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)

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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 problem asks us to assess whether the statement about conditional variances makes sense. 
Option 1: Make sense. Here, we are comparing Var Wed[ RT_th + RF_fri ] to Var Wed[ RT_th ]. Intuitively, the two-day cumulative return from Thursday to Friday encompasses two sources of randomness (the Thursday return and the Friday return) rather than just the single Thursday return. In general, the variance of a sum of random variables is Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y). Even when conditioning on information available as of Wednesday, the second day’s return (Friday in this setup) can still contribute additional variability, and any positive or nonzero covariance between Thursday and Friday will typically push Var(Thu+Fri|Wed) above Var(Thu|Wed). Therefore, it is reasonable to expect that the conditional variance of the two-day cumulative return is greater than the conditional variance of just the first day’s return, especially if there is any uncertainty remaining about Friday and any positive autocorrelation between days. 
Option 2: Make no sense. This would imply that adding Friday into the two-day sum does not increase uncertainty given Wednesday, or even reduces it. That would require either zero variance on Friday, perfect negative covariance canceling the variance from Thursday, or some conditioning that eliminates all added uncertainty on Friday. In typical financial return processes, these special coincidences are unlikely: Friday’s return generally has its own variability and can be correlated with Thursday’s return, making Var(Thu+Fri|Wed) at least as large as Var(Thu|Wed), and often larger. Hence this option would rely on atypical or degenerate assumptions and does not align with the standard variance formula for a sum. 
In summary, the more plausible interpretation is that the two-day conditional variance should be at least as large as the single-day conditional variance, and often larger, given additional source(s) of randomness and potential covariance between days.

BU.230.730.51.FA25 Final Exam- Requires Respondus LockDown Browser

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