Chapter 12, Problem 21PAA

To determine

To explain: The better investment between the two.

Answer to Problem 21PAA

The better investment plan will be to invest in first fast food chain..

Explanation of Solution

From the two fast food chains, investment must be made in the franchise from where the expected value of profit is more and the risk associated with the investment is less.

The expected value of a variable is the sum of probabilities that different profits would be there multiplied with the resulting payoffs.

The aggregate 10 year profit by both the fast food giants is,

  $E\left(\pi \right)=q_1{\pi }_1+q_2{\pi }_2+q_3{\pi }_3$

Here, the profit is $E$ the probability is $q$ and the resulting payoffs is $\pi$ .

Here, $q_1+q_2+q_3+\mathrm{...}+q_n=1$

For first fast food chain the expected profit is,

  $\begin{array}{c}{E}_M\left(\pi \right)=q_1{\pi }_1+q_2{\pi }_2+q_3{\pi }_3\\ =\left(0.025\right)\left(48\right)+\left(0.95\right)\left(8\right)+\left(0.025\right)\left(-48\right)\\ =7.6\end{array}$

For second fast food chain the expected profit is,

  $\begin{array}{c}{E}_S\left(\pi \right)=q_1{\pi }_1+q_2{\pi }_2+q_3{\pi }_3\\ =\left(0.025\right)\left(48\right)+\left(0.95\right)\left(8\right)+\left(0.025\right)\left(-48\right)\\ =7.6\end{array}$

The variance is,

  ${\sigma }^2=q_1{\left[{\pi }_1-E\left(\pi \right)\right]}^2+q_2{\left[{\pi }_2-E\left(\pi \right)\right]}^2+q_3{\left[{\pi }_3-E\left(\pi \right)\right]}^2$

Here, the variance is ${\sigma }^2$ , the profit is $E$ the probability is $q$ and the resulting payoffs is $\pi$ .

For first fast food chain the variance is,

  $\begin{array}{c}{\sigma }^2=q_1{\left[{\pi }_1-E\left(\pi \right)\right]}^2+q_2{\left[{\pi }_2-E\left(\pi \right)\right]}^2+q_3{\left[{\pi }_3-E\left(\pi \right)\right]}^2\\ =0.25{\left[16-7.6\right]}^2+0.5{\left[8-7.6\right]}^2+0.25{\left[-1.6-7.6\right]}^2\\ =38.88\end{array}$

For second fast food chain the variance is,

  $\begin{array}{c}{\sigma }^2=q_1{\left[{\pi }_1-E\left(\pi \right)\right]}^2+q_2{\left[{\pi }_2-E\left(\pi \right)\right]}^2+q_3{\left[{\pi }_3-E\left(\pi \right)\right]}^2\\ =0.25{\left[48-7.6\right]}^2+0.5{\left[8-7.6\right]}^2+0.25{\left[-48-7.6\right]}^2\\ =118.24\end{array}$

The expected profit for both is same, the risk associated with the investment in Subs is m ore than risk involved in McDonald’s on the account of higher variance in the former option

Hence, the better investment plan will be to invest in first fast food chain.

Economics Concept Introduction

The aggregate 10 year profit by both the fast food giants is,

  $E\left(\pi \right)=q_1{\pi }_1+q_2{\pi }_2+q_3{\pi }_3$

Here, the profit is $E$ the probability is $q$ and the resulting payoffs is $\pi$ .