It is Wednesday evening now, you want to know the cumulative risk from now to the end of the week (Thursday and Friday). You get the following responses from analysts, which statements make sense or not?

Conditional variance of the two day cumulative return is greater than just the first day's (Thursday)

𝑉
𝑎
𝑟
𝑊
𝑒
𝑑
[
𝑅
𝑇
ℎ
𝑢
+
𝑅
𝐹
𝑟
𝑖
]

𝑉
𝑎
𝑟
𝑊
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𝑑
[
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]

Question

It is Wednesday evening now, you want to know the cumulative risk from now to the end of the week (Thursday and Friday). You get the following responses from analysts, which statements make sense or not?

Conditional variance of the two day cumulative return is greater than just the first day's (Thursday)

𝑉
𝑎
𝑟
𝑊
𝑒
𝑑
[
𝑅
𝑇
ℎ
𝑢
+
𝑅
𝐹
𝑟
𝑖
]
>
𝑉
𝑎
𝑟
𝑊
𝑒
𝑑
[
𝑅
𝑇
ℎ
𝑢
]

BlackTom AI · Accepted Answer

Question restatement: The scenario asks about the conditional variance of the two-day cumulative return from now (Wednesday evening) through the end of the week (Thursday and Friday), and presents the statement: Var[Wed -> (Thu + Fri)] > Var[Wed -> Thu]. Answer options are:
- Make sense
- Make no sense

Option 1: Make sense
- In general, the variance of a sum can be larger than the variance of a single component because Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y). When you consider the two-day cumulative return (Thu + Fri) as the sum of two daily returns, its variance depends on Var(Thu), Var(Fri), and the covariance between Thu and Fri. If Fri contributes additional uncertainty and is not perfectly negatively correlated with Thu, the two-day variance can exceed the variance of just Thursday, especially if Fri’s variance is substantial or if Cov(Thu,Fri) is positive. Thus, under common scenarios (positive or modest negative covariance with limited magnitude, and nonzero Fri variance), the two-day variance is able to be larger than the Thursday variance, which aligns with the intuition that aggregating uncertainty over more days tends to increase total risk. A caveat is that if Fri is highly negatively correlated with Thu and the Fri variance is large in magnitude, the combined variance could be reduced, so the statement is not universally guaranteed but is plausible in many real-world settings.

Option 2: Make no sense
- If one insisted that two-day variance must always be greater than Thursday variance, that would ignore the mathematical fact that adding a second random component with its own variance and potential covariance can either increase or, in some contrived cases with strong negative correlation and specific variance-covariance structure, decrease the total variance. Therefore, declaring it as an absolute inequality in all cases would be incorrect. The conditional variance given information up to Wednesday also depends on what is known at that time; conditioning can reduce uncertainty if there is informative content about the future, but it can also leave open future variability due to unresolved randomness in Fri’s return.

In summary, the statement’s correctness is context-dependent. The two-day variance Var(Thu + Fri) relative to Var(Thu) can be larger in many practical situations, but it is not guaranteed in all possible joint distributions of Thu and Fri. The analysis hinges on Var(Thu), Var(Fri), and Cov(Thu,Fri), and on what information is conditioned on Wednesday evening. Line of thinking for the options highlights: (a) the additive-variance formula, (b) the role of covariance, (c) the effect of conditioning, and (d) edge cases where negative dependence could reverse the inequality. Regardless of the exact distribution, the claim cannot be deemed universally true or universally false without specifying Var(Fri) and Cov(Thu,Fri).

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

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