Question

Possible outcomes for three investment alternatives and their probabilities of occurrence are given next.

Alternative 1 Alternative 2 Alternative 3

Outcomes Probability Outcomes Probability Outcomes Probability

Failure ................... 50 0.2 90 0.3 95 0.2

Acceptable ........... 90 0.4 190 0.3 215 0.6

Successful ............ 135 0.4 225 0.4 380 0.2

Rank the three alternatives in terms of risk from lowest to highest (compute the coefficient of variation).

Short answer

Table showing the Ranks
	Coefficient of variation	Rank
Alternative 1	0.3209	Rank 1
Alternative 2	0.5687	Rank 2
Alternative 3	0.9954	Rank 3

Step-by-step solution

Compute expected value

$\begin{array}{c}\mathrm{Expected}\mathrm{value}\mathrm{Alternative}1=\mathrm{Respective}\mathrm{sales}\times \mathrm{Respective}\mathrm{probability}\\ \mathbf{=}(\mathbf{50}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{20})\mathbf{+}(\mathbf{90}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{40})\mathbf{+}(\mathbf{135}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{40})\\ =10+36+54\\ =100\mathrm{units}\end{array}$

$\begin{array}{c}\mathrm{Expected}\mathrm{value}\mathrm{Alternative}2=\mathrm{Respective}\mathrm{sales}\times \mathrm{Respective}\mathrm{probability}\\ \mathbf{=}(\mathbf{90}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{30})\mathbf{+}(\mathbf{190}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{30})\mathbf{+}(\mathbf{225}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{40})\\ =27+57+90\\ =174\mathrm{units}\end{array}$

$\begin{array}{c}\mathrm{Expected}\mathrm{value}\mathrm{Alternative}3=\mathrm{Respective}\mathrm{sales}\times \mathrm{Respective}\mathrm{probability}\\ \mathbf{=}(\mathbf{95}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{20})\mathbf{+}(\mathbf{215}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{60})\mathbf{+}(\mathbf{380}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{20})\\ =19+129+76\\ =224\mathrm{units}\end{array}$

Computation of total probability

Alternative 1

Probability	Sales	Probability*(sales-Expected sales)2	Total
0.20	50	0.20*(50-100)2	500
0.40	90	0.40*(90-100)2	40
0.40	135	0.40*(135-100)2	490
			1030

Alternative 2

Probability	Sales	Probability*(sales-Expected sales)2	Total
0.30	90	0.30*(90-174)2	2116.8
0.30	190	0.30*(190-174)2	76.8
0.40	225	0.40*(225-174)2	1040.4
			3234

Alternative 3

Probability	Sales	Probability*(sales-Expected sales)2	Total
0.20	95	0.20*(95-224)2	4992.3
0.60	215	0.60*(215-224)2	48.6
0.20	380	0.20*(380-224)2	4867.2
			9908.1

Computation of Standard deviation ( approx ) 

$\begin{array}{c}\mathbf{Standard}\mathbf{}\mathbf{deviation}\mathbf{}\mathbf{Alternative}\mathbf{1}\mathbf{=}\sqrt{\begin{array}{l}\mathbf{Probability}\mathbf{\times }{(\mathbf{sales}\mathbf{-}\mathbf{Expectedsales})}^2\end{array}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{1030}\\ \mathbf{=}\mathbf{32}\mathbf{.}\mathbf{09}\mathbf{units}\end{array}$

$\begin{array}{c}\mathbf{Standard}\mathbf{}\mathbf{deviation}\mathbf{}\mathbf{Alternative}\mathbf{2}\mathbf{=}\sqrt{\begin{array}{l}\mathbf{Probability}\mathbf{\times }{(\mathbf{sales}\mathbf{-}\mathbf{Expected}\mathbf{}\mathbf{sales})}^2\end{array}}\\ \mathbf{=}\sqrt{3234}\\ \mathbf{=}\mathbf{56}\mathbf{.}\mathbf{87}\mathbf{units}\end{array}$

$\begin{array}{c}\mathbf{Standard}\mathbf{}\mathbf{deviation}\mathbf{}\mathbf{Alternative}\mathbf{}\mathbf{3}\mathbf{=}\sqrt{\begin{array}{l}\mathbf{Probability}\mathbf{\times }{(\mathbf{sales}\mathbf{-}\mathbf{Expected}\mathbf{}\mathbf{sales})}^2\end{array}}\\ \mathbf{=}\sqrt{\mathbf{9908}\mathbf{.}\mathbf{1}}\\ \mathbf{=}\mathbf{99}\mathbf{.}\mathbf{54}\mathbf{units}\end{array}$

Computation of co-efficient of variation ( approx.)

$\begin{array}{c}\mathbf{Coefficient}\mathbf{}\mathbf{of}\mathbf{}\mathbf{variation}\mathbf{}\mathbf{Alternative}\mathbf{2}\mathbf{=}\frac{\mathrm{Standard}\mathrm{deviation}}{\mathrm{Expected}\mathrm{sales}}\\ \mathbf{=}\frac{\mathbf{56}\mathbf{.}\mathbf{87}}{100}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5687}\end{array}$

$\begin{array}{c}\mathbf{Coefficient}\mathbf{}\mathbf{of}\mathbf{}\mathbf{variation}\mathbf{}\mathbf{Alternative}\mathbf{3}\mathbf{=}\frac{\mathrm{Standard}\mathrm{deviation}}{\mathrm{Expected}\mathrm{sales}}\\ \mathbf{=}\frac{\mathbf{99}\mathbf{.}\mathbf{54}}{100}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9954}\end{array}$