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

The question discusses the conditional variance Var tomorrow's return and a proposed conditional expectation for tomorrow's return. 
Option 1: 'Make sense' — This would claim that knowing Var tomorrow's return equals 0.0004 implies a specific conditional mean, namely 2% or -2%. In probability, the conditional variance provides information about the spread of R_{t+1} around its conditional mean E[R_{t+1}|F_t], but it does not pin down the value of the conditional mean itself. Therefore, deducing that the conditional expectation is exactly 2% or -2% from the variance alone is invalid; the mean could be any value, and the variance alone does not determine it. Thus this option is conceptually flawed. 
Option 2: 'Make no sense' — This correctly identifies that variance and its numerical value do not specify, nor constrain, the conditional expectation. The relationship Var(R_{t+1}|F_t) = E[(R_{t+1} - E[R_{t+1}|F_t])^2 | F_t] = 0.0004 tells us about dispersion around the mean but not about the mean itself. Even if the standard deviation equals 0.02, that only describes the typical deviation from the mean, not the mean's location or sign. Therefore, asserting that the conditional expectation is exactly 2% or -2% based solely on the variance is not justified. 
In summary, the given conditional variance value informs us about the variability around the conditional mean but does not determine the conditional mean itself, so the statement that the conditional expectation is 2% or -2% does not make sense. The correct interpretation aligns with the idea that the mean is not fixed by the variance alone, and any definite claim about the mean requires additional information about E[R_{t+1}|F_t].

BU.230.730.52.SP25 Final Exam- Requires Respondus LockDown Browser

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%.

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