Modern Portfolio Theory

Everybody already knew not to put all the eggs in one basket. What nobody could do before 1952 was say by how much the second basket helped.

Harry Markowitz's paper "Portfolio Selection", published in the Journal of Finance in 1952, supplied the arithmetic. He showed that the variance of returns is a meaningful measure of portfolio risk, and derived the formula for the variance of a portfolio — which turned out to depend not just on how risky each holding is, but on how the holdings move relative to each other. He was awarded the Nobel Prize in Economics for the work in 1990.

The consequence is the one line worth carrying out of the chapter: a portfolio of risky assets can be less risky than the assets it contains. Not less risky than the riskiest — less risky than the weighted average. Nothing was hedged; the combination did it.

Portfolio return is easy — it is simply the weighted average of the expected returns of the holdings.

Weights 0.2, 0.1, 0.3, 0.4 on assets returning 9%, 12%, 15%, 18%
E(Rp) = 0.018 + 0.012 + 0.045 + 0.072 = 14.7%

Portfolio risk is not the weighted average, and that asymmetry is the entire theory. Three things drive it:

1. the weights of the investments,
2. the risk of each investment, and
3. the co-movement between every pair of them.

The third term is covariance, standardised into a correlation coefficient running from −1 to +1. At +1 the two move in perfectly linear step and there is no diversification benefit at all. Below +1, some of the variation cancels.

MPT rests on a specific set of assumptions: investors maximise return for a given level of risk, view each alternative as a probability distribution of returns over a holding period, maximise one-period expected utility, decide solely on expected return and risk, and estimate portfolio risk from the variability of the constituent assets' expected returns.

Plot every feasible combination and the outer upper edge — highest return for each level of risk, lowest risk for each level of return — is the efficient-frontier. Pick the point on it that suits the investor's constraints and you have the optimum portfolio.

E(Rp) = Σ Wᵢ × E(Rᵢ)

σp = √[ Σ Wᵢ²σᵢ²  +  Σ Σ Wᵢ Wⱼ Covᵢⱼ ]      where Covᵢⱼ = rᵢⱼ σᵢ σⱼ

For two assets:
σp = √[ W₁²σ₁² + W₂²σ₂² + 2 W₁ W₂ r₁₂ σ₁ σ₂ ]

A family office splits Rs 10 crore equally between two funds. Each is expected to return 15% with a standard deviation of 20%. Identical on every visible measure — so does the split achieve anything?

Expected return is 15% either way. Risk depends entirely on correlation:

σp = √[ 0.5²(0.20²) + 0.5²(0.20²) + 2(0.5)(0.5)(r)(0.20)(0.20) ]
   = √[ 0.01 + 0.01 + 0.02r ]

The same 15% expected return is available at anywhere between 20% and 0% risk, and the only thing that changed was how the two funds move together.

In rupee terms, a one-standard-deviation bad year on Rs 10 crore costs roughly Rs 50 lakh at 20% risk and about Rs 10 lakh at 10% risk, for the identical expected outcome.

This is why an allocator asks a Category III AIF manager for the strategy's correlation with the NIFTY 50 before asking about returns. A fund returning 15% at a correlation of 0.2 is worth more to the portfolio than one returning 17% at a correlation of 0.9 — and MPT is what lets you say that with a number rather than a feeling.

Chapter 3 is built on this, sections 3.1 through 3.9, and Chapter 3 is one of the more heavily examined chapters in the paper. Expect the historical facts (Markowitz, 1952, "Portfolio Selection", Journal of Finance, Nobel 1990), the list of MPT assumptions, the weighted-average portfolio return computation, the correlation range of −1 to +1, and the standard question: if correlation between two holdings falls, what happens to portfolio risk? It decreases. Chapter 3 then builds Capital Market Theory and capm on top of this foundation.

• Portfolio return is a weighted average; portfolio risk is not. This is the single most examined idea in the chapter, and the reason diversification works at all.
• Diversification benefit needs correlation below +1, not below zero. Even +0.5 helps materially. Only at exactly +1 does the benefit vanish.
• Correlation and covariance are not the same. Covariance has messy units; correlation is covariance divided by the two standard deviations, and so always sits between −1 and +1.
• Adding more holdings is not automatically diversifying. Twenty funds all running the same long-only Indian equity strategy are close to one fund. What matters is the correlation, not the count.
• MPT does not remove all risk — only the diversifiable part. Economy-wide systematic-risk remains however many assets you hold.
• The inputs are estimates. For 50 securities the model needs 1,225 correlation estimates, by (n² − n) ÷ 2. Error in those inputs is estimation risk, and the workbook flags it as a real limitation of the optimisation.