Converting Prices Into Matched Return Series

Why Price Data Must Be Prepared Carefully

A beta analysis uses paired observations: for each period, one return for an asset and one return for a market proxy. The calculations that follow depend on those pairs being comparable. A price change alone may omit dividends, a calendar mismatch may compare different holding periods, and a missing quote may create a return that was never observed. Data preparation is therefore part of the financial analysis, not just a formatting step.

This tutorial develops a consistent way to convert prices into paired returns. It focuses on the data choices that affect the resulting observations: cash distributions, adjusted prices, dates, missing values, and return conventions. The procedures are aligned with the SOA VEE Accounting and Finance syllabus.

Price Returns and Total Returns

A price return answers, “By what proportion did the quoted price change?” It is appropriate when the intended measure is price appreciation alone or when distributions are not part of the chosen data series. A total return answers, “What return did an investor receive from the asset, including its distribution?” It is generally the relevant measure when comparing the performance of dividend-paying assets with a market total-return series.

The total-return formula follows from the investor's one-period gain. An investor begins with one share worth \(P_{t-1}\). At the end of the period, that share is worth \(P_t\), and the investor has received cash \(D_t\). The dollar gain is therefore \(P_t+D_t-P_{t-1}\). Dividing this gain by the initial investment gives the return:

This convention treats the distribution as cash received during the interval. If the intended convention assumes reinvestment at the distribution date, the ending wealth can differ because the reinvested cash earns a return over the rest of the interval. A total-return index or a properly constructed adjusted-price series may incorporate that reinvestment convention. Use the data provider's definition rather than assuming all “adjusted” series are built identically.

For a market proxy, the same distinction applies. A price index records changes in index prices or levels, while a total-return index also reflects distributions under its stated convention. Pairing an asset total return with a market price return mixes definitions. It may be a deliberate choice in a specific analysis, but it should not happen accidentally.

Adjusted Prices and Corporate Actions

A provider may publish adjusted prices that account for some or all distributions and corporate actions. If \(A_t\) is a consistently adjusted price series, the simple return implied by that series is \(A_t/A_{t-1}-1\). When the adjustment includes dividends under a total-return convention, this ratio can represent a total return. When it only adjusts for splits, it remains a price return apart from the mechanical split correction. The label “adjusted close” by itself is not a complete definition.

A stock split changes the number of shares and quoted price without, by itself, changing the investor's wealth. For example, in a two-for-one split, each old share becomes two shares and the per-share price is approximately halved. An unadjusted price ratio would show an artificial loss. A correctly split-adjusted series removes this mechanical discontinuity. Dividends and splits are different: a split changes share units, while a dividend is a cash distribution.

Before calculating returns from a vendor series, establish what it adjusts for, whether the adjustments are applied consistently over the full history, and whether the market series uses a compatible convention. If a reliable adjusted total-return series is used, do not add the dividend again; that would count the same distribution twice. If using unadjusted prices, include cash distributions explicitly when total returns are required.

Matching Dates and Return Intervals

A return is attached to an interval, not merely to the date printed beside it. If a price is observed on date \(t-1\) and again on date \(t\), the return covers the period from the first date through the second. The asset and market observations form a pair only when both returns cover the same start and end dates and use compatible frequencies and conventions.

A practical construction begins with the dates on which the selected frequency is to be measured, such as month-end. For each interval between consecutive selected dates, calculate the asset return and market return using those same endpoints. If one series has a quote on a different day, decide on a consistent date rule—such as the last available trading close on or before month-end—and apply it to both series. A holiday can mean the most recent trading date is not the calendar month-end; that does not justify using different rules for the asset and market.

With a missing observation, do not automatically carry the previous price forward. Forward-filling creates an artificial zero return for one interval and shifts the eventual price movement into another interval. Nor should one calculate an asset return over two months and pair it with a one-month market return. If a needed endpoint is unavailable, omit the affected pair or adopt a clearly defined alternative interval for both series. When interval lengths differ, keep that fact explicit; do not describe a multi-month return as a monthly return.

A matched series should also use a consistent currency and return definition. If an asset is quoted in one currency and the market proxy in another, currency movements can affect the comparison. A calculation must state whether returns are local-currency or converted to a common currency. Similarly, choose simple returns or another return convention consistently; do not mix them within a series.

Common Errors and Final Checks

The most frequent data-preparation errors are simple to state but can materially change the observations supplied to a later analysis. Before using a paired series, check the following:

• Omitting distributions unintentionally. A price-only return is not a total return when a cash distribution occurred. Decide which measure is intended.
• Counting dividends twice. Do not add a cash dividend to a price series whose adjustment already includes that dividend.
• Assuming every adjusted series means total return. Confirm whether the adjustment covers dividends, splits, or both.
• Pairing different dates. A return from January 31 to February 28 cannot be paired with a return from February 1 to February 28 and treated as the same interval.
• Filling missing prices without justification. A carried-forward value can create false zero returns; an interpolated value is not an observed price.
• Mixing return definitions or units. Keep frequency, simple-versus-other return convention, currency, and total-versus-price basis consistent.

These checks prepare the data; they do not calculate or interpret beta. Once each observation is genuinely paired, the resulting series can be used in subsequent risk analysis without confusing data construction with the statistical calculation.