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Portfolio & Asset Pricingintermediate

Return and Risk

Every investment is judged on two numbers, its return and its risk. The holding-period return combines income and capital gain over a horizon, and we summarise a series of returns with either the arithmetic mean or the compounding geometric mean. Risk is measured by the variance or its square root, the standard deviation, of those returns. Because investors dislike uncertainty, a risk-averse investor demands a higher expected return to bear higher risk, an idea captured formally by a utility function.

Why it matters

Return tells you how much you made. Risk tells you how reliably you made it. The arithmetic mean answers the question of what you expect next period, while the geometric mean answers what you actually grew at over many periods, and the gap between them widens as returns get more volatile. Standard deviation is the everyday language of risk because it puts the spread of outcomes back into percentage-point units. Risk aversion is simply the observation that a sure dollar feels better than a coin-flip averaging a dollar, so risky assets have to pay you to hold them.

Formulas

Holding-period return
R=P1−P0+D1P0R = \dfrac{P_1 - P_0 + D_1}{P_0}
Capital gain plus income, divided by the price paid. P0P_0 and P1P_1 are start and end prices and D1D_1 is the cash distribution over the period.
Geometric mean return
RG=[ ∏t=1T(1+Rt)]1/T−1R_G = \left[\,\prod_{t=1}^{T}(1 + R_t)\right]^{1/T} - 1
The constant per-period rate that reproduces the total compounded growth. It is always at or below the arithmetic mean, and the gap grows with volatility.
Variance and standard deviation of returns
σ2=1T−1∑t=1T ⁣(Rt−Rˉ)2,σ=σ2\sigma^2 = \dfrac{1}{T-1}\sum_{t=1}^{T}\!\left(R_t - \bar{R}\right)^2, \qquad \sigma = \sqrt{\sigma^2}
Average squared deviation from the mean return, with σ\sigma reported in the same percentage units as the returns themselves.

Worked examples

Scenario

A share is bought at A$50, pays a A$2 dividend, and is sold a year later at A$54. Find the holding-period return.

Solution

The capital gain is 54 minus 50, or 4 dollars, and income is 2 dollars, so R=(4+2)/50=0.12R=(4+2)/50=0.12, a return of 12%. Of that, 8% is price appreciation and 4% is dividend yield, which is why a total-return view matters rather than price alone.

Scenario

A fund returns +50%+50\% then −50%-50\% over two years. Compare its arithmetic and geometric mean returns.

Solution

The arithmetic mean is (50%+(−50%))/2=0%(50\% + (-50\%))/2 = 0\%, but A$100 grows to V=100×1.5×0.5=75V=100\times 1.5\times 0.5=75 dollars, a true loss. The geometric mean is 1.5×0.5−1≈−13.4%\sqrt{1.5\times 0.5}-1\approx -13.4\% per year. The arithmetic mean overstates realised growth whenever returns are volatile, so use the geometric mean to describe past performance.

Common mistakes

  • ✗The arithmetic and geometric mean are interchangeable. They answer different questions, and the geometric mean is always lower once returns vary, so quoting the arithmetic mean as a track record flatters a volatile fund.
  • ✗Standard deviation only counts downside losses. Variance treats upside and downside deviations symmetrically, so a fund with large positive surprises also records a high standard deviation.
  • ✗A higher expected return is always better regardless of risk. A risk-averse investor weighs return against risk through utility, so a higher mean paired with far more variance can leave the investor worse off.
  • ✗Risk aversion means refusing all risky assets. Risk aversion only means demanding compensation for risk, and a risk-averse investor still holds risky assets when the expected reward is sufficient.

Revision bullets

  • •Holding-period return adds capital gain and income over the price paid
  • •Arithmetic mean for the expected next period, geometric mean for realised growth
  • •Geometric mean sits at or below the arithmetic mean, gap rising with volatility
  • •Variance and standard deviation measure the spread of returns
  • •Risk aversion means demanding more expected return for more risk, framed by utility

Quick check

A stock falls then rises by the same percentage each period. Which statement about its mean returns is correct?

What does it mean to say an investor is risk-averse?

Connected topics

DiversificationPortfolio VarianceCapital AllocationCAPM & SMLPerformance Eval

Sources

  1. Brailsford, Heaney & Bilson (2015), Ch. 6
    Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
    Defines holding-period return, arithmetic and geometric means, variance as risk, and risk-averse utility.
  2. Bodie, Kane & Marcus (2021), Ch. 5
    Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
    Reference treatment of returns, risk measurement, and risk aversion in a utility framework.
How to cite this page
Dr. Phil's Quant Lab. (2026). Return and Risk. Derivatives Atlas. https://phucnguyenvan.com/concept/im-return-and-risk
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