Expected return and variance

Two formulas. The first is unremarkable and the second is the reason this subject exists.

Expected return is a weighted average

$E(R_p) = \sum_{i=1}^{n} w_i \, E(R_i)$

This is exact, not an approximation. It follows from expectation being linear — the property the Quantitative methods subject derives — which holds whatever the assets are, however they are related, and whatever shape their distributions have.

Two consequences that are worth saying out loud, because people expect more from diversification than it gives:

Diversifying cannot raise your expected return above your best holding's. A portfolio's expected return is pinned between the lowest and highest of its components. If you want more expected return you must hold more of something with more expected return, and that is the only way.

Nothing is gained or lost in combining. Mix a 7% expectation and an 11% expectation 40/60 and you get exactly 9.4%. There is no interaction term.

So if risk behaved the same way, portfolio construction would not be a subject — you would rank everything by expected return, buy the top one, and be done. Chapter 1 is Markowitz's argument for why that is not what anyone does.

Variance is not

$\operatorname{Var}(R_p) = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i w_j \sigma_{ij}$

where $\sigma_{ij}$ is the covariance between assets $i$ and $j$, and $\sigma_{ii} = \sigma_i^2$ is asset $i$'s own variance.

Read the double sum before moving on. With $n$ assets there are $n$ variance terms on the diagonal and $n^2 - n$ covariance terms off it. For a twenty-holding portfolio that is 20 variances and 380 covariances — so by sheer count, a portfolio's risk is mostly about how its holdings relate to each other, not about how risky they are individually.

Markowitz says this was the point at which the approach became credible to him:

Variance (or, equivalently, standard deviation), came to mind as a measure of risk of the portfolio. The fact that the variance of the portfolio, that is the variance of a weighted sum, involved all covariance terms added to the plausibility of the approach.

That is the subject's central claim, in its author's words. Risk measured this way is automatically a property of combinations, because the mathematics of a weighted sum forces every pair into the answer.

The two-asset case, written out

With two assets the double sum has four terms, two of which are identical:

$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\,\rho_{12}\,\sigma_1\sigma_2$

using $\sigma_{12} = \rho_{12}\sigma_1\sigma_2$.

The first two terms are what you would guess. The third is the one that does the work, and chapter 3 is entirely about it.

Notice what the third term is not. It is not a correction or a refinement. At $\rho = -1$ it can cancel the other two exactly and take portfolio risk to zero with two risky assets. Nothing in the expected-return formula can do anything remotely like that.

Why variance, and the honest objection

Variance is a strange choice at first sight: it counts an unexpectedly good year as risk.

The case for it is mathematical, and it is strong. Variance of a sum decomposes into terms involving every pair, and those terms are computable from data. No other candidate measure combines so cleanly. Downside measures do not aggregate like this, which is why the subject that can be built on variance is so much larger than the subject that can be built on anything else.

The case against it is empirical, and that is strong too. The Measuring your return subject sets out the defects in detail: variance punishes upside, assumes thinner tails than markets have, can be flattered by illiquidity, and says nothing about the worst case. Chapter 9 picks up the ratios built on it, and chapter 12 asks what survives the objections.

The honest position, held throughout this subject: variance is used because it is tractable, it is a decent proxy for what people experience as risk most of the time, and it fails exactly when it matters most. Knowing which of those is operating at a given moment is most of the skill.

What the numbers actually look like

The Nifty 50's own factsheet gives both quantities for a real portfolio, measured to 30 September 2026:

Horizon	Annualised total return	Annualised standard deviation
5 years	6.38%	13.82%
Since inception	12.12%	22.39%

Two things to take from this before the theory goes further.

The standard deviation is roughly twice the return, over both horizons. That ratio is the thing the whole subject is trying to improve, and chapter 9 names it.

The five-year window is a poor one. Indian equities returned 6.38% a year over it, against a policy repo rate of 5.50%. Any example built on recent five-year numbers will make equities look barely worth owning, and any built on since-inception numbers will make them look wonderful. Both are real, and the gap between them is a warning about estimating these inputs that chapter 12 returns to.

Working the problem

Equity: $\mu = 11\%$, $\sigma = 14\%$. Debt: $\mu = 7\%$, $\sigma = 4\%$. Correlation 0.2. Weights 60/40.

(These are illustrative assumptions chosen to be round, not forecasts.)

Expected return.

$E(R_p) = 0.6 \times 11 + 0.4 \times 7 = 6.6 + 2.8 = 9.40\%$

And the weighted average of the two holdings' expected returns is 9.40%. Identical, necessarily — it is the same calculation.

Variance.

$\sigma_p^2 = (0.6)^2(14)^2 + (0.4)^2(4)^2 + 2(0.6)(0.4)(0.2)(14)(4)$

$= 70.56 + 2.56 + 5.376 = 78.496$

$\sigma_p = \sqrt{78.496} = 8.86\%$

Now the comparison. The weighted average of the two standard deviations is

$0.6 \times 14 + 0.4 \times 4 = 8.4 + 1.6 = 10.00\%$

The portfolio's actual risk is 8.86%, not 10.00%. You get the full 9.40% of expected return and 1.14 percentage points less risk than the weighted average of the parts.

Where the 1.14 points came from. Nowhere — no asset gave it up. It is a consequence of the two holdings not moving together, and it is available to anyone who holds both. The Risk subject calls diversification "the only thing in investing that improves one side of the trade without costing the other"; this is that sentence as arithmetic.

How the gap depends on the correlation, holding everything else fixed:

Correlation	Portfolio standard deviation	Below the weighted average by
+1.0	10.000%	0.000
+0.5	9.304%	0.696
+0.2	8.860%	1.140
0.0	8.551%	1.449
−0.3	8.066%	1.934

The expected return is 9.40% in every row. Only the risk column moves. That is the asymmetry, and it is the entire reason a portfolio is worth thinking about as a portfolio.

And the limiting case tells you what the weighted average was. At $\rho = +1$ the portfolio's risk is exactly 10.00% — the weighted average. So the weighted average is not a neutral benchmark; it is the answer you get when the holdings move in lockstep, which is the worst case rather than the normal one.

The point

A portfolio's expected return is the weighted average of its holdings' expected returns, exactly, with no interaction — so diversification can never raise expected return above the best holding's. Its variance is a double sum over every pair, dominated by covariance terms rather than by individual variances, which is why Markowitz found the measure plausible in the first place. In the two-asset case the cross term $2w_1w_2\rho\sigma_1\sigma_2$ is the whole difference: a 60/40 mix of 14% and 4% volatility assets at a correlation of 0.2 has a standard deviation of 8.86% rather than the 10.00% weighted average, with no expected return given up. The weighted average is what you would get if the holdings moved together perfectly.