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
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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

The question asks us to assess whether the conditional variance of a two-day cumulative return (Thursday + Friday) is greater than the variance of the single-day return on Thursday, given information up to Wednesday.
Option 1: 'Make sense' — This can be true in general. Once you condition on information available through Wednesday, the two-day ahead return R_Thu + R_Fri contains uncertainty from both Thursday and Friday. The variance of the sum typically incorporates two sources of variability: Var(R_Thu | F_t) and Var(R_Fri | F_t), plus twice the covariance term 2 Cov(R_Thu, R_Fri | F_t). Unless the second day contributed no uncertainty or was perfectly negatively correlated in just the right way to cancel the first day’s variance, the sum’s conditional variance should not shrink below the variance of the first day alone. In most realistic settings, adding another uncertain component increases or at least does not decrease overall uncertainty, so Var(R_Thu+R_Fri | F_t) ≥ Var(R_Thu | F_t), with strict > under standard assumptions (positive variance for Fri or nonzero covariance). Therefore, the statement that the two-day conditional variance is greater than the Thursday single-day variance makes sense.
Option 2: 'Make no sense' — This would be incorrect under typical financial variance properties. If the Friday return carried zero variance or were perfectly predictable given Wednesday, you could imagine the two-day variance being equal or even smaller in a degenerate case. However, with ordinary uncertain future days, adding another uncertain component (Friday) cannot reduce total uncertainty below that of Thursday alone. Hence this option would rely on unrealistic or special-case degeneracy, not the general situation.
In summary, the plausible and standard reasoning supports that the two-day conditional variance is greater than the one-day variance, making 'Make sense' the reasonable interpretation in ordinary contexts. The alternative would require unusual or degenerate assumptions about Friday’s return, which are not typically assumed in a normal analysis.

BU.230.730.53.SP25 Final Exam- Requires Respondus LockDown Browser

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