Point in time financial data and sampling

Definition A timestamp records when something happened, and one observation in a financial dataset usually carries several. The economic date, or reference period, says which period the number describes. The availability time $a_j$ of observation $j$ is the first time a user of the data could have read its value, including any publication and processing delay. A revision time is the availability time of a later replacement value. A trade in a backtest adds the decision time $t$ and the execution time $e$, and it is consistent with the timestamps only if

for every observation $j$ that the decision uses, and if the fill price could actually have been traded at time $e$. A sampling rule says when observations are taken. Clock-time sampling takes them at fixed calendar intervals, trade-time sampling at each trade and event-time sampling after each event of a specified kind.

Definition Fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and observations $j = 1, \dots, J$, each with a real value $v_j$ and an availability time $a_j$, both random variables, so that a fixed release date is a special case. The information set at decision time $t$ is the sigma algebra

generated by whether each observation has arrived by each time $s \le t$ and by the value of each observation that has arrived. The indicator separates an observation that arrived with value $0$ from one that has not arrived. If $s \le t$, then $\mathcal{F}_s \subseteq \mathcal{F}_t$, so the information sets form a filtration. A feature $X_t$ is admissible at $t$ when

that is, when $X_t$ is $\mathcal{F}_t$-measurable. When $\mathcal{F}_t$ is generated by a finite partition of $\Omega$, this means that $X_t$ is constant on each cell of the partition. The sigma algebras and partitions used here are those of Probability spaces and conditional information.

Definition A data revision is a later value for the same item, the same variable for the same entity and economic period, that replaces a value published earlier. Write the versions of one record as pairs $(a^{(k)}, v^{(k)})$ for $k = 1, \dots, K$, with availability times $a^{(1)} < a^{(2)} < \dots < a^{(K)}$. For $t \ge a^{(1)}$ the as-of value is

and for $t < a^{(1)}$ the record is not yet available. When the versions and their availability times are among the observations that generate $\mathcal{F}_t$, the as-of value is $\mathcal{F}_t$-measurable. The final value $v^{(K)}$ used at a time $t < a^{(K)}$ is in general not, because it is information that arrived later.

Definition A dataset is point-in-time if, for every time $t$ in its range, it can reproduce exactly what a user could have known at $t$. For that it stores each value with its economic period, its availability time and its version, keeps revisions as new rows instead of overwriting old ones, and records the membership of every security universe with dated entries and exits. The as-of view at $t$, the latest version of every record whose availability time is at most $t$, then generates the information set $\mathcal{F}_t$ of the recorded observations. A dataset that keeps only the latest revision, or lists only the securities that exist today, cannot reproduce $\mathcal{F}_t$, so a study built on it uses a different information set from the one it describes.

Definition Fix an initial universe $U_0$ of $N$ securities chosen at time $0$ with information in $\mathcal{F}_0$, invest $W_0/N$ in each, and hold to time $T$ without rebalancing. Let $R_i$ be the total return of holding $i$. It includes the cash or replacement securities a holder receives if the security is delisted, merged or liquidated, and it equals $-1$ if the holding becomes worthless. Terminal wealth is $\sum_i (W_0/N)(1 + R_i)$, so the portfolio return is exactly

Survivorship bias is the error from computing a return or another statistic only over the subset of securities that still exist at $T$. Survival is a future event that can depend on the return itself. In a probability model for the return $R$ of one security and the event $S$ that it survives to $T$, with $\mathbb{E}[\lvert R \rvert] < \infty$ and $\mathbb{P}(S) > 0$,

because $\mathbb{E}[R \mid S] = \mathbb{E}[R \mathbf{1}_{S}]/\mathbb{P}(S)$. So the bias has the sign of the covariance between return and survival, and it can be positive, negative or zero. When $\mathbb{P}$ is the uniform draw of one name from $U_0$, the mean $\mathbb{E}[R]$ is $R_{U_0}$ and $\mathbb{E}[R \mid S]$ is the survivor average. A strategy that trades a changing universe is a different and equally legitimate object, but it needs a dated membership rule $U_t$ and an account of the trades made when membership changes.

Definition A corporate action changes the shares or the cash attached to a holding, for example a split, a reverse split, a cash dividend, a spin-off or a merger, and a corporate action adjustment makes a return measure the change in the holder’s wealth through it. Consider $n_0$ shares bought at the ex-dividend price $P_0$ and held without trading to time $1$, with $n_1$ shares after the corporate actions, the price $P_1$ per share at time $1$, and cash distributions $C_1$ held as cash without interest. The total return is

For a $k$-for-$1$ split, $n_1 = k n_0$, where $k < 1$ describes a reverse split. With a dividend $d$ per post-split share paid after the split, $C_1 = k n_0 d$ and

which compares post-split amounts with the adjusted starting price $P_0/k$. The split-adjusted price return $k P_1 / P_0 - 1$ leaves out the dividend, and the raw price return $P_1/P_0 - 1$ compares shares that represent different fractions of the company. Reinvesting the dividends instead of holding the cash would need their reinvestment dates and prices as further data.