Basics of Risk and Return

Every investment decision balances two questions: how much might I earn? and how much could it hurt? This guide builds both answers from first principles—starting with a single rupee and ending with the ratios professionals use to compare portfolio managers.

One small dataset runs through most of the guide, so you can check every number yourself in Excel or Python.

The running example

Fictional annual returns chosen for clarity. Bond fund B is deliberately built to rise when equities fall, to make diversification visible.

What is a return?

A return measures what you gained (or lost) per rupee invested over a period. It has two parts: the change in price and any income received.

Watch the period. A 12% return over three months and 12% over three years are very different achievements. Always state the horizon, and annualise before comparing (section 05).

Arithmetic mean return

The arithmetic mean is the simple average of periodic returns. It answers: what is a typical one-year return?

The arithmetic mean is the right input for forecasting a single future period and for most portfolio mathematics. But it overstates what a buy-and-hold investor actually earned over several periods—as the next section shows.

Geometric mean return and CAGR

Returns compound: each year’s return applies to a different starting amount. The geometric mean is the constant rate that would produce the same ending wealth. When applied to start and end values, it is called the compound annual growth rate (CAGR).

Why the geometric mean is lower: volatility drag

Losses hurt more than equal gains help. A useful approximation links the two averages:

For fund A: 9.00% − 2.43%/2 ≈ 7.79%, close to the exact 8.08%. Bond fund B, with tiny volatility, has almost no gap (6.40% vs 6.38%).

Log (continuously compounded) returns

A log return converts a simple return into its continuously compounded equivalent. Its great advantage: log returns add up over time.

Annualising returns and risk

Data often comes monthly or daily, but decisions are annual. Means scale with time; standard deviations scale with the square root of time (assuming independent periods).

Risk of a single asset: variance and standard deviation

Risk is the uncertainty of outcomes. The most common measure is how widely returns scatter around their mean.

Reading the number: if returns were roughly normal, about two years in three would fall within one standard deviation of the mean—for fund A, between about −6.6% and +24.6%. Real markets have fatter tails, so treat this as a rough guide.

Why n − 1? With a sample, dividing by n − 1 corrects the tendency to underestimate the true variance. Excel’s VAR.S and STDEV.S do this; VAR.P and STDEV.P divide by n.

Beyond standard deviation

• Downside deviation: counts only returns below a target (e.g., 0%). Investors fear losses, not upside surprises.
• Maximum drawdown: the largest peak-to-trough fall. Fund A’s worst was −10% (₹1,20,000 → ₹1,08,000).
• Value at Risk: a loss threshold at a confidence level — see our VaR & Expected Shortfall guide.

Two assets: covariance, correlation and portfolio risk

Once you hold two assets, what matters is not only how risky each one is, but how they move together.

Portfolio return and risk

The general pattern (σ_A = 20%, σ_B = 8%, weights 60/40):

Multiple assets: weighted returns and the covariance matrix

With n assets, the portfolio return is still a weighted average. Risk needs every pairwise covariance, which is neatly organised in a covariance matrix Σ.

Illustration: 50% equity, 30% bonds, 20% gold.

Why this matters: as you add assets, the individual variances matter less and the covariances matter more. In a large portfolio, risk is driven mostly by how assets co-move—which is why the market-wide component cannot be diversified away.

Systematic risk and beta

Total risk has two parts. Unsystematic (specific) risk—a factory fire, a failed product—can be diversified away. Systematic (market) risk—interest rates, recessions—cannot. Investors are rewarded mainly for bearing systematic risk.

CAPM: the required return

With a 6.5% risk-free rate and a 12% expected market return, a fund with β = 1.2 must earn 6.5% + 1.2 × 5.5% = 13.1% just to compensate for its market risk.

Risk-adjusted performance: Sharpe, Treynor, Jensen’s alpha and more

A higher return is not impressive if it came with much higher risk. Risk-adjusted measures put funds on a common footing.

Illustration: which fund is best?

Assume R_f = 6.5%, market return 12%, market volatility 16%.

• Sharpe and M² pick Y: it delivers the most excess return per unit of total volatility.
• Treynor and Jensen pick Z: after adjusting only for market risk, Z adds the most. But Z carries a lot of non-market risk (σ = 28%), which matters if you hold it alone.
• All three beat the market on every measure here—in practice, many funds do not.

Excel and Python toolkit

import numpy as np

A = np.array([0.20, -0.10, 0.25, -0.05, 0.15])   # equity fund
B = np.array([0.04,  0.09, 0.05,  0.08, 0.06])   # bond fund
M = np.array([0.15, -0.08, 0.20, -0.02, 0.12])   # market

arith = A.mean()                                  # 0.0900
geo   = np.prod(1 + A) ** (1 / len(A)) - 1        # 0.0808
logr  = np.log(1 + A)                             # sums to 0.3886
sd    = A.std(ddof=1)                             # 0.1557
corr  = np.corrcoef(A, B)[0, 1]                   # -0.95
beta  = np.cov(A, M, ddof=1)[0, 1] / M.var(ddof=1)  # 1.31

w     = np.array([0.6, 0.4])
cov   = np.cov(np.vstack([A, B]), ddof=1)
sd_p  = np.sqrt(w @ cov @ w)                      # 0.0856

rf = 0.065
sharpe = (arith - rf) / sd                        # 0.16

Common mistakes to avoid

• Quoting the arithmetic mean as the return an investor actually earned over many years.
• Annualising volatility by multiplying by 12 instead of √12.
• Averaging log returns across assets to get a portfolio return (use simple returns across assets).
• Treating correlation as stable—correlations often rise towards +1 in a crisis, exactly when diversification is needed. See Copulas & Tail Dependence.
• Mixing periods: monthly returns with an annual risk-free rate in a Sharpe ratio.
• Ranking funds on one ratio over one short period, or against the wrong benchmark.
• Forgetting that beta and correlation describe linear co-movement only.

Frequently asked questions