What a portfolio is, mathematically

The Risk subject argued for diversification and the Measuring your return subject measured what a portfolio did. This subject asks the question underneath both: what is a portfolio, such that those arguments work?

The definition

A portfolio of $n$ holdings is a set of weights:

$w_1, w_2, \ldots, w_n \qquad \text{with} \qquad \sum_{i=1}^{n} w_i = 1$

where $w_i$ is the share of total value held in asset $i$.

Three consequences follow immediately, and they are the reason for stating it this way.

Rupees do not appear. A portfolio with ₹10 lakh and one with ₹10 crore in the same proportions are the same portfolio. Everything in this subject is scale-free, which is why the same results apply to a first SIP and to a pension fund.

The weights are not chosen independently. They sum to one, so raising one lowers the others. You cannot add a holding; you can only change a set of proportions. A decision to buy is always also a decision to hold less of everything else.

Weights move without you. They are ratios of values, so they change whenever any price changes. A portfolio left alone does not stay the same portfolio — which is chapter 11's subject.

Negative weights

A weight can be below zero. $w_i < 0$ means a short position: you have sold something you do not own, or borrowed to buy more of something else.

The constraint still holds. If $w_{\text{equity}} = 1.5$ and $w_{\text{cash}} = -0.5$, they sum to one, and the portfolio is an equity holding one and a half times your capital, financed by borrowing half of it.

The Option pricing subject's replicating portfolio was exactly this — 0.5 units of the index with $w < 0$ in a loan. The two subjects are using the same object, and that is not a coincidence: a replicating portfolio is just a portfolio with its weights chosen to match a payoff instead of to suit an investor.

Most individuals face $w_i \ge 0$, since shorting requires a facility and a margin account. That restriction matters for chapter 5, where it bends the shape of the result.

Why one number is not enough

Here is the argument the subject rests on, and Markowitz states it in his own account of how the theory began.

Start from the natural idea that an asset is worth the present value of what it pays. Then:

if the investor is concerned only with the expected values of securities, the investor must also be only interested in the expected value of the portfolio. To maximize the expected value of a portfolio, one need only invest in one security — the security with maximum expected return

Follow that through. If the only thing that matters is expected return, then the best portfolio has $w = 1$ on whichever asset has the highest expected return and zero everywhere else. Not a large weight — the entire portfolio.

And the same holds with certainty rather than expectation: an investor "who knew future returns with certainty would invest in only one security, namely the one with the highest future return."

But nobody does this. Markowitz's conclusion is blunt: action based on expected return only "must be rejected as descriptive of actual or rational investment behavior."

But diversification is a common and reasonable investment practice. Why?

To reduce uncertainty!

So a second dimension is forced. Not because anyone finds it attractive, but because a one-dimensional theory predicts behaviour nobody exhibits, including people who are obviously not being foolish. Chapter 2 supplies the second dimension; and as Markowitz says, risk and return "should be measured for the portfolio as a whole."

That last phrase is the thesis of the subject. Not for each holding, separately, and added up. For the whole.

What a real set of weights looks like

An index is a published portfolio, so it makes the abstraction concrete. The Nifty 50 is "computed using free float market capitalization method", meaning each company's weight is its freely tradeable market value as a share of the total.

As of 30 September 2026 its largest holdings run 10.38%, 9.05%, 7.58%, 5.10%, 4.20% and down to 2.52% for the tenth. By sector, financial services alone is 37.45%, with oil, gas and consumable fuels at 9.39% and information technology at 7.52%.

Two things worth noticing before the mathematics starts.

"Diversified" is a matter of degree, not a yes or no. Fifty companies sounds like a lot, and over a third of the money is in one sector. Chapter 3 gives the arithmetic for saying how much diversification that actually buys.

The weights were not chosen. They are an output of market values, so the index's concentration in financial services is a fact about Indian market capitalisation rather than a decision anybody took. A market-cap index holds more of whatever has gone up, which is a property worth knowing before chapter 6 treats such an index as a theoretical object.

The notation, once

Collecting what the rest of the subject uses:

Symbol	Meaning
$w_i$	weight of asset $i$
$R_i$	return of asset $i$ — a random variable, not a number
$E(R_i)$ or $\mu_i$	its expected return
$\sigma_i$	its standard deviation of return
$\sigma_{ij}$	covariance between assets $i$ and $j$
$\rho_{ij}$	their correlation
$R_p, \mu_p, \sigma_p$	the same quantities for the portfolio

$R_i$ being a random variable is the thing to hold onto. The Quantitative methods subject's chapter on random variables and expectation is the background, and the whole of portfolio theory is the algebra of combining them.

Working the problem

₹6 lakh index fund, ₹3 lakh in one company, ₹1 lakh liquid fund.

The weights. Total is ₹10 lakh, so

$w_{\text{index}} = 0.60, \qquad w_{\text{company}} = 0.30, \qquad w_{\text{liquid}} = 0.10$

and they sum to 1, as they must.

Now the company doubles and you do nothing. The holding goes from ₹3 lakh to ₹6 lakh, so the total is ₹13 lakh and the weights are:

$w_{\text{index}} = \frac{6}{13} = 0.462, \qquad w_{\text{company}} = \frac{6}{13} = 0.462, \qquad w_{\text{liquid}} = \frac{1}{13} = 0.077$

Three observations, in increasing order of importance.

You took no action and your portfolio changed. The single-company weight went from 30% to 46%, and the liquid buffer from 10% to under 8%.

It changed in the direction of more risk. The holding that grew is the undiversified one, so success concentrated the portfolio. This is the general case, not a special one: whatever rises gets a bigger weight, so a portfolio left alone drifts towards whatever has recently done well — and, through the same mechanism, towards whatever is most volatile.

Doing nothing is a decision. You now hold a 46% single-company position. If you would not have chosen that position deliberately, you are holding it by default — and the Risk subject's chapter on concentration is about what that costs. Chapter 11 is about the alternative and what the alternative costs.

One honest note on the arithmetic. The liquid fund's weight fell from 10% to 7.7% without losing a rupee. Weights are relative, so a holding can shrink in the portfolio while growing in value — which is why an emergency fund sized as a percentage behaves differently from one sized in months of expenses, as the Risk subject sets it.

The point

A portfolio is a set of weights summing to one, which makes it scale-free, makes every purchase simultaneously a decision to hold less of everything else, and makes the weights drift whenever prices move. Negative weights are shorts and borrowing — the same object the Option pricing subject replicates with. The reason a second dimension is needed is Markowitz's: if only expected return mattered, the best portfolio would hold one security entirely, which no sensible investor does, so risk has to enter and has to be measured for the portfolio as a whole rather than holding by holding.