If the conditional variance for tomorrow's return is 0.0004,
𝑉
𝑎
𝑅
𝑡
(
𝑅
𝑡
+
1
)
=
(
0.02
)
2
=
0.0004
, then the conditional expectation for tomorrow's return is 2% or -2%.

Question

If the conditional variance for tomorrow's return is 0.0004, 
𝑉
𝑎
𝑅
𝑡
(
𝑅
𝑡
+
1
)
=
(
0.02
)
2
=
0.0004
, then the conditional expectation for tomorrow's return is 2% or -2%.

BlackTom AI · Accepted Answer

Question restatement: If the conditional variance for tomorrow's return is 0.0004, V_t(R_{t+1}) = (0.02)^2 = 0.0004, then the conditional expectation for tomorrow's return is 2% or -2%.

Option 1: "Make sense". This option would imply that knowing the conditional variance alone is enough to pin down the conditional expectation as either +2% or -2%. In probability and finance, the variance of a random variable tells us about the dispersion around its mean, but it does not determine the mean itself. The conditional mean (expectation) could be any value consistent with the information available; variance being 0.0004 merely tells us how spread out R_{t+1} is around its conditional mean, not what that mean is. Therefore, concluding that the conditional expectation must be exactly ±2% is unfounded. This makes the statement false.

Option 2: "Make no sense". This option correctly captures the misinterpretation in Option 1: the given conditional variance equals (0.02)^2, but that equality does not uniquely specify the conditional mean. The mean could be 0, 1%, -0.5%, or any other value, depending on other information such as the conditional distribution or model assumptions. Thus asserting that the conditional expectation is exactly 2% or -2% is not warranted by the variance alone. This makes the statement true.

Why the others are misleading: If someone assumes that variance equals the square of a potential mean, they are conflating dispersion with central tendency. Variance being 0.0004 only confirms a certain level of uncertainty around whatever the mean is; it does not identify the mean or its sign. In short, variance does not constrain the conditional expectation to ±2% without extra information. The reasoning rests on the fundamental separation between a distribution’s mean and its variance, not on a direct equivalence between them.

If the conditional variance for tomorrow's return is 0.0004, 𝑉 𝑎 𝑅 𝑡 ( 𝑅 𝑡 + 1 ) = ( 0.02 ) 2 = 0.0004 , then the conditional expectation for tomorrow's return is 2% or -2%.单项选择题

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