Returns and Risk: Measuring Performance and Volatility in Financial Markets

The concept of return is central to finance. Returns allow us to measure the performance of an investment and compare it across different assets. Understanding the nuances of returns is crucial for making informed investment decisions.

Consider buying a share of a company, say STV, at time $t_0$ and selling it at time $t_1$. If the price of the stock at $t_0$ is $P_0$ and at $t_1$ is $P_1$, the arithmetic return over this period is given by:

This formula measures the percentage change in the stock price over the period.

If the company pays a dividend $D_1$ at time $t_1$, the return formula becomes:

This gives a more complete picture of the investment’s performance. While commissions and fees technically affect returns, they are typically small relative to price gains and dividends, so we will ignore them for simplicity.

Total return in dollars can also be calculated as:

For this course, we will mostly focus on percent returns, but “total return” and “return” will be used interchangeably. When there are no dividends, both formulas are equivalent.

An alternative approach uses logarithmic returns:

where $ln$ is the natural logarithm. Log returns are often used in finance because they are time-additive, which simplifies calculations for multi-period returns.

Dividends can also be incorporated in log returns:

Example:

• Buy a stock on Jan 2, 2024, for $125.
• On Feb 28, 2024, it is worth $153.

The arithmetic return is:

The log return is:

Log returns are slightly lower than arithmetic returns for positive returns, and the difference increases with larger returns. For small returns, the two measures converge.

Which method should you use? It depends on how you want to aggregate returns over time.

Suppose we invest $X_0$ dollars in an asset and observe returns over multiple periods:

• Arithmetic returns measure the simple percent change between two points in time.
• Log returns assume continuous compounding, making them additive across multiple periods:

For long-term or multi-period analysis, log returns are often preferred because they accurately capture compounded growth. Arithmetic returns are simpler and often sufficient for short-term analysis or when compounding is not a major concern.

Suppose instead we remove profits at the end of each period, so that at the start of every subperiod, the investment is reset to its original amount $X_0$. In this case, the cumulative return is calculated differently:

Here, we use a different symbol $(R_{rebalance})$ to distinguish it from the compounded total return. This method is called rebalancing:

• If a subperiod return is positive, we take out the gain (spend it or reinvest elsewhere).
• If the return is negative, we add funds to bring the investment back to $X_0$.

Rebalancing is a useful concept because it mirrors situations where an investor maintains a constant investment base across periods rather than letting gains compound within the same account.

To understand why log returns are often preferred, consider the cumulative return with continuous compounding:

Taking the natural logarithm of both sides:

If we define the log return of a subperiod as $r_i = \ln(1 + R_i)$, this becomes:

In other words, the total return under continuous compounding equals the sum of log returns across subperiods:

This additive property is one of the main reasons log returns are favored in finance.

For small returns $(|R_i| < 0.1)$ arithmetic and log returns produce almost identical results:

Thus, for short-term or low-volatility assets, either method works well.

Another reason to prefer log returns is statistical: if prices follow a log-normal distribution, then log returns are normally distributed, which simplifies many quantitative finance models. This property is particularly important in derivative pricing, which will be explored in more detail in the advanced course.

When working with financial data, the timeframe over which returns are calculated matters. Suppose we have daily closing prices for an asset. Calculating daily returns is straightforward, but what if we want weekly, monthly, or quarterly returns?

• Weekly returns: Use closing prices on a specific day of the week (e.g., Friday). The day chosen is arbitrary; Monday or Wednesday works the same.
• Monthly returns: Use the last trading day of each month. Months vary in length, so this standardization ensures consistency.
• Quarterly returns: Commonly use March 31, June 30, September 30, and December 31.
• Annual returns: Typically from December 31 to December 31 of the following year.

Important: The granularity of your data limits the timeframes you can compute. For instance, if you have only monthly prices, weekly or daily returns cannot be derived.

• Stock markets are usually open Monday to Friday. Weekends and holidays have no official data.
• Data providers may carry forward Friday’s closing price to fill missing weekend days. This can artificially reduce volatility in daily returns.
• Cryptocurrencies trade 24/7, so they do not have this issue.
• When comparing stocks and crypto, always account for differences in data availability.

Also, note that if we have $n$ daily closing prices, we can calculate only $n-1$ daily returns, since returns are based on price changes between consecutive days.

Similarly, computing weekly or monthly returns reduces the number of available observations. Reliable statistical modeling requires sufficient data points, so longer timeframes or fewer observations may limit analysis.

It can be tricky to compare returns from different periods:

• Example: A trader boasts 3% monthly returns, while another claims 20% annual returns. Which investment performed better?

To make returns comparable, investors and funds often annualize returns.

Example:

• Assets in Fund X: $100,000 on July 1, $110,000 on December 31.
• Six-month return: $\frac{110,000 - 100,000}{100,000} = 0.10 = 10\%$.
• Annualized return (extrapolating 6 months to 12 months): $10\% \times 2 = 20\%$.

Caution: Annualized returns are not actual returns; they are hypothetical projections.

• Example: Buy a stock for $2, sell the next day for $2.20 → 10% daily return.
• Annualized (assuming 365 days): 10% × 365 = 3,650%!

Clearly, a single day’s return cannot predict long-term performance. Actual returns fluctuate, and daily luck does not equal skill.

Key takeaway: Always distinguish between:

1. Actual annual returns: Based on prices exactly one year apart.
2. Annualized returns: Extrapolated from shorter periods, which may not reflect reality.

• Standards like GIPS (Global Investment Performance Standards) restrict reporting of annualized returns for partial periods.
• Example: A fund started Nov 1, 2023, reports a 2-month return of 4%. Annualized (using compounding) → 26.53%.
• According to GIPS, only the 4% partial-period return should be reported. Presenting 26.53% is misleading.

Investor guidance: Always check:

• Whether returns are annualized or actual
• The time period used for reporting
• If short-term returns are cherry-picked to appear more impressive

This awareness helps avoid being misled by exaggerated performance claims.

A savvy investor who understands the difference between actual performance and annualized or extrapolated returns would often start by de-annualizing any presented figures.

For example, if a firm reports a 26.53% annualized return for a 2-month period, the actual 2-month performance can be calculated as:

This gives the true performance over the actual time the investment was held.

Another useful measure is the holding period return (HPR), which simply answers: “How much did I make on this investment?”

• Suppose you invested $100,000 and later sold the investment for $170,000.

• The holding period return is:

• Note: This return does not account for the length of time you held the investment; it only measures the raw gain.

A more comprehensive measure is the net return, which accounts for all inflows and outflows, including transaction costs and dividends.

Example:

• Buy stock: $100 + $2 commission
• Dividend received: $7
• Sell stock: $115

The net return formula is:

• Applying it:

Note: Commissions, fees, and other costs reduce returns, and while taxes also affect returns, they are not included here due to variability across countries.

So far, we’ve focused on expected returns – what we anticipate earning on an investment. But investing in risky assets introduces uncertainty: the actual return may differ from the expected return.

• Expected return can be thought of as the average outcome from a set of possible outcomes.
• But averages alone don’t tell the full story. We also want to know: How far are actual outcomes likely to deviate from the average?

This leads us to volatility, measured by the standard deviation of returns.

• Standard deviation quantifies the dispersion of returns around the expected value.
• A higher standard deviation means returns are more spread out and less predictable, while a lower standard deviation implies returns are more consistent.

In short:

• Expected return tells us what we hope to earn.
• Volatility tells us how uncertain that outcome is.

Understanding both metrics is essential for evaluating the risk-return tradeoff of any investment.

Volatility measures the fluctuations in asset prices or returns over time. While standard deviation is the most commonly used measure, volatility is a broader concept that captures the unpredictability of markets.

Financial markets are complex systems with many participants who have different objectives, risk tolerances, and time horizons. These differences, combined with factors like news, cash flow needs, and tax considerations, drive price movements.

• Some traders react quickly to news, while others take time to analyze and decide.
• Algorithms can respond to headlines instantaneously, creating sudden market swings.
• Investors with longer-term horizons may wait for confirmation before acting.

Because participants act differently and at different speeds, volatility is an emergent property of the market. News and information do not affect all participants equally, which is one reason why volatility can change over time.

• Price Range

A simple way to measure volatility is Zprice range**, the difference between the high and low prices within a trading period.

Example:

• Friday: High = $105, Low = $99 → Price range = $6
• Monday: High = $106, Low = $104 → Price range = $2

Using price range, Friday was more volatile than Monday.

Key points:

• Price range is measured in units of price, not returns.

• You need high and low prices for the period; closing prices alone are insufficient.

• Intraday price range represents the maximum possible gain for a day trader who bought at the low and sold at the high.

• Standard Deviation of Returns

Another approach uses returns instead of prices.

• Calculate daily returns over a period (e.g., 21 trading days ≈ 1 month).
• Compute the standard deviation of these returns.

Interpretation:

• High standard deviation → more volatile stock

• Low standard deviation → less price fluctuation

• Units: Returns (percentage), which differs from the price-range measure.

• Example: Average daily return = 0.2%, standard deviation = 0.5% → assuming normal distribution: 68% of daily returns fall between -0.3% and 0.7%

• ** Variance**

Variance is the average squared deviation from the mean.

• Returns above the mean → positive deviation
• Returns below the mean → negative deviation

We are interested in the magnitude of fluctuations, not direction. Squaring deviations eliminates negative signs, giving variance:

• Standard deviation is the square root of variance, restoring the original units of return.

Important:

• Variance and standard deviation are related but not interchangeable.
• Standard deviation is typically used in discussions of volatility because it is in the same units as returns.
• Example: Standard deviation = 20% → Variance = 0.2² = 4%

We’ve discussed returns and volatility separately. But how can we use both together to make better investment decisions?

Ideally, we want high returns with low volatility.

• High returns → higher profits
• Low volatility → more predictability and lower risk of losses

One way to combine these is the coefficient of variation (CV):

• Volatility is the numerator, return is the denominator.
• Intuition: How much risk (volatility) do you incur to achieve a given unit of return?

Example:

• Stock A: CV = 50% → 50 units of volatility per unit of return
• Stock B: CV = 25% → 25 units of volatility per unit of return

Interpretation: Stock B is “cheaper” in terms of risk – it requires less volatility to achieve the same return. A lower CV indicates a more efficient tradeoff between risk and return.

Caveat: This assumes:

• Data samples are representative of future performance
• Calculations are accurate

Instead of dividing volatility by return, we can flip the formula:

• Interpretation: Stock C has higher return per unit of risk, so it is preferred.

Key difference from CV:

• CV: Lower number is better (volatility per unit of return)
• Sharpe Ratio: Higher number is better (return per unit of volatility)

This approach allows investors to compare assets across different stocks or even asset classes, capturing both expected return and risk in a single measure.