A coherent risk measure satisfies the property of sub$-$additivity. Suppose a stock whose random return can be approximately summarized by the following

density function:

$f(r)=\{\begin{array}{ccc}-r^2+0\mathrm{.}5r+0\mathrm{.}763& if& -0\mathrm{.}659r<mn\; id="EditorElement\_95329">1</mn><mn\; id="EditorElement\_95330">.</mn><mn\; id="EditorElement\_95331">1</mn><mn\; id="EditorElement\_95332">5</mn><mn\; id="EditorElement\_95333">9</mn>\\ 0& if\text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

where ${\displaystyle \underset{-0\mathrm{.}659}{\overset{1\mathrm{.}159}{}}}f(r)dr$~~$1$

Considering a target of $50\%,$ what is the shortfall probability?

a$.$ The shortfall probability is somewhere between $66\%$ and $72\%.$

b$.$ The shortfall probability is somewhere between $42\%$ and $48\%.$

c$.$ The shortfall probability is somewhere between $18\%$ and $24\%.$

d$.$ The shortfall probability is somewhere between $34\%$ and $40\%.$

e$.$ The shortfall probability is somewhere between $10\%$ and $16\%.$What does sub$-$additivity imply?

Question $4$Answer

a$.$

The risk measure for a combined portfolio should be smaller than the product of the risk measures of each portfolio.

b$.$

The risk measure for a combined portfolio should not be greater than the product of the risk measures of each portfolio.

c$.$

The risk measure for a combined portfolio should be greater than the sum of the risk measures of each portfolio.

d$.$

The risk measure for a combined portfolio should not be greater than the sum of the risk measures of each portfolio.

e$.$

The risk measure for a combined portfolio should be smaller than the difference of the risk measures of each portfolio.