This is for your practice!
Consider three possible investments, with returns denoted as X, Y, and Z, respectively, and probability distributions outlined in the tables below.
\[\begin{array}{c|c}X&14,000\\\hline P(X=x)&1\end{array}\]
Investment X is what we’d call a “sure thing,” with a guaranteed return of $14,000: there is no risk involved at all.
\[\begin{array}{c|c|c}Y&0&1,000,000\\\hline P(X=x)&0.98&0.02\end{array}\]
Investment Y is extremely risky, with a high probability (98) of no gain at all, contrasted by a slight probability (02) of “making a killing” with a return of a million dollars.
\[\begin{array}{c|c|c}Z&10,000&20,000\\\hline P(X=x)&0.5&0.5\end{array}\]
Investment Z is somewhere in between: there is an equal chance for either a return that’s on the low side or a return that’s on the high side.
If you only consider the mean return on each investment, would you prefer X, Y, or Z? The means for X, Y, and Z are calculated as follows:
\[\mu_X = 14000(1) = 14000\]
\[\mu_Y = 0(0. 98) + 1000000(0. 02) = 20000\]
\[\mu_Z = 10000(0. 5) + 20000(0. 5) = 15000\]
Clearly, the mean return for Y is highest, and so investment in Y would seem to be preferable.
Now consider the standard deviations, and consider which investment you’d prefer-X, Y, or Z.
The standard deviations are:
\[\sigma_X^2 = (14000-14000)^2\cdot 1 = 0\] \[\sigma_X=0\]
\[\sigma_Y^2 = (0-20000)^2(0.98) + (1, 000, 000 - 20000)^2(0. 02) = 1.96 × 10^{10}\] \[\sigma_Y=140,000\]
\[\sigma_Z^2=(10000-15000)^2\cdot 0.5+(20000-15000)^2\cdot 0.5 =25,000,000\] \[\sigma_Z=5000\]
Granted, the mean returns suggest that investment X is least profitable and investment Y is most profitable. On the other hand, the standard deviations are telling us that the return for X is a sure thing; for Y, the remote chance of making a huge profit is offset by a high risk of losing the investment entirely; for Z, there is a modest amount of risk involved. If you can’t afford to lose any money, then investment X would be the way to go. If you have enough assets to take a chance, then investment Y would be worthwhile. In particular, if a large company routinely makes many such investments, then in the long run there will occasionally be such enormous gains that the company is willing to absorb many smaller losses. Investment Z represents the middle ground, somewhere between the other two.