Question

Compute the expected return, standard deviation, and value at risk for each of the following investments:

Investment (A): Pays $800 three-fourths of the time and a $1,200 loss otherwise. Investment (B): Pays $1,000 loss half of the time and a $1,600 gain otherwise.

State which investment will be preferred by each of the following investors, and explain why.

(i) a risk-neutral investor

(ii) an investor who seeks to avoid the worst-case scenario.

(iii) a risk-averse investor.

Answer 1

Investment (A)
Expected return = 0.25(-$1200) + 0.75($800) = $300
Standard Deviation = sqrt[0.25 (-1000 - 300)2 + 0.75 (800 - 300)2 ] = 750000 = 866
Value at Risk = -$1200
Investment (B)
Expected return = 0.5(-$1000) + 0.5($2000) = $500
Standard Deviation = sqrt[0.5 (-1000 - 500)2 + 0.5 (1600 - 500)2 ] = sqrt[1690000] = 1300
Value at Risk = -$1000

(I) The risk-neutral investor is unconcerned between these two speculations since they pay the equivalent anticipated return.
(ii) The investor who seeks to avoid the worst-case scenario will choose Investment (B) because it has the lower value at risk.
(iii) The risk-averse investor will prefer Investment (A) because it has a lower standard deviation. This proposes there is less vulnerability about the normal return with respect to Investment (B).