Perfect Positive Correlation

The correlation coefficient runs from −1 to +1. At +1, the workbook says, the returns of the two assets move together in a completely linear manner.

That has a blunt consequence for portfolios. Combining two perfectly positively correlated assets gives no benefit. The portfolio's risk is simply the weighted average of the two assets' risks, and no mix can have lower risk than the less risky asset or higher return than the higher-return asset.

In the workbook's words, there is no possibility for risk diversification when two assets with perfect positive correlation are combined — "a very fundamental understanding in Modern Portfolio Theory".

Portfolio variance for two assets (Chapter 15, section 15.7.1):

σp² = wA²σA² + wB²σB² + 2·wA·wB·ρAB·σA·σB

Put ρ = +1 and the expression becomes a perfect square:

σp² = (wA·σA + wB·σB)²   →   σp = wA·σA + wB·σB

So portfolio risk is exactly the weighted average of the individual risks.

Graphically (section 15.8). All combinations lie on a straight line joining the two securities on a risk (x-axis) / return (y-axis) chart. As soon as correlation is below +1, the line bulges to the left: portfolio risk is less than the weighted average. "As long as the correlation is less than perfect 1, benefits of diversification occur. Lower the correlation, higher the benefits."

In Chapter 20 (section 20.3.2). Portfolio standard deviation is not the weighted average of individual standard deviations — except when the investments have perfect positive correlation, which the workbook calls practically an impossibility.

In capital market theory. Every portfolio on the capital market line combines the risk-free asset with the same market portfolio M, and its risk is simply Wmp × σmp (Chapter 16, section 16.3) — which is why Chapter 16's sample question 4 asks whether all CML portfolios are perfectly positively correlated.

Illustrative figures. ₹10 lakh is split 60:40 between two funds.

Expected return (any correlation) = 0.6 × 10 + 0.4 × 14 = 6 + 5.6 = 11.6%

Risk at ρ = +1 = 0.6 × 12 + 0.4 × 20 = 7.2 + 8 = 15.2% — exactly the weighted average.

Risk at ρ = 0.5, for comparison: σp² = 0.36 × 144 + 0.16 × 400 + 2 × 0.6 × 0.4 × 0.5 × 12 × 20 = 51.84 + 64 + 57.6 = 173.44 → σp = √173.44 ≈ 13.17%

Same funds, same weights, same 11.6% return — but about 2 percentage points less risk once correlation falls from +1 to 0.5. At +1, that saving is zero.

The workbook's Table 15.4 uses two securities with 15% return, 5% risk each, 50:50 weights and ρ = 0.5: portfolio variance = 0.000625 + 0.000625 + 0.000625 = 0.001875, so σp ≈ 4.33% — below the 5% it would be at ρ = +1.

Chapter 15 (Introduction to Modern Portfolio Theory), sections 15.7–15.8, defines perfect positive correlation and shows the straight-line risk-return plot with no diversification benefit. Chapter 9 (section 9.2) introduces the −1 to +1 scale; Chapter 20 (section 20.3.2) repeats that portfolio risk equals the weighted average only at perfect positive correlation; Chapter 16's sample questions ask about correlation among CML portfolios and between a completely diversified portfolio and the market portfolio.

• At +1, portfolio risk = weighted average of individual risks. At anything less, it is lower.
• Portfolio return is a weighted average at every correlation. Correlation changes risk, not expected return.
• Perfect positive correlation gives no diversification benefit — not "maximum" benefit. Maximum benefit is at −1.
• Straight line on the risk-return chart at +1; a leftward bulge below +1.
• The workbook calls perfect positive correlation practically an impossibility among real investments (Chapter 20).