Chapter 11.3, Problem 5GP

To determine

To find: The better investment of $\$10000$ from the given two funds.

Answer to Problem 5GP

The investment should be done in Fund C.

Explanation of Solution

The expected value of Fund C is calculated as:

  $\begin{array}{c}E\left(X\right)=0.3\left(1000\right)+0.4\left(500\right)+0.2\left(-100\right)+0.1\left(-100\right)\\ =470\end{array}$

The expected value of Fund D is calculated as:

  $\begin{array}{c}E\left(X\right)=0.4\left(1000\right)+0.3\left(600\right)+1.5\left(100\right)+1.5\left(-100\right)\\ =580\end{array}$

The standard deviation of Fund C is calculated as:

Profit $X$	$P\left(X\right)$	${\left(X-E\left(X\right)\right)}^2$	${\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)$
$1000$	$0.3$	${\left(1000-470\right)}^2=280900$	$280900\cdot 0.3=84270$
$500$	$0.4$	${\left(500-470\right)}^2=900$	$900\cdot 0.4=360$
$-100$	$0.2$	${\left(-100-470\right)}^2=324900$	$324900\cdot 0.2=64980$
$-100$	$0.1$	${\left(-100-470\right)}^2=324900$	$324900\cdot 0.1=32490$
			${\displaystyle \sum {\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)}=182100$

So the standard deviation of Fund C is as:

  $\begin{array}{c}\sigma =\sqrt{{\displaystyle \sum {\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)}}\\ =\sqrt{182100}\\ =426.73\end{array}$

The standard deviation of Fund D is calculated as:

Profit $X$	$P\left(X\right)$	${\left(X-E\left(X\right)\right)}^2$	${\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)$
$1000$	$0.4$	${\left(1000-580\right)}^2=176400$	$176400\cdot 0.4=70560$
$600$	$0.3$	${\left(600-580\right)}^2=400$	$400\cdot 0.3=120$
$100$	$1.5$	${\left(100-580\right)}^2=230400$	$230400\cdot 1.5=345600$
$200$	$1.5$	${\left(200-580\right)}^2=144400$	$144400\cdot 1.5=216600$
			${\displaystyle \sum {\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)}=632880$

So the standard deviation of Fund C is as:

  $\begin{array}{c}\sigma =\sqrt{{\displaystyle \sum {\left(X-E\left(X\right)\right)}^2\cdot P\left(X\right)}}\\ =\sqrt{632880}\\ =795.537\end{array}$

While the expected value of fund D is more than the expected value of Fund C, the standard deviation of Fund D is almost twice the standard deviation of Fund C. This means the expected value for Fund D will have about twice the variability than the Fund C and will be riskier with a greater chances of gains and losses.

Therefore the investment should be done in Fund C.