7.6billion \\ ExpectedvalueofSubsFranchise: \\ E\left(x\right)=2.5\%\left(48M\right)+95\%\left(8M\right)+2.5\%(1-48M) \\ E\left(x\right)=0.025\left(48M\right)+0.95\left(8M\right)+0.025(1-48M) \\ E\left(x\right)=1.2+7.6-1.2 \\ =7.6billion \end{gathered}$$

Thus, the expected value for both McDonald’s and Subs Franchise is $7.6 billion.

Variance Equation 

As a result, in order to compute the standard deviation, the variance must be calculated first, using the following equation:

${\text{σ}}^2{\text{=q}}_{\text{1}}{\text{(x}}_{\text{1}}{\text{- E[x])}}^{\text{2}}{\text{q}}_{\text{2}}{\text{(x}}_{\text{2}}{\text{- E[x])}}^{\text{2}}\text{+}........{\text{+q}}_{\text{n}}{\text{(x}}_{\text{n}}{\text{- E[x])}}^{\text{2}}$

Calculating variance and standard deviation

The preceding equation is changed with the following data for variance and standard deviation for each investment:

The risk of investing in McDonald’s is:

$$\begin{gathered} {\sigma }^2=25\%(\$16-\$7.6{)}^2+50\%(\$8-7.6{)}^2+25\% (\$-1.6-\$7.6{)}^2 \\ {\sigma }^2=0.25(8.4{)}^2+0.5(0.4{)}^2+0.25(-9.2{)}^2 \\ {\sigma }^2=17.64+0.08+21.16 \\ {\sigma }^2=38.88 \end{gathered}$$

The standard deviation is evaluated as:

$$\begin{gathered} \sigma =\sqrt{38.88} \\ =6.24 \end{gathered}$$

The risk of investing the Sub Franchise is:

$$\begin{gathered} {\sigma }^2=2.5\%(\$480-\$7.6{)}^2+95\%(\$8-7.6{)}^2+2.5\%(\$-48-\$7.6{)}^2 \\ {\sigma }^2=0.025(40.4{)}^2+0.95(0.4{)}^2+0.025(-55.6{)}^2 \\ {\sigma }^2=\left(40.804+0.152+77.284\right) \\ {\sigma }^2=118.24 \end{gathered}$$

The standard deviation is evaluated as:

$$\begin{gathered} \sigma =\sqrt{118.24} \\ =10.87 \end{gathered}$$

Based on the findings, it can be determined that both investments would provide the same $7.6 billion projected return in ten years; but, the McDonalds investment has a smaller standard deviation than the Subs Franchise, making it less hazardous.

Therefore, McDonald's has a smaller standard deviation than the Subs Franchise, making it a safer investment.