An investor can choose between two investments. Investment P gives a gain of Rs. 4000 with probability 0.5 and a loss of Rs. 1000 with probability 0.5. Investment Q gives a gain of Rs. 2000 with probability 0.8 and a loss of Rs. 500 with probability 0.2. Which of the following are correct?
(A) Both investments have the same expected gain of Rs. 1500.
(B) The standard deviation of the gain from P is Rs. 2500.
(C) The standard deviation of the gain from Q is Rs. 1000.
(D) Q is the riskier investment, since the standard deviation of its gain is larger.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
Let X and Y be the gains (in Rs.) from P and Q, a loss being negative.
\(E(X) = 4000 \times 0.5 + (-1000) \times 0.5 = 1500\)
\(E(X^{2}) = 16000000 \times 0.5 + 1000000 \times 0.5 = 8500000\)
\(\mathrm{Var}(X) = 8500000 - 1500^{2} = 6250000\), so SD = 2500
\(E(Y) = 2000 \times 0.8 + (-500) \times 0.2 = 1500\)
\(E(Y^{2}) = 4000000 \times 0.8 + 250000 \times 0.2 = 3250000\)
\(\mathrm{Var}(Y) = 3250000 - 1500^{2} = 1000000\), so SD = 1000
So both expected gains are Rs. 1500, and the standard deviations are Rs. 2500 for P and Rs. 1000 for Q. The larger standard deviation means the greater risk.
Checking the statements
(A) Both investments have the same expected gain of Rs. 1500 — Correct. E(X) = E(Y) = 1500.
(B) The standard deviation of the gain from P is Rs. 2500 — Correct. \(\sqrt{6250000} = 2500\).
(C) The standard deviation of the gain from Q is Rs. 1000 — Correct. \(\sqrt{1000000} = 1000\).
(D) Q is the riskier investment, since the standard deviation of its gain is larger — Incorrect. P has the larger standard deviation (2500 against 1000), so P is the riskier one.