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

We begin by restating the problem in our own words. The question compares the conditional variance of the two-day cumulative return (Thursday + Friday) to the variance of the single-day return on Thursday, given information available on Wednesday. The two answer options are:
- Make sense
- Make no sense

Now, let's analyze each option in turn, keeping in mind the formula for the variance of a sum and how conditioning can affect it.

Option 1: Make sense
- If we consider Var(Thu + Fri | Wed), we can use the variance of a sum rule: Var(Thu + Fri | Wed) = Var(Thu | Wed) + Var(Fri | Wed) + 2 Cov(Thu, Fri | Wed).
- Under typical modeling assumptions in finance (positive variance components and non-negative or modestly positive autocovariance between consecutive daily returns), Var(Fri | Wed) is positive and Cov(Thu, Fri | Wed) is often non-negative or not excessively negative. In such common cases, the sum of Var(Thu | Wed) and Var(Fri | Wed) plus twice a non-negative covariance would tend to be larger than Var(Thu | Wed) alone. In other words, adding Friday’s return introduces additional uncertainty (variance) unless there is a perfectly negative and large covariance that cancels it out, which is generally unlikely in practice.
- Therefore, in many realistic scenarios, the conditional variance of the two-day cumulative return can be greater than the conditional variance of just the first day, making the statement “Make sense” reasonable and aligned with standard intuition about diversification of time horizons and accumulation of risk.
- A nuance to keep in mind: if Fri’s returns are highly negatively correlated with Thu’s returns in a way that Cov(Thu, Fri | Wed) is strongly negative, it could reduce Var(Thu + Fri | Wed) below Var(Thu | Wed). However, such strong negative cross-day dependence is not the typical or default case one would assume without specific evidence, so the statement is sensible under common modeling settings.

Option 2: Make no sense
- This would assert that Var(Thu + Fri | Wed) is always less than or equal to Var(Thu | Wed) or never exceeds it, which would require a sufficiently large negative Cov(Thu, Fri | Wed) to offset the additional positive Var(Fri | Wed). While mathematically possible in some contrived or extreme parameterizations, it is not the norm in standard financial time-series models where daily returns exhibit positive variance components and at least non-negative or moderate covariances across adjacent days.
- Therefore, as a blanket claim, saying the two-day conditional variance is never greater than the one-day variance is too strong and generally incorrect given typical return dynamics.

In summary, option 1 (Make sense) is commonly tenable because adding the second day tends to increase total risk through the addition of Fri’s variance and potential positive or modestly positive covariance, barring unusual strong negative cross-day dependence. Option 2 asserts a universal impossibility which does not hold under all feasible models, making it less defensible. Since the question presents the statement as a scenario to evaluate, the reasoning supports interpreting “Make sense” as the more plausible stance in standard contexts, with the caveat that special cases with strong negative cross-day dependence could violate it.

BU.230.730.81.SP25 Final Exam- Requires Respondus LockDown Browser

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