Question

A company has the investment opportunities. The following table of results gives us data related to each of them. Using the expected value techniques, the standard deviation and the coefficient of variation determine which of them the company should select

            Inversion # 1                                                             Inversion # 2

States of Nature Economic	P	Results	VE	States of Nature Economic	P	Results	VE
Bonanza	.30	5,000		Bonanza	.30	4,000
Normal	.50	2,000		Normal	.50	2,000
Recession	.20	1,000		Recession	.20	1,500
		VE				VE

Answer 1

The information on investment opportunities that have been summed up under two options ‘Inversion #1’ and ‘Inversion #2’, can be utilized to infer which investment option is better than the other. This can be done with the help of Expected Value (Mean), the standard deviation and finally, the coefficient of variation. The given information represents discrete random variables with probabilities ‘P’ and Results ‘x’.

The formulae for our required value are as follows:

    where

From the given table, the required value can be calculated:

Inversion #1

States of Nature (Economic)	P(x)	x	xP(x)	(x-u)2	(x-u)2.P(x)
Bonanza	0.30	5000	1500	5290000	1587000
Normal	0.50	2000	1000	490000	245000
Recession	0.20	1000	200	2890000	578000
Total			2700		2410000

Therefore E(x) = u = 1500+1000+200 = 2700.This is the mean or the expected value of inversion #1

The variance of the above data is 2410000. Therefore the Standard deviation is (2410000)^0.5 or

Standard deviation (SD) = 1552.417.

Thus the coefficient of variation is: SD/Mean * 100 = (1552.417/2700) *100 = 57.496% ~ 57.5%

The same process is repeated for Inversion #2 in the following way:

States of Nature (Economic)	P(x)	x	xP(x)	(x-u)2	(x-u)2.P(x)
Bonanza	0.30	4000	1200	2250000	675000
Normal	0.50	2000	1000	250000	125000
Recession	0.20	1500	300	1000000	200000
Total			2500		1000000

Therefore E(x) = u = 1200+1000+300 = 2500.This is the mean or the expected value of inversion #1

The variance of the above data is 1000000. Therefore the Standard deviation is (2410000)^0.5 or

Standard deviation (SD) = 1000

Thus the coefficient of variation is: SD/Mean * 100 = (1000/2500)*100 = 40%.

It is this Coefficient of variation that helps a rational investor decide on which investment to go for. Since Inversion #1 has a higher coefficient of variation, it is more volatile in comparison to Inversion #2. Thus, Inversion #1 is more risky and prone for more ups and downs than Inversion #2. Assuming that the investor is risk averse, he would ideally choose Inversion #2 with a low degree of volatility (risk) and higher return.

It is important to note that, while the probabilities associated with different states of economic nature are the same for both the Inversions, it is the difference in return that makes all the difference in this particular example.