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 links a conditional variance value to a claim about the conditional mean of tomorrow's return, so we should unpack what variance tells us and what it does not. 
Option 1: "Make sense". This would imply that knowing Var_t(R_{t+1}) = 0.0004 directly pins down the conditional expectation to be +/- 2%. However, variance measures dispersion around the conditional mean, not the mean itself. A variance of 0.0004 only tells us the spread around the mean is 0.02, i.e., the standard deviation is 0.02; it does not specify the actual average return. Therefore, claiming the mean is exactly 2% or -2% is unsupported by the given information, so this option is not valid. 
Option 2: "Make no sense". This option correctly reflects that the conditional mean is not determined by the conditional variance alone. Without additional information about the conditional mean (e.g., from a model specification like E_t[R_{t+1}] = some value or sign constraints), the mean could be any value, and the sign could be positive, negative, or zero. The fact that Var_t(R_{t+1}) equals 0.0004 only conveys the uncertainty around the mean, not the mean itself, so asserting the mean must be exactly ±2% is incorrect. 
In short, variance and mean are distinct characteristics of a distribution: variance quantifies variability around the mean, while the mean is the central tendency that variance does not itself determine. Consequently, the statement about a ±2% conditional expectation is unfounded given only the variance value.}

BU.230.730.81.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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