What you will learn
- Compound returns and annualize volatility correctly.
- Calculate two-asset portfolio volatility at different correlations.
Measuring return
A period return is the change in value plus any income, divided by the starting value. Total return includes dividends or interest while price return does not, so using price return alone understates an income-paying asset.
Returns compound by multiplication, not addition. A gain of 50% followed by a loss of 50% leaves 1.5 × 0.5 = 0.75, a 25% loss, which is why large losses are hard to recover.
Volatility and annualizing
Volatility is the standard deviation of returns and measures how widely they scatter around their average. It describes variability, not direction, and because it is estimated from a sample it changes with the period chosen.
To annualize, multiply by the square root of periods per year: about √252 for daily data and √12 for monthly data. This assumes returns are independent over time, which holds only approximately.
Correlation and diversification
Correlation ranges from -1 to +1 and measures how similarly two return series move. It reflects direction of co-movement, not size, and it is unstable, tending to rise when markets fall together in stress.
Combining assets with correlation below 1 gives a portfolio volatility below the weighted average of their individual volatilities. This is diversification. It reduces asset-specific risk but cannot remove risk shared by the whole market.
Worked example: Two assets, different correlations
Two fictional assets each have 20% volatility and are held 50/50. Portfolio variance is 0.02 + 0.02 × correlation. At correlation 1 volatility is √0.04 = 20.0%; at 0.5 it is √0.03 = 17.3%; at 0 it is √0.02 = 14.1%. The same two holdings carry less risk when they move less in step, though in a crisis correlations tend to rise and the benefit shrinks.
Check your understanding
An asset gains 40% in year one and loses 40% in year two. What is the two-year return? Separately, what does a daily volatility of 1.2% become when annualized with 252 trading days?
Show answer and explanation
The two-year result is 1.4 × 0.6 = 0.84, a 16% loss, not zero. Annualized volatility is 1.2% × √252 ≈ 1.2% × 15.87 ≈ 19.0%. Both answers show why averaging percentages naively can mislead.