Expected Return and Variance — The Risk-Return Duo That Actually Talks Back

"If expected return is the promise, variance is the sneaky fine print." — Your slightly unhinged finance TA

This lesson builds on our earlier chat about arithmetic vs geometric returns and the hands-on work you've been doing in Excel/Python for returns and portfolio analytics. We also lean on the visual intuition you got from risk–return plots. Now we answer: how do you compute expected return and variance, what do they mean, and why your backtests keep lying to you if you’re sloppy about estimators and probability.

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What is expected return (and why it matters)

Expected return is the probability-weighted average of possible outcomes — the mean of the return distribution. It’s what the market on average gives you next period (if your probability model is right). In symbols:

For a discrete set of returns r_i with probabilities p_i:

E[R] = Σ p_i * r_i

For empirical returns (the usual case), we use the arithmetic mean as the one-period expected return estimate:

E[R] ≈ (1/N) * Σ r_t

Reminder from the arithmetic vs geometric conversation: use the arithmetic mean to estimate the expected next-period return. Use the geometric mean when you care about long-run compounded growth.

Practical Excel:

• Arithmetic mean: =AVERAGE(range)
• If you have probability weights: =SUMPRODUCT(returns_range, probs_range)

Python (pandas):

import numpy as np
expected_return = np.mean(returns_array)
# with weights/probabilities
expected_return = np.dot(returns_array, probs_array)

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What is variance (and why it’s the risk metric everyone quotes but not everyone understands)

Variance measures the expected squared deviation from the mean. It’s the spread — how wildly returns dance around the expected value.

Mathematically:

Var(R) = E[(R - E[R])^2]

The square root of variance is the standard deviation (a.k.a. volatility), which brings units back to % returns. Use variance when you do algebra (because it behaves nicely); use standard deviation when you want interpretable units.

Sample formulas (empirical data):

• Sample variance (unbiased): s^2 = (1/(N-1)) * Σ (r_t - r̄)^2
• Population variance (if you knew the true distribution): σ^2 = (1/N) * Σ (r_t - μ)^2

Excel:

• Sample SD: =STDEV.S(range)
• Sample variance: =VAR.S(range)

Python:

sample_var = np.var(returns_array, ddof=1)  # ddof=1 gives unbiased sample variance
sample_std = np.std(returns_array, ddof=1)

Why N-1? Because using the sample mean r̄ selects a parameter from the same data, reducing degrees of freedom — N-1 corrects bias.

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Quick numeric example (discrete probabilities)

Imagine a toy asset with returns: -10%, 0%, +10% each with probability 1/3.

Expected return:

E[R] = (1/3)(-0.10) + (1/3)(0) + (1/3)(0.10) = 0

Variance:

Var(R) = (1/3)((-0.10 - 0)^2 + 0^2 + (0.10 - 0)^2) = (1/3)(0.01 + 0 + 0.01) = 0.006666...

SD ≈ 0.08165 = 8.165%

Interpretation: zero expected return, but reasonable volatility. Not every asset with zero mean is safe; variance tells the rest of the story.

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Portfolio expected return and variance — yes, it’s linear and then not

Expected portfolio return is blissfully linear:

E[R_p] = Σ w_i * E[R_i]  = w'μ

Portfolio variance is not linear because of covariances:

Var(R_p) = w'Σw = Σ_i Σ_j w_i w_j Cov(R_i, R_j)

That Cov(R_i,R_j) term is the secret sauce of diversification. Two assets with high individual variance can produce a low-variance portfolio if their covariance is negative.

Excel hints:

• Compute expected returns vector: =AVERAGE(range_for_asset1), etc.
• Covariance matrix: =COVARIANCE.S(range1, range2) for each cell.
• Portfolio variance: =MMULT(TRANSPOSE(weights_range), MMULT(cov_matrix_range, weights_range))

Python (numpy):

import numpy as np
w = np.array([0.6, 0.4])
mu = np.array([0.08, 0.12])
Sigma = np.array([[0.04, 0.006], [0.006, 0.09]])
E_portfolio = w.dot(mu)
Var_portfolio = w.dot(Sigma).dot(w)
Std_portfolio = np.sqrt(Var_portfolio)

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Practical caveats — the traps that make your backtest a liar

• Estimation error: sample means and covariances are noisy, especially for small N and many assets. Your “optimal” weights will bake in noise.
• Non-normal returns: variance captures dispersion but not skewness/kurtosis (fat tails). Two assets with same mean & variance can be worlds apart in tail risk.
• Time-varying volatility: volatility clusters; past variance ≠ future variance unless stationary.
• Look-ahead and survivorship bias: if your historical sample ignores dead funds, your mean/variance estimates will be too rosy — remember our backtesting pitfalls.
• Arithmetic vs geometric misuse: using geometric mean as expected return for next period undercounts; arithmetic mean is the right estimator for single-period expectation.

Small, important truth: expected return and variance are models, not gospel. Treat them like imperfect maps — useful until you meet reality.

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How this ties to visualization and decisions

You’ve already plotted mean vs standard deviation in risk–return visuals. Use the expected return on the y-axis and SD on the x-axis to place assets/portfolios. Points left/up are superior (higher return, lower risk). But don’t forget error bars — the uncertainty in your estimated expected return and variance is often large; plot confidence intervals or run bootstrap resamples.

Practical metric: Sharpe ratio = (E[R] - R_f) / σ. It’s a normalized tradeoff of return per unit of risk. But remember: it assumes returns are symmetric and uses sample σ that might be unstable.

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Exercises to lock it in (do these in Excel or Python)

1. Compute arithmetic and geometric mean of a 10-year monthly return series. Explain which you’d use for next-month expected return vs long-run growth.
2. Build a covariance matrix for 4 assets. For random weights, compute portfolio E[R] and Var(R). Repeat 10,000 times and plot the resulting cloud on mean–SD axes (you'll get the efficient frontier emerging).
3. Bootstrap the sample mean 1,000 times to get a confidence interval for expected return. How wide is it? How confident are you, really?

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Key takeaways (read these like commandments)

• Expected return = probability-weighted average. Use arithmetic mean for single-period expectation.
• Variance = expected squared deviation. SD is its interpretable square-root cousin.
• Portfolio expected return is linear; variance depends on covariances. Diversification lives in covariance terms.
• Estimators lie when data is thin or biased. Watch out for estimation error, non-normal tails, and backtest pitfalls you’ve already met.

Final nugget: Expected return and variance give you the first two terms of the story. If you want a full novel, add skewness, kurtosis, regime shifts, and — crucially — humility.

Version note: this lesson assumes you’ve done hands-on return calculations in Excel/Python and seen risk–return plots; it’s meant to take those tools from "I can plot" to "I can interpret and survive the statistical nastiness." Good luck, and don’t trust a model that smiles too much.